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WIMP dark matter and unitarity-conserving inflation via a gauge singlet scalar

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JCAP11(2015)015

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JCAP11(2015)015

ournal of Cosmology and Astroparticle PhysicsAn IOP and SISSA journalJ

WIMP dark matter andunitarity-conserving inflation via agauge singlet scalar

Felix Kahlhoefera and John McDonaldb

aDESYNotkestrasse 85 Hamburg 22607 GermanybDepartment of Physics Lancaster UniversityLancaster LA1 4YB United Kingdom

E-mail felixkahlhoeferdesyde jmcdonaldlancasteracuk

Received July 27 2015Accepted October 10 2015Published November 9 2015

Abstract A gauge singlet scalar with non-minimal coupling to gravity can drive inflation andlater freeze out to become cold dark matter We explore this idea by revisiting inflation in thesinglet direction (S-inflation) and Higgs Portal Dark Matter in light of the Higgs discoverylimits from LUX and observations by Planck We show that large regions of parameter spaceremain viable so that successful inflation is possible and the dark matter relic abundancecan be reproduced Moreover the scalar singlet can stabilise the electroweak vacuum andat the same time overcome the problem of unitarity-violation during inflation encounteredby Higgs Inflation provided the singlet is a real scalar The 2-σ Planck upper bound on nsimposes that the singlet mass is below 2 TeV so that almost the entire allowed parameterrange can be probed by XENON1T

Keywords dark matter theory inflation cosmology of theories beyond the SM

ArXiv ePrint 150703600

ccopy 2015 IOP Publishing Ltd and Sissa Medialab srl doi1010881475-7516201511015

JCAP11(2015)015

Contents

1 Introduction 1

2 The S-inflation model 3

3 Singlet scalar as dark matter 531 Relic abundance 532 Direct detection constraints 633 Invisible Higgs decays 634 Other constraints 7

4 Renormalisation group evolution and theoretical constraints 841 Metastability 942 Examples 943 Perturbativity 10

5 Unitarity-violation during inflation 1151 Regime A large field values 1352 Regime B medium field values 1453 Regime C small field values 1454 Discussion 14

6 Results 1561 The high-mass region 1662 The low-mass region 17

7 Conclusions 19

A RG equations 21

1 Introduction

There is strong evidence in support of the idea that the Universe underwent a period ofprimordial inflation In particular the observation of adiabatic density perturbations with aspectral index which deviates from unity by a few percent [1] is consistent with the genericprediction of scalar field inflation models However the identity of the scalar field responsiblefor inflation remains unknown Another unsolved problem of similar importance for cosmol-ogy is the nature of dark matter (DM) While it is possible to explain DM by the additionof a new particle there is presently no experimental evidence for its existence or its identity

The most studied proposal is that DM is a thermal relic weakly-interacting massiveparticle (WIMP) WIMPs typically have annihilation cross-sections comparable to the valuerequired to reproduce the observed density of DM the so-called ldquoWIMP miraclerdquo Neverthe-less the non-observation of any new weak-scale particles at the LHC beyond the StandardModel (SM) places strong constraints on many models for WIMPs such as in supersymmetricextensions of the SM The absence of new particles may indeed indicate that any extensionof the SM to include WIMP DM should be rather minimal In the present work we therefore

ndash 1 ndash

JCAP11(2015)015

focus on a particularly simple extension of the SM namely an additional gauge singlet scalarwhich is arguably one of the most minimal models of DM [2ndash4]

A similar issue arises from recent constraints on inflation In fact the non-observationof non-Gaussianity by Planck [5] suggests that the inflation model should also be minimal inthe sense of being due to a single scalar field The absence of evidence for new physics thenraises the question of whether the inflaton scalar can be part of the SM or a minimal extensionof the SM The former possibility is realized by Higgs Inflation [6] which is a version of thenon-minimally coupled scalar field inflation model of Salopek Bond and Bardeen (SBB) [7]with the scalar field identified with the Higgs boson A good example for the latter optionare gauge singlet scalar extensions of the SM because the DM particle can also provide awell-motivated candidate for the scalar of the SBB model In other words in these modelsthe same scalar particle drives inflation and later freezes out to become cold DM

The resulting gauge singlet inflation model was first considered in [8] where it was calledS-inflation (see also [9])1 All non-minimally coupled scalar field inflation models based onthe SBB model are identical at the classical level but differ once quantum corrections to theinflaton potential are included These result in characteristic deviations of the spectral indexfrom its classical value which have been extensively studied in both Higgs Inflation [6 11ndash15]and S-inflation [16]

Since the original studies were performed the mass of the Higgs boson [17] and thePlanck results for the inflation observables [1] have become known In addition direct DMdetection experiments such as LUX [18] have imposed stronger bounds on gauge singletscalar DM [19ndash25] This new data has important implications for these models in particularfor S-inflation which can be tested in Higgs physics and DM searches The main objectiveof the present paper is to compare the S-inflation model with the latest results from CMBobservations and direct DM detection experiments

We will demonstrate that mdash in spite of its simplicity mdash the model still has a large viableparameter space where the predictions for inflation are consistent with all current constraintsand the observed DM relic abundance can be reproduced In addition we observe that thismodel can solve the potential problem that the electroweak vacuum may be metastablebecause the singlet gives a positive contribution to the running of the quartic Higgs couplingIntriguingly the relevant parameter range can be almost completely tested by XENON1T

Another important aspect of our study is perturbative unitarity-violation which maybe a significant problem for Higgs Inflation Since Higgs boson scattering via graviton ex-change violates unitarity at high energies [26 27] one might be worried that the theory iseither incomplete or that perturbation theory breaks down so that unitarity is only conservednon-perturbatively [28ndash31] In both cases there can be important modification of the infla-ton potential due to new physics or strong-coupling effects Indeed in conventional HiggsInflation the unitarity-violation scale is of the same magnitude as the Higgs field duringinflation [14 32] placing in doubt the predictions of the model or even its viability

In contrast we will show that S-inflation has sufficient freedom to evade this problemprovided that the DM scalar is specifically a real singlet By choosing suitable values for thenon-minimal couplings at the Planck scale it is possible for the unitarity-violation scale to bemuch larger than the inflaton field throughout inflation so that the predictions of the modelare robust Therefore in addition to providing a minimal candidate for WIMP DM theextension of the SM by a non-minimally coupled real gauge singlet scalar can also accountfor inflation while having a consistent scale of unitarity-violation

1The case of singlet DM added to Higgs Inflation was considered in [10]

ndash 2 ndash

JCAP11(2015)015

The paper is organized as follows In section 2 we review the real gauge singlet scalar ex-tension of the SM and the S-inflation model We estimate the predictions of the model for thespectral index ns and discuss the effect of constraints from inflation on the model parameterspace Section 3 considers the DM phenomenology of the model and the implications fromDM searches In section 4 we discuss how to connect these two aspects via renormalisationgroup evolution and which constraints follow from electroweak vacuum stability and pertur-bativity The scale of unitarity-violation during inflation and the consistency of S-inflationare discussed in section 5 Finally we present our results in section 6 and our conclusions insection 7 Additional details are provided in the appendix

2 The S-inflation model

S-inflation is a version of the non-minimally coupled inflation model of [7] in which the scalarfield is identified with the gauge singlet scalar responsible for thermal relic cold DM In thepresent work we focus on the case of a real singlet scalar s In the Jordan frame which isthe standard frame for interpreting measurements and calculating radiative corrections theaction for this model is

SJ =

int radicminusg d4x

[LSM + (partmicroH)dagger (partmicroH) +

1

2partmicros part

micros

minusm2

PR

2minus ξhHdaggerH Rminus 1

2ξs s

2Rminus V (s2 HdaggerH)

] (21)

where LSM is the SM Lagrangian density minus the purely Higgs doublet terms mP is thereduced Planck mass and

V (s2 HdaggerH) = λh

[(HdaggerH

)minus v2

2

]2+

1

2λhs s

2HdaggerH +1

4λs s

4 +1

2m2s0 s

2 (22)

with v = 246 GeV the vacuum expectation value of the Higgs field Writing H = (h+v 0)radic

2with a real scalar h we obtain

V (s2 h) = V (h) +1

2m2s s

2 +1

4λs s

4 +1

2λhs v h s

2 +1

4λhs h

2 s2 (23)

where we have introduced the physical singlet mass m2s = m2

s0 + λhs v22

In order to calculate the observables predicted by inflation we perform a conformaltransformation to the Einstein frame where the non-minimal coupling to gravity disappearsIn the case that s 6= 0 and h = 0 this transformation is defined by

gmicroν = Ω2 gmicroν Ω2 = 1 +ξs s

2

m2P

(24)

The transformation yields

SE =

int radicminusg d4x

[LSM +

1

2

(1

Ω2+

6 ξ2s s2

m2P Ω4

)gmicroνpartmicrospartνsminus

m2P R

2minus V (s 0)

Ω4

] (25)

ndash 3 ndash

JCAP11(2015)015

where R is the Ricci scalar with respect to gmicroν We can then rescale the field using

dχsds

=

radicΩ2 + 6 ξ2s s

2m2P

Ω4 (26)

which gives

SE =

int radicminusg d4x

(LSM minus

m2P R

2+

1

2gmicroνpartmicroχspartνχs minus U(χs 0)

) (27)

with

U(χs 0) =λs s

4(χs)

4 Ω4 (28)

The relationship between s and χs is determined by the solution to eq (26) In particularfor s mP

radicξs the Einstein frame potential is

U(χs 0) =λsm

4P

4 ξ2s

(1 + exp

(minus 2χsradic

6mP

))minus2 (29)

This is sufficiently flat at large χs to support slow-roll inflationAn analogous expression is obtained for the potential along the h-direction In both

cases the Einstein frame potential is proportional to λφξ2φ where φ = s or h Therefore

the minimum of the potential at large s and h will be very close to h = 0 and inflation willnaturally occur along the s-direction if λsξ

2s λhξ

2h which is true for example if ξs ξh

and λs sim λhIn the following inflation is always considered to be in the direction of s with h = 0

The conventional analysis of inflation can then be performed in the Einstein frame Afterinflation the Jordan and Einstein frames will be indistinguishable since ξs s

2 m2P and

so Ω rarr 1 Therefore the curvature perturbation spectrum calculated in the Einstein framebecomes equal to that observed in the physical Jordan frame at late times

The classical (tree-level) predictions for the spectral index and tensor-to-scalar ratioare [33]

ntrees asymp 1minus 2

Nminus 3

2N2+O

(1

N3

)= 0965 (210)

rtree asymp 12

N2+O

(1

ξsN2

)= 36times 10minus3 (211)

while the field during inflation is

s2Nasymp 4m2

P N(3 ξs) (212)

In the equations above N is the number of e-foldings as defined in the Einstein framewhich differs from that in the Jordan frame by N asymp N + ln(1

radicN) [8] and we have used

N = 582 The classical predictions are in good agreement with the most recent Planck

2Reheating in S-inflation occurs via stochastic resonance to Higgs bosons through the coupling λhs Itwas shown in [16] that this process is very efficient and makes quite precise predictions for the reheatingtemperature and the number of e-foldings of inflation with 57 N 60 at the WMAP pivot scale This inturn allows for quite precise predictions of the inflation observables

ndash 4 ndash

JCAP11(2015)015

values ns = 09677plusmn 00060 (68 confidence level (CL) Planck TT + lowP + lensing) andr0002 lt 011 (95 CL Planck TT + lowP + lensing) [1]

The classical predictions for S-inflation are the same as those of any model based onthe SBB model Differences between S-inflation and other models do however arise fromquantum corrections to the effective potential To include these corrections we calculatethe RG evolution of the various couplings as a function of the renormalisation scale micro (seesection 4 for details) We can then obtain the renormalisation group (RG)-improved effectivepotential for s in the Jordan frame by replacing the couplings in eq (23) by the runningcouplings and setting micro equal to the value of the field For h = 0 (and neglecting the singletmass term ms sim 1 TeV s) this approach yields

VRG(s2 0) =λs(s) s

4

4 (213)

The RG-improved potential can then be transformed into the Einstein frame in order tocalculate the observables predicted by inflation3 The inflationary parameters are calculatedusing the methods discussed in [16] In particular the Einstein frame slow-roll parametersare given by

ε =m2

P

2

(1

U

dU

dχs

)2

η =m2

P

U

d2U

dχ2s

ξ2 =m4

P

U2

dU

dχs

d3U

dχ3s

(214)

3 Singlet scalar as dark matter

Let us now turn to the phenomenology of the singlet scalar in the present Universe andat energies well below the scale of inflation [19 24 35] Most importantly the assumedZ2 symmetry ensures the stability of the scalar so that it can potentially account for theobserved abundance of DM [2 3] If the mass of the singlet is comparable to the electroweakscale the singlet is a typical WIMP which obtains its relic abundance from thermal freeze-out Indeed at low energies where the effects of the non-minimal coupling to gravity arenegligible our model becomes identical to what is often referred to as Higgs Portal DarkMatter [21 22 36] because all interactions of the singlet with SM particles are mediatedby the Higgs In this section we review the constraints on these models and determine theparameter space allowed by the most recent experimental results In the process we pointout several discrepancies in the literature and resolve the resulting confusion

31 Relic abundance

The calculation of the relic abundance of singlet scalars is discussed in detail in [19] Threekinds of processes are relevant for the annihilation of singlets into SM states annihilation

3In [34] it was proposed to use of the Einstein frame for the computation of quantum corrections TheJordan frame analysis is however easier to implement correctly being a straightforward extension of theStandard Model analysis

ndash 5 ndash

JCAP11(2015)015

into SM fermions annihilation into SM gauge bosons and annihilation into two Higgs par-ticles The first kind dominates as long as ms lt mW while for larger masses the secondkind gives the largest contribution Notably all of these processes can proceed via an s-channel Higgs boson leading to a resonant enhancement of the annihilation cross-sectionand a corresponding suppression of the DM relic abundance for ms sim mh24

For the present work we calculate the singlet abundance using micrOMEGAs 3 [37]which numerically solves the Boltzmann equation while calculating the Higgs width in aself-consistent way It is then straightforward to numerically find the coupling λhs that givesΩs h

2 = 01197 in order to reproduce the value of the DM density ΩDM h2 = 01197plusmn 00022determined by Planck (TT + lowP 68 CL) [38] For example we find λhs asymp 008 forms = 300 GeV and λhs asymp 030 for ms = 1000 GeV These values agree with the ones foundin [19 23] but disagree with [20ndash22 24] by a factor of 2 after accounting for the differentconventions5

32 Direct detection constraints

The strongest constraints on λhs stem from DM direct detection experiments since thesinglet-Higgs coupling induces spin-independent interactions between the DM particle andnuclei The scattering cross-section at zero momentum transfer is given by [19]

σSI =λ2hsf

2N

4π

micro2r m2n

m4hm

2s

(31)

where mn is the neutron mass micror = (msmn)(ms + mn) is the reduced mass and fN isthe effective Higgs-nucleon coupling6 In terms of the light-quark matrix elements fNTq theeffective coupling can be written as

fN =

2

9+

7

9

sumq=uds

fNTq

(32)

The values of fNTq can either be determined phenomenologically from baryon masses andmeson-baryon scattering data or computed within lattice QCD A comparison of the differentmethods was recently performed in [19] and we adopt their result of fN = 030 for the effectivecoupling

The scattering cross-section given above can be directly compared to the bound obtainedfrom the LUX experiment [18] Indeed as shown in figure 1 LUX is typically sensitive to thesame range of values for λhs as what is implied by the relic density constraint Specificallythe LUX bound excludes the mass ranges 57 GeV lt ms lt 526 GeV and 645 GeV lt ms lt928 GeV

33 Invisible Higgs decays

Direct detection experiments cannot constrain singlet scalars with a mass of a few GeV orless since such particles would deposit too little energy in the detector to be observable This

4The process ssrarr hh also receives a contribution from t-channel singlet exchange which gives a relevantcontribution if λhs is large compared to λh

5The Higgs-singlet coupling is called λhS in [19] λHS in [20] λhSS in [21 22] λDM in [23] and a2 in [24]The respective conventions are captured by λhs = λhS = λHS2 = λhSS2 = λDM2 = 2 a2

6Note that [24] uses an approximate expression valid for ms mn such that micro2r asymp m2

n

ndash 6 ndash

JCAP11(2015)015

5 10 50 100 500 100010-4

0001

0010

0100

1

ms [GeV]

λhs

Figure 1 Excluded parameter regions from LUX (red dotted) and searches for invisible Higgs decays(blue dashed) compared to the coupling implied by the relic density constraint (green solid)

parameter region can however be efficiently constrained by considering how the Higgs-singletcoupling λhs would modify the branching ratios of the SM Higgs boson The partial decaywidth for hrarr ss is given by7

Γ(hrarr ss) =λ2hsv

2

32πmh

radic1minus 4m2

s

m2h

(33)

This theoretical prediction can be compared to the experimental bound on invisibleHiggs decays from the LHC Direct searches for invisible Higgs decays in the vector bosonfusion channel give BR(h rarr inv) 029 [39] A somewhat stronger bound can be obtainedfrom the observation that in our model there are no additional contributions to the Higgsproduction cross-section and no modifications of the partial decay widths of the Higgs bosoninto SM final states Therefore the presence of an invisible decay channel leads to an overallreduction of the signal strength in visible channels A global fit of all observed decay channels(combined with the bounds on invisible Higgs decays) then gives BR(hrarr inv) 026 [40]

Crucially the bound from invisible Higgs decays becomes independent of the singletmass for ms mh2 Invisible Higgs decays will therefore provide the strongest constraintsfor small singlet masses Indeed this constraint rules out the entire mass region where directdetection experiments lose sensitivity (see figure 1) As a result only two mass regions remainviable a low-mass region 526 GeV lt ms lt 645 GeV and a high-mass region ms amp 93 GeV

34 Other constraints

It has been pointed out recently [24] that bounds on γ-ray lines from Fermi-LAT [41] ruleout the parameter region where ms is slightly above mh2 To be safe from this constraintwe will focus on the mass range 526 GeV lt ms lt 624 GeV which we shall refer to as thelow-mass region For the high-mass region on the other hand there are no strong constraintsfrom indirect detection Moreover collider searches for singlet scalars with ms gt mh2 areextremely challenging [20ndash23 35] and consequently there are no relevant bounds from theLHC for the high-mass region [42]

The most significant improvements in sensitivity in the near future are expected tocome from direct detection experiments Indeed XENON1T [43] is expected to improve

7This equation agrees with [4 19ndash22 35] but disagrees with [23]

ndash 7 ndash

JCAP11(2015)015

upon current LUX constraints on the DM scattering cross-section by a factor of about 50and will therefore be able to probe the high-mass region up to ms asymp 4 TeV As we will showin S-inflation singlet masses larger than about 2 TeV are excluded by the Planck 2-σ upperbound on ns and perturbativity XENON1T will therefore be able to probe the entire high-mass region relevant for singlet inflation Similarly XENON1T can also further constrain thelow-mass region and potentially probe singlet masses in the range 53 GeV lt ms lt 57 GeV

4 Renormalisation group evolution and theoretical constraints

In order to connect the inflationary observables of our model to the measured SM parametersand the DM phenomenology discussed in the previous section we need to calculate theevolution of all couplings under the RG equations [8 10 12 14 44ndash47] Existing analyseshave considered the RG equations for the SM at two-loop order and the contributions of thesinglet sector and non-minimal coupling at one-loop order (see also the appendix) [8 10] Toexamine the issue of vacuum stability we improve the accuracy of our analysis further byincorporating the three-loop RG equations for the SM gauge couplings [48] and the leadingorder three-loop corrections to the RG equations for λh and yt [49]8

When considering large field values for either s or h the RG equations are modifiedbecause there is a suppression of scalar propagators This suppression is captured by insertinga factor

cφ =1 +

ξφ φ2

mP

1 + (6ξφ + 1)ξφ φ2

mP

(41)

with φ = s (φ = h) for each s (h) propagating in a loop [8] The changes in the RGequations for large values of the Higgs field have been discussed in detail in [47] Themodifications resulting from large singlet field values can be found in [8] and are reviewedin the appendix Note that when considering S-inflation such that s h we can set thesuppression factor ch = 1

We determine the values of the SM parameters at micro = mt following [46] Using themost recent values from the Particle Data Group [50]

mt = (1732plusmn 09) GeV mH = (12509plusmn 024) GeV αS(mZ) = 01185plusmn 00006 (42)

we obtain at micro = mt

yt = 0936plusmn 0005 λh = 01260plusmn 00014 gS = 1164plusmn 0003 (43)

Unless explicitly stated otherwise we will use the central values for all calculations belowFor given couplings at micro = mt we then use the public code RGErun 207 [51] to calculatethe couplings at higher scales

In contrast to the remaining couplings we fix the non-minimal couplings ξh and ξs atmicro = mP In order to obtain the correct amplitude of the scalar power spectrum we require

U

ε= (000271mP)4 (44)

where U and ε are the potential and the first slow-roll parameter in the Einstein frameat the beginning of inflation as defined in eq (28) and eq (214) respectively Imposing

8We thank Kyle Allison for sharing his numerical implementation of these equations

ndash 8 ndash

JCAP11(2015)015

equation (44) allows us to determine ξs at the scale of inflation once all other parametershave been fixed Note that since the value of s at the beginning of inflation also depends onξs equation (44) can only be solved numerically We then iteratively determine the requiredvalue of ξs at the electroweak scale such that RG evolution yields the desired value at thescale of inflation

The coupling ξh plays a very limited role for the phenomenology of our model because wedo not consider the case of large Higgs field values for inflation As a result our predictionsfor the inflationary observables show only a very mild dependence on ξh so that ξh canessentially be chosen arbitrarily Nevertheless it is not possible to simply set this parameterto zero since radiative corrections induce a mixing between ξs and ξh Moreover we willsee below that the value of ξh plays an important role for determining whether our modelviolates unitarity below the scale of inflation As with ξs we fix ξh at mP and then determineiteratively the required value of ξh at the electroweak scale9

41 Metastability

We now discuss various theoretical constraints related to the RG evolution of the parame-ters in our model It is a well-known fact that for the central values of the measured SMparameters the electroweak vacuum becomes metastable at high scales because the quarticHiggs coupling λh runs to negative values (see eg [46]) This metastability is not in anyobvious way a problem as the lifetime of the electroweak vacuum is well above the age ofthe Universe [52] (note however that this estimate may potentially be spoiled by effectsfrom Planck-scale higher-dimensional operators [53ndash55]) However one may speculate thata stable electroweak vacuum is necessary for a consistent theory for example if the vacuumenergy relative to the absolute minimum is a physical energy density leading to inflation Itis therefore an interesting aspect of singlet extensions of the SM that scalar singlets give apositive contribution to the running of λh [56 57]

βλh = βSMλh +1

32π2c2s λ

2hs (45)

In fact it was shown in [58] that for the case of a minimally-coupled singlet λhs can bechosen such that the electroweak vacuum remains stable all the way up to the Planck scaleand at the same time (for appropriate choices of the singlet mass ms) the singlet obtains athermal relic density compatible with the observed DM abundance10

In order to study electroweak vacuum stability we need to consider the potential in theh-direction with s = 011 Vacuum stability then requires that λh(micro) gt 0 for micro up to mPIn the present study we consider both the case where λhs is sufficiently large to stabilisethe electroweak vacuum and the case where λhs only increases the lifetime of the metastablevacuum but does not render it completely stable We focus throughout on the case whereλhs is positive

42 Examples

Figure 2 shows an example for the evolution of scalar couplings (left) and the non-minimalcouplings (right) under the RG equations discussed above Solid lines correspond to the case

9The running of ξh between mP and the scale of inflation is completely negligible since the relevantdiagrams are strongly suppressed for large values of s

10Note that if the singlet mixes with the Higgs there will be additional threshold effects at micro = ms fromintegrating out the singlet [59] In the setup we consider however this effect is not important [60]

11A more detailed study of the potential in general directions with both s 6= 0 and h 6= 0 (along the linesof [61]) is beyond the scope of the present work

ndash 9 ndash

JCAP11(2015)015

1000 106 109 1012 1015 10180001

00050010

00500100

05001

μ [GeV]

Couplingstrength

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

λh

λhs

λs

1000 106 109 1012 1015 101810

100

1000

104

105

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

ξh

ξs

cs

Figure 2 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplingsξh and ξs (right) as a function of the renormalisation scale micro for a typical parameter point in thehigh-mass region Solid lines show the running in the s-direction while dotted lines correspond to therunning in the h-direction In the right panel we also show the suppression factor cs which modifiesthe running in the s-direction at large field value

s h which is relevant for inflation while dotted lines correspond to h s which is relevantfor vacuum stability We fix the scalar couplings at the weak scale choosing λhs = 025 andλs = 001 such that the observed relic abundance can be reproduced for ms = 835 GeV Weconsider λs λhs in which case the value of ξs necessary to obtain the correct amplitude ofthe scalar power spectrum is reduced For our choice we find ξs sim 104 Note however thatwhile λhs exhibits only moderate running λs grows significantly with the renormalisationscale micro because its β-function contains a term proportional to λ2hs Choosing even smallervalues of λs at the weak scale will therefore not significantly reduce its value at the scale ofinflation nor the corresponding value of ξs

An important observation from figure 2 is that λh does not run negative and hencethe electroweak vacuum remains stable all the way up to the Planck scale The additionalcontribution from the singlet scalar is sufficient to ensure λh gt 10minus3 for all renormalisationscales up to mP For field values s amp 1015 GeV h the singlet propagator is suppressedleading to a visible kink in the running of λh We show the propagator suppression factorcs in the right panel of figure 2 One can clearly see how this suppression factor affects therunning of ξh which becomes nearly constant for s amp 1015 GeV In this particular examplewe have chosen ξh(mP) = 1000 (for s h) This choice together with λs 1 implies thatthe running of ξs(micro) from mt to mP is negligible

In the low-mass region we are interested in much smaller values of λhs typically below10minus2 A particular example is shown in figure 3 (left) for the representative choice λhs(mt) =0002 and λs = 00005 which yields the observed relic abundance for ms asymp 57 GeV Weobserve that if λs and λhs are both small at the electroweak scale these couplings exhibitonly very little running up to the scale of inflation For the same reason it is impossible toinfluence the running of the Higgs couplings sufficiently to prevent λhs from running negativeat around 1011 GeV

If λhs is small we can obtain the correct scalar power spectrum amplitude with a muchsmaller value of ξs during inflation For the specific case considered in figure 3 we findξs sim 103 For these values of ξs and λhs the loop-induced corrections to ξh are very smalland hence this coupling changes only very slightly under RG evolution

43 Perturbativity

In order for our calculation of the running couplings and the radiative corrections to thepotential to be reliable we must require that all couplings remain perturbative up to the

ndash 10 ndash

JCAP11(2015)015

1000 106 109 1012 1015 101810-4

0001

0010

0100

1

μ [GeV]

Couplingstrength

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

λh

λhs

λs

1000 106 109 1012 1015 101801

1

10

100

1000

104

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

ξh

ξs

cs

Figure 3 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplings ξhand ξs (right) as a function of the renormalisation scale micro for a typical parameter point in the low-mass region In the right panel we also show the suppression factor cs which modifies the runningat large field value Note that λh runs negative for micro amp 1011 GeV

scale of inflation which is typically 1017ndash1018 GeV This requirement is easily satisfied forthe SM couplings but needs to be checked explicitly for the couplings of the singlet whichcan grow significantly with increasing renormalisation scale micro We follow [58 62] and usethe requirement of perturbative unitarity to impose an upper bound on the scalar couplingsThis procedure gives

λs lt4π

3and λhs lt 8π (46)

As we will see below the non-minimal coupling ξs can be much larger than unity withoutinvalidating a perturbative calculation Nevertheless if ξs and ξh are both very large pro-cesses involving both couplings may violate perturbative unitarity implying that there maybe new physics or strong coupling below the scale of inflation We will now discuss this issuein more detail

5 Unitarity-violation during inflation

In this section we estimate the scale of perturbative unitarity-violation as a function of thebackground inflaton field Note that by ldquounitarity-violation scalerdquo we mean the scale at whichperturbation theory in scalar scattering breaks down so this may in fact indicate the onsetof unitarity-conserving scattering in a strongly-coupled regime [28 31] We will considerunitarity-violation in the scattering of scalar particles corresponding to perturbations aboutthe background field In the case of a real scalar s and the fields of the Higgs doublet Hthere are two distinct scattering processes we need to consider (i) δs h1 harr δs h1 and (ii)h1h2 harr h1h2 where δs is the perturbation about the background s field and h1 and h2are two of the Higgs doublet scalars Scattering with the other scalars in H is equivalentto these two processes We will use dimensional analysis to estimate the scale of unitarity-violation by determining the leading-order processes in the Einstein frame which result inunitarity-violating scattering

It will be sufficient to consider the Einstein frame Lagrangian for two real scalar fieldsφi where mdash using the notation of [30] mdash φi stands for either s or a component of the Higgsdoublet Unitarity-violation requires that there are two different scalars in the scatteringprocess since in the case of a single scalar there is a cancellation between s- t- and u-channelamplitudes [63]

ndash 11 ndash

JCAP11(2015)015

Since unitarity-violating scattering in the Jordan frame is due to graviton exchange viathe non-minimal coupling to R we cat set V = 0 The Einstein frame action for two realscalars is then of the form

SE =

intd4xradicminusg

Lii +sumiltj

Lij minus1

2m2

P R

(51)

where

Lii =1

2

Ω2 +6 ξ2i φ

2i

m2P

Ω4

gmicroν partmicroφi partνφi and Lij =6 ξi ξj φi φj g

microν partmicroφi partνφjm2

P Ω4(52)

with

Ω2 = 1 +ξj φ

2j

m2P

(53)

The interaction terms proportional to ξi ξj are responsible for the dominant unitarity-viola-tion in scattering cross-sections calculated in the Einstein frame These interactions are theEinstein frame analogue of scalar scattering via graviton exchange in the Jordan frame due tothe non-minimal coupling To obtain the scale of unitarity-violation in terms of the physicalenergy defined in the Jordan frame we first canonically normalize the fields in the Einsteinframe then estimate the magnitude of the scattering matrix element and finally transformthe unitarity-violation scale in the Einstein frame back to that in the Jordan frame

In the following we will denote the inflaton by φ1 (equiv s) which we expand about thebackground field ie φ1 = φ1 + δφ1 The Higgs doublet scalars are denoted by φ2 (equiv h1)and φ3 (equiv h2) The corresponding canonically normalized scattering fields in the Einsteinframe are then defined to be ϕ1 ϕ2 and ϕ3 Once we have determined the interactions ofthe canonically normalised fields we use dimensional analysis to estimate the scale of tree-level unitarity-violation For this purpose we introduce appropriate factors of E to makethe coefficient of the interaction terms in L dimensionless Energy scales E in the Einsteinframe are related to the ones in the Jordan frame via E = EΩ where during inflation

Ω2 ξ1φ21m

2P asymp N 1 Unitarity conservation implies that the matrix element for any

2 harr 2 scattering process should be smaller than O(1) so we can determine the scale ofunitarity-violation (denoted by Λ) by determining the value of E that saturates this boundThis was demonstrated explicitly in [8] by comparing the dimensional estimate with theexact value from the full scattering amplitude

In the following we consider three regimes for φ1 each of which leads to a differentform of the Lagrangian and the scattering amplitudes

bull Regime A large field values In this regime we have Ω gt 1 (implying that ξ1 φ21m

2P gt 1)

and 6 ξ21 φ21m

2P gt 1

bull Regime B medium field values In this regime we have approximately Ω asymp 1 (implying

that ξ1 φ21m

2P lt 1) but still 6 ξ21 φ

21m

2P gt 1

bull Regime C small field values Finally we consider 6 ξ21 φ21m

2P lt 1 which in particular

implies Ω asymp 1

ndash 12 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

JCAP11(2015)015

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

focus on a particularly simple extension of the SM namely an additional gauge singlet scalarwhich is arguably one of the most minimal models of DM [2ndash4]

A similar issue arises from recent constraints on inflation In fact the non-observationof non-Gaussianity by Planck [5] suggests that the inflation model should also be minimal inthe sense of being due to a single scalar field The absence of evidence for new physics thenraises the question of whether the inflaton scalar can be part of the SM or a minimal extensionof the SM The former possibility is realized by Higgs Inflation [6] which is a version of thenon-minimally coupled scalar field inflation model of Salopek Bond and Bardeen (SBB) [7]with the scalar field identified with the Higgs boson A good example for the latter optionare gauge singlet scalar extensions of the SM because the DM particle can also provide awell-motivated candidate for the scalar of the SBB model In other words in these modelsthe same scalar particle drives inflation and later freezes out to become cold DM

The resulting gauge singlet inflation model was first considered in [8] where it was calledS-inflation (see also [9])1 All non-minimally coupled scalar field inflation models based onthe SBB model are identical at the classical level but differ once quantum corrections to theinflaton potential are included These result in characteristic deviations of the spectral indexfrom its classical value which have been extensively studied in both Higgs Inflation [6 11ndash15]and S-inflation [16]

Since the original studies were performed the mass of the Higgs boson [17] and thePlanck results for the inflation observables [1] have become known In addition direct DMdetection experiments such as LUX [18] have imposed stronger bounds on gauge singletscalar DM [19ndash25] This new data has important implications for these models in particularfor S-inflation which can be tested in Higgs physics and DM searches The main objectiveof the present paper is to compare the S-inflation model with the latest results from CMBobservations and direct DM detection experiments

We will demonstrate that mdash in spite of its simplicity mdash the model still has a large viableparameter space where the predictions for inflation are consistent with all current constraintsand the observed DM relic abundance can be reproduced In addition we observe that thismodel can solve the potential problem that the electroweak vacuum may be metastablebecause the singlet gives a positive contribution to the running of the quartic Higgs couplingIntriguingly the relevant parameter range can be almost completely tested by XENON1T

Another important aspect of our study is perturbative unitarity-violation which maybe a significant problem for Higgs Inflation Since Higgs boson scattering via graviton ex-change violates unitarity at high energies [26 27] one might be worried that the theory iseither incomplete or that perturbation theory breaks down so that unitarity is only conservednon-perturbatively [28ndash31] In both cases there can be important modification of the infla-ton potential due to new physics or strong-coupling effects Indeed in conventional HiggsInflation the unitarity-violation scale is of the same magnitude as the Higgs field duringinflation [14 32] placing in doubt the predictions of the model or even its viability

In contrast we will show that S-inflation has sufficient freedom to evade this problemprovided that the DM scalar is specifically a real singlet By choosing suitable values for thenon-minimal couplings at the Planck scale it is possible for the unitarity-violation scale to bemuch larger than the inflaton field throughout inflation so that the predictions of the modelare robust Therefore in addition to providing a minimal candidate for WIMP DM theextension of the SM by a non-minimally coupled real gauge singlet scalar can also accountfor inflation while having a consistent scale of unitarity-violation

1The case of singlet DM added to Higgs Inflation was considered in [10]

ndash 2 ndash

JCAP11(2015)015

The paper is organized as follows In section 2 we review the real gauge singlet scalar ex-tension of the SM and the S-inflation model We estimate the predictions of the model for thespectral index ns and discuss the effect of constraints from inflation on the model parameterspace Section 3 considers the DM phenomenology of the model and the implications fromDM searches In section 4 we discuss how to connect these two aspects via renormalisationgroup evolution and which constraints follow from electroweak vacuum stability and pertur-bativity The scale of unitarity-violation during inflation and the consistency of S-inflationare discussed in section 5 Finally we present our results in section 6 and our conclusions insection 7 Additional details are provided in the appendix

2 The S-inflation model

S-inflation is a version of the non-minimally coupled inflation model of [7] in which the scalarfield is identified with the gauge singlet scalar responsible for thermal relic cold DM In thepresent work we focus on the case of a real singlet scalar s In the Jordan frame which isthe standard frame for interpreting measurements and calculating radiative corrections theaction for this model is

SJ =

int radicminusg d4x

[LSM + (partmicroH)dagger (partmicroH) +

1

2partmicros part

micros

minusm2

PR

2minus ξhHdaggerH Rminus 1

2ξs s

2Rminus V (s2 HdaggerH)

] (21)

where LSM is the SM Lagrangian density minus the purely Higgs doublet terms mP is thereduced Planck mass and

V (s2 HdaggerH) = λh

[(HdaggerH

)minus v2

2

]2+

1

2λhs s

2HdaggerH +1

4λs s

4 +1

2m2s0 s

2 (22)

with v = 246 GeV the vacuum expectation value of the Higgs field Writing H = (h+v 0)radic

2with a real scalar h we obtain

V (s2 h) = V (h) +1

2m2s s

2 +1

4λs s

4 +1

2λhs v h s

2 +1

4λhs h

2 s2 (23)

where we have introduced the physical singlet mass m2s = m2

s0 + λhs v22

In order to calculate the observables predicted by inflation we perform a conformaltransformation to the Einstein frame where the non-minimal coupling to gravity disappearsIn the case that s 6= 0 and h = 0 this transformation is defined by

gmicroν = Ω2 gmicroν Ω2 = 1 +ξs s

2

m2P

(24)

The transformation yields

SE =

int radicminusg d4x

[LSM +

1

2

(1

Ω2+

6 ξ2s s2

m2P Ω4

)gmicroνpartmicrospartνsminus

m2P R

2minus V (s 0)

Ω4

] (25)

ndash 3 ndash

JCAP11(2015)015

where R is the Ricci scalar with respect to gmicroν We can then rescale the field using

dχsds

=

radicΩ2 + 6 ξ2s s

2m2P

Ω4 (26)

which gives

SE =

int radicminusg d4x

(LSM minus

m2P R

2+

1

2gmicroνpartmicroχspartνχs minus U(χs 0)

) (27)

with

U(χs 0) =λs s

4(χs)

4 Ω4 (28)

The relationship between s and χs is determined by the solution to eq (26) In particularfor s mP

radicξs the Einstein frame potential is

U(χs 0) =λsm

4P

4 ξ2s

(1 + exp

(minus 2χsradic

6mP

))minus2 (29)

This is sufficiently flat at large χs to support slow-roll inflationAn analogous expression is obtained for the potential along the h-direction In both

cases the Einstein frame potential is proportional to λφξ2φ where φ = s or h Therefore

the minimum of the potential at large s and h will be very close to h = 0 and inflation willnaturally occur along the s-direction if λsξ

2s λhξ

2h which is true for example if ξs ξh

and λs sim λhIn the following inflation is always considered to be in the direction of s with h = 0

The conventional analysis of inflation can then be performed in the Einstein frame Afterinflation the Jordan and Einstein frames will be indistinguishable since ξs s

2 m2P and

so Ω rarr 1 Therefore the curvature perturbation spectrum calculated in the Einstein framebecomes equal to that observed in the physical Jordan frame at late times

The classical (tree-level) predictions for the spectral index and tensor-to-scalar ratioare [33]

ntrees asymp 1minus 2

Nminus 3

2N2+O

(1

N3

)= 0965 (210)

rtree asymp 12

N2+O

(1

ξsN2

)= 36times 10minus3 (211)

while the field during inflation is

s2Nasymp 4m2

P N(3 ξs) (212)

In the equations above N is the number of e-foldings as defined in the Einstein framewhich differs from that in the Jordan frame by N asymp N + ln(1

radicN) [8] and we have used

N = 582 The classical predictions are in good agreement with the most recent Planck

2Reheating in S-inflation occurs via stochastic resonance to Higgs bosons through the coupling λhs Itwas shown in [16] that this process is very efficient and makes quite precise predictions for the reheatingtemperature and the number of e-foldings of inflation with 57 N 60 at the WMAP pivot scale This inturn allows for quite precise predictions of the inflation observables

ndash 4 ndash

JCAP11(2015)015

values ns = 09677plusmn 00060 (68 confidence level (CL) Planck TT + lowP + lensing) andr0002 lt 011 (95 CL Planck TT + lowP + lensing) [1]

The classical predictions for S-inflation are the same as those of any model based onthe SBB model Differences between S-inflation and other models do however arise fromquantum corrections to the effective potential To include these corrections we calculatethe RG evolution of the various couplings as a function of the renormalisation scale micro (seesection 4 for details) We can then obtain the renormalisation group (RG)-improved effectivepotential for s in the Jordan frame by replacing the couplings in eq (23) by the runningcouplings and setting micro equal to the value of the field For h = 0 (and neglecting the singletmass term ms sim 1 TeV s) this approach yields

VRG(s2 0) =λs(s) s

4

4 (213)

The RG-improved potential can then be transformed into the Einstein frame in order tocalculate the observables predicted by inflation3 The inflationary parameters are calculatedusing the methods discussed in [16] In particular the Einstein frame slow-roll parametersare given by

ε =m2

P

2

(1

U

dU

dχs

)2

η =m2

P

U

d2U

dχ2s

ξ2 =m4

P

U2

dU

dχs

d3U

dχ3s

(214)

3 Singlet scalar as dark matter

Let us now turn to the phenomenology of the singlet scalar in the present Universe andat energies well below the scale of inflation [19 24 35] Most importantly the assumedZ2 symmetry ensures the stability of the scalar so that it can potentially account for theobserved abundance of DM [2 3] If the mass of the singlet is comparable to the electroweakscale the singlet is a typical WIMP which obtains its relic abundance from thermal freeze-out Indeed at low energies where the effects of the non-minimal coupling to gravity arenegligible our model becomes identical to what is often referred to as Higgs Portal DarkMatter [21 22 36] because all interactions of the singlet with SM particles are mediatedby the Higgs In this section we review the constraints on these models and determine theparameter space allowed by the most recent experimental results In the process we pointout several discrepancies in the literature and resolve the resulting confusion

31 Relic abundance

The calculation of the relic abundance of singlet scalars is discussed in detail in [19] Threekinds of processes are relevant for the annihilation of singlets into SM states annihilation

3In [34] it was proposed to use of the Einstein frame for the computation of quantum corrections TheJordan frame analysis is however easier to implement correctly being a straightforward extension of theStandard Model analysis

ndash 5 ndash

JCAP11(2015)015

into SM fermions annihilation into SM gauge bosons and annihilation into two Higgs par-ticles The first kind dominates as long as ms lt mW while for larger masses the secondkind gives the largest contribution Notably all of these processes can proceed via an s-channel Higgs boson leading to a resonant enhancement of the annihilation cross-sectionand a corresponding suppression of the DM relic abundance for ms sim mh24

For the present work we calculate the singlet abundance using micrOMEGAs 3 [37]which numerically solves the Boltzmann equation while calculating the Higgs width in aself-consistent way It is then straightforward to numerically find the coupling λhs that givesΩs h

2 = 01197 in order to reproduce the value of the DM density ΩDM h2 = 01197plusmn 00022determined by Planck (TT + lowP 68 CL) [38] For example we find λhs asymp 008 forms = 300 GeV and λhs asymp 030 for ms = 1000 GeV These values agree with the ones foundin [19 23] but disagree with [20ndash22 24] by a factor of 2 after accounting for the differentconventions5

32 Direct detection constraints

The strongest constraints on λhs stem from DM direct detection experiments since thesinglet-Higgs coupling induces spin-independent interactions between the DM particle andnuclei The scattering cross-section at zero momentum transfer is given by [19]

σSI =λ2hsf

2N

4π

micro2r m2n

m4hm

2s

(31)

where mn is the neutron mass micror = (msmn)(ms + mn) is the reduced mass and fN isthe effective Higgs-nucleon coupling6 In terms of the light-quark matrix elements fNTq theeffective coupling can be written as

fN =

2

9+

7

9

sumq=uds

fNTq

(32)

The values of fNTq can either be determined phenomenologically from baryon masses andmeson-baryon scattering data or computed within lattice QCD A comparison of the differentmethods was recently performed in [19] and we adopt their result of fN = 030 for the effectivecoupling

The scattering cross-section given above can be directly compared to the bound obtainedfrom the LUX experiment [18] Indeed as shown in figure 1 LUX is typically sensitive to thesame range of values for λhs as what is implied by the relic density constraint Specificallythe LUX bound excludes the mass ranges 57 GeV lt ms lt 526 GeV and 645 GeV lt ms lt928 GeV

33 Invisible Higgs decays

Direct detection experiments cannot constrain singlet scalars with a mass of a few GeV orless since such particles would deposit too little energy in the detector to be observable This

4The process ssrarr hh also receives a contribution from t-channel singlet exchange which gives a relevantcontribution if λhs is large compared to λh

5The Higgs-singlet coupling is called λhS in [19] λHS in [20] λhSS in [21 22] λDM in [23] and a2 in [24]The respective conventions are captured by λhs = λhS = λHS2 = λhSS2 = λDM2 = 2 a2

6Note that [24] uses an approximate expression valid for ms mn such that micro2r asymp m2

n

ndash 6 ndash

JCAP11(2015)015

5 10 50 100 500 100010-4

0001

0010

0100

1

ms [GeV]

λhs

Figure 1 Excluded parameter regions from LUX (red dotted) and searches for invisible Higgs decays(blue dashed) compared to the coupling implied by the relic density constraint (green solid)

parameter region can however be efficiently constrained by considering how the Higgs-singletcoupling λhs would modify the branching ratios of the SM Higgs boson The partial decaywidth for hrarr ss is given by7

Γ(hrarr ss) =λ2hsv

2

32πmh

radic1minus 4m2

s

m2h

(33)

This theoretical prediction can be compared to the experimental bound on invisibleHiggs decays from the LHC Direct searches for invisible Higgs decays in the vector bosonfusion channel give BR(h rarr inv) 029 [39] A somewhat stronger bound can be obtainedfrom the observation that in our model there are no additional contributions to the Higgsproduction cross-section and no modifications of the partial decay widths of the Higgs bosoninto SM final states Therefore the presence of an invisible decay channel leads to an overallreduction of the signal strength in visible channels A global fit of all observed decay channels(combined with the bounds on invisible Higgs decays) then gives BR(hrarr inv) 026 [40]

Crucially the bound from invisible Higgs decays becomes independent of the singletmass for ms mh2 Invisible Higgs decays will therefore provide the strongest constraintsfor small singlet masses Indeed this constraint rules out the entire mass region where directdetection experiments lose sensitivity (see figure 1) As a result only two mass regions remainviable a low-mass region 526 GeV lt ms lt 645 GeV and a high-mass region ms amp 93 GeV

34 Other constraints

It has been pointed out recently [24] that bounds on γ-ray lines from Fermi-LAT [41] ruleout the parameter region where ms is slightly above mh2 To be safe from this constraintwe will focus on the mass range 526 GeV lt ms lt 624 GeV which we shall refer to as thelow-mass region For the high-mass region on the other hand there are no strong constraintsfrom indirect detection Moreover collider searches for singlet scalars with ms gt mh2 areextremely challenging [20ndash23 35] and consequently there are no relevant bounds from theLHC for the high-mass region [42]

The most significant improvements in sensitivity in the near future are expected tocome from direct detection experiments Indeed XENON1T [43] is expected to improve

7This equation agrees with [4 19ndash22 35] but disagrees with [23]

ndash 7 ndash

JCAP11(2015)015

upon current LUX constraints on the DM scattering cross-section by a factor of about 50and will therefore be able to probe the high-mass region up to ms asymp 4 TeV As we will showin S-inflation singlet masses larger than about 2 TeV are excluded by the Planck 2-σ upperbound on ns and perturbativity XENON1T will therefore be able to probe the entire high-mass region relevant for singlet inflation Similarly XENON1T can also further constrain thelow-mass region and potentially probe singlet masses in the range 53 GeV lt ms lt 57 GeV

4 Renormalisation group evolution and theoretical constraints

In order to connect the inflationary observables of our model to the measured SM parametersand the DM phenomenology discussed in the previous section we need to calculate theevolution of all couplings under the RG equations [8 10 12 14 44ndash47] Existing analyseshave considered the RG equations for the SM at two-loop order and the contributions of thesinglet sector and non-minimal coupling at one-loop order (see also the appendix) [8 10] Toexamine the issue of vacuum stability we improve the accuracy of our analysis further byincorporating the three-loop RG equations for the SM gauge couplings [48] and the leadingorder three-loop corrections to the RG equations for λh and yt [49]8

When considering large field values for either s or h the RG equations are modifiedbecause there is a suppression of scalar propagators This suppression is captured by insertinga factor

cφ =1 +

ξφ φ2

mP

1 + (6ξφ + 1)ξφ φ2

mP

(41)

with φ = s (φ = h) for each s (h) propagating in a loop [8] The changes in the RGequations for large values of the Higgs field have been discussed in detail in [47] Themodifications resulting from large singlet field values can be found in [8] and are reviewedin the appendix Note that when considering S-inflation such that s h we can set thesuppression factor ch = 1

We determine the values of the SM parameters at micro = mt following [46] Using themost recent values from the Particle Data Group [50]

mt = (1732plusmn 09) GeV mH = (12509plusmn 024) GeV αS(mZ) = 01185plusmn 00006 (42)

we obtain at micro = mt

yt = 0936plusmn 0005 λh = 01260plusmn 00014 gS = 1164plusmn 0003 (43)

Unless explicitly stated otherwise we will use the central values for all calculations belowFor given couplings at micro = mt we then use the public code RGErun 207 [51] to calculatethe couplings at higher scales

In contrast to the remaining couplings we fix the non-minimal couplings ξh and ξs atmicro = mP In order to obtain the correct amplitude of the scalar power spectrum we require

U

ε= (000271mP)4 (44)

where U and ε are the potential and the first slow-roll parameter in the Einstein frameat the beginning of inflation as defined in eq (28) and eq (214) respectively Imposing

8We thank Kyle Allison for sharing his numerical implementation of these equations

ndash 8 ndash

JCAP11(2015)015

equation (44) allows us to determine ξs at the scale of inflation once all other parametershave been fixed Note that since the value of s at the beginning of inflation also depends onξs equation (44) can only be solved numerically We then iteratively determine the requiredvalue of ξs at the electroweak scale such that RG evolution yields the desired value at thescale of inflation

The coupling ξh plays a very limited role for the phenomenology of our model because wedo not consider the case of large Higgs field values for inflation As a result our predictionsfor the inflationary observables show only a very mild dependence on ξh so that ξh canessentially be chosen arbitrarily Nevertheless it is not possible to simply set this parameterto zero since radiative corrections induce a mixing between ξs and ξh Moreover we willsee below that the value of ξh plays an important role for determining whether our modelviolates unitarity below the scale of inflation As with ξs we fix ξh at mP and then determineiteratively the required value of ξh at the electroweak scale9

41 Metastability

We now discuss various theoretical constraints related to the RG evolution of the parame-ters in our model It is a well-known fact that for the central values of the measured SMparameters the electroweak vacuum becomes metastable at high scales because the quarticHiggs coupling λh runs to negative values (see eg [46]) This metastability is not in anyobvious way a problem as the lifetime of the electroweak vacuum is well above the age ofthe Universe [52] (note however that this estimate may potentially be spoiled by effectsfrom Planck-scale higher-dimensional operators [53ndash55]) However one may speculate thata stable electroweak vacuum is necessary for a consistent theory for example if the vacuumenergy relative to the absolute minimum is a physical energy density leading to inflation Itis therefore an interesting aspect of singlet extensions of the SM that scalar singlets give apositive contribution to the running of λh [56 57]

βλh = βSMλh +1

32π2c2s λ

2hs (45)

In fact it was shown in [58] that for the case of a minimally-coupled singlet λhs can bechosen such that the electroweak vacuum remains stable all the way up to the Planck scaleand at the same time (for appropriate choices of the singlet mass ms) the singlet obtains athermal relic density compatible with the observed DM abundance10

In order to study electroweak vacuum stability we need to consider the potential in theh-direction with s = 011 Vacuum stability then requires that λh(micro) gt 0 for micro up to mPIn the present study we consider both the case where λhs is sufficiently large to stabilisethe electroweak vacuum and the case where λhs only increases the lifetime of the metastablevacuum but does not render it completely stable We focus throughout on the case whereλhs is positive

42 Examples

Figure 2 shows an example for the evolution of scalar couplings (left) and the non-minimalcouplings (right) under the RG equations discussed above Solid lines correspond to the case

9The running of ξh between mP and the scale of inflation is completely negligible since the relevantdiagrams are strongly suppressed for large values of s

10Note that if the singlet mixes with the Higgs there will be additional threshold effects at micro = ms fromintegrating out the singlet [59] In the setup we consider however this effect is not important [60]

11A more detailed study of the potential in general directions with both s 6= 0 and h 6= 0 (along the linesof [61]) is beyond the scope of the present work

ndash 9 ndash

JCAP11(2015)015

1000 106 109 1012 1015 10180001

00050010

00500100

05001

μ [GeV]

Couplingstrength

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

λh

λhs

λs

1000 106 109 1012 1015 101810

100

1000

104

105

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

ξh

ξs

cs

Figure 2 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplingsξh and ξs (right) as a function of the renormalisation scale micro for a typical parameter point in thehigh-mass region Solid lines show the running in the s-direction while dotted lines correspond to therunning in the h-direction In the right panel we also show the suppression factor cs which modifiesthe running in the s-direction at large field value

s h which is relevant for inflation while dotted lines correspond to h s which is relevantfor vacuum stability We fix the scalar couplings at the weak scale choosing λhs = 025 andλs = 001 such that the observed relic abundance can be reproduced for ms = 835 GeV Weconsider λs λhs in which case the value of ξs necessary to obtain the correct amplitude ofthe scalar power spectrum is reduced For our choice we find ξs sim 104 Note however thatwhile λhs exhibits only moderate running λs grows significantly with the renormalisationscale micro because its β-function contains a term proportional to λ2hs Choosing even smallervalues of λs at the weak scale will therefore not significantly reduce its value at the scale ofinflation nor the corresponding value of ξs

An important observation from figure 2 is that λh does not run negative and hencethe electroweak vacuum remains stable all the way up to the Planck scale The additionalcontribution from the singlet scalar is sufficient to ensure λh gt 10minus3 for all renormalisationscales up to mP For field values s amp 1015 GeV h the singlet propagator is suppressedleading to a visible kink in the running of λh We show the propagator suppression factorcs in the right panel of figure 2 One can clearly see how this suppression factor affects therunning of ξh which becomes nearly constant for s amp 1015 GeV In this particular examplewe have chosen ξh(mP) = 1000 (for s h) This choice together with λs 1 implies thatthe running of ξs(micro) from mt to mP is negligible

In the low-mass region we are interested in much smaller values of λhs typically below10minus2 A particular example is shown in figure 3 (left) for the representative choice λhs(mt) =0002 and λs = 00005 which yields the observed relic abundance for ms asymp 57 GeV Weobserve that if λs and λhs are both small at the electroweak scale these couplings exhibitonly very little running up to the scale of inflation For the same reason it is impossible toinfluence the running of the Higgs couplings sufficiently to prevent λhs from running negativeat around 1011 GeV

If λhs is small we can obtain the correct scalar power spectrum amplitude with a muchsmaller value of ξs during inflation For the specific case considered in figure 3 we findξs sim 103 For these values of ξs and λhs the loop-induced corrections to ξh are very smalland hence this coupling changes only very slightly under RG evolution

43 Perturbativity

In order for our calculation of the running couplings and the radiative corrections to thepotential to be reliable we must require that all couplings remain perturbative up to the

ndash 10 ndash

JCAP11(2015)015

1000 106 109 1012 1015 101810-4

0001

0010

0100

1

μ [GeV]

Couplingstrength

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

λh

λhs

λs

1000 106 109 1012 1015 101801

1

10

100

1000

104

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

ξh

ξs

cs

Figure 3 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplings ξhand ξs (right) as a function of the renormalisation scale micro for a typical parameter point in the low-mass region In the right panel we also show the suppression factor cs which modifies the runningat large field value Note that λh runs negative for micro amp 1011 GeV

scale of inflation which is typically 1017ndash1018 GeV This requirement is easily satisfied forthe SM couplings but needs to be checked explicitly for the couplings of the singlet whichcan grow significantly with increasing renormalisation scale micro We follow [58 62] and usethe requirement of perturbative unitarity to impose an upper bound on the scalar couplingsThis procedure gives

λs lt4π

3and λhs lt 8π (46)

As we will see below the non-minimal coupling ξs can be much larger than unity withoutinvalidating a perturbative calculation Nevertheless if ξs and ξh are both very large pro-cesses involving both couplings may violate perturbative unitarity implying that there maybe new physics or strong coupling below the scale of inflation We will now discuss this issuein more detail

5 Unitarity-violation during inflation

In this section we estimate the scale of perturbative unitarity-violation as a function of thebackground inflaton field Note that by ldquounitarity-violation scalerdquo we mean the scale at whichperturbation theory in scalar scattering breaks down so this may in fact indicate the onsetof unitarity-conserving scattering in a strongly-coupled regime [28 31] We will considerunitarity-violation in the scattering of scalar particles corresponding to perturbations aboutthe background field In the case of a real scalar s and the fields of the Higgs doublet Hthere are two distinct scattering processes we need to consider (i) δs h1 harr δs h1 and (ii)h1h2 harr h1h2 where δs is the perturbation about the background s field and h1 and h2are two of the Higgs doublet scalars Scattering with the other scalars in H is equivalentto these two processes We will use dimensional analysis to estimate the scale of unitarity-violation by determining the leading-order processes in the Einstein frame which result inunitarity-violating scattering

It will be sufficient to consider the Einstein frame Lagrangian for two real scalar fieldsφi where mdash using the notation of [30] mdash φi stands for either s or a component of the Higgsdoublet Unitarity-violation requires that there are two different scalars in the scatteringprocess since in the case of a single scalar there is a cancellation between s- t- and u-channelamplitudes [63]

ndash 11 ndash

JCAP11(2015)015

Since unitarity-violating scattering in the Jordan frame is due to graviton exchange viathe non-minimal coupling to R we cat set V = 0 The Einstein frame action for two realscalars is then of the form

SE =

intd4xradicminusg

Lii +sumiltj

Lij minus1

2m2

P R

(51)

where

Lii =1

2

Ω2 +6 ξ2i φ

2i

m2P

Ω4

gmicroν partmicroφi partνφi and Lij =6 ξi ξj φi φj g

microν partmicroφi partνφjm2

P Ω4(52)

with

Ω2 = 1 +ξj φ

2j

m2P

(53)

The interaction terms proportional to ξi ξj are responsible for the dominant unitarity-viola-tion in scattering cross-sections calculated in the Einstein frame These interactions are theEinstein frame analogue of scalar scattering via graviton exchange in the Jordan frame due tothe non-minimal coupling To obtain the scale of unitarity-violation in terms of the physicalenergy defined in the Jordan frame we first canonically normalize the fields in the Einsteinframe then estimate the magnitude of the scattering matrix element and finally transformthe unitarity-violation scale in the Einstein frame back to that in the Jordan frame

In the following we will denote the inflaton by φ1 (equiv s) which we expand about thebackground field ie φ1 = φ1 + δφ1 The Higgs doublet scalars are denoted by φ2 (equiv h1)and φ3 (equiv h2) The corresponding canonically normalized scattering fields in the Einsteinframe are then defined to be ϕ1 ϕ2 and ϕ3 Once we have determined the interactions ofthe canonically normalised fields we use dimensional analysis to estimate the scale of tree-level unitarity-violation For this purpose we introduce appropriate factors of E to makethe coefficient of the interaction terms in L dimensionless Energy scales E in the Einsteinframe are related to the ones in the Jordan frame via E = EΩ where during inflation

Ω2 ξ1φ21m

2P asymp N 1 Unitarity conservation implies that the matrix element for any

2 harr 2 scattering process should be smaller than O(1) so we can determine the scale ofunitarity-violation (denoted by Λ) by determining the value of E that saturates this boundThis was demonstrated explicitly in [8] by comparing the dimensional estimate with theexact value from the full scattering amplitude

In the following we consider three regimes for φ1 each of which leads to a differentform of the Lagrangian and the scattering amplitudes

bull Regime A large field values In this regime we have Ω gt 1 (implying that ξ1 φ21m

2P gt 1)

and 6 ξ21 φ21m

2P gt 1

bull Regime B medium field values In this regime we have approximately Ω asymp 1 (implying

that ξ1 φ21m

2P lt 1) but still 6 ξ21 φ

21m

2P gt 1

bull Regime C small field values Finally we consider 6 ξ21 φ21m

2P lt 1 which in particular

implies Ω asymp 1

ndash 12 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

JCAP11(2015)015

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

The paper is organized as follows In section 2 we review the real gauge singlet scalar ex-tension of the SM and the S-inflation model We estimate the predictions of the model for thespectral index ns and discuss the effect of constraints from inflation on the model parameterspace Section 3 considers the DM phenomenology of the model and the implications fromDM searches In section 4 we discuss how to connect these two aspects via renormalisationgroup evolution and which constraints follow from electroweak vacuum stability and pertur-bativity The scale of unitarity-violation during inflation and the consistency of S-inflationare discussed in section 5 Finally we present our results in section 6 and our conclusions insection 7 Additional details are provided in the appendix

2 The S-inflation model

S-inflation is a version of the non-minimally coupled inflation model of [7] in which the scalarfield is identified with the gauge singlet scalar responsible for thermal relic cold DM In thepresent work we focus on the case of a real singlet scalar s In the Jordan frame which isthe standard frame for interpreting measurements and calculating radiative corrections theaction for this model is

SJ =

int radicminusg d4x

[LSM + (partmicroH)dagger (partmicroH) +

1

2partmicros part

micros

minusm2

PR

2minus ξhHdaggerH Rminus 1

2ξs s

2Rminus V (s2 HdaggerH)

] (21)

where LSM is the SM Lagrangian density minus the purely Higgs doublet terms mP is thereduced Planck mass and

V (s2 HdaggerH) = λh

[(HdaggerH

)minus v2

2

]2+

1

2λhs s

2HdaggerH +1

4λs s

4 +1

2m2s0 s

2 (22)

with v = 246 GeV the vacuum expectation value of the Higgs field Writing H = (h+v 0)radic

2with a real scalar h we obtain

V (s2 h) = V (h) +1

2m2s s

2 +1

4λs s

4 +1

2λhs v h s

2 +1

4λhs h

2 s2 (23)

where we have introduced the physical singlet mass m2s = m2

s0 + λhs v22

In order to calculate the observables predicted by inflation we perform a conformaltransformation to the Einstein frame where the non-minimal coupling to gravity disappearsIn the case that s 6= 0 and h = 0 this transformation is defined by

gmicroν = Ω2 gmicroν Ω2 = 1 +ξs s

2

m2P

(24)

The transformation yields

SE =

int radicminusg d4x

[LSM +

1

2

(1

Ω2+

6 ξ2s s2

m2P Ω4

)gmicroνpartmicrospartνsminus

m2P R

2minus V (s 0)

Ω4

] (25)

ndash 3 ndash

JCAP11(2015)015

where R is the Ricci scalar with respect to gmicroν We can then rescale the field using

dχsds

=

radicΩ2 + 6 ξ2s s

2m2P

Ω4 (26)

which gives

SE =

int radicminusg d4x

(LSM minus

m2P R

2+

1

2gmicroνpartmicroχspartνχs minus U(χs 0)

) (27)

with

U(χs 0) =λs s

4(χs)

4 Ω4 (28)

The relationship between s and χs is determined by the solution to eq (26) In particularfor s mP

radicξs the Einstein frame potential is

U(χs 0) =λsm

4P

4 ξ2s

(1 + exp

(minus 2χsradic

6mP

))minus2 (29)

This is sufficiently flat at large χs to support slow-roll inflationAn analogous expression is obtained for the potential along the h-direction In both

cases the Einstein frame potential is proportional to λφξ2φ where φ = s or h Therefore

the minimum of the potential at large s and h will be very close to h = 0 and inflation willnaturally occur along the s-direction if λsξ

2s λhξ

2h which is true for example if ξs ξh

and λs sim λhIn the following inflation is always considered to be in the direction of s with h = 0

The conventional analysis of inflation can then be performed in the Einstein frame Afterinflation the Jordan and Einstein frames will be indistinguishable since ξs s

2 m2P and

so Ω rarr 1 Therefore the curvature perturbation spectrum calculated in the Einstein framebecomes equal to that observed in the physical Jordan frame at late times

The classical (tree-level) predictions for the spectral index and tensor-to-scalar ratioare [33]

ntrees asymp 1minus 2

Nminus 3

2N2+O

(1

N3

)= 0965 (210)

rtree asymp 12

N2+O

(1

ξsN2

)= 36times 10minus3 (211)

while the field during inflation is

s2Nasymp 4m2

P N(3 ξs) (212)

In the equations above N is the number of e-foldings as defined in the Einstein framewhich differs from that in the Jordan frame by N asymp N + ln(1

radicN) [8] and we have used

N = 582 The classical predictions are in good agreement with the most recent Planck

2Reheating in S-inflation occurs via stochastic resonance to Higgs bosons through the coupling λhs Itwas shown in [16] that this process is very efficient and makes quite precise predictions for the reheatingtemperature and the number of e-foldings of inflation with 57 N 60 at the WMAP pivot scale This inturn allows for quite precise predictions of the inflation observables

ndash 4 ndash

JCAP11(2015)015

values ns = 09677plusmn 00060 (68 confidence level (CL) Planck TT + lowP + lensing) andr0002 lt 011 (95 CL Planck TT + lowP + lensing) [1]

The classical predictions for S-inflation are the same as those of any model based onthe SBB model Differences between S-inflation and other models do however arise fromquantum corrections to the effective potential To include these corrections we calculatethe RG evolution of the various couplings as a function of the renormalisation scale micro (seesection 4 for details) We can then obtain the renormalisation group (RG)-improved effectivepotential for s in the Jordan frame by replacing the couplings in eq (23) by the runningcouplings and setting micro equal to the value of the field For h = 0 (and neglecting the singletmass term ms sim 1 TeV s) this approach yields

VRG(s2 0) =λs(s) s

4

4 (213)

The RG-improved potential can then be transformed into the Einstein frame in order tocalculate the observables predicted by inflation3 The inflationary parameters are calculatedusing the methods discussed in [16] In particular the Einstein frame slow-roll parametersare given by

ε =m2

P

2

(1

U

dU

dχs

)2

η =m2

P

U

d2U

dχ2s

ξ2 =m4

P

U2

dU

dχs

d3U

dχ3s

(214)

3 Singlet scalar as dark matter

Let us now turn to the phenomenology of the singlet scalar in the present Universe andat energies well below the scale of inflation [19 24 35] Most importantly the assumedZ2 symmetry ensures the stability of the scalar so that it can potentially account for theobserved abundance of DM [2 3] If the mass of the singlet is comparable to the electroweakscale the singlet is a typical WIMP which obtains its relic abundance from thermal freeze-out Indeed at low energies where the effects of the non-minimal coupling to gravity arenegligible our model becomes identical to what is often referred to as Higgs Portal DarkMatter [21 22 36] because all interactions of the singlet with SM particles are mediatedby the Higgs In this section we review the constraints on these models and determine theparameter space allowed by the most recent experimental results In the process we pointout several discrepancies in the literature and resolve the resulting confusion

31 Relic abundance

The calculation of the relic abundance of singlet scalars is discussed in detail in [19] Threekinds of processes are relevant for the annihilation of singlets into SM states annihilation

3In [34] it was proposed to use of the Einstein frame for the computation of quantum corrections TheJordan frame analysis is however easier to implement correctly being a straightforward extension of theStandard Model analysis

ndash 5 ndash

JCAP11(2015)015

into SM fermions annihilation into SM gauge bosons and annihilation into two Higgs par-ticles The first kind dominates as long as ms lt mW while for larger masses the secondkind gives the largest contribution Notably all of these processes can proceed via an s-channel Higgs boson leading to a resonant enhancement of the annihilation cross-sectionand a corresponding suppression of the DM relic abundance for ms sim mh24

For the present work we calculate the singlet abundance using micrOMEGAs 3 [37]which numerically solves the Boltzmann equation while calculating the Higgs width in aself-consistent way It is then straightforward to numerically find the coupling λhs that givesΩs h

2 = 01197 in order to reproduce the value of the DM density ΩDM h2 = 01197plusmn 00022determined by Planck (TT + lowP 68 CL) [38] For example we find λhs asymp 008 forms = 300 GeV and λhs asymp 030 for ms = 1000 GeV These values agree with the ones foundin [19 23] but disagree with [20ndash22 24] by a factor of 2 after accounting for the differentconventions5

32 Direct detection constraints

The strongest constraints on λhs stem from DM direct detection experiments since thesinglet-Higgs coupling induces spin-independent interactions between the DM particle andnuclei The scattering cross-section at zero momentum transfer is given by [19]

σSI =λ2hsf

2N

4π

micro2r m2n

m4hm

2s

(31)

where mn is the neutron mass micror = (msmn)(ms + mn) is the reduced mass and fN isthe effective Higgs-nucleon coupling6 In terms of the light-quark matrix elements fNTq theeffective coupling can be written as

fN =

2

9+

7

9

sumq=uds

fNTq

(32)

The values of fNTq can either be determined phenomenologically from baryon masses andmeson-baryon scattering data or computed within lattice QCD A comparison of the differentmethods was recently performed in [19] and we adopt their result of fN = 030 for the effectivecoupling

The scattering cross-section given above can be directly compared to the bound obtainedfrom the LUX experiment [18] Indeed as shown in figure 1 LUX is typically sensitive to thesame range of values for λhs as what is implied by the relic density constraint Specificallythe LUX bound excludes the mass ranges 57 GeV lt ms lt 526 GeV and 645 GeV lt ms lt928 GeV

33 Invisible Higgs decays

Direct detection experiments cannot constrain singlet scalars with a mass of a few GeV orless since such particles would deposit too little energy in the detector to be observable This

4The process ssrarr hh also receives a contribution from t-channel singlet exchange which gives a relevantcontribution if λhs is large compared to λh

5The Higgs-singlet coupling is called λhS in [19] λHS in [20] λhSS in [21 22] λDM in [23] and a2 in [24]The respective conventions are captured by λhs = λhS = λHS2 = λhSS2 = λDM2 = 2 a2

6Note that [24] uses an approximate expression valid for ms mn such that micro2r asymp m2

n

ndash 6 ndash

JCAP11(2015)015

5 10 50 100 500 100010-4

0001

0010

0100

1

ms [GeV]

λhs

Figure 1 Excluded parameter regions from LUX (red dotted) and searches for invisible Higgs decays(blue dashed) compared to the coupling implied by the relic density constraint (green solid)

parameter region can however be efficiently constrained by considering how the Higgs-singletcoupling λhs would modify the branching ratios of the SM Higgs boson The partial decaywidth for hrarr ss is given by7

Γ(hrarr ss) =λ2hsv

2

32πmh

radic1minus 4m2

s

m2h

(33)

This theoretical prediction can be compared to the experimental bound on invisibleHiggs decays from the LHC Direct searches for invisible Higgs decays in the vector bosonfusion channel give BR(h rarr inv) 029 [39] A somewhat stronger bound can be obtainedfrom the observation that in our model there are no additional contributions to the Higgsproduction cross-section and no modifications of the partial decay widths of the Higgs bosoninto SM final states Therefore the presence of an invisible decay channel leads to an overallreduction of the signal strength in visible channels A global fit of all observed decay channels(combined with the bounds on invisible Higgs decays) then gives BR(hrarr inv) 026 [40]

Crucially the bound from invisible Higgs decays becomes independent of the singletmass for ms mh2 Invisible Higgs decays will therefore provide the strongest constraintsfor small singlet masses Indeed this constraint rules out the entire mass region where directdetection experiments lose sensitivity (see figure 1) As a result only two mass regions remainviable a low-mass region 526 GeV lt ms lt 645 GeV and a high-mass region ms amp 93 GeV

34 Other constraints

It has been pointed out recently [24] that bounds on γ-ray lines from Fermi-LAT [41] ruleout the parameter region where ms is slightly above mh2 To be safe from this constraintwe will focus on the mass range 526 GeV lt ms lt 624 GeV which we shall refer to as thelow-mass region For the high-mass region on the other hand there are no strong constraintsfrom indirect detection Moreover collider searches for singlet scalars with ms gt mh2 areextremely challenging [20ndash23 35] and consequently there are no relevant bounds from theLHC for the high-mass region [42]

The most significant improvements in sensitivity in the near future are expected tocome from direct detection experiments Indeed XENON1T [43] is expected to improve

7This equation agrees with [4 19ndash22 35] but disagrees with [23]

ndash 7 ndash

JCAP11(2015)015

upon current LUX constraints on the DM scattering cross-section by a factor of about 50and will therefore be able to probe the high-mass region up to ms asymp 4 TeV As we will showin S-inflation singlet masses larger than about 2 TeV are excluded by the Planck 2-σ upperbound on ns and perturbativity XENON1T will therefore be able to probe the entire high-mass region relevant for singlet inflation Similarly XENON1T can also further constrain thelow-mass region and potentially probe singlet masses in the range 53 GeV lt ms lt 57 GeV

4 Renormalisation group evolution and theoretical constraints

In order to connect the inflationary observables of our model to the measured SM parametersand the DM phenomenology discussed in the previous section we need to calculate theevolution of all couplings under the RG equations [8 10 12 14 44ndash47] Existing analyseshave considered the RG equations for the SM at two-loop order and the contributions of thesinglet sector and non-minimal coupling at one-loop order (see also the appendix) [8 10] Toexamine the issue of vacuum stability we improve the accuracy of our analysis further byincorporating the three-loop RG equations for the SM gauge couplings [48] and the leadingorder three-loop corrections to the RG equations for λh and yt [49]8

When considering large field values for either s or h the RG equations are modifiedbecause there is a suppression of scalar propagators This suppression is captured by insertinga factor

cφ =1 +

ξφ φ2

mP

1 + (6ξφ + 1)ξφ φ2

mP

(41)

with φ = s (φ = h) for each s (h) propagating in a loop [8] The changes in the RGequations for large values of the Higgs field have been discussed in detail in [47] Themodifications resulting from large singlet field values can be found in [8] and are reviewedin the appendix Note that when considering S-inflation such that s h we can set thesuppression factor ch = 1

We determine the values of the SM parameters at micro = mt following [46] Using themost recent values from the Particle Data Group [50]

mt = (1732plusmn 09) GeV mH = (12509plusmn 024) GeV αS(mZ) = 01185plusmn 00006 (42)

we obtain at micro = mt

yt = 0936plusmn 0005 λh = 01260plusmn 00014 gS = 1164plusmn 0003 (43)

Unless explicitly stated otherwise we will use the central values for all calculations belowFor given couplings at micro = mt we then use the public code RGErun 207 [51] to calculatethe couplings at higher scales

In contrast to the remaining couplings we fix the non-minimal couplings ξh and ξs atmicro = mP In order to obtain the correct amplitude of the scalar power spectrum we require

U

ε= (000271mP)4 (44)

where U and ε are the potential and the first slow-roll parameter in the Einstein frameat the beginning of inflation as defined in eq (28) and eq (214) respectively Imposing

8We thank Kyle Allison for sharing his numerical implementation of these equations

ndash 8 ndash

JCAP11(2015)015

equation (44) allows us to determine ξs at the scale of inflation once all other parametershave been fixed Note that since the value of s at the beginning of inflation also depends onξs equation (44) can only be solved numerically We then iteratively determine the requiredvalue of ξs at the electroweak scale such that RG evolution yields the desired value at thescale of inflation

The coupling ξh plays a very limited role for the phenomenology of our model because wedo not consider the case of large Higgs field values for inflation As a result our predictionsfor the inflationary observables show only a very mild dependence on ξh so that ξh canessentially be chosen arbitrarily Nevertheless it is not possible to simply set this parameterto zero since radiative corrections induce a mixing between ξs and ξh Moreover we willsee below that the value of ξh plays an important role for determining whether our modelviolates unitarity below the scale of inflation As with ξs we fix ξh at mP and then determineiteratively the required value of ξh at the electroweak scale9

41 Metastability

We now discuss various theoretical constraints related to the RG evolution of the parame-ters in our model It is a well-known fact that for the central values of the measured SMparameters the electroweak vacuum becomes metastable at high scales because the quarticHiggs coupling λh runs to negative values (see eg [46]) This metastability is not in anyobvious way a problem as the lifetime of the electroweak vacuum is well above the age ofthe Universe [52] (note however that this estimate may potentially be spoiled by effectsfrom Planck-scale higher-dimensional operators [53ndash55]) However one may speculate thata stable electroweak vacuum is necessary for a consistent theory for example if the vacuumenergy relative to the absolute minimum is a physical energy density leading to inflation Itis therefore an interesting aspect of singlet extensions of the SM that scalar singlets give apositive contribution to the running of λh [56 57]

βλh = βSMλh +1

32π2c2s λ

2hs (45)

In fact it was shown in [58] that for the case of a minimally-coupled singlet λhs can bechosen such that the electroweak vacuum remains stable all the way up to the Planck scaleand at the same time (for appropriate choices of the singlet mass ms) the singlet obtains athermal relic density compatible with the observed DM abundance10

In order to study electroweak vacuum stability we need to consider the potential in theh-direction with s = 011 Vacuum stability then requires that λh(micro) gt 0 for micro up to mPIn the present study we consider both the case where λhs is sufficiently large to stabilisethe electroweak vacuum and the case where λhs only increases the lifetime of the metastablevacuum but does not render it completely stable We focus throughout on the case whereλhs is positive

42 Examples

Figure 2 shows an example for the evolution of scalar couplings (left) and the non-minimalcouplings (right) under the RG equations discussed above Solid lines correspond to the case

9The running of ξh between mP and the scale of inflation is completely negligible since the relevantdiagrams are strongly suppressed for large values of s

10Note that if the singlet mixes with the Higgs there will be additional threshold effects at micro = ms fromintegrating out the singlet [59] In the setup we consider however this effect is not important [60]

11A more detailed study of the potential in general directions with both s 6= 0 and h 6= 0 (along the linesof [61]) is beyond the scope of the present work

ndash 9 ndash

JCAP11(2015)015

1000 106 109 1012 1015 10180001

00050010

00500100

05001

μ [GeV]

Couplingstrength

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

λh

λhs

λs

1000 106 109 1012 1015 101810

100

1000

104

105

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

ξh

ξs

cs

Figure 2 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplingsξh and ξs (right) as a function of the renormalisation scale micro for a typical parameter point in thehigh-mass region Solid lines show the running in the s-direction while dotted lines correspond to therunning in the h-direction In the right panel we also show the suppression factor cs which modifiesthe running in the s-direction at large field value

s h which is relevant for inflation while dotted lines correspond to h s which is relevantfor vacuum stability We fix the scalar couplings at the weak scale choosing λhs = 025 andλs = 001 such that the observed relic abundance can be reproduced for ms = 835 GeV Weconsider λs λhs in which case the value of ξs necessary to obtain the correct amplitude ofthe scalar power spectrum is reduced For our choice we find ξs sim 104 Note however thatwhile λhs exhibits only moderate running λs grows significantly with the renormalisationscale micro because its β-function contains a term proportional to λ2hs Choosing even smallervalues of λs at the weak scale will therefore not significantly reduce its value at the scale ofinflation nor the corresponding value of ξs

An important observation from figure 2 is that λh does not run negative and hencethe electroweak vacuum remains stable all the way up to the Planck scale The additionalcontribution from the singlet scalar is sufficient to ensure λh gt 10minus3 for all renormalisationscales up to mP For field values s amp 1015 GeV h the singlet propagator is suppressedleading to a visible kink in the running of λh We show the propagator suppression factorcs in the right panel of figure 2 One can clearly see how this suppression factor affects therunning of ξh which becomes nearly constant for s amp 1015 GeV In this particular examplewe have chosen ξh(mP) = 1000 (for s h) This choice together with λs 1 implies thatthe running of ξs(micro) from mt to mP is negligible

In the low-mass region we are interested in much smaller values of λhs typically below10minus2 A particular example is shown in figure 3 (left) for the representative choice λhs(mt) =0002 and λs = 00005 which yields the observed relic abundance for ms asymp 57 GeV Weobserve that if λs and λhs are both small at the electroweak scale these couplings exhibitonly very little running up to the scale of inflation For the same reason it is impossible toinfluence the running of the Higgs couplings sufficiently to prevent λhs from running negativeat around 1011 GeV

If λhs is small we can obtain the correct scalar power spectrum amplitude with a muchsmaller value of ξs during inflation For the specific case considered in figure 3 we findξs sim 103 For these values of ξs and λhs the loop-induced corrections to ξh are very smalland hence this coupling changes only very slightly under RG evolution

43 Perturbativity

In order for our calculation of the running couplings and the radiative corrections to thepotential to be reliable we must require that all couplings remain perturbative up to the

ndash 10 ndash

JCAP11(2015)015

1000 106 109 1012 1015 101810-4

0001

0010

0100

1

μ [GeV]

Couplingstrength

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

λh

λhs

λs

1000 106 109 1012 1015 101801

1

10

100

1000

104

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

ξh

ξs

cs

Figure 3 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplings ξhand ξs (right) as a function of the renormalisation scale micro for a typical parameter point in the low-mass region In the right panel we also show the suppression factor cs which modifies the runningat large field value Note that λh runs negative for micro amp 1011 GeV

scale of inflation which is typically 1017ndash1018 GeV This requirement is easily satisfied forthe SM couplings but needs to be checked explicitly for the couplings of the singlet whichcan grow significantly with increasing renormalisation scale micro We follow [58 62] and usethe requirement of perturbative unitarity to impose an upper bound on the scalar couplingsThis procedure gives

λs lt4π

3and λhs lt 8π (46)

As we will see below the non-minimal coupling ξs can be much larger than unity withoutinvalidating a perturbative calculation Nevertheless if ξs and ξh are both very large pro-cesses involving both couplings may violate perturbative unitarity implying that there maybe new physics or strong coupling below the scale of inflation We will now discuss this issuein more detail

5 Unitarity-violation during inflation

In this section we estimate the scale of perturbative unitarity-violation as a function of thebackground inflaton field Note that by ldquounitarity-violation scalerdquo we mean the scale at whichperturbation theory in scalar scattering breaks down so this may in fact indicate the onsetof unitarity-conserving scattering in a strongly-coupled regime [28 31] We will considerunitarity-violation in the scattering of scalar particles corresponding to perturbations aboutthe background field In the case of a real scalar s and the fields of the Higgs doublet Hthere are two distinct scattering processes we need to consider (i) δs h1 harr δs h1 and (ii)h1h2 harr h1h2 where δs is the perturbation about the background s field and h1 and h2are two of the Higgs doublet scalars Scattering with the other scalars in H is equivalentto these two processes We will use dimensional analysis to estimate the scale of unitarity-violation by determining the leading-order processes in the Einstein frame which result inunitarity-violating scattering

It will be sufficient to consider the Einstein frame Lagrangian for two real scalar fieldsφi where mdash using the notation of [30] mdash φi stands for either s or a component of the Higgsdoublet Unitarity-violation requires that there are two different scalars in the scatteringprocess since in the case of a single scalar there is a cancellation between s- t- and u-channelamplitudes [63]

ndash 11 ndash

JCAP11(2015)015

Since unitarity-violating scattering in the Jordan frame is due to graviton exchange viathe non-minimal coupling to R we cat set V = 0 The Einstein frame action for two realscalars is then of the form

SE =

intd4xradicminusg

Lii +sumiltj

Lij minus1

2m2

P R

(51)

where

Lii =1

2

Ω2 +6 ξ2i φ

2i

m2P

Ω4

gmicroν partmicroφi partνφi and Lij =6 ξi ξj φi φj g

microν partmicroφi partνφjm2

P Ω4(52)

with

Ω2 = 1 +ξj φ

2j

m2P

(53)

The interaction terms proportional to ξi ξj are responsible for the dominant unitarity-viola-tion in scattering cross-sections calculated in the Einstein frame These interactions are theEinstein frame analogue of scalar scattering via graviton exchange in the Jordan frame due tothe non-minimal coupling To obtain the scale of unitarity-violation in terms of the physicalenergy defined in the Jordan frame we first canonically normalize the fields in the Einsteinframe then estimate the magnitude of the scattering matrix element and finally transformthe unitarity-violation scale in the Einstein frame back to that in the Jordan frame

In the following we will denote the inflaton by φ1 (equiv s) which we expand about thebackground field ie φ1 = φ1 + δφ1 The Higgs doublet scalars are denoted by φ2 (equiv h1)and φ3 (equiv h2) The corresponding canonically normalized scattering fields in the Einsteinframe are then defined to be ϕ1 ϕ2 and ϕ3 Once we have determined the interactions ofthe canonically normalised fields we use dimensional analysis to estimate the scale of tree-level unitarity-violation For this purpose we introduce appropriate factors of E to makethe coefficient of the interaction terms in L dimensionless Energy scales E in the Einsteinframe are related to the ones in the Jordan frame via E = EΩ where during inflation

Ω2 ξ1φ21m

2P asymp N 1 Unitarity conservation implies that the matrix element for any

2 harr 2 scattering process should be smaller than O(1) so we can determine the scale ofunitarity-violation (denoted by Λ) by determining the value of E that saturates this boundThis was demonstrated explicitly in [8] by comparing the dimensional estimate with theexact value from the full scattering amplitude

In the following we consider three regimes for φ1 each of which leads to a differentform of the Lagrangian and the scattering amplitudes

bull Regime A large field values In this regime we have Ω gt 1 (implying that ξ1 φ21m

2P gt 1)

and 6 ξ21 φ21m

2P gt 1

bull Regime B medium field values In this regime we have approximately Ω asymp 1 (implying

that ξ1 φ21m

2P lt 1) but still 6 ξ21 φ

21m

2P gt 1

bull Regime C small field values Finally we consider 6 ξ21 φ21m

2P lt 1 which in particular

implies Ω asymp 1

ndash 12 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

JCAP11(2015)015

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

where R is the Ricci scalar with respect to gmicroν We can then rescale the field using

dχsds

=

radicΩ2 + 6 ξ2s s

2m2P

Ω4 (26)

which gives

SE =

int radicminusg d4x

(LSM minus

m2P R

2+

1

2gmicroνpartmicroχspartνχs minus U(χs 0)

) (27)

with

U(χs 0) =λs s

4(χs)

4 Ω4 (28)

The relationship between s and χs is determined by the solution to eq (26) In particularfor s mP

radicξs the Einstein frame potential is

U(χs 0) =λsm

4P

4 ξ2s

(1 + exp

(minus 2χsradic

6mP

))minus2 (29)

This is sufficiently flat at large χs to support slow-roll inflationAn analogous expression is obtained for the potential along the h-direction In both

cases the Einstein frame potential is proportional to λφξ2φ where φ = s or h Therefore

the minimum of the potential at large s and h will be very close to h = 0 and inflation willnaturally occur along the s-direction if λsξ

2s λhξ

2h which is true for example if ξs ξh

and λs sim λhIn the following inflation is always considered to be in the direction of s with h = 0

The conventional analysis of inflation can then be performed in the Einstein frame Afterinflation the Jordan and Einstein frames will be indistinguishable since ξs s

2 m2P and

so Ω rarr 1 Therefore the curvature perturbation spectrum calculated in the Einstein framebecomes equal to that observed in the physical Jordan frame at late times

The classical (tree-level) predictions for the spectral index and tensor-to-scalar ratioare [33]

ntrees asymp 1minus 2

Nminus 3

2N2+O

(1

N3

)= 0965 (210)

rtree asymp 12

N2+O

(1

ξsN2

)= 36times 10minus3 (211)

while the field during inflation is

s2Nasymp 4m2

P N(3 ξs) (212)

In the equations above N is the number of e-foldings as defined in the Einstein framewhich differs from that in the Jordan frame by N asymp N + ln(1

radicN) [8] and we have used

N = 582 The classical predictions are in good agreement with the most recent Planck

2Reheating in S-inflation occurs via stochastic resonance to Higgs bosons through the coupling λhs Itwas shown in [16] that this process is very efficient and makes quite precise predictions for the reheatingtemperature and the number of e-foldings of inflation with 57 N 60 at the WMAP pivot scale This inturn allows for quite precise predictions of the inflation observables

ndash 4 ndash

JCAP11(2015)015

values ns = 09677plusmn 00060 (68 confidence level (CL) Planck TT + lowP + lensing) andr0002 lt 011 (95 CL Planck TT + lowP + lensing) [1]

The classical predictions for S-inflation are the same as those of any model based onthe SBB model Differences between S-inflation and other models do however arise fromquantum corrections to the effective potential To include these corrections we calculatethe RG evolution of the various couplings as a function of the renormalisation scale micro (seesection 4 for details) We can then obtain the renormalisation group (RG)-improved effectivepotential for s in the Jordan frame by replacing the couplings in eq (23) by the runningcouplings and setting micro equal to the value of the field For h = 0 (and neglecting the singletmass term ms sim 1 TeV s) this approach yields

VRG(s2 0) =λs(s) s

4

4 (213)

The RG-improved potential can then be transformed into the Einstein frame in order tocalculate the observables predicted by inflation3 The inflationary parameters are calculatedusing the methods discussed in [16] In particular the Einstein frame slow-roll parametersare given by

ε =m2

P

2

(1

U

dU

dχs

)2

η =m2

P

U

d2U

dχ2s

ξ2 =m4

P

U2

dU

dχs

d3U

dχ3s

(214)

3 Singlet scalar as dark matter

Let us now turn to the phenomenology of the singlet scalar in the present Universe andat energies well below the scale of inflation [19 24 35] Most importantly the assumedZ2 symmetry ensures the stability of the scalar so that it can potentially account for theobserved abundance of DM [2 3] If the mass of the singlet is comparable to the electroweakscale the singlet is a typical WIMP which obtains its relic abundance from thermal freeze-out Indeed at low energies where the effects of the non-minimal coupling to gravity arenegligible our model becomes identical to what is often referred to as Higgs Portal DarkMatter [21 22 36] because all interactions of the singlet with SM particles are mediatedby the Higgs In this section we review the constraints on these models and determine theparameter space allowed by the most recent experimental results In the process we pointout several discrepancies in the literature and resolve the resulting confusion

31 Relic abundance

The calculation of the relic abundance of singlet scalars is discussed in detail in [19] Threekinds of processes are relevant for the annihilation of singlets into SM states annihilation

3In [34] it was proposed to use of the Einstein frame for the computation of quantum corrections TheJordan frame analysis is however easier to implement correctly being a straightforward extension of theStandard Model analysis

ndash 5 ndash

JCAP11(2015)015

into SM fermions annihilation into SM gauge bosons and annihilation into two Higgs par-ticles The first kind dominates as long as ms lt mW while for larger masses the secondkind gives the largest contribution Notably all of these processes can proceed via an s-channel Higgs boson leading to a resonant enhancement of the annihilation cross-sectionand a corresponding suppression of the DM relic abundance for ms sim mh24

For the present work we calculate the singlet abundance using micrOMEGAs 3 [37]which numerically solves the Boltzmann equation while calculating the Higgs width in aself-consistent way It is then straightforward to numerically find the coupling λhs that givesΩs h

2 = 01197 in order to reproduce the value of the DM density ΩDM h2 = 01197plusmn 00022determined by Planck (TT + lowP 68 CL) [38] For example we find λhs asymp 008 forms = 300 GeV and λhs asymp 030 for ms = 1000 GeV These values agree with the ones foundin [19 23] but disagree with [20ndash22 24] by a factor of 2 after accounting for the differentconventions5

32 Direct detection constraints

The strongest constraints on λhs stem from DM direct detection experiments since thesinglet-Higgs coupling induces spin-independent interactions between the DM particle andnuclei The scattering cross-section at zero momentum transfer is given by [19]

σSI =λ2hsf

2N

4π

micro2r m2n

m4hm

2s

(31)

where mn is the neutron mass micror = (msmn)(ms + mn) is the reduced mass and fN isthe effective Higgs-nucleon coupling6 In terms of the light-quark matrix elements fNTq theeffective coupling can be written as

fN =

2

9+

7

9

sumq=uds

fNTq

(32)

The values of fNTq can either be determined phenomenologically from baryon masses andmeson-baryon scattering data or computed within lattice QCD A comparison of the differentmethods was recently performed in [19] and we adopt their result of fN = 030 for the effectivecoupling

The scattering cross-section given above can be directly compared to the bound obtainedfrom the LUX experiment [18] Indeed as shown in figure 1 LUX is typically sensitive to thesame range of values for λhs as what is implied by the relic density constraint Specificallythe LUX bound excludes the mass ranges 57 GeV lt ms lt 526 GeV and 645 GeV lt ms lt928 GeV

33 Invisible Higgs decays

Direct detection experiments cannot constrain singlet scalars with a mass of a few GeV orless since such particles would deposit too little energy in the detector to be observable This

4The process ssrarr hh also receives a contribution from t-channel singlet exchange which gives a relevantcontribution if λhs is large compared to λh

5The Higgs-singlet coupling is called λhS in [19] λHS in [20] λhSS in [21 22] λDM in [23] and a2 in [24]The respective conventions are captured by λhs = λhS = λHS2 = λhSS2 = λDM2 = 2 a2

6Note that [24] uses an approximate expression valid for ms mn such that micro2r asymp m2

n

ndash 6 ndash

JCAP11(2015)015

5 10 50 100 500 100010-4

0001

0010

0100

1

ms [GeV]

λhs

Figure 1 Excluded parameter regions from LUX (red dotted) and searches for invisible Higgs decays(blue dashed) compared to the coupling implied by the relic density constraint (green solid)

parameter region can however be efficiently constrained by considering how the Higgs-singletcoupling λhs would modify the branching ratios of the SM Higgs boson The partial decaywidth for hrarr ss is given by7

Γ(hrarr ss) =λ2hsv

2

32πmh

radic1minus 4m2

s

m2h

(33)

This theoretical prediction can be compared to the experimental bound on invisibleHiggs decays from the LHC Direct searches for invisible Higgs decays in the vector bosonfusion channel give BR(h rarr inv) 029 [39] A somewhat stronger bound can be obtainedfrom the observation that in our model there are no additional contributions to the Higgsproduction cross-section and no modifications of the partial decay widths of the Higgs bosoninto SM final states Therefore the presence of an invisible decay channel leads to an overallreduction of the signal strength in visible channels A global fit of all observed decay channels(combined with the bounds on invisible Higgs decays) then gives BR(hrarr inv) 026 [40]

Crucially the bound from invisible Higgs decays becomes independent of the singletmass for ms mh2 Invisible Higgs decays will therefore provide the strongest constraintsfor small singlet masses Indeed this constraint rules out the entire mass region where directdetection experiments lose sensitivity (see figure 1) As a result only two mass regions remainviable a low-mass region 526 GeV lt ms lt 645 GeV and a high-mass region ms amp 93 GeV

34 Other constraints

It has been pointed out recently [24] that bounds on γ-ray lines from Fermi-LAT [41] ruleout the parameter region where ms is slightly above mh2 To be safe from this constraintwe will focus on the mass range 526 GeV lt ms lt 624 GeV which we shall refer to as thelow-mass region For the high-mass region on the other hand there are no strong constraintsfrom indirect detection Moreover collider searches for singlet scalars with ms gt mh2 areextremely challenging [20ndash23 35] and consequently there are no relevant bounds from theLHC for the high-mass region [42]

The most significant improvements in sensitivity in the near future are expected tocome from direct detection experiments Indeed XENON1T [43] is expected to improve

7This equation agrees with [4 19ndash22 35] but disagrees with [23]

ndash 7 ndash

JCAP11(2015)015

upon current LUX constraints on the DM scattering cross-section by a factor of about 50and will therefore be able to probe the high-mass region up to ms asymp 4 TeV As we will showin S-inflation singlet masses larger than about 2 TeV are excluded by the Planck 2-σ upperbound on ns and perturbativity XENON1T will therefore be able to probe the entire high-mass region relevant for singlet inflation Similarly XENON1T can also further constrain thelow-mass region and potentially probe singlet masses in the range 53 GeV lt ms lt 57 GeV

4 Renormalisation group evolution and theoretical constraints

In order to connect the inflationary observables of our model to the measured SM parametersand the DM phenomenology discussed in the previous section we need to calculate theevolution of all couplings under the RG equations [8 10 12 14 44ndash47] Existing analyseshave considered the RG equations for the SM at two-loop order and the contributions of thesinglet sector and non-minimal coupling at one-loop order (see also the appendix) [8 10] Toexamine the issue of vacuum stability we improve the accuracy of our analysis further byincorporating the three-loop RG equations for the SM gauge couplings [48] and the leadingorder three-loop corrections to the RG equations for λh and yt [49]8

When considering large field values for either s or h the RG equations are modifiedbecause there is a suppression of scalar propagators This suppression is captured by insertinga factor

cφ =1 +

ξφ φ2

mP

1 + (6ξφ + 1)ξφ φ2

mP

(41)

with φ = s (φ = h) for each s (h) propagating in a loop [8] The changes in the RGequations for large values of the Higgs field have been discussed in detail in [47] Themodifications resulting from large singlet field values can be found in [8] and are reviewedin the appendix Note that when considering S-inflation such that s h we can set thesuppression factor ch = 1

We determine the values of the SM parameters at micro = mt following [46] Using themost recent values from the Particle Data Group [50]

mt = (1732plusmn 09) GeV mH = (12509plusmn 024) GeV αS(mZ) = 01185plusmn 00006 (42)

we obtain at micro = mt

yt = 0936plusmn 0005 λh = 01260plusmn 00014 gS = 1164plusmn 0003 (43)

Unless explicitly stated otherwise we will use the central values for all calculations belowFor given couplings at micro = mt we then use the public code RGErun 207 [51] to calculatethe couplings at higher scales

In contrast to the remaining couplings we fix the non-minimal couplings ξh and ξs atmicro = mP In order to obtain the correct amplitude of the scalar power spectrum we require

U

ε= (000271mP)4 (44)

where U and ε are the potential and the first slow-roll parameter in the Einstein frameat the beginning of inflation as defined in eq (28) and eq (214) respectively Imposing

8We thank Kyle Allison for sharing his numerical implementation of these equations

ndash 8 ndash

JCAP11(2015)015

equation (44) allows us to determine ξs at the scale of inflation once all other parametershave been fixed Note that since the value of s at the beginning of inflation also depends onξs equation (44) can only be solved numerically We then iteratively determine the requiredvalue of ξs at the electroweak scale such that RG evolution yields the desired value at thescale of inflation

The coupling ξh plays a very limited role for the phenomenology of our model because wedo not consider the case of large Higgs field values for inflation As a result our predictionsfor the inflationary observables show only a very mild dependence on ξh so that ξh canessentially be chosen arbitrarily Nevertheless it is not possible to simply set this parameterto zero since radiative corrections induce a mixing between ξs and ξh Moreover we willsee below that the value of ξh plays an important role for determining whether our modelviolates unitarity below the scale of inflation As with ξs we fix ξh at mP and then determineiteratively the required value of ξh at the electroweak scale9

41 Metastability

We now discuss various theoretical constraints related to the RG evolution of the parame-ters in our model It is a well-known fact that for the central values of the measured SMparameters the electroweak vacuum becomes metastable at high scales because the quarticHiggs coupling λh runs to negative values (see eg [46]) This metastability is not in anyobvious way a problem as the lifetime of the electroweak vacuum is well above the age ofthe Universe [52] (note however that this estimate may potentially be spoiled by effectsfrom Planck-scale higher-dimensional operators [53ndash55]) However one may speculate thata stable electroweak vacuum is necessary for a consistent theory for example if the vacuumenergy relative to the absolute minimum is a physical energy density leading to inflation Itis therefore an interesting aspect of singlet extensions of the SM that scalar singlets give apositive contribution to the running of λh [56 57]

βλh = βSMλh +1

32π2c2s λ

2hs (45)

In fact it was shown in [58] that for the case of a minimally-coupled singlet λhs can bechosen such that the electroweak vacuum remains stable all the way up to the Planck scaleand at the same time (for appropriate choices of the singlet mass ms) the singlet obtains athermal relic density compatible with the observed DM abundance10

In order to study electroweak vacuum stability we need to consider the potential in theh-direction with s = 011 Vacuum stability then requires that λh(micro) gt 0 for micro up to mPIn the present study we consider both the case where λhs is sufficiently large to stabilisethe electroweak vacuum and the case where λhs only increases the lifetime of the metastablevacuum but does not render it completely stable We focus throughout on the case whereλhs is positive

42 Examples

Figure 2 shows an example for the evolution of scalar couplings (left) and the non-minimalcouplings (right) under the RG equations discussed above Solid lines correspond to the case

9The running of ξh between mP and the scale of inflation is completely negligible since the relevantdiagrams are strongly suppressed for large values of s

10Note that if the singlet mixes with the Higgs there will be additional threshold effects at micro = ms fromintegrating out the singlet [59] In the setup we consider however this effect is not important [60]

11A more detailed study of the potential in general directions with both s 6= 0 and h 6= 0 (along the linesof [61]) is beyond the scope of the present work

ndash 9 ndash

JCAP11(2015)015

1000 106 109 1012 1015 10180001

00050010

00500100

05001

μ [GeV]

Couplingstrength

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

λh

λhs

λs

1000 106 109 1012 1015 101810

100

1000

104

105

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

ξh

ξs

cs

Figure 2 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplingsξh and ξs (right) as a function of the renormalisation scale micro for a typical parameter point in thehigh-mass region Solid lines show the running in the s-direction while dotted lines correspond to therunning in the h-direction In the right panel we also show the suppression factor cs which modifiesthe running in the s-direction at large field value

s h which is relevant for inflation while dotted lines correspond to h s which is relevantfor vacuum stability We fix the scalar couplings at the weak scale choosing λhs = 025 andλs = 001 such that the observed relic abundance can be reproduced for ms = 835 GeV Weconsider λs λhs in which case the value of ξs necessary to obtain the correct amplitude ofthe scalar power spectrum is reduced For our choice we find ξs sim 104 Note however thatwhile λhs exhibits only moderate running λs grows significantly with the renormalisationscale micro because its β-function contains a term proportional to λ2hs Choosing even smallervalues of λs at the weak scale will therefore not significantly reduce its value at the scale ofinflation nor the corresponding value of ξs

An important observation from figure 2 is that λh does not run negative and hencethe electroweak vacuum remains stable all the way up to the Planck scale The additionalcontribution from the singlet scalar is sufficient to ensure λh gt 10minus3 for all renormalisationscales up to mP For field values s amp 1015 GeV h the singlet propagator is suppressedleading to a visible kink in the running of λh We show the propagator suppression factorcs in the right panel of figure 2 One can clearly see how this suppression factor affects therunning of ξh which becomes nearly constant for s amp 1015 GeV In this particular examplewe have chosen ξh(mP) = 1000 (for s h) This choice together with λs 1 implies thatthe running of ξs(micro) from mt to mP is negligible

In the low-mass region we are interested in much smaller values of λhs typically below10minus2 A particular example is shown in figure 3 (left) for the representative choice λhs(mt) =0002 and λs = 00005 which yields the observed relic abundance for ms asymp 57 GeV Weobserve that if λs and λhs are both small at the electroweak scale these couplings exhibitonly very little running up to the scale of inflation For the same reason it is impossible toinfluence the running of the Higgs couplings sufficiently to prevent λhs from running negativeat around 1011 GeV

If λhs is small we can obtain the correct scalar power spectrum amplitude with a muchsmaller value of ξs during inflation For the specific case considered in figure 3 we findξs sim 103 For these values of ξs and λhs the loop-induced corrections to ξh are very smalland hence this coupling changes only very slightly under RG evolution

43 Perturbativity

In order for our calculation of the running couplings and the radiative corrections to thepotential to be reliable we must require that all couplings remain perturbative up to the

ndash 10 ndash

JCAP11(2015)015

1000 106 109 1012 1015 101810-4

0001

0010

0100

1

μ [GeV]

Couplingstrength

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

λh

λhs

λs

1000 106 109 1012 1015 101801

1

10

100

1000

104

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

ξh

ξs

cs

Figure 3 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplings ξhand ξs (right) as a function of the renormalisation scale micro for a typical parameter point in the low-mass region In the right panel we also show the suppression factor cs which modifies the runningat large field value Note that λh runs negative for micro amp 1011 GeV

scale of inflation which is typically 1017ndash1018 GeV This requirement is easily satisfied forthe SM couplings but needs to be checked explicitly for the couplings of the singlet whichcan grow significantly with increasing renormalisation scale micro We follow [58 62] and usethe requirement of perturbative unitarity to impose an upper bound on the scalar couplingsThis procedure gives

λs lt4π

3and λhs lt 8π (46)

As we will see below the non-minimal coupling ξs can be much larger than unity withoutinvalidating a perturbative calculation Nevertheless if ξs and ξh are both very large pro-cesses involving both couplings may violate perturbative unitarity implying that there maybe new physics or strong coupling below the scale of inflation We will now discuss this issuein more detail

5 Unitarity-violation during inflation

In this section we estimate the scale of perturbative unitarity-violation as a function of thebackground inflaton field Note that by ldquounitarity-violation scalerdquo we mean the scale at whichperturbation theory in scalar scattering breaks down so this may in fact indicate the onsetof unitarity-conserving scattering in a strongly-coupled regime [28 31] We will considerunitarity-violation in the scattering of scalar particles corresponding to perturbations aboutthe background field In the case of a real scalar s and the fields of the Higgs doublet Hthere are two distinct scattering processes we need to consider (i) δs h1 harr δs h1 and (ii)h1h2 harr h1h2 where δs is the perturbation about the background s field and h1 and h2are two of the Higgs doublet scalars Scattering with the other scalars in H is equivalentto these two processes We will use dimensional analysis to estimate the scale of unitarity-violation by determining the leading-order processes in the Einstein frame which result inunitarity-violating scattering

It will be sufficient to consider the Einstein frame Lagrangian for two real scalar fieldsφi where mdash using the notation of [30] mdash φi stands for either s or a component of the Higgsdoublet Unitarity-violation requires that there are two different scalars in the scatteringprocess since in the case of a single scalar there is a cancellation between s- t- and u-channelamplitudes [63]

ndash 11 ndash

JCAP11(2015)015

Since unitarity-violating scattering in the Jordan frame is due to graviton exchange viathe non-minimal coupling to R we cat set V = 0 The Einstein frame action for two realscalars is then of the form

SE =

intd4xradicminusg

Lii +sumiltj

Lij minus1

2m2

P R

(51)

where

Lii =1

2

Ω2 +6 ξ2i φ

2i

m2P

Ω4

gmicroν partmicroφi partνφi and Lij =6 ξi ξj φi φj g

microν partmicroφi partνφjm2

P Ω4(52)

with

Ω2 = 1 +ξj φ

2j

m2P

(53)

The interaction terms proportional to ξi ξj are responsible for the dominant unitarity-viola-tion in scattering cross-sections calculated in the Einstein frame These interactions are theEinstein frame analogue of scalar scattering via graviton exchange in the Jordan frame due tothe non-minimal coupling To obtain the scale of unitarity-violation in terms of the physicalenergy defined in the Jordan frame we first canonically normalize the fields in the Einsteinframe then estimate the magnitude of the scattering matrix element and finally transformthe unitarity-violation scale in the Einstein frame back to that in the Jordan frame

In the following we will denote the inflaton by φ1 (equiv s) which we expand about thebackground field ie φ1 = φ1 + δφ1 The Higgs doublet scalars are denoted by φ2 (equiv h1)and φ3 (equiv h2) The corresponding canonically normalized scattering fields in the Einsteinframe are then defined to be ϕ1 ϕ2 and ϕ3 Once we have determined the interactions ofthe canonically normalised fields we use dimensional analysis to estimate the scale of tree-level unitarity-violation For this purpose we introduce appropriate factors of E to makethe coefficient of the interaction terms in L dimensionless Energy scales E in the Einsteinframe are related to the ones in the Jordan frame via E = EΩ where during inflation

Ω2 ξ1φ21m

2P asymp N 1 Unitarity conservation implies that the matrix element for any

2 harr 2 scattering process should be smaller than O(1) so we can determine the scale ofunitarity-violation (denoted by Λ) by determining the value of E that saturates this boundThis was demonstrated explicitly in [8] by comparing the dimensional estimate with theexact value from the full scattering amplitude

In the following we consider three regimes for φ1 each of which leads to a differentform of the Lagrangian and the scattering amplitudes

bull Regime A large field values In this regime we have Ω gt 1 (implying that ξ1 φ21m

2P gt 1)

and 6 ξ21 φ21m

2P gt 1

bull Regime B medium field values In this regime we have approximately Ω asymp 1 (implying

that ξ1 φ21m

2P lt 1) but still 6 ξ21 φ

21m

2P gt 1

bull Regime C small field values Finally we consider 6 ξ21 φ21m

2P lt 1 which in particular

implies Ω asymp 1

ndash 12 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

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1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

values ns = 09677plusmn 00060 (68 confidence level (CL) Planck TT + lowP + lensing) andr0002 lt 011 (95 CL Planck TT + lowP + lensing) [1]

The classical predictions for S-inflation are the same as those of any model based onthe SBB model Differences between S-inflation and other models do however arise fromquantum corrections to the effective potential To include these corrections we calculatethe RG evolution of the various couplings as a function of the renormalisation scale micro (seesection 4 for details) We can then obtain the renormalisation group (RG)-improved effectivepotential for s in the Jordan frame by replacing the couplings in eq (23) by the runningcouplings and setting micro equal to the value of the field For h = 0 (and neglecting the singletmass term ms sim 1 TeV s) this approach yields

VRG(s2 0) =λs(s) s

4

4 (213)

The RG-improved potential can then be transformed into the Einstein frame in order tocalculate the observables predicted by inflation3 The inflationary parameters are calculatedusing the methods discussed in [16] In particular the Einstein frame slow-roll parametersare given by

ε =m2

P

2

(1

U

dU

dχs

)2

η =m2

P

U

d2U

dχ2s

ξ2 =m4

P

U2

dU

dχs

d3U

dχ3s

(214)

3 Singlet scalar as dark matter

Let us now turn to the phenomenology of the singlet scalar in the present Universe andat energies well below the scale of inflation [19 24 35] Most importantly the assumedZ2 symmetry ensures the stability of the scalar so that it can potentially account for theobserved abundance of DM [2 3] If the mass of the singlet is comparable to the electroweakscale the singlet is a typical WIMP which obtains its relic abundance from thermal freeze-out Indeed at low energies where the effects of the non-minimal coupling to gravity arenegligible our model becomes identical to what is often referred to as Higgs Portal DarkMatter [21 22 36] because all interactions of the singlet with SM particles are mediatedby the Higgs In this section we review the constraints on these models and determine theparameter space allowed by the most recent experimental results In the process we pointout several discrepancies in the literature and resolve the resulting confusion

31 Relic abundance

The calculation of the relic abundance of singlet scalars is discussed in detail in [19] Threekinds of processes are relevant for the annihilation of singlets into SM states annihilation

3In [34] it was proposed to use of the Einstein frame for the computation of quantum corrections TheJordan frame analysis is however easier to implement correctly being a straightforward extension of theStandard Model analysis

ndash 5 ndash

JCAP11(2015)015

into SM fermions annihilation into SM gauge bosons and annihilation into two Higgs par-ticles The first kind dominates as long as ms lt mW while for larger masses the secondkind gives the largest contribution Notably all of these processes can proceed via an s-channel Higgs boson leading to a resonant enhancement of the annihilation cross-sectionand a corresponding suppression of the DM relic abundance for ms sim mh24

For the present work we calculate the singlet abundance using micrOMEGAs 3 [37]which numerically solves the Boltzmann equation while calculating the Higgs width in aself-consistent way It is then straightforward to numerically find the coupling λhs that givesΩs h

2 = 01197 in order to reproduce the value of the DM density ΩDM h2 = 01197plusmn 00022determined by Planck (TT + lowP 68 CL) [38] For example we find λhs asymp 008 forms = 300 GeV and λhs asymp 030 for ms = 1000 GeV These values agree with the ones foundin [19 23] but disagree with [20ndash22 24] by a factor of 2 after accounting for the differentconventions5

32 Direct detection constraints

The strongest constraints on λhs stem from DM direct detection experiments since thesinglet-Higgs coupling induces spin-independent interactions between the DM particle andnuclei The scattering cross-section at zero momentum transfer is given by [19]

σSI =λ2hsf

2N

4π

micro2r m2n

m4hm

2s

(31)

where mn is the neutron mass micror = (msmn)(ms + mn) is the reduced mass and fN isthe effective Higgs-nucleon coupling6 In terms of the light-quark matrix elements fNTq theeffective coupling can be written as

fN =

2

9+

7

9

sumq=uds

fNTq

(32)

The values of fNTq can either be determined phenomenologically from baryon masses andmeson-baryon scattering data or computed within lattice QCD A comparison of the differentmethods was recently performed in [19] and we adopt their result of fN = 030 for the effectivecoupling

The scattering cross-section given above can be directly compared to the bound obtainedfrom the LUX experiment [18] Indeed as shown in figure 1 LUX is typically sensitive to thesame range of values for λhs as what is implied by the relic density constraint Specificallythe LUX bound excludes the mass ranges 57 GeV lt ms lt 526 GeV and 645 GeV lt ms lt928 GeV

33 Invisible Higgs decays

Direct detection experiments cannot constrain singlet scalars with a mass of a few GeV orless since such particles would deposit too little energy in the detector to be observable This

4The process ssrarr hh also receives a contribution from t-channel singlet exchange which gives a relevantcontribution if λhs is large compared to λh

5The Higgs-singlet coupling is called λhS in [19] λHS in [20] λhSS in [21 22] λDM in [23] and a2 in [24]The respective conventions are captured by λhs = λhS = λHS2 = λhSS2 = λDM2 = 2 a2

6Note that [24] uses an approximate expression valid for ms mn such that micro2r asymp m2

n

ndash 6 ndash

JCAP11(2015)015

5 10 50 100 500 100010-4

0001

0010

0100

1

ms [GeV]

λhs

Figure 1 Excluded parameter regions from LUX (red dotted) and searches for invisible Higgs decays(blue dashed) compared to the coupling implied by the relic density constraint (green solid)

parameter region can however be efficiently constrained by considering how the Higgs-singletcoupling λhs would modify the branching ratios of the SM Higgs boson The partial decaywidth for hrarr ss is given by7

Γ(hrarr ss) =λ2hsv

2

32πmh

radic1minus 4m2

s

m2h

(33)

This theoretical prediction can be compared to the experimental bound on invisibleHiggs decays from the LHC Direct searches for invisible Higgs decays in the vector bosonfusion channel give BR(h rarr inv) 029 [39] A somewhat stronger bound can be obtainedfrom the observation that in our model there are no additional contributions to the Higgsproduction cross-section and no modifications of the partial decay widths of the Higgs bosoninto SM final states Therefore the presence of an invisible decay channel leads to an overallreduction of the signal strength in visible channels A global fit of all observed decay channels(combined with the bounds on invisible Higgs decays) then gives BR(hrarr inv) 026 [40]

Crucially the bound from invisible Higgs decays becomes independent of the singletmass for ms mh2 Invisible Higgs decays will therefore provide the strongest constraintsfor small singlet masses Indeed this constraint rules out the entire mass region where directdetection experiments lose sensitivity (see figure 1) As a result only two mass regions remainviable a low-mass region 526 GeV lt ms lt 645 GeV and a high-mass region ms amp 93 GeV

34 Other constraints

It has been pointed out recently [24] that bounds on γ-ray lines from Fermi-LAT [41] ruleout the parameter region where ms is slightly above mh2 To be safe from this constraintwe will focus on the mass range 526 GeV lt ms lt 624 GeV which we shall refer to as thelow-mass region For the high-mass region on the other hand there are no strong constraintsfrom indirect detection Moreover collider searches for singlet scalars with ms gt mh2 areextremely challenging [20ndash23 35] and consequently there are no relevant bounds from theLHC for the high-mass region [42]

The most significant improvements in sensitivity in the near future are expected tocome from direct detection experiments Indeed XENON1T [43] is expected to improve

7This equation agrees with [4 19ndash22 35] but disagrees with [23]

ndash 7 ndash

JCAP11(2015)015

upon current LUX constraints on the DM scattering cross-section by a factor of about 50and will therefore be able to probe the high-mass region up to ms asymp 4 TeV As we will showin S-inflation singlet masses larger than about 2 TeV are excluded by the Planck 2-σ upperbound on ns and perturbativity XENON1T will therefore be able to probe the entire high-mass region relevant for singlet inflation Similarly XENON1T can also further constrain thelow-mass region and potentially probe singlet masses in the range 53 GeV lt ms lt 57 GeV

4 Renormalisation group evolution and theoretical constraints

In order to connect the inflationary observables of our model to the measured SM parametersand the DM phenomenology discussed in the previous section we need to calculate theevolution of all couplings under the RG equations [8 10 12 14 44ndash47] Existing analyseshave considered the RG equations for the SM at two-loop order and the contributions of thesinglet sector and non-minimal coupling at one-loop order (see also the appendix) [8 10] Toexamine the issue of vacuum stability we improve the accuracy of our analysis further byincorporating the three-loop RG equations for the SM gauge couplings [48] and the leadingorder three-loop corrections to the RG equations for λh and yt [49]8

When considering large field values for either s or h the RG equations are modifiedbecause there is a suppression of scalar propagators This suppression is captured by insertinga factor

cφ =1 +

ξφ φ2

mP

1 + (6ξφ + 1)ξφ φ2

mP

(41)

with φ = s (φ = h) for each s (h) propagating in a loop [8] The changes in the RGequations for large values of the Higgs field have been discussed in detail in [47] Themodifications resulting from large singlet field values can be found in [8] and are reviewedin the appendix Note that when considering S-inflation such that s h we can set thesuppression factor ch = 1

We determine the values of the SM parameters at micro = mt following [46] Using themost recent values from the Particle Data Group [50]

mt = (1732plusmn 09) GeV mH = (12509plusmn 024) GeV αS(mZ) = 01185plusmn 00006 (42)

we obtain at micro = mt

yt = 0936plusmn 0005 λh = 01260plusmn 00014 gS = 1164plusmn 0003 (43)

Unless explicitly stated otherwise we will use the central values for all calculations belowFor given couplings at micro = mt we then use the public code RGErun 207 [51] to calculatethe couplings at higher scales

In contrast to the remaining couplings we fix the non-minimal couplings ξh and ξs atmicro = mP In order to obtain the correct amplitude of the scalar power spectrum we require

U

ε= (000271mP)4 (44)

where U and ε are the potential and the first slow-roll parameter in the Einstein frameat the beginning of inflation as defined in eq (28) and eq (214) respectively Imposing

8We thank Kyle Allison for sharing his numerical implementation of these equations

ndash 8 ndash

JCAP11(2015)015

equation (44) allows us to determine ξs at the scale of inflation once all other parametershave been fixed Note that since the value of s at the beginning of inflation also depends onξs equation (44) can only be solved numerically We then iteratively determine the requiredvalue of ξs at the electroweak scale such that RG evolution yields the desired value at thescale of inflation

The coupling ξh plays a very limited role for the phenomenology of our model because wedo not consider the case of large Higgs field values for inflation As a result our predictionsfor the inflationary observables show only a very mild dependence on ξh so that ξh canessentially be chosen arbitrarily Nevertheless it is not possible to simply set this parameterto zero since radiative corrections induce a mixing between ξs and ξh Moreover we willsee below that the value of ξh plays an important role for determining whether our modelviolates unitarity below the scale of inflation As with ξs we fix ξh at mP and then determineiteratively the required value of ξh at the electroweak scale9

41 Metastability

We now discuss various theoretical constraints related to the RG evolution of the parame-ters in our model It is a well-known fact that for the central values of the measured SMparameters the electroweak vacuum becomes metastable at high scales because the quarticHiggs coupling λh runs to negative values (see eg [46]) This metastability is not in anyobvious way a problem as the lifetime of the electroweak vacuum is well above the age ofthe Universe [52] (note however that this estimate may potentially be spoiled by effectsfrom Planck-scale higher-dimensional operators [53ndash55]) However one may speculate thata stable electroweak vacuum is necessary for a consistent theory for example if the vacuumenergy relative to the absolute minimum is a physical energy density leading to inflation Itis therefore an interesting aspect of singlet extensions of the SM that scalar singlets give apositive contribution to the running of λh [56 57]

βλh = βSMλh +1

32π2c2s λ

2hs (45)

In fact it was shown in [58] that for the case of a minimally-coupled singlet λhs can bechosen such that the electroweak vacuum remains stable all the way up to the Planck scaleand at the same time (for appropriate choices of the singlet mass ms) the singlet obtains athermal relic density compatible with the observed DM abundance10

In order to study electroweak vacuum stability we need to consider the potential in theh-direction with s = 011 Vacuum stability then requires that λh(micro) gt 0 for micro up to mPIn the present study we consider both the case where λhs is sufficiently large to stabilisethe electroweak vacuum and the case where λhs only increases the lifetime of the metastablevacuum but does not render it completely stable We focus throughout on the case whereλhs is positive

42 Examples

Figure 2 shows an example for the evolution of scalar couplings (left) and the non-minimalcouplings (right) under the RG equations discussed above Solid lines correspond to the case

9The running of ξh between mP and the scale of inflation is completely negligible since the relevantdiagrams are strongly suppressed for large values of s

10Note that if the singlet mixes with the Higgs there will be additional threshold effects at micro = ms fromintegrating out the singlet [59] In the setup we consider however this effect is not important [60]

11A more detailed study of the potential in general directions with both s 6= 0 and h 6= 0 (along the linesof [61]) is beyond the scope of the present work

ndash 9 ndash

JCAP11(2015)015

1000 106 109 1012 1015 10180001

00050010

00500100

05001

μ [GeV]

Couplingstrength

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

λh

λhs

λs

1000 106 109 1012 1015 101810

100

1000

104

105

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

ξh

ξs

cs

Figure 2 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplingsξh and ξs (right) as a function of the renormalisation scale micro for a typical parameter point in thehigh-mass region Solid lines show the running in the s-direction while dotted lines correspond to therunning in the h-direction In the right panel we also show the suppression factor cs which modifiesthe running in the s-direction at large field value

s h which is relevant for inflation while dotted lines correspond to h s which is relevantfor vacuum stability We fix the scalar couplings at the weak scale choosing λhs = 025 andλs = 001 such that the observed relic abundance can be reproduced for ms = 835 GeV Weconsider λs λhs in which case the value of ξs necessary to obtain the correct amplitude ofthe scalar power spectrum is reduced For our choice we find ξs sim 104 Note however thatwhile λhs exhibits only moderate running λs grows significantly with the renormalisationscale micro because its β-function contains a term proportional to λ2hs Choosing even smallervalues of λs at the weak scale will therefore not significantly reduce its value at the scale ofinflation nor the corresponding value of ξs

An important observation from figure 2 is that λh does not run negative and hencethe electroweak vacuum remains stable all the way up to the Planck scale The additionalcontribution from the singlet scalar is sufficient to ensure λh gt 10minus3 for all renormalisationscales up to mP For field values s amp 1015 GeV h the singlet propagator is suppressedleading to a visible kink in the running of λh We show the propagator suppression factorcs in the right panel of figure 2 One can clearly see how this suppression factor affects therunning of ξh which becomes nearly constant for s amp 1015 GeV In this particular examplewe have chosen ξh(mP) = 1000 (for s h) This choice together with λs 1 implies thatthe running of ξs(micro) from mt to mP is negligible

In the low-mass region we are interested in much smaller values of λhs typically below10minus2 A particular example is shown in figure 3 (left) for the representative choice λhs(mt) =0002 and λs = 00005 which yields the observed relic abundance for ms asymp 57 GeV Weobserve that if λs and λhs are both small at the electroweak scale these couplings exhibitonly very little running up to the scale of inflation For the same reason it is impossible toinfluence the running of the Higgs couplings sufficiently to prevent λhs from running negativeat around 1011 GeV

If λhs is small we can obtain the correct scalar power spectrum amplitude with a muchsmaller value of ξs during inflation For the specific case considered in figure 3 we findξs sim 103 For these values of ξs and λhs the loop-induced corrections to ξh are very smalland hence this coupling changes only very slightly under RG evolution

43 Perturbativity

In order for our calculation of the running couplings and the radiative corrections to thepotential to be reliable we must require that all couplings remain perturbative up to the

ndash 10 ndash

JCAP11(2015)015

1000 106 109 1012 1015 101810-4

0001

0010

0100

1

μ [GeV]

Couplingstrength

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

λh

λhs

λs

1000 106 109 1012 1015 101801

1

10

100

1000

104

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

ξh

ξs

cs

Figure 3 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplings ξhand ξs (right) as a function of the renormalisation scale micro for a typical parameter point in the low-mass region In the right panel we also show the suppression factor cs which modifies the runningat large field value Note that λh runs negative for micro amp 1011 GeV

scale of inflation which is typically 1017ndash1018 GeV This requirement is easily satisfied forthe SM couplings but needs to be checked explicitly for the couplings of the singlet whichcan grow significantly with increasing renormalisation scale micro We follow [58 62] and usethe requirement of perturbative unitarity to impose an upper bound on the scalar couplingsThis procedure gives

λs lt4π

3and λhs lt 8π (46)

As we will see below the non-minimal coupling ξs can be much larger than unity withoutinvalidating a perturbative calculation Nevertheless if ξs and ξh are both very large pro-cesses involving both couplings may violate perturbative unitarity implying that there maybe new physics or strong coupling below the scale of inflation We will now discuss this issuein more detail

5 Unitarity-violation during inflation

In this section we estimate the scale of perturbative unitarity-violation as a function of thebackground inflaton field Note that by ldquounitarity-violation scalerdquo we mean the scale at whichperturbation theory in scalar scattering breaks down so this may in fact indicate the onsetof unitarity-conserving scattering in a strongly-coupled regime [28 31] We will considerunitarity-violation in the scattering of scalar particles corresponding to perturbations aboutthe background field In the case of a real scalar s and the fields of the Higgs doublet Hthere are two distinct scattering processes we need to consider (i) δs h1 harr δs h1 and (ii)h1h2 harr h1h2 where δs is the perturbation about the background s field and h1 and h2are two of the Higgs doublet scalars Scattering with the other scalars in H is equivalentto these two processes We will use dimensional analysis to estimate the scale of unitarity-violation by determining the leading-order processes in the Einstein frame which result inunitarity-violating scattering

It will be sufficient to consider the Einstein frame Lagrangian for two real scalar fieldsφi where mdash using the notation of [30] mdash φi stands for either s or a component of the Higgsdoublet Unitarity-violation requires that there are two different scalars in the scatteringprocess since in the case of a single scalar there is a cancellation between s- t- and u-channelamplitudes [63]

ndash 11 ndash

JCAP11(2015)015

Since unitarity-violating scattering in the Jordan frame is due to graviton exchange viathe non-minimal coupling to R we cat set V = 0 The Einstein frame action for two realscalars is then of the form

SE =

intd4xradicminusg

Lii +sumiltj

Lij minus1

2m2

P R

(51)

where

Lii =1

2

Ω2 +6 ξ2i φ

2i

m2P

Ω4

gmicroν partmicroφi partνφi and Lij =6 ξi ξj φi φj g

microν partmicroφi partνφjm2

P Ω4(52)

with

Ω2 = 1 +ξj φ

2j

m2P

(53)

The interaction terms proportional to ξi ξj are responsible for the dominant unitarity-viola-tion in scattering cross-sections calculated in the Einstein frame These interactions are theEinstein frame analogue of scalar scattering via graviton exchange in the Jordan frame due tothe non-minimal coupling To obtain the scale of unitarity-violation in terms of the physicalenergy defined in the Jordan frame we first canonically normalize the fields in the Einsteinframe then estimate the magnitude of the scattering matrix element and finally transformthe unitarity-violation scale in the Einstein frame back to that in the Jordan frame

In the following we will denote the inflaton by φ1 (equiv s) which we expand about thebackground field ie φ1 = φ1 + δφ1 The Higgs doublet scalars are denoted by φ2 (equiv h1)and φ3 (equiv h2) The corresponding canonically normalized scattering fields in the Einsteinframe are then defined to be ϕ1 ϕ2 and ϕ3 Once we have determined the interactions ofthe canonically normalised fields we use dimensional analysis to estimate the scale of tree-level unitarity-violation For this purpose we introduce appropriate factors of E to makethe coefficient of the interaction terms in L dimensionless Energy scales E in the Einsteinframe are related to the ones in the Jordan frame via E = EΩ where during inflation

Ω2 ξ1φ21m

2P asymp N 1 Unitarity conservation implies that the matrix element for any

2 harr 2 scattering process should be smaller than O(1) so we can determine the scale ofunitarity-violation (denoted by Λ) by determining the value of E that saturates this boundThis was demonstrated explicitly in [8] by comparing the dimensional estimate with theexact value from the full scattering amplitude

In the following we consider three regimes for φ1 each of which leads to a differentform of the Lagrangian and the scattering amplitudes

bull Regime A large field values In this regime we have Ω gt 1 (implying that ξ1 φ21m

2P gt 1)

and 6 ξ21 φ21m

2P gt 1

bull Regime B medium field values In this regime we have approximately Ω asymp 1 (implying

that ξ1 φ21m

2P lt 1) but still 6 ξ21 φ

21m

2P gt 1

bull Regime C small field values Finally we consider 6 ξ21 φ21m

2P lt 1 which in particular

implies Ω asymp 1

ndash 12 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

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1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

into SM fermions annihilation into SM gauge bosons and annihilation into two Higgs par-ticles The first kind dominates as long as ms lt mW while for larger masses the secondkind gives the largest contribution Notably all of these processes can proceed via an s-channel Higgs boson leading to a resonant enhancement of the annihilation cross-sectionand a corresponding suppression of the DM relic abundance for ms sim mh24

For the present work we calculate the singlet abundance using micrOMEGAs 3 [37]which numerically solves the Boltzmann equation while calculating the Higgs width in aself-consistent way It is then straightforward to numerically find the coupling λhs that givesΩs h

2 = 01197 in order to reproduce the value of the DM density ΩDM h2 = 01197plusmn 00022determined by Planck (TT + lowP 68 CL) [38] For example we find λhs asymp 008 forms = 300 GeV and λhs asymp 030 for ms = 1000 GeV These values agree with the ones foundin [19 23] but disagree with [20ndash22 24] by a factor of 2 after accounting for the differentconventions5

32 Direct detection constraints

The strongest constraints on λhs stem from DM direct detection experiments since thesinglet-Higgs coupling induces spin-independent interactions between the DM particle andnuclei The scattering cross-section at zero momentum transfer is given by [19]

σSI =λ2hsf

2N

4π

micro2r m2n

m4hm

2s

(31)

where mn is the neutron mass micror = (msmn)(ms + mn) is the reduced mass and fN isthe effective Higgs-nucleon coupling6 In terms of the light-quark matrix elements fNTq theeffective coupling can be written as

fN =

2

9+

7

9

sumq=uds

fNTq

(32)

The values of fNTq can either be determined phenomenologically from baryon masses andmeson-baryon scattering data or computed within lattice QCD A comparison of the differentmethods was recently performed in [19] and we adopt their result of fN = 030 for the effectivecoupling

The scattering cross-section given above can be directly compared to the bound obtainedfrom the LUX experiment [18] Indeed as shown in figure 1 LUX is typically sensitive to thesame range of values for λhs as what is implied by the relic density constraint Specificallythe LUX bound excludes the mass ranges 57 GeV lt ms lt 526 GeV and 645 GeV lt ms lt928 GeV

33 Invisible Higgs decays

Direct detection experiments cannot constrain singlet scalars with a mass of a few GeV orless since such particles would deposit too little energy in the detector to be observable This

4The process ssrarr hh also receives a contribution from t-channel singlet exchange which gives a relevantcontribution if λhs is large compared to λh

5The Higgs-singlet coupling is called λhS in [19] λHS in [20] λhSS in [21 22] λDM in [23] and a2 in [24]The respective conventions are captured by λhs = λhS = λHS2 = λhSS2 = λDM2 = 2 a2

6Note that [24] uses an approximate expression valid for ms mn such that micro2r asymp m2

n

ndash 6 ndash

JCAP11(2015)015

5 10 50 100 500 100010-4

0001

0010

0100

1

ms [GeV]

λhs

Figure 1 Excluded parameter regions from LUX (red dotted) and searches for invisible Higgs decays(blue dashed) compared to the coupling implied by the relic density constraint (green solid)

parameter region can however be efficiently constrained by considering how the Higgs-singletcoupling λhs would modify the branching ratios of the SM Higgs boson The partial decaywidth for hrarr ss is given by7

Γ(hrarr ss) =λ2hsv

2

32πmh

radic1minus 4m2

s

m2h

(33)

This theoretical prediction can be compared to the experimental bound on invisibleHiggs decays from the LHC Direct searches for invisible Higgs decays in the vector bosonfusion channel give BR(h rarr inv) 029 [39] A somewhat stronger bound can be obtainedfrom the observation that in our model there are no additional contributions to the Higgsproduction cross-section and no modifications of the partial decay widths of the Higgs bosoninto SM final states Therefore the presence of an invisible decay channel leads to an overallreduction of the signal strength in visible channels A global fit of all observed decay channels(combined with the bounds on invisible Higgs decays) then gives BR(hrarr inv) 026 [40]

Crucially the bound from invisible Higgs decays becomes independent of the singletmass for ms mh2 Invisible Higgs decays will therefore provide the strongest constraintsfor small singlet masses Indeed this constraint rules out the entire mass region where directdetection experiments lose sensitivity (see figure 1) As a result only two mass regions remainviable a low-mass region 526 GeV lt ms lt 645 GeV and a high-mass region ms amp 93 GeV

34 Other constraints

It has been pointed out recently [24] that bounds on γ-ray lines from Fermi-LAT [41] ruleout the parameter region where ms is slightly above mh2 To be safe from this constraintwe will focus on the mass range 526 GeV lt ms lt 624 GeV which we shall refer to as thelow-mass region For the high-mass region on the other hand there are no strong constraintsfrom indirect detection Moreover collider searches for singlet scalars with ms gt mh2 areextremely challenging [20ndash23 35] and consequently there are no relevant bounds from theLHC for the high-mass region [42]

The most significant improvements in sensitivity in the near future are expected tocome from direct detection experiments Indeed XENON1T [43] is expected to improve

7This equation agrees with [4 19ndash22 35] but disagrees with [23]

ndash 7 ndash

JCAP11(2015)015

upon current LUX constraints on the DM scattering cross-section by a factor of about 50and will therefore be able to probe the high-mass region up to ms asymp 4 TeV As we will showin S-inflation singlet masses larger than about 2 TeV are excluded by the Planck 2-σ upperbound on ns and perturbativity XENON1T will therefore be able to probe the entire high-mass region relevant for singlet inflation Similarly XENON1T can also further constrain thelow-mass region and potentially probe singlet masses in the range 53 GeV lt ms lt 57 GeV

4 Renormalisation group evolution and theoretical constraints

In order to connect the inflationary observables of our model to the measured SM parametersand the DM phenomenology discussed in the previous section we need to calculate theevolution of all couplings under the RG equations [8 10 12 14 44ndash47] Existing analyseshave considered the RG equations for the SM at two-loop order and the contributions of thesinglet sector and non-minimal coupling at one-loop order (see also the appendix) [8 10] Toexamine the issue of vacuum stability we improve the accuracy of our analysis further byincorporating the three-loop RG equations for the SM gauge couplings [48] and the leadingorder three-loop corrections to the RG equations for λh and yt [49]8

When considering large field values for either s or h the RG equations are modifiedbecause there is a suppression of scalar propagators This suppression is captured by insertinga factor

cφ =1 +

ξφ φ2

mP

1 + (6ξφ + 1)ξφ φ2

mP

(41)

with φ = s (φ = h) for each s (h) propagating in a loop [8] The changes in the RGequations for large values of the Higgs field have been discussed in detail in [47] Themodifications resulting from large singlet field values can be found in [8] and are reviewedin the appendix Note that when considering S-inflation such that s h we can set thesuppression factor ch = 1

We determine the values of the SM parameters at micro = mt following [46] Using themost recent values from the Particle Data Group [50]

mt = (1732plusmn 09) GeV mH = (12509plusmn 024) GeV αS(mZ) = 01185plusmn 00006 (42)

we obtain at micro = mt

yt = 0936plusmn 0005 λh = 01260plusmn 00014 gS = 1164plusmn 0003 (43)

Unless explicitly stated otherwise we will use the central values for all calculations belowFor given couplings at micro = mt we then use the public code RGErun 207 [51] to calculatethe couplings at higher scales

In contrast to the remaining couplings we fix the non-minimal couplings ξh and ξs atmicro = mP In order to obtain the correct amplitude of the scalar power spectrum we require

U

ε= (000271mP)4 (44)

where U and ε are the potential and the first slow-roll parameter in the Einstein frameat the beginning of inflation as defined in eq (28) and eq (214) respectively Imposing

8We thank Kyle Allison for sharing his numerical implementation of these equations

ndash 8 ndash

JCAP11(2015)015

equation (44) allows us to determine ξs at the scale of inflation once all other parametershave been fixed Note that since the value of s at the beginning of inflation also depends onξs equation (44) can only be solved numerically We then iteratively determine the requiredvalue of ξs at the electroweak scale such that RG evolution yields the desired value at thescale of inflation

The coupling ξh plays a very limited role for the phenomenology of our model because wedo not consider the case of large Higgs field values for inflation As a result our predictionsfor the inflationary observables show only a very mild dependence on ξh so that ξh canessentially be chosen arbitrarily Nevertheless it is not possible to simply set this parameterto zero since radiative corrections induce a mixing between ξs and ξh Moreover we willsee below that the value of ξh plays an important role for determining whether our modelviolates unitarity below the scale of inflation As with ξs we fix ξh at mP and then determineiteratively the required value of ξh at the electroweak scale9

41 Metastability

We now discuss various theoretical constraints related to the RG evolution of the parame-ters in our model It is a well-known fact that for the central values of the measured SMparameters the electroweak vacuum becomes metastable at high scales because the quarticHiggs coupling λh runs to negative values (see eg [46]) This metastability is not in anyobvious way a problem as the lifetime of the electroweak vacuum is well above the age ofthe Universe [52] (note however that this estimate may potentially be spoiled by effectsfrom Planck-scale higher-dimensional operators [53ndash55]) However one may speculate thata stable electroweak vacuum is necessary for a consistent theory for example if the vacuumenergy relative to the absolute minimum is a physical energy density leading to inflation Itis therefore an interesting aspect of singlet extensions of the SM that scalar singlets give apositive contribution to the running of λh [56 57]

βλh = βSMλh +1

32π2c2s λ

2hs (45)

In fact it was shown in [58] that for the case of a minimally-coupled singlet λhs can bechosen such that the electroweak vacuum remains stable all the way up to the Planck scaleand at the same time (for appropriate choices of the singlet mass ms) the singlet obtains athermal relic density compatible with the observed DM abundance10

In order to study electroweak vacuum stability we need to consider the potential in theh-direction with s = 011 Vacuum stability then requires that λh(micro) gt 0 for micro up to mPIn the present study we consider both the case where λhs is sufficiently large to stabilisethe electroweak vacuum and the case where λhs only increases the lifetime of the metastablevacuum but does not render it completely stable We focus throughout on the case whereλhs is positive

42 Examples

Figure 2 shows an example for the evolution of scalar couplings (left) and the non-minimalcouplings (right) under the RG equations discussed above Solid lines correspond to the case

9The running of ξh between mP and the scale of inflation is completely negligible since the relevantdiagrams are strongly suppressed for large values of s

10Note that if the singlet mixes with the Higgs there will be additional threshold effects at micro = ms fromintegrating out the singlet [59] In the setup we consider however this effect is not important [60]

11A more detailed study of the potential in general directions with both s 6= 0 and h 6= 0 (along the linesof [61]) is beyond the scope of the present work

ndash 9 ndash

JCAP11(2015)015

1000 106 109 1012 1015 10180001

00050010

00500100

05001

μ [GeV]

Couplingstrength

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

λh

λhs

λs

1000 106 109 1012 1015 101810

100

1000

104

105

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

ξh

ξs

cs

Figure 2 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplingsξh and ξs (right) as a function of the renormalisation scale micro for a typical parameter point in thehigh-mass region Solid lines show the running in the s-direction while dotted lines correspond to therunning in the h-direction In the right panel we also show the suppression factor cs which modifiesthe running in the s-direction at large field value

s h which is relevant for inflation while dotted lines correspond to h s which is relevantfor vacuum stability We fix the scalar couplings at the weak scale choosing λhs = 025 andλs = 001 such that the observed relic abundance can be reproduced for ms = 835 GeV Weconsider λs λhs in which case the value of ξs necessary to obtain the correct amplitude ofthe scalar power spectrum is reduced For our choice we find ξs sim 104 Note however thatwhile λhs exhibits only moderate running λs grows significantly with the renormalisationscale micro because its β-function contains a term proportional to λ2hs Choosing even smallervalues of λs at the weak scale will therefore not significantly reduce its value at the scale ofinflation nor the corresponding value of ξs

An important observation from figure 2 is that λh does not run negative and hencethe electroweak vacuum remains stable all the way up to the Planck scale The additionalcontribution from the singlet scalar is sufficient to ensure λh gt 10minus3 for all renormalisationscales up to mP For field values s amp 1015 GeV h the singlet propagator is suppressedleading to a visible kink in the running of λh We show the propagator suppression factorcs in the right panel of figure 2 One can clearly see how this suppression factor affects therunning of ξh which becomes nearly constant for s amp 1015 GeV In this particular examplewe have chosen ξh(mP) = 1000 (for s h) This choice together with λs 1 implies thatthe running of ξs(micro) from mt to mP is negligible

In the low-mass region we are interested in much smaller values of λhs typically below10minus2 A particular example is shown in figure 3 (left) for the representative choice λhs(mt) =0002 and λs = 00005 which yields the observed relic abundance for ms asymp 57 GeV Weobserve that if λs and λhs are both small at the electroweak scale these couplings exhibitonly very little running up to the scale of inflation For the same reason it is impossible toinfluence the running of the Higgs couplings sufficiently to prevent λhs from running negativeat around 1011 GeV

If λhs is small we can obtain the correct scalar power spectrum amplitude with a muchsmaller value of ξs during inflation For the specific case considered in figure 3 we findξs sim 103 For these values of ξs and λhs the loop-induced corrections to ξh are very smalland hence this coupling changes only very slightly under RG evolution

43 Perturbativity

In order for our calculation of the running couplings and the radiative corrections to thepotential to be reliable we must require that all couplings remain perturbative up to the

ndash 10 ndash

JCAP11(2015)015

1000 106 109 1012 1015 101810-4

0001

0010

0100

1

μ [GeV]

Couplingstrength

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

λh

λhs

λs

1000 106 109 1012 1015 101801

1

10

100

1000

104

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

ξh

ξs

cs

Figure 3 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplings ξhand ξs (right) as a function of the renormalisation scale micro for a typical parameter point in the low-mass region In the right panel we also show the suppression factor cs which modifies the runningat large field value Note that λh runs negative for micro amp 1011 GeV

scale of inflation which is typically 1017ndash1018 GeV This requirement is easily satisfied forthe SM couplings but needs to be checked explicitly for the couplings of the singlet whichcan grow significantly with increasing renormalisation scale micro We follow [58 62] and usethe requirement of perturbative unitarity to impose an upper bound on the scalar couplingsThis procedure gives

λs lt4π

3and λhs lt 8π (46)

As we will see below the non-minimal coupling ξs can be much larger than unity withoutinvalidating a perturbative calculation Nevertheless if ξs and ξh are both very large pro-cesses involving both couplings may violate perturbative unitarity implying that there maybe new physics or strong coupling below the scale of inflation We will now discuss this issuein more detail

5 Unitarity-violation during inflation

In this section we estimate the scale of perturbative unitarity-violation as a function of thebackground inflaton field Note that by ldquounitarity-violation scalerdquo we mean the scale at whichperturbation theory in scalar scattering breaks down so this may in fact indicate the onsetof unitarity-conserving scattering in a strongly-coupled regime [28 31] We will considerunitarity-violation in the scattering of scalar particles corresponding to perturbations aboutthe background field In the case of a real scalar s and the fields of the Higgs doublet Hthere are two distinct scattering processes we need to consider (i) δs h1 harr δs h1 and (ii)h1h2 harr h1h2 where δs is the perturbation about the background s field and h1 and h2are two of the Higgs doublet scalars Scattering with the other scalars in H is equivalentto these two processes We will use dimensional analysis to estimate the scale of unitarity-violation by determining the leading-order processes in the Einstein frame which result inunitarity-violating scattering

It will be sufficient to consider the Einstein frame Lagrangian for two real scalar fieldsφi where mdash using the notation of [30] mdash φi stands for either s or a component of the Higgsdoublet Unitarity-violation requires that there are two different scalars in the scatteringprocess since in the case of a single scalar there is a cancellation between s- t- and u-channelamplitudes [63]

ndash 11 ndash

JCAP11(2015)015

Since unitarity-violating scattering in the Jordan frame is due to graviton exchange viathe non-minimal coupling to R we cat set V = 0 The Einstein frame action for two realscalars is then of the form

SE =

intd4xradicminusg

Lii +sumiltj

Lij minus1

2m2

P R

(51)

where

Lii =1

2

Ω2 +6 ξ2i φ

2i

m2P

Ω4

gmicroν partmicroφi partνφi and Lij =6 ξi ξj φi φj g

microν partmicroφi partνφjm2

P Ω4(52)

with

Ω2 = 1 +ξj φ

2j

m2P

(53)

The interaction terms proportional to ξi ξj are responsible for the dominant unitarity-viola-tion in scattering cross-sections calculated in the Einstein frame These interactions are theEinstein frame analogue of scalar scattering via graviton exchange in the Jordan frame due tothe non-minimal coupling To obtain the scale of unitarity-violation in terms of the physicalenergy defined in the Jordan frame we first canonically normalize the fields in the Einsteinframe then estimate the magnitude of the scattering matrix element and finally transformthe unitarity-violation scale in the Einstein frame back to that in the Jordan frame

In the following we will denote the inflaton by φ1 (equiv s) which we expand about thebackground field ie φ1 = φ1 + δφ1 The Higgs doublet scalars are denoted by φ2 (equiv h1)and φ3 (equiv h2) The corresponding canonically normalized scattering fields in the Einsteinframe are then defined to be ϕ1 ϕ2 and ϕ3 Once we have determined the interactions ofthe canonically normalised fields we use dimensional analysis to estimate the scale of tree-level unitarity-violation For this purpose we introduce appropriate factors of E to makethe coefficient of the interaction terms in L dimensionless Energy scales E in the Einsteinframe are related to the ones in the Jordan frame via E = EΩ where during inflation

Ω2 ξ1φ21m

2P asymp N 1 Unitarity conservation implies that the matrix element for any

2 harr 2 scattering process should be smaller than O(1) so we can determine the scale ofunitarity-violation (denoted by Λ) by determining the value of E that saturates this boundThis was demonstrated explicitly in [8] by comparing the dimensional estimate with theexact value from the full scattering amplitude

In the following we consider three regimes for φ1 each of which leads to a differentform of the Lagrangian and the scattering amplitudes

bull Regime A large field values In this regime we have Ω gt 1 (implying that ξ1 φ21m

2P gt 1)

and 6 ξ21 φ21m

2P gt 1

bull Regime B medium field values In this regime we have approximately Ω asymp 1 (implying

that ξ1 φ21m

2P lt 1) but still 6 ξ21 φ

21m

2P gt 1

bull Regime C small field values Finally we consider 6 ξ21 φ21m

2P lt 1 which in particular

implies Ω asymp 1

ndash 12 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

JCAP11(2015)015

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

5 10 50 100 500 100010-4

0001

0010

0100

1

ms [GeV]

λhs

Figure 1 Excluded parameter regions from LUX (red dotted) and searches for invisible Higgs decays(blue dashed) compared to the coupling implied by the relic density constraint (green solid)

parameter region can however be efficiently constrained by considering how the Higgs-singletcoupling λhs would modify the branching ratios of the SM Higgs boson The partial decaywidth for hrarr ss is given by7

Γ(hrarr ss) =λ2hsv

2

32πmh

radic1minus 4m2

s

m2h

(33)

This theoretical prediction can be compared to the experimental bound on invisibleHiggs decays from the LHC Direct searches for invisible Higgs decays in the vector bosonfusion channel give BR(h rarr inv) 029 [39] A somewhat stronger bound can be obtainedfrom the observation that in our model there are no additional contributions to the Higgsproduction cross-section and no modifications of the partial decay widths of the Higgs bosoninto SM final states Therefore the presence of an invisible decay channel leads to an overallreduction of the signal strength in visible channels A global fit of all observed decay channels(combined with the bounds on invisible Higgs decays) then gives BR(hrarr inv) 026 [40]

Crucially the bound from invisible Higgs decays becomes independent of the singletmass for ms mh2 Invisible Higgs decays will therefore provide the strongest constraintsfor small singlet masses Indeed this constraint rules out the entire mass region where directdetection experiments lose sensitivity (see figure 1) As a result only two mass regions remainviable a low-mass region 526 GeV lt ms lt 645 GeV and a high-mass region ms amp 93 GeV

34 Other constraints

It has been pointed out recently [24] that bounds on γ-ray lines from Fermi-LAT [41] ruleout the parameter region where ms is slightly above mh2 To be safe from this constraintwe will focus on the mass range 526 GeV lt ms lt 624 GeV which we shall refer to as thelow-mass region For the high-mass region on the other hand there are no strong constraintsfrom indirect detection Moreover collider searches for singlet scalars with ms gt mh2 areextremely challenging [20ndash23 35] and consequently there are no relevant bounds from theLHC for the high-mass region [42]

The most significant improvements in sensitivity in the near future are expected tocome from direct detection experiments Indeed XENON1T [43] is expected to improve

7This equation agrees with [4 19ndash22 35] but disagrees with [23]

ndash 7 ndash

JCAP11(2015)015

upon current LUX constraints on the DM scattering cross-section by a factor of about 50and will therefore be able to probe the high-mass region up to ms asymp 4 TeV As we will showin S-inflation singlet masses larger than about 2 TeV are excluded by the Planck 2-σ upperbound on ns and perturbativity XENON1T will therefore be able to probe the entire high-mass region relevant for singlet inflation Similarly XENON1T can also further constrain thelow-mass region and potentially probe singlet masses in the range 53 GeV lt ms lt 57 GeV

4 Renormalisation group evolution and theoretical constraints

In order to connect the inflationary observables of our model to the measured SM parametersand the DM phenomenology discussed in the previous section we need to calculate theevolution of all couplings under the RG equations [8 10 12 14 44ndash47] Existing analyseshave considered the RG equations for the SM at two-loop order and the contributions of thesinglet sector and non-minimal coupling at one-loop order (see also the appendix) [8 10] Toexamine the issue of vacuum stability we improve the accuracy of our analysis further byincorporating the three-loop RG equations for the SM gauge couplings [48] and the leadingorder three-loop corrections to the RG equations for λh and yt [49]8

When considering large field values for either s or h the RG equations are modifiedbecause there is a suppression of scalar propagators This suppression is captured by insertinga factor

cφ =1 +

ξφ φ2

mP

1 + (6ξφ + 1)ξφ φ2

mP

(41)

with φ = s (φ = h) for each s (h) propagating in a loop [8] The changes in the RGequations for large values of the Higgs field have been discussed in detail in [47] Themodifications resulting from large singlet field values can be found in [8] and are reviewedin the appendix Note that when considering S-inflation such that s h we can set thesuppression factor ch = 1

We determine the values of the SM parameters at micro = mt following [46] Using themost recent values from the Particle Data Group [50]

mt = (1732plusmn 09) GeV mH = (12509plusmn 024) GeV αS(mZ) = 01185plusmn 00006 (42)

we obtain at micro = mt

yt = 0936plusmn 0005 λh = 01260plusmn 00014 gS = 1164plusmn 0003 (43)

Unless explicitly stated otherwise we will use the central values for all calculations belowFor given couplings at micro = mt we then use the public code RGErun 207 [51] to calculatethe couplings at higher scales

In contrast to the remaining couplings we fix the non-minimal couplings ξh and ξs atmicro = mP In order to obtain the correct amplitude of the scalar power spectrum we require

U

ε= (000271mP)4 (44)

where U and ε are the potential and the first slow-roll parameter in the Einstein frameat the beginning of inflation as defined in eq (28) and eq (214) respectively Imposing

8We thank Kyle Allison for sharing his numerical implementation of these equations

ndash 8 ndash

JCAP11(2015)015

equation (44) allows us to determine ξs at the scale of inflation once all other parametershave been fixed Note that since the value of s at the beginning of inflation also depends onξs equation (44) can only be solved numerically We then iteratively determine the requiredvalue of ξs at the electroweak scale such that RG evolution yields the desired value at thescale of inflation

The coupling ξh plays a very limited role for the phenomenology of our model because wedo not consider the case of large Higgs field values for inflation As a result our predictionsfor the inflationary observables show only a very mild dependence on ξh so that ξh canessentially be chosen arbitrarily Nevertheless it is not possible to simply set this parameterto zero since radiative corrections induce a mixing between ξs and ξh Moreover we willsee below that the value of ξh plays an important role for determining whether our modelviolates unitarity below the scale of inflation As with ξs we fix ξh at mP and then determineiteratively the required value of ξh at the electroweak scale9

41 Metastability

We now discuss various theoretical constraints related to the RG evolution of the parame-ters in our model It is a well-known fact that for the central values of the measured SMparameters the electroweak vacuum becomes metastable at high scales because the quarticHiggs coupling λh runs to negative values (see eg [46]) This metastability is not in anyobvious way a problem as the lifetime of the electroweak vacuum is well above the age ofthe Universe [52] (note however that this estimate may potentially be spoiled by effectsfrom Planck-scale higher-dimensional operators [53ndash55]) However one may speculate thata stable electroweak vacuum is necessary for a consistent theory for example if the vacuumenergy relative to the absolute minimum is a physical energy density leading to inflation Itis therefore an interesting aspect of singlet extensions of the SM that scalar singlets give apositive contribution to the running of λh [56 57]

βλh = βSMλh +1

32π2c2s λ

2hs (45)

In fact it was shown in [58] that for the case of a minimally-coupled singlet λhs can bechosen such that the electroweak vacuum remains stable all the way up to the Planck scaleand at the same time (for appropriate choices of the singlet mass ms) the singlet obtains athermal relic density compatible with the observed DM abundance10

In order to study electroweak vacuum stability we need to consider the potential in theh-direction with s = 011 Vacuum stability then requires that λh(micro) gt 0 for micro up to mPIn the present study we consider both the case where λhs is sufficiently large to stabilisethe electroweak vacuum and the case where λhs only increases the lifetime of the metastablevacuum but does not render it completely stable We focus throughout on the case whereλhs is positive

42 Examples

Figure 2 shows an example for the evolution of scalar couplings (left) and the non-minimalcouplings (right) under the RG equations discussed above Solid lines correspond to the case

9The running of ξh between mP and the scale of inflation is completely negligible since the relevantdiagrams are strongly suppressed for large values of s

10Note that if the singlet mixes with the Higgs there will be additional threshold effects at micro = ms fromintegrating out the singlet [59] In the setup we consider however this effect is not important [60]

11A more detailed study of the potential in general directions with both s 6= 0 and h 6= 0 (along the linesof [61]) is beyond the scope of the present work

ndash 9 ndash

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1000 106 109 1012 1015 10180001

00050010

00500100

05001

μ [GeV]

Couplingstrength

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

λh

λhs

λs

1000 106 109 1012 1015 101810

100

1000

104

105

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

ξh

ξs

cs

Figure 2 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplingsξh and ξs (right) as a function of the renormalisation scale micro for a typical parameter point in thehigh-mass region Solid lines show the running in the s-direction while dotted lines correspond to therunning in the h-direction In the right panel we also show the suppression factor cs which modifiesthe running in the s-direction at large field value

s h which is relevant for inflation while dotted lines correspond to h s which is relevantfor vacuum stability We fix the scalar couplings at the weak scale choosing λhs = 025 andλs = 001 such that the observed relic abundance can be reproduced for ms = 835 GeV Weconsider λs λhs in which case the value of ξs necessary to obtain the correct amplitude ofthe scalar power spectrum is reduced For our choice we find ξs sim 104 Note however thatwhile λhs exhibits only moderate running λs grows significantly with the renormalisationscale micro because its β-function contains a term proportional to λ2hs Choosing even smallervalues of λs at the weak scale will therefore not significantly reduce its value at the scale ofinflation nor the corresponding value of ξs

An important observation from figure 2 is that λh does not run negative and hencethe electroweak vacuum remains stable all the way up to the Planck scale The additionalcontribution from the singlet scalar is sufficient to ensure λh gt 10minus3 for all renormalisationscales up to mP For field values s amp 1015 GeV h the singlet propagator is suppressedleading to a visible kink in the running of λh We show the propagator suppression factorcs in the right panel of figure 2 One can clearly see how this suppression factor affects therunning of ξh which becomes nearly constant for s amp 1015 GeV In this particular examplewe have chosen ξh(mP) = 1000 (for s h) This choice together with λs 1 implies thatthe running of ξs(micro) from mt to mP is negligible

In the low-mass region we are interested in much smaller values of λhs typically below10minus2 A particular example is shown in figure 3 (left) for the representative choice λhs(mt) =0002 and λs = 00005 which yields the observed relic abundance for ms asymp 57 GeV Weobserve that if λs and λhs are both small at the electroweak scale these couplings exhibitonly very little running up to the scale of inflation For the same reason it is impossible toinfluence the running of the Higgs couplings sufficiently to prevent λhs from running negativeat around 1011 GeV

If λhs is small we can obtain the correct scalar power spectrum amplitude with a muchsmaller value of ξs during inflation For the specific case considered in figure 3 we findξs sim 103 For these values of ξs and λhs the loop-induced corrections to ξh are very smalland hence this coupling changes only very slightly under RG evolution

43 Perturbativity

In order for our calculation of the running couplings and the radiative corrections to thepotential to be reliable we must require that all couplings remain perturbative up to the

ndash 10 ndash

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1000 106 109 1012 1015 101810-4

0001

0010

0100

1

μ [GeV]

Couplingstrength

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

λh

λhs

λs

1000 106 109 1012 1015 101801

1

10

100

1000

104

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

ξh

ξs

cs

Figure 3 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplings ξhand ξs (right) as a function of the renormalisation scale micro for a typical parameter point in the low-mass region In the right panel we also show the suppression factor cs which modifies the runningat large field value Note that λh runs negative for micro amp 1011 GeV

scale of inflation which is typically 1017ndash1018 GeV This requirement is easily satisfied forthe SM couplings but needs to be checked explicitly for the couplings of the singlet whichcan grow significantly with increasing renormalisation scale micro We follow [58 62] and usethe requirement of perturbative unitarity to impose an upper bound on the scalar couplingsThis procedure gives

λs lt4π

3and λhs lt 8π (46)

As we will see below the non-minimal coupling ξs can be much larger than unity withoutinvalidating a perturbative calculation Nevertheless if ξs and ξh are both very large pro-cesses involving both couplings may violate perturbative unitarity implying that there maybe new physics or strong coupling below the scale of inflation We will now discuss this issuein more detail

5 Unitarity-violation during inflation

In this section we estimate the scale of perturbative unitarity-violation as a function of thebackground inflaton field Note that by ldquounitarity-violation scalerdquo we mean the scale at whichperturbation theory in scalar scattering breaks down so this may in fact indicate the onsetof unitarity-conserving scattering in a strongly-coupled regime [28 31] We will considerunitarity-violation in the scattering of scalar particles corresponding to perturbations aboutthe background field In the case of a real scalar s and the fields of the Higgs doublet Hthere are two distinct scattering processes we need to consider (i) δs h1 harr δs h1 and (ii)h1h2 harr h1h2 where δs is the perturbation about the background s field and h1 and h2are two of the Higgs doublet scalars Scattering with the other scalars in H is equivalentto these two processes We will use dimensional analysis to estimate the scale of unitarity-violation by determining the leading-order processes in the Einstein frame which result inunitarity-violating scattering

It will be sufficient to consider the Einstein frame Lagrangian for two real scalar fieldsφi where mdash using the notation of [30] mdash φi stands for either s or a component of the Higgsdoublet Unitarity-violation requires that there are two different scalars in the scatteringprocess since in the case of a single scalar there is a cancellation between s- t- and u-channelamplitudes [63]

ndash 11 ndash

JCAP11(2015)015

Since unitarity-violating scattering in the Jordan frame is due to graviton exchange viathe non-minimal coupling to R we cat set V = 0 The Einstein frame action for two realscalars is then of the form

SE =

intd4xradicminusg

Lii +sumiltj

Lij minus1

2m2

P R

(51)

where

Lii =1

2

Ω2 +6 ξ2i φ

2i

m2P

Ω4

gmicroν partmicroφi partνφi and Lij =6 ξi ξj φi φj g

microν partmicroφi partνφjm2

P Ω4(52)

with

Ω2 = 1 +ξj φ

2j

m2P

(53)

The interaction terms proportional to ξi ξj are responsible for the dominant unitarity-viola-tion in scattering cross-sections calculated in the Einstein frame These interactions are theEinstein frame analogue of scalar scattering via graviton exchange in the Jordan frame due tothe non-minimal coupling To obtain the scale of unitarity-violation in terms of the physicalenergy defined in the Jordan frame we first canonically normalize the fields in the Einsteinframe then estimate the magnitude of the scattering matrix element and finally transformthe unitarity-violation scale in the Einstein frame back to that in the Jordan frame

In the following we will denote the inflaton by φ1 (equiv s) which we expand about thebackground field ie φ1 = φ1 + δφ1 The Higgs doublet scalars are denoted by φ2 (equiv h1)and φ3 (equiv h2) The corresponding canonically normalized scattering fields in the Einsteinframe are then defined to be ϕ1 ϕ2 and ϕ3 Once we have determined the interactions ofthe canonically normalised fields we use dimensional analysis to estimate the scale of tree-level unitarity-violation For this purpose we introduce appropriate factors of E to makethe coefficient of the interaction terms in L dimensionless Energy scales E in the Einsteinframe are related to the ones in the Jordan frame via E = EΩ where during inflation

Ω2 ξ1φ21m

2P asymp N 1 Unitarity conservation implies that the matrix element for any

2 harr 2 scattering process should be smaller than O(1) so we can determine the scale ofunitarity-violation (denoted by Λ) by determining the value of E that saturates this boundThis was demonstrated explicitly in [8] by comparing the dimensional estimate with theexact value from the full scattering amplitude

In the following we consider three regimes for φ1 each of which leads to a differentform of the Lagrangian and the scattering amplitudes

bull Regime A large field values In this regime we have Ω gt 1 (implying that ξ1 φ21m

2P gt 1)

and 6 ξ21 φ21m

2P gt 1

bull Regime B medium field values In this regime we have approximately Ω asymp 1 (implying

that ξ1 φ21m

2P lt 1) but still 6 ξ21 φ

21m

2P gt 1

bull Regime C small field values Finally we consider 6 ξ21 φ21m

2P lt 1 which in particular

implies Ω asymp 1

ndash 12 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

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1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

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27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

upon current LUX constraints on the DM scattering cross-section by a factor of about 50and will therefore be able to probe the high-mass region up to ms asymp 4 TeV As we will showin S-inflation singlet masses larger than about 2 TeV are excluded by the Planck 2-σ upperbound on ns and perturbativity XENON1T will therefore be able to probe the entire high-mass region relevant for singlet inflation Similarly XENON1T can also further constrain thelow-mass region and potentially probe singlet masses in the range 53 GeV lt ms lt 57 GeV

4 Renormalisation group evolution and theoretical constraints

In order to connect the inflationary observables of our model to the measured SM parametersand the DM phenomenology discussed in the previous section we need to calculate theevolution of all couplings under the RG equations [8 10 12 14 44ndash47] Existing analyseshave considered the RG equations for the SM at two-loop order and the contributions of thesinglet sector and non-minimal coupling at one-loop order (see also the appendix) [8 10] Toexamine the issue of vacuum stability we improve the accuracy of our analysis further byincorporating the three-loop RG equations for the SM gauge couplings [48] and the leadingorder three-loop corrections to the RG equations for λh and yt [49]8

When considering large field values for either s or h the RG equations are modifiedbecause there is a suppression of scalar propagators This suppression is captured by insertinga factor

cφ =1 +

ξφ φ2

mP

1 + (6ξφ + 1)ξφ φ2

mP

(41)

with φ = s (φ = h) for each s (h) propagating in a loop [8] The changes in the RGequations for large values of the Higgs field have been discussed in detail in [47] Themodifications resulting from large singlet field values can be found in [8] and are reviewedin the appendix Note that when considering S-inflation such that s h we can set thesuppression factor ch = 1

We determine the values of the SM parameters at micro = mt following [46] Using themost recent values from the Particle Data Group [50]

mt = (1732plusmn 09) GeV mH = (12509plusmn 024) GeV αS(mZ) = 01185plusmn 00006 (42)

we obtain at micro = mt

yt = 0936plusmn 0005 λh = 01260plusmn 00014 gS = 1164plusmn 0003 (43)

Unless explicitly stated otherwise we will use the central values for all calculations belowFor given couplings at micro = mt we then use the public code RGErun 207 [51] to calculatethe couplings at higher scales

In contrast to the remaining couplings we fix the non-minimal couplings ξh and ξs atmicro = mP In order to obtain the correct amplitude of the scalar power spectrum we require

U

ε= (000271mP)4 (44)

where U and ε are the potential and the first slow-roll parameter in the Einstein frameat the beginning of inflation as defined in eq (28) and eq (214) respectively Imposing

8We thank Kyle Allison for sharing his numerical implementation of these equations

ndash 8 ndash

JCAP11(2015)015

equation (44) allows us to determine ξs at the scale of inflation once all other parametershave been fixed Note that since the value of s at the beginning of inflation also depends onξs equation (44) can only be solved numerically We then iteratively determine the requiredvalue of ξs at the electroweak scale such that RG evolution yields the desired value at thescale of inflation

The coupling ξh plays a very limited role for the phenomenology of our model because wedo not consider the case of large Higgs field values for inflation As a result our predictionsfor the inflationary observables show only a very mild dependence on ξh so that ξh canessentially be chosen arbitrarily Nevertheless it is not possible to simply set this parameterto zero since radiative corrections induce a mixing between ξs and ξh Moreover we willsee below that the value of ξh plays an important role for determining whether our modelviolates unitarity below the scale of inflation As with ξs we fix ξh at mP and then determineiteratively the required value of ξh at the electroweak scale9

41 Metastability

We now discuss various theoretical constraints related to the RG evolution of the parame-ters in our model It is a well-known fact that for the central values of the measured SMparameters the electroweak vacuum becomes metastable at high scales because the quarticHiggs coupling λh runs to negative values (see eg [46]) This metastability is not in anyobvious way a problem as the lifetime of the electroweak vacuum is well above the age ofthe Universe [52] (note however that this estimate may potentially be spoiled by effectsfrom Planck-scale higher-dimensional operators [53ndash55]) However one may speculate thata stable electroweak vacuum is necessary for a consistent theory for example if the vacuumenergy relative to the absolute minimum is a physical energy density leading to inflation Itis therefore an interesting aspect of singlet extensions of the SM that scalar singlets give apositive contribution to the running of λh [56 57]

βλh = βSMλh +1

32π2c2s λ

2hs (45)

In fact it was shown in [58] that for the case of a minimally-coupled singlet λhs can bechosen such that the electroweak vacuum remains stable all the way up to the Planck scaleand at the same time (for appropriate choices of the singlet mass ms) the singlet obtains athermal relic density compatible with the observed DM abundance10

In order to study electroweak vacuum stability we need to consider the potential in theh-direction with s = 011 Vacuum stability then requires that λh(micro) gt 0 for micro up to mPIn the present study we consider both the case where λhs is sufficiently large to stabilisethe electroweak vacuum and the case where λhs only increases the lifetime of the metastablevacuum but does not render it completely stable We focus throughout on the case whereλhs is positive

42 Examples

Figure 2 shows an example for the evolution of scalar couplings (left) and the non-minimalcouplings (right) under the RG equations discussed above Solid lines correspond to the case

9The running of ξh between mP and the scale of inflation is completely negligible since the relevantdiagrams are strongly suppressed for large values of s

10Note that if the singlet mixes with the Higgs there will be additional threshold effects at micro = ms fromintegrating out the singlet [59] In the setup we consider however this effect is not important [60]

11A more detailed study of the potential in general directions with both s 6= 0 and h 6= 0 (along the linesof [61]) is beyond the scope of the present work

ndash 9 ndash

JCAP11(2015)015

1000 106 109 1012 1015 10180001

00050010

00500100

05001

μ [GeV]

Couplingstrength

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

λh

λhs

λs

1000 106 109 1012 1015 101810

100

1000

104

105

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

ξh

ξs

cs

Figure 2 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplingsξh and ξs (right) as a function of the renormalisation scale micro for a typical parameter point in thehigh-mass region Solid lines show the running in the s-direction while dotted lines correspond to therunning in the h-direction In the right panel we also show the suppression factor cs which modifiesthe running in the s-direction at large field value

s h which is relevant for inflation while dotted lines correspond to h s which is relevantfor vacuum stability We fix the scalar couplings at the weak scale choosing λhs = 025 andλs = 001 such that the observed relic abundance can be reproduced for ms = 835 GeV Weconsider λs λhs in which case the value of ξs necessary to obtain the correct amplitude ofthe scalar power spectrum is reduced For our choice we find ξs sim 104 Note however thatwhile λhs exhibits only moderate running λs grows significantly with the renormalisationscale micro because its β-function contains a term proportional to λ2hs Choosing even smallervalues of λs at the weak scale will therefore not significantly reduce its value at the scale ofinflation nor the corresponding value of ξs

An important observation from figure 2 is that λh does not run negative and hencethe electroweak vacuum remains stable all the way up to the Planck scale The additionalcontribution from the singlet scalar is sufficient to ensure λh gt 10minus3 for all renormalisationscales up to mP For field values s amp 1015 GeV h the singlet propagator is suppressedleading to a visible kink in the running of λh We show the propagator suppression factorcs in the right panel of figure 2 One can clearly see how this suppression factor affects therunning of ξh which becomes nearly constant for s amp 1015 GeV In this particular examplewe have chosen ξh(mP) = 1000 (for s h) This choice together with λs 1 implies thatthe running of ξs(micro) from mt to mP is negligible

In the low-mass region we are interested in much smaller values of λhs typically below10minus2 A particular example is shown in figure 3 (left) for the representative choice λhs(mt) =0002 and λs = 00005 which yields the observed relic abundance for ms asymp 57 GeV Weobserve that if λs and λhs are both small at the electroweak scale these couplings exhibitonly very little running up to the scale of inflation For the same reason it is impossible toinfluence the running of the Higgs couplings sufficiently to prevent λhs from running negativeat around 1011 GeV

If λhs is small we can obtain the correct scalar power spectrum amplitude with a muchsmaller value of ξs during inflation For the specific case considered in figure 3 we findξs sim 103 For these values of ξs and λhs the loop-induced corrections to ξh are very smalland hence this coupling changes only very slightly under RG evolution

43 Perturbativity

In order for our calculation of the running couplings and the radiative corrections to thepotential to be reliable we must require that all couplings remain perturbative up to the

ndash 10 ndash

JCAP11(2015)015

1000 106 109 1012 1015 101810-4

0001

0010

0100

1

μ [GeV]

Couplingstrength

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

λh

λhs

λs

1000 106 109 1012 1015 101801

1

10

100

1000

104

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

ξh

ξs

cs

Figure 3 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplings ξhand ξs (right) as a function of the renormalisation scale micro for a typical parameter point in the low-mass region In the right panel we also show the suppression factor cs which modifies the runningat large field value Note that λh runs negative for micro amp 1011 GeV

scale of inflation which is typically 1017ndash1018 GeV This requirement is easily satisfied forthe SM couplings but needs to be checked explicitly for the couplings of the singlet whichcan grow significantly with increasing renormalisation scale micro We follow [58 62] and usethe requirement of perturbative unitarity to impose an upper bound on the scalar couplingsThis procedure gives

λs lt4π

3and λhs lt 8π (46)

As we will see below the non-minimal coupling ξs can be much larger than unity withoutinvalidating a perturbative calculation Nevertheless if ξs and ξh are both very large pro-cesses involving both couplings may violate perturbative unitarity implying that there maybe new physics or strong coupling below the scale of inflation We will now discuss this issuein more detail

5 Unitarity-violation during inflation

In this section we estimate the scale of perturbative unitarity-violation as a function of thebackground inflaton field Note that by ldquounitarity-violation scalerdquo we mean the scale at whichperturbation theory in scalar scattering breaks down so this may in fact indicate the onsetof unitarity-conserving scattering in a strongly-coupled regime [28 31] We will considerunitarity-violation in the scattering of scalar particles corresponding to perturbations aboutthe background field In the case of a real scalar s and the fields of the Higgs doublet Hthere are two distinct scattering processes we need to consider (i) δs h1 harr δs h1 and (ii)h1h2 harr h1h2 where δs is the perturbation about the background s field and h1 and h2are two of the Higgs doublet scalars Scattering with the other scalars in H is equivalentto these two processes We will use dimensional analysis to estimate the scale of unitarity-violation by determining the leading-order processes in the Einstein frame which result inunitarity-violating scattering

It will be sufficient to consider the Einstein frame Lagrangian for two real scalar fieldsφi where mdash using the notation of [30] mdash φi stands for either s or a component of the Higgsdoublet Unitarity-violation requires that there are two different scalars in the scatteringprocess since in the case of a single scalar there is a cancellation between s- t- and u-channelamplitudes [63]

ndash 11 ndash

JCAP11(2015)015

Since unitarity-violating scattering in the Jordan frame is due to graviton exchange viathe non-minimal coupling to R we cat set V = 0 The Einstein frame action for two realscalars is then of the form

SE =

intd4xradicminusg

Lii +sumiltj

Lij minus1

2m2

P R

(51)

where

Lii =1

2

Ω2 +6 ξ2i φ

2i

m2P

Ω4

gmicroν partmicroφi partνφi and Lij =6 ξi ξj φi φj g

microν partmicroφi partνφjm2

P Ω4(52)

with

Ω2 = 1 +ξj φ

2j

m2P

(53)

The interaction terms proportional to ξi ξj are responsible for the dominant unitarity-viola-tion in scattering cross-sections calculated in the Einstein frame These interactions are theEinstein frame analogue of scalar scattering via graviton exchange in the Jordan frame due tothe non-minimal coupling To obtain the scale of unitarity-violation in terms of the physicalenergy defined in the Jordan frame we first canonically normalize the fields in the Einsteinframe then estimate the magnitude of the scattering matrix element and finally transformthe unitarity-violation scale in the Einstein frame back to that in the Jordan frame

In the following we will denote the inflaton by φ1 (equiv s) which we expand about thebackground field ie φ1 = φ1 + δφ1 The Higgs doublet scalars are denoted by φ2 (equiv h1)and φ3 (equiv h2) The corresponding canonically normalized scattering fields in the Einsteinframe are then defined to be ϕ1 ϕ2 and ϕ3 Once we have determined the interactions ofthe canonically normalised fields we use dimensional analysis to estimate the scale of tree-level unitarity-violation For this purpose we introduce appropriate factors of E to makethe coefficient of the interaction terms in L dimensionless Energy scales E in the Einsteinframe are related to the ones in the Jordan frame via E = EΩ where during inflation

Ω2 ξ1φ21m

2P asymp N 1 Unitarity conservation implies that the matrix element for any

2 harr 2 scattering process should be smaller than O(1) so we can determine the scale ofunitarity-violation (denoted by Λ) by determining the value of E that saturates this boundThis was demonstrated explicitly in [8] by comparing the dimensional estimate with theexact value from the full scattering amplitude

In the following we consider three regimes for φ1 each of which leads to a differentform of the Lagrangian and the scattering amplitudes

bull Regime A large field values In this regime we have Ω gt 1 (implying that ξ1 φ21m

2P gt 1)

and 6 ξ21 φ21m

2P gt 1

bull Regime B medium field values In this regime we have approximately Ω asymp 1 (implying

that ξ1 φ21m

2P lt 1) but still 6 ξ21 φ

21m

2P gt 1

bull Regime C small field values Finally we consider 6 ξ21 φ21m

2P lt 1 which in particular

implies Ω asymp 1

ndash 12 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

JCAP11(2015)015

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

equation (44) allows us to determine ξs at the scale of inflation once all other parametershave been fixed Note that since the value of s at the beginning of inflation also depends onξs equation (44) can only be solved numerically We then iteratively determine the requiredvalue of ξs at the electroweak scale such that RG evolution yields the desired value at thescale of inflation

The coupling ξh plays a very limited role for the phenomenology of our model because wedo not consider the case of large Higgs field values for inflation As a result our predictionsfor the inflationary observables show only a very mild dependence on ξh so that ξh canessentially be chosen arbitrarily Nevertheless it is not possible to simply set this parameterto zero since radiative corrections induce a mixing between ξs and ξh Moreover we willsee below that the value of ξh plays an important role for determining whether our modelviolates unitarity below the scale of inflation As with ξs we fix ξh at mP and then determineiteratively the required value of ξh at the electroweak scale9

41 Metastability

We now discuss various theoretical constraints related to the RG evolution of the parame-ters in our model It is a well-known fact that for the central values of the measured SMparameters the electroweak vacuum becomes metastable at high scales because the quarticHiggs coupling λh runs to negative values (see eg [46]) This metastability is not in anyobvious way a problem as the lifetime of the electroweak vacuum is well above the age ofthe Universe [52] (note however that this estimate may potentially be spoiled by effectsfrom Planck-scale higher-dimensional operators [53ndash55]) However one may speculate thata stable electroweak vacuum is necessary for a consistent theory for example if the vacuumenergy relative to the absolute minimum is a physical energy density leading to inflation Itis therefore an interesting aspect of singlet extensions of the SM that scalar singlets give apositive contribution to the running of λh [56 57]

βλh = βSMλh +1

32π2c2s λ

2hs (45)

In fact it was shown in [58] that for the case of a minimally-coupled singlet λhs can bechosen such that the electroweak vacuum remains stable all the way up to the Planck scaleand at the same time (for appropriate choices of the singlet mass ms) the singlet obtains athermal relic density compatible with the observed DM abundance10

In order to study electroweak vacuum stability we need to consider the potential in theh-direction with s = 011 Vacuum stability then requires that λh(micro) gt 0 for micro up to mPIn the present study we consider both the case where λhs is sufficiently large to stabilisethe electroweak vacuum and the case where λhs only increases the lifetime of the metastablevacuum but does not render it completely stable We focus throughout on the case whereλhs is positive

42 Examples

Figure 2 shows an example for the evolution of scalar couplings (left) and the non-minimalcouplings (right) under the RG equations discussed above Solid lines correspond to the case

9The running of ξh between mP and the scale of inflation is completely negligible since the relevantdiagrams are strongly suppressed for large values of s

10Note that if the singlet mixes with the Higgs there will be additional threshold effects at micro = ms fromintegrating out the singlet [59] In the setup we consider however this effect is not important [60]

11A more detailed study of the potential in general directions with both s 6= 0 and h 6= 0 (along the linesof [61]) is beyond the scope of the present work

ndash 9 ndash

JCAP11(2015)015

1000 106 109 1012 1015 10180001

00050010

00500100

05001

μ [GeV]

Couplingstrength

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

λh

λhs

λs

1000 106 109 1012 1015 101810

100

1000

104

105

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

ξh

ξs

cs

Figure 2 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplingsξh and ξs (right) as a function of the renormalisation scale micro for a typical parameter point in thehigh-mass region Solid lines show the running in the s-direction while dotted lines correspond to therunning in the h-direction In the right panel we also show the suppression factor cs which modifiesthe running in the s-direction at large field value

s h which is relevant for inflation while dotted lines correspond to h s which is relevantfor vacuum stability We fix the scalar couplings at the weak scale choosing λhs = 025 andλs = 001 such that the observed relic abundance can be reproduced for ms = 835 GeV Weconsider λs λhs in which case the value of ξs necessary to obtain the correct amplitude ofthe scalar power spectrum is reduced For our choice we find ξs sim 104 Note however thatwhile λhs exhibits only moderate running λs grows significantly with the renormalisationscale micro because its β-function contains a term proportional to λ2hs Choosing even smallervalues of λs at the weak scale will therefore not significantly reduce its value at the scale ofinflation nor the corresponding value of ξs

An important observation from figure 2 is that λh does not run negative and hencethe electroweak vacuum remains stable all the way up to the Planck scale The additionalcontribution from the singlet scalar is sufficient to ensure λh gt 10minus3 for all renormalisationscales up to mP For field values s amp 1015 GeV h the singlet propagator is suppressedleading to a visible kink in the running of λh We show the propagator suppression factorcs in the right panel of figure 2 One can clearly see how this suppression factor affects therunning of ξh which becomes nearly constant for s amp 1015 GeV In this particular examplewe have chosen ξh(mP) = 1000 (for s h) This choice together with λs 1 implies thatthe running of ξs(micro) from mt to mP is negligible

In the low-mass region we are interested in much smaller values of λhs typically below10minus2 A particular example is shown in figure 3 (left) for the representative choice λhs(mt) =0002 and λs = 00005 which yields the observed relic abundance for ms asymp 57 GeV Weobserve that if λs and λhs are both small at the electroweak scale these couplings exhibitonly very little running up to the scale of inflation For the same reason it is impossible toinfluence the running of the Higgs couplings sufficiently to prevent λhs from running negativeat around 1011 GeV

If λhs is small we can obtain the correct scalar power spectrum amplitude with a muchsmaller value of ξs during inflation For the specific case considered in figure 3 we findξs sim 103 For these values of ξs and λhs the loop-induced corrections to ξh are very smalland hence this coupling changes only very slightly under RG evolution

43 Perturbativity

In order for our calculation of the running couplings and the radiative corrections to thepotential to be reliable we must require that all couplings remain perturbative up to the

ndash 10 ndash

JCAP11(2015)015

1000 106 109 1012 1015 101810-4

0001

0010

0100

1

μ [GeV]

Couplingstrength

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

λh

λhs

λs

1000 106 109 1012 1015 101801

1

10

100

1000

104

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

ξh

ξs

cs

Figure 3 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplings ξhand ξs (right) as a function of the renormalisation scale micro for a typical parameter point in the low-mass region In the right panel we also show the suppression factor cs which modifies the runningat large field value Note that λh runs negative for micro amp 1011 GeV

scale of inflation which is typically 1017ndash1018 GeV This requirement is easily satisfied forthe SM couplings but needs to be checked explicitly for the couplings of the singlet whichcan grow significantly with increasing renormalisation scale micro We follow [58 62] and usethe requirement of perturbative unitarity to impose an upper bound on the scalar couplingsThis procedure gives

λs lt4π

3and λhs lt 8π (46)

As we will see below the non-minimal coupling ξs can be much larger than unity withoutinvalidating a perturbative calculation Nevertheless if ξs and ξh are both very large pro-cesses involving both couplings may violate perturbative unitarity implying that there maybe new physics or strong coupling below the scale of inflation We will now discuss this issuein more detail

5 Unitarity-violation during inflation

In this section we estimate the scale of perturbative unitarity-violation as a function of thebackground inflaton field Note that by ldquounitarity-violation scalerdquo we mean the scale at whichperturbation theory in scalar scattering breaks down so this may in fact indicate the onsetof unitarity-conserving scattering in a strongly-coupled regime [28 31] We will considerunitarity-violation in the scattering of scalar particles corresponding to perturbations aboutthe background field In the case of a real scalar s and the fields of the Higgs doublet Hthere are two distinct scattering processes we need to consider (i) δs h1 harr δs h1 and (ii)h1h2 harr h1h2 where δs is the perturbation about the background s field and h1 and h2are two of the Higgs doublet scalars Scattering with the other scalars in H is equivalentto these two processes We will use dimensional analysis to estimate the scale of unitarity-violation by determining the leading-order processes in the Einstein frame which result inunitarity-violating scattering

It will be sufficient to consider the Einstein frame Lagrangian for two real scalar fieldsφi where mdash using the notation of [30] mdash φi stands for either s or a component of the Higgsdoublet Unitarity-violation requires that there are two different scalars in the scatteringprocess since in the case of a single scalar there is a cancellation between s- t- and u-channelamplitudes [63]

ndash 11 ndash

JCAP11(2015)015

Since unitarity-violating scattering in the Jordan frame is due to graviton exchange viathe non-minimal coupling to R we cat set V = 0 The Einstein frame action for two realscalars is then of the form

SE =

intd4xradicminusg

Lii +sumiltj

Lij minus1

2m2

P R

(51)

where

Lii =1

2

Ω2 +6 ξ2i φ

2i

m2P

Ω4

gmicroν partmicroφi partνφi and Lij =6 ξi ξj φi φj g

microν partmicroφi partνφjm2

P Ω4(52)

with

Ω2 = 1 +ξj φ

2j

m2P

(53)

The interaction terms proportional to ξi ξj are responsible for the dominant unitarity-viola-tion in scattering cross-sections calculated in the Einstein frame These interactions are theEinstein frame analogue of scalar scattering via graviton exchange in the Jordan frame due tothe non-minimal coupling To obtain the scale of unitarity-violation in terms of the physicalenergy defined in the Jordan frame we first canonically normalize the fields in the Einsteinframe then estimate the magnitude of the scattering matrix element and finally transformthe unitarity-violation scale in the Einstein frame back to that in the Jordan frame

In the following we will denote the inflaton by φ1 (equiv s) which we expand about thebackground field ie φ1 = φ1 + δφ1 The Higgs doublet scalars are denoted by φ2 (equiv h1)and φ3 (equiv h2) The corresponding canonically normalized scattering fields in the Einsteinframe are then defined to be ϕ1 ϕ2 and ϕ3 Once we have determined the interactions ofthe canonically normalised fields we use dimensional analysis to estimate the scale of tree-level unitarity-violation For this purpose we introduce appropriate factors of E to makethe coefficient of the interaction terms in L dimensionless Energy scales E in the Einsteinframe are related to the ones in the Jordan frame via E = EΩ where during inflation

Ω2 ξ1φ21m

2P asymp N 1 Unitarity conservation implies that the matrix element for any

2 harr 2 scattering process should be smaller than O(1) so we can determine the scale ofunitarity-violation (denoted by Λ) by determining the value of E that saturates this boundThis was demonstrated explicitly in [8] by comparing the dimensional estimate with theexact value from the full scattering amplitude

In the following we consider three regimes for φ1 each of which leads to a differentform of the Lagrangian and the scattering amplitudes

bull Regime A large field values In this regime we have Ω gt 1 (implying that ξ1 φ21m

2P gt 1)

and 6 ξ21 φ21m

2P gt 1

bull Regime B medium field values In this regime we have approximately Ω asymp 1 (implying

that ξ1 φ21m

2P lt 1) but still 6 ξ21 φ

21m

2P gt 1

bull Regime C small field values Finally we consider 6 ξ21 φ21m

2P lt 1 which in particular

implies Ω asymp 1

ndash 12 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

JCAP11(2015)015

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

1000 106 109 1012 1015 10180001

00050010

00500100

05001

μ [GeV]

Couplingstrength

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

λh

λhs

λs

1000 106 109 1012 1015 101810

100

1000

104

105

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

ξh

ξs

cs

Figure 2 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplingsξh and ξs (right) as a function of the renormalisation scale micro for a typical parameter point in thehigh-mass region Solid lines show the running in the s-direction while dotted lines correspond to therunning in the h-direction In the right panel we also show the suppression factor cs which modifiesthe running in the s-direction at large field value

s h which is relevant for inflation while dotted lines correspond to h s which is relevantfor vacuum stability We fix the scalar couplings at the weak scale choosing λhs = 025 andλs = 001 such that the observed relic abundance can be reproduced for ms = 835 GeV Weconsider λs λhs in which case the value of ξs necessary to obtain the correct amplitude ofthe scalar power spectrum is reduced For our choice we find ξs sim 104 Note however thatwhile λhs exhibits only moderate running λs grows significantly with the renormalisationscale micro because its β-function contains a term proportional to λ2hs Choosing even smallervalues of λs at the weak scale will therefore not significantly reduce its value at the scale ofinflation nor the corresponding value of ξs

An important observation from figure 2 is that λh does not run negative and hencethe electroweak vacuum remains stable all the way up to the Planck scale The additionalcontribution from the singlet scalar is sufficient to ensure λh gt 10minus3 for all renormalisationscales up to mP For field values s amp 1015 GeV h the singlet propagator is suppressedleading to a visible kink in the running of λh We show the propagator suppression factorcs in the right panel of figure 2 One can clearly see how this suppression factor affects therunning of ξh which becomes nearly constant for s amp 1015 GeV In this particular examplewe have chosen ξh(mP) = 1000 (for s h) This choice together with λs 1 implies thatthe running of ξs(micro) from mt to mP is negligible

In the low-mass region we are interested in much smaller values of λhs typically below10minus2 A particular example is shown in figure 3 (left) for the representative choice λhs(mt) =0002 and λs = 00005 which yields the observed relic abundance for ms asymp 57 GeV Weobserve that if λs and λhs are both small at the electroweak scale these couplings exhibitonly very little running up to the scale of inflation For the same reason it is impossible toinfluence the running of the Higgs couplings sufficiently to prevent λhs from running negativeat around 1011 GeV

If λhs is small we can obtain the correct scalar power spectrum amplitude with a muchsmaller value of ξs during inflation For the specific case considered in figure 3 we findξs sim 103 For these values of ξs and λhs the loop-induced corrections to ξh are very smalland hence this coupling changes only very slightly under RG evolution

43 Perturbativity

In order for our calculation of the running couplings and the radiative corrections to thepotential to be reliable we must require that all couplings remain perturbative up to the

ndash 10 ndash

JCAP11(2015)015

1000 106 109 1012 1015 101810-4

0001

0010

0100

1

μ [GeV]

Couplingstrength

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

λh

λhs

λs

1000 106 109 1012 1015 101801

1

10

100

1000

104

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

ξh

ξs

cs

Figure 3 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplings ξhand ξs (right) as a function of the renormalisation scale micro for a typical parameter point in the low-mass region In the right panel we also show the suppression factor cs which modifies the runningat large field value Note that λh runs negative for micro amp 1011 GeV

scale of inflation which is typically 1017ndash1018 GeV This requirement is easily satisfied forthe SM couplings but needs to be checked explicitly for the couplings of the singlet whichcan grow significantly with increasing renormalisation scale micro We follow [58 62] and usethe requirement of perturbative unitarity to impose an upper bound on the scalar couplingsThis procedure gives

λs lt4π

3and λhs lt 8π (46)

As we will see below the non-minimal coupling ξs can be much larger than unity withoutinvalidating a perturbative calculation Nevertheless if ξs and ξh are both very large pro-cesses involving both couplings may violate perturbative unitarity implying that there maybe new physics or strong coupling below the scale of inflation We will now discuss this issuein more detail

5 Unitarity-violation during inflation

In this section we estimate the scale of perturbative unitarity-violation as a function of thebackground inflaton field Note that by ldquounitarity-violation scalerdquo we mean the scale at whichperturbation theory in scalar scattering breaks down so this may in fact indicate the onsetof unitarity-conserving scattering in a strongly-coupled regime [28 31] We will considerunitarity-violation in the scattering of scalar particles corresponding to perturbations aboutthe background field In the case of a real scalar s and the fields of the Higgs doublet Hthere are two distinct scattering processes we need to consider (i) δs h1 harr δs h1 and (ii)h1h2 harr h1h2 where δs is the perturbation about the background s field and h1 and h2are two of the Higgs doublet scalars Scattering with the other scalars in H is equivalentto these two processes We will use dimensional analysis to estimate the scale of unitarity-violation by determining the leading-order processes in the Einstein frame which result inunitarity-violating scattering

It will be sufficient to consider the Einstein frame Lagrangian for two real scalar fieldsφi where mdash using the notation of [30] mdash φi stands for either s or a component of the Higgsdoublet Unitarity-violation requires that there are two different scalars in the scatteringprocess since in the case of a single scalar there is a cancellation between s- t- and u-channelamplitudes [63]

ndash 11 ndash

JCAP11(2015)015

Since unitarity-violating scattering in the Jordan frame is due to graviton exchange viathe non-minimal coupling to R we cat set V = 0 The Einstein frame action for two realscalars is then of the form

SE =

intd4xradicminusg

Lii +sumiltj

Lij minus1

2m2

P R

(51)

where

Lii =1

2

Ω2 +6 ξ2i φ

2i

m2P

Ω4

gmicroν partmicroφi partνφi and Lij =6 ξi ξj φi φj g

microν partmicroφi partνφjm2

P Ω4(52)

with

Ω2 = 1 +ξj φ

2j

m2P

(53)

The interaction terms proportional to ξi ξj are responsible for the dominant unitarity-viola-tion in scattering cross-sections calculated in the Einstein frame These interactions are theEinstein frame analogue of scalar scattering via graviton exchange in the Jordan frame due tothe non-minimal coupling To obtain the scale of unitarity-violation in terms of the physicalenergy defined in the Jordan frame we first canonically normalize the fields in the Einsteinframe then estimate the magnitude of the scattering matrix element and finally transformthe unitarity-violation scale in the Einstein frame back to that in the Jordan frame

In the following we will denote the inflaton by φ1 (equiv s) which we expand about thebackground field ie φ1 = φ1 + δφ1 The Higgs doublet scalars are denoted by φ2 (equiv h1)and φ3 (equiv h2) The corresponding canonically normalized scattering fields in the Einsteinframe are then defined to be ϕ1 ϕ2 and ϕ3 Once we have determined the interactions ofthe canonically normalised fields we use dimensional analysis to estimate the scale of tree-level unitarity-violation For this purpose we introduce appropriate factors of E to makethe coefficient of the interaction terms in L dimensionless Energy scales E in the Einsteinframe are related to the ones in the Jordan frame via E = EΩ where during inflation

Ω2 ξ1φ21m

2P asymp N 1 Unitarity conservation implies that the matrix element for any

2 harr 2 scattering process should be smaller than O(1) so we can determine the scale ofunitarity-violation (denoted by Λ) by determining the value of E that saturates this boundThis was demonstrated explicitly in [8] by comparing the dimensional estimate with theexact value from the full scattering amplitude

In the following we consider three regimes for φ1 each of which leads to a differentform of the Lagrangian and the scattering amplitudes

bull Regime A large field values In this regime we have Ω gt 1 (implying that ξ1 φ21m

2P gt 1)

and 6 ξ21 φ21m

2P gt 1

bull Regime B medium field values In this regime we have approximately Ω asymp 1 (implying

that ξ1 φ21m

2P lt 1) but still 6 ξ21 φ

21m

2P gt 1

bull Regime C small field values Finally we consider 6 ξ21 φ21m

2P lt 1 which in particular

implies Ω asymp 1

ndash 12 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

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1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

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27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

1000 106 109 1012 1015 101810-4

0001

0010

0100

1

μ [GeV]

Couplingstrength

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

λh

λhs

λs

1000 106 109 1012 1015 101801

1

10

100

1000

104

0

025

05

075

1

125

15

μ [GeV]

Couplingstrength

Suppressionfactor

λhs(mt)=0002 λs(mt)=00005 ξh(mP)=1

ξh

ξs

cs

Figure 3 Running of the scalar couplings λh λs and λhs (left) and of the non-minimal couplings ξhand ξs (right) as a function of the renormalisation scale micro for a typical parameter point in the low-mass region In the right panel we also show the suppression factor cs which modifies the runningat large field value Note that λh runs negative for micro amp 1011 GeV

scale of inflation which is typically 1017ndash1018 GeV This requirement is easily satisfied forthe SM couplings but needs to be checked explicitly for the couplings of the singlet whichcan grow significantly with increasing renormalisation scale micro We follow [58 62] and usethe requirement of perturbative unitarity to impose an upper bound on the scalar couplingsThis procedure gives

λs lt4π

3and λhs lt 8π (46)

As we will see below the non-minimal coupling ξs can be much larger than unity withoutinvalidating a perturbative calculation Nevertheless if ξs and ξh are both very large pro-cesses involving both couplings may violate perturbative unitarity implying that there maybe new physics or strong coupling below the scale of inflation We will now discuss this issuein more detail

5 Unitarity-violation during inflation

In this section we estimate the scale of perturbative unitarity-violation as a function of thebackground inflaton field Note that by ldquounitarity-violation scalerdquo we mean the scale at whichperturbation theory in scalar scattering breaks down so this may in fact indicate the onsetof unitarity-conserving scattering in a strongly-coupled regime [28 31] We will considerunitarity-violation in the scattering of scalar particles corresponding to perturbations aboutthe background field In the case of a real scalar s and the fields of the Higgs doublet Hthere are two distinct scattering processes we need to consider (i) δs h1 harr δs h1 and (ii)h1h2 harr h1h2 where δs is the perturbation about the background s field and h1 and h2are two of the Higgs doublet scalars Scattering with the other scalars in H is equivalentto these two processes We will use dimensional analysis to estimate the scale of unitarity-violation by determining the leading-order processes in the Einstein frame which result inunitarity-violating scattering

It will be sufficient to consider the Einstein frame Lagrangian for two real scalar fieldsφi where mdash using the notation of [30] mdash φi stands for either s or a component of the Higgsdoublet Unitarity-violation requires that there are two different scalars in the scatteringprocess since in the case of a single scalar there is a cancellation between s- t- and u-channelamplitudes [63]

ndash 11 ndash

JCAP11(2015)015

Since unitarity-violating scattering in the Jordan frame is due to graviton exchange viathe non-minimal coupling to R we cat set V = 0 The Einstein frame action for two realscalars is then of the form

SE =

intd4xradicminusg

Lii +sumiltj

Lij minus1

2m2

P R

(51)

where

Lii =1

2

Ω2 +6 ξ2i φ

2i

m2P

Ω4

gmicroν partmicroφi partνφi and Lij =6 ξi ξj φi φj g

microν partmicroφi partνφjm2

P Ω4(52)

with

Ω2 = 1 +ξj φ

2j

m2P

(53)

The interaction terms proportional to ξi ξj are responsible for the dominant unitarity-viola-tion in scattering cross-sections calculated in the Einstein frame These interactions are theEinstein frame analogue of scalar scattering via graviton exchange in the Jordan frame due tothe non-minimal coupling To obtain the scale of unitarity-violation in terms of the physicalenergy defined in the Jordan frame we first canonically normalize the fields in the Einsteinframe then estimate the magnitude of the scattering matrix element and finally transformthe unitarity-violation scale in the Einstein frame back to that in the Jordan frame

In the following we will denote the inflaton by φ1 (equiv s) which we expand about thebackground field ie φ1 = φ1 + δφ1 The Higgs doublet scalars are denoted by φ2 (equiv h1)and φ3 (equiv h2) The corresponding canonically normalized scattering fields in the Einsteinframe are then defined to be ϕ1 ϕ2 and ϕ3 Once we have determined the interactions ofthe canonically normalised fields we use dimensional analysis to estimate the scale of tree-level unitarity-violation For this purpose we introduce appropriate factors of E to makethe coefficient of the interaction terms in L dimensionless Energy scales E in the Einsteinframe are related to the ones in the Jordan frame via E = EΩ where during inflation

Ω2 ξ1φ21m

2P asymp N 1 Unitarity conservation implies that the matrix element for any

2 harr 2 scattering process should be smaller than O(1) so we can determine the scale ofunitarity-violation (denoted by Λ) by determining the value of E that saturates this boundThis was demonstrated explicitly in [8] by comparing the dimensional estimate with theexact value from the full scattering amplitude

In the following we consider three regimes for φ1 each of which leads to a differentform of the Lagrangian and the scattering amplitudes

bull Regime A large field values In this regime we have Ω gt 1 (implying that ξ1 φ21m

2P gt 1)

and 6 ξ21 φ21m

2P gt 1

bull Regime B medium field values In this regime we have approximately Ω asymp 1 (implying

that ξ1 φ21m

2P lt 1) but still 6 ξ21 φ

21m

2P gt 1

bull Regime C small field values Finally we consider 6 ξ21 φ21m

2P lt 1 which in particular

implies Ω asymp 1

ndash 12 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

JCAP11(2015)015

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

Since unitarity-violating scattering in the Jordan frame is due to graviton exchange viathe non-minimal coupling to R we cat set V = 0 The Einstein frame action for two realscalars is then of the form

SE =

intd4xradicminusg

Lii +sumiltj

Lij minus1

2m2

P R

(51)

where

Lii =1

2

Ω2 +6 ξ2i φ

2i

m2P

Ω4

gmicroν partmicroφi partνφi and Lij =6 ξi ξj φi φj g

microν partmicroφi partνφjm2

P Ω4(52)

with

Ω2 = 1 +ξj φ

2j

m2P

(53)

The interaction terms proportional to ξi ξj are responsible for the dominant unitarity-viola-tion in scattering cross-sections calculated in the Einstein frame These interactions are theEinstein frame analogue of scalar scattering via graviton exchange in the Jordan frame due tothe non-minimal coupling To obtain the scale of unitarity-violation in terms of the physicalenergy defined in the Jordan frame we first canonically normalize the fields in the Einsteinframe then estimate the magnitude of the scattering matrix element and finally transformthe unitarity-violation scale in the Einstein frame back to that in the Jordan frame

In the following we will denote the inflaton by φ1 (equiv s) which we expand about thebackground field ie φ1 = φ1 + δφ1 The Higgs doublet scalars are denoted by φ2 (equiv h1)and φ3 (equiv h2) The corresponding canonically normalized scattering fields in the Einsteinframe are then defined to be ϕ1 ϕ2 and ϕ3 Once we have determined the interactions ofthe canonically normalised fields we use dimensional analysis to estimate the scale of tree-level unitarity-violation For this purpose we introduce appropriate factors of E to makethe coefficient of the interaction terms in L dimensionless Energy scales E in the Einsteinframe are related to the ones in the Jordan frame via E = EΩ where during inflation

Ω2 ξ1φ21m

2P asymp N 1 Unitarity conservation implies that the matrix element for any

2 harr 2 scattering process should be smaller than O(1) so we can determine the scale ofunitarity-violation (denoted by Λ) by determining the value of E that saturates this boundThis was demonstrated explicitly in [8] by comparing the dimensional estimate with theexact value from the full scattering amplitude

In the following we consider three regimes for φ1 each of which leads to a differentform of the Lagrangian and the scattering amplitudes

bull Regime A large field values In this regime we have Ω gt 1 (implying that ξ1 φ21m

2P gt 1)

and 6 ξ21 φ21m

2P gt 1

bull Regime B medium field values In this regime we have approximately Ω asymp 1 (implying

that ξ1 φ21m

2P lt 1) but still 6 ξ21 φ

21m

2P gt 1

bull Regime C small field values Finally we consider 6 ξ21 φ21m

2P lt 1 which in particular

implies Ω asymp 1

ndash 12 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

JCAP11(2015)015

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

51 Regime A large field values

In this case the canonically normalized fields are ϕ1 =radic

6mP δφ1φ1 and ϕ23 = φ23Ω Theinteraction leading to unitarity-violation in δs h1 scattering is

L sup 6 ξ1 ξ2m2

P Ω4(φ1 + δφ1)φ2 g

microν partmicroδφ1 partνφ2 (54)

This results in a 3-point and a 4-point interaction After rescaling to canonically normalizedfields the 3-point interaction is

L supradic

6 ξ2mP

ϕ2 gmicroν partmicroδϕ1 partνϕ2 (55)

This interaction can mediate ϕ1ϕ2 harr ϕ1ϕ2 scattering at energy E via ϕ2 exchange with amatrix element given dimensionally by |M| sim E2 ξ22m

2P Unitarity is violated once |M| sim 1

therefore the unitarity-violation scale in the Einstein frame is

Λ(3)12 sim

mP

ξ2 (56)

where the superscript (3) denotes unitarity-violation due to the 3-point interaction In the

Jordan frame Λ(3)12 = ΩΛ

(3)12 where Ω asymp

radicξ1 φ1mP therefore

Λ(3)12 sim

radicξ1ξ2

φ1 (57)

Similarly the 4-point interaction has an Einstein frame matrix element given by |M| simξ2 E

2m2P therefore the scale of unitarity-violation in the Jordan frame is

Λ(4)12 sim

radicξ1ξ2φ1 (58)

For ξ2 gt 1 this is larger than Λ(3)12 therefore Λ

(3)12 is the dominant scale of unitarity-violation

In general these estimates of the unitarity-violation scales are valid if the scalars can be

considered massless which will be true if φ1 lt Λ(3)12 ie for

radicξ1 gt ξ2

In the case of Higgs scattering ϕ2ϕ3 harr ϕ2ϕ3 there is only the 4-point interactionfollowing from

L sup 6 ξ2 ξ3m2

P Ω4φ2 φ3 g

microν partmicroφ2 partνφ3 (59)

As the canonically normalized Higgs fields are in general given by ϕ23 = φ23Ω the unitarity-violation scale in the Einstein frame is generally Λ23 sim mP

radicξ2 ξ3 equiv mPξ2 (since ξ2 = ξ3 if

both scalars are part of the Higgs doublet) On translating the energy to the Jordan framethe unitarity-violation scale becomes

Λ23 simradicξ1ξ2

φ1 (510)

which is the same expression as for Λ(3)12

ndash 13 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

JCAP11(2015)015

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

52 Regime B medium field values

In this case the canonically normalized fields are ϕ1 =radic

6ξ1φ1δφ1mP and ϕ2 = φ2 SinceΩ = 1 the energies are the same in the Einstein and Jordan frames Using the same procedureas before we find

Λ(3)12 sim

mP

ξ2 (511)

and

Λ(4)12 sim

radicξ1ξ2φ1 (512)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (513)

53 Regime C small field values

In this case the Einstein and Jordan frames are completely equivalent Therefore

Λ(3)12 sim

m2P

6 ξ1ξ2φ1 (514)

and

Λ(4)12 sim

mPradicξ1ξ2

(515)

For Higgs scattering we obtain

Λ23 simmP

ξ2 (516)

54 Discussion

In summary for ξs gt ξh the smallest (and so dominant) scale of unitarity-violation in eachregime is given by

A Λ(3)sh sim Λh sim

radicξsξh

s

B Λ(3)sh sim Λh sim

mP

ξh

C Λ(4)sh sim

mPradicξsξh

(517)

where Λ(3)sh equiv Λ

(3)12 Λh equiv Λ23 ξs equiv ξ1 and ξh equiv ξ2

In the case of Higgs Inflation the scale of unitarity-violation is obtained as above butwith ξs set equal to ξh since the inflaton is now a component of H During inflation Λ asympφ1radicξh with ξh sim 105 Since this energy scale is less than φ1 the gauge bosons become

massive and only the physical Higgs scalar takes part in scattering Since unitarity-violationrequires more than one massless non-minimally coupled scalar there is no unitarity-violationat energies less than φ1 Unitarity-violation therefore occurs at Λ asymp mW (φ1) asymp φ1 ie theunitarity-violation scale is essentially equal to the Higgs field value during inflation [14 32]As a result either the new physics associated with unitarising the theory or strong coupling

ndash 14 ndash

JCAP11(2015)015

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1000

Regime C B A

Beginningofinflation

Unitarityviolation

1011 1013 1015 1017 10191011

1013

1015

1017

1019

s [GeV]

Λ[GeV

]

λhs(mt)=025 λs(mt)=001 ξh(mP)=1

Regime C B A

Beginningofinflation

Figure 4 The scale of unitarity-violation Λ as a function of the field value s (blue) for two differentchoices of parameters The orange dashed line indicates the condition Λ gt s which must be satisfiedin order to avoid unitarity-violation In the left panel (with λh(mP) = 500) unitarity is violated fors amp 1016 GeV In the right panel (with λh(mP) = 50) no unitarity-violation occurs up to the scale ofinflation Note that in the right panel ξh(micro) runs negative for micro 5times 1013 GeV

effects are expected to significantly modify the effective potential during inflation It isuncertain in this case whether inflation is even possible and its predictions are unclear

In S-inflation on the other hand it is possible to ensure that Λ s provided ξh issufficiently small compared to ξs at the scale of inflation This is illustrated in figure 4 forξh(mP) = 1000 (left) and ξh(mP) = 1 (right) taking λhs(mt) = 025 λs(mt) = 001 andξs(mP) asymp 104 as above Both panels show the scale of unitarity-violation Λ as a functionof the field value s For ξh(mP) = 1000 we observe that Λ lt s for s amp 1016 GeV which issignificantly smaller than the field value at the beginning of inflation For ξh(mP) = 1 onthe other hand Λ always remains larger than s In this case it is reasonable to assume thatnew physics in the form of additional particles with mass of order Λ (or strong couplingeffects12) will have only a small effect on the effective potential at the scale micro = s

The right panel of figure 4 exhibits another new feature we find Λ rarr infin for s sim1015 GeV The reason is that as already observed in figure 2 ξh(mP) exhibits a strongrunning for micro lt 1015 GeV Consequently if we fix ξh to a rather small value at the Planckscale eg ξh(mP) = 1 ξh(micro) will run negative at lower scales During the transition ξh(micro)will be very small and hence the scale of unitarity-violation can be very large

It should be emphasized that the advantage of S-inflation with respect to unitarity-violation is only obtained if the singlet is a real scalar In the case of a complex singlet thereal and imaginary parts of the scalar both have the same non-minimal coupling ξs Thereforewe would have ξ1 = ξ2 in the above analysis and the unitarity-violation scale would becomethe same as in Higgs Inflation Therefore the requirement that unitarity is not violatedduring inflation predicts that the DM scalar is a real singlet scalar

6 Results

In this section we combine the experimental and theoretical constraints discussed above andpresent the viable parameter space for our model Out of the five free parameters (λs λhs

12In unitarisation by strong coupling Λ is automatically field dependent and equal to the scale at whichthe potential is expected to change In unitarisation by new particles on the other hand Λ is only an upperbound on the masses of the new particles Moreover the masses need not be field-dependent in order tounitarise the theory Therefore strong coupling is more naturally compatible with the scale of inflation

ndash 15 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

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52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

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27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Planckexcluded

(95CL)

ns - nstree

0

0005

0010

0015

0020

27 28 29 30 31 32 33 34-40

-35

-30

-25

-20

-15

-10

-05

-08 -07 -06 -05 -04 -03 -02

log10 msGeV

log10λs

log10 λhs

Non-perturbative couplings

Metastablevacuum

Nounitarityviolation

Planckexcluded

(95CL)

log10 ξs

30

35

40

45

50

Figure 5 Predictions for inflation in the high-mass region (500GeV lt ms lt 2500GeV) as a functionof ms and λs for fixed ξh(mP) = 100 Shown are the deviations from the tree-level predictionsntrees = 0965 (left) as well as the value of ξs at the beginning of inflation (right) For each value of ms

the coupling λhs has been fixed by the relic density requirement as shown on the top of each panelThe grey shaded region indicates the parameter region where couplings become non-perturbativebelow the scale of inflation and the purple shaded region indicates the parameter space where λhruns negative below the scale of inflation leading to a metastable vacuum We furthermore showthe parameter region excluded by the upper bound on ns from Planck at 95 CL (shaded in blue)The green line in the right panel indicates the value of ξs where unitarity is violated at the scale ofinflation

ms ξs and ξh) we can eliminate λhs (or ms) by requiring the model to yield the observedDM abundance (see figure 1) and ξs by imposing the correct amplitude of the scalar powerspectrum In the following we always ensure that these two basic requirements are satisfiedand then consider additional constraints in terms of the remaining parameters λs ms (orλhs) and ξh We begin with a detailed discussion of the high-mass region and then turn tothe low-mass region

61 The high-mass region

Let us for the moment fix ξh(mP) = 100 and study how the predictions depend on λs and ms

(or alternatively λhs) The left panel of figure 5 shows the predicted value of ns comparedto the tree-level estimate ntrees = 0965 We find that in our model ns is always slightlylarger than the tree-level estimate but the differences are typically ∆ns lt 001 Only forλhs gt 05 corresponding to ms amp 2 TeV do the differences grow so large that the model canbe excluded by the Planck 2-σ bound ns lt 098 In the same parameter region we find thelargest differences between the tree-level predictions of r and α (see section 2) and the valuepredicted in our model However we find these deviations to be negligibly small In particularour model predicts r lt 001 everywhere ie the tensor-to-scalar ratio would be very difficultto observe in the near future13 Figure 5 also shows the parameter region excluded by therequirements that all couplings remain perturbative up to the scale of inflation (shaded ingrey) This constraint requires λs 03 for small values of λhs and becomes more severewith increasing λhs

13Next generation CMB satellites such as PIXIE [64] and LiteBIRD [65] plan to measure r to an accuracyof δr lt 0001 This would be sufficient to detect the tensor-to-scalar ratio in our model

ndash 16 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

Finally we also show the parameter region where λhs is too small to prevent λh fromrunning to negative values in the h-direction (shaded in purple) While this is not fatal forthe model (the electroweak vacuum remains metastable with a lifetime that is longer than theone predicted for the SM alone) this constraint may be physically significant depending uponthe interpretation of the energy of the metastable state We note however that the boundfrom metastability depends very sensitively on the assumed values of the SM parametersat the electroweak scale Indeed it is still possible within experimental uncertainties (at95 CL) that the electroweak vacuum is completely stable even in the absence of any newphysics [46]

For the currently preferred values of the SM parameters we find the interesting pa-rameter region to be 02 λhs 06 corresponding roughly to 700 GeV ms 2 TeVVery significantly this entire range of masses and couplings can potentially be probed byXENON1T

To study the predictions of inflation mdash and in particular the scale of unitarity-violationmdash in more detail we show in the right panel of figure 5 the value of ξs (at the scale ofinflation) required by the scalar power spectrum amplitude We typically find values around104 although values as large as 105 become necessary as λs comes close to the perturbativebound Since we have fixed ξh(mP) = 100 in this plot such large values of ξs imply thatradicξsξh gt 1 which in turn means that the scale of unitarity-violation is larger than the

field value s at the beginning of inflation Conversely if both λs and λhs are small ξscan be significantly below 104 such that unitarity is violated below the scale of inflationThe parameters for which the scale of unitarity-violation is equal to the scale of inflation isindicated by a green line

Let us now turn to the dependence of our results on the value of ξh(mP) For thispurpose we fix λs = 001 and consider the effect of varying ξh(mP) in the range 0 le ξh(mP) le1000 We find that neither the constraint from Planck nor the bounds from metastability andperturbativity depend strongly on ξh(mP) Nevertheless as discussed in section 5 ξh(mP)does play a crucial role for the scale of unitarity-violation We therefore show in the left panelof figure 6 the scale of unitarity-violation at the beginning of inflation divided by the fieldvalue s at the beginning of inflation This ratio is to be larger than unity in order to avoidunitarity-violation As indicated by the green line this requirement implies ξh(mP) 150 inthe parameter region of interest

As discussed in section 4 small values of ξh(mP) imply that ξh(micro) will run to negativevalues for micro rarr mt To conclude our discussion of the high-mass region we therefore showin the right panel of figure 6 the magnitude of ξh(mt) as a function of ms and ξh(mP) Thethick black line indicates the transition between ξh(mt) gt 0 and ξh(mt) lt 0 By comparingthis plot with the one to the left we conclude that within the parameter region that avoidsunitarity-violation ξh(mt) necessarily becomes negative

62 The low-mass region

We study the predictions for the low-mass region in figure 7 The top row shows ∆ns andξs at the beginning of inflation as a function of λs and λhs for ξh(mP) = 1 These plotsare analogous to the ones for the high-mass region in figure 5 The crucial observationis that unless λs λhs radiative corrections to the inflationary potential are completelynegligible because any contribution proportional to λs is suppressed by powers of cs 1during inflation Consequently in most of the low-mass region ns is identical to its tree-levelvalue We furthermore find that as expected the tensor-to-scalar ratio and the running of

ndash 17 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

No unitarity violation

ln Λs

-2

-1

0

1

2

27 28 29 30 31 32 33 340

200

400

600

800

1000-08 -07 -06 -05 -04 -03 -02

log10 msGeV

ξ h(m

P)

log10 λhs

Planckexcluded

(95CL)

Metastablevacuum

ξh(mt) gt 0

ξh(mt) lt 0

log10 |ξh(mt)|

20

25

30

35

40

Figure 6 Left the scale of unitarity-violation Λ compared to the field value at the beginning ofinflation s as a function of ξh(mP) and ms (or λhs) for λs = 001 In order to avoid unitarity-violationwe require log Λs gt 0 Right the corresponding value of ξh at the electroweak scale (micro = mt) Theshaded regions correspond to the same constraints as in figure 5

the spectral index are both unobservably small Since radiative corrections play such a smallrole the value of ξs required from inflation depends almost exclusively on λs As a result itis easily possible to have ξs lt 1000 during inflation for λs lt 10minus3 and ξs lt 100 for λs lt 10minus5

Since ξs can be much smaller in the low-mass region than in the high-mass region it isnatural to also choose a very small value for ξh In fact for typical values in the low-massregion ξs sim 103 and λs sim λhs sim 10minus3 the loop-induced contribution to ξh is ∆ξh lt 10minus2 sothat it is technically natural to have ξh 1 One then obtains

radicξsξh 1 and hence the

scale of unitarity-violation is well above the scale of inflation It is therefore possible withoutdifficulty to solve the issue of unitarity-violation in the low-mass region

This conclusion is illustrated in the bottom row of figure 7 which should be comparedto figure 6 from the high-mass case except that we keep ξh(mP) = 1 fixed and vary λsinstead The bottom-left plot clearly shows that (for our choice of ξh) the scale of unitarity-violation is always well above the field value at the beginning of inflation so that the problemof unitarity-violation can easily be solved in the low-mass region In addition if the scalarcouplings are sufficiently small ξh will have negligible running from the Planck scale down tothe weak scale It is therefore easily possible to set eg ξh(mP) = 1 and still have ξh(mt) gt 0as illustrated in the bottom-right panel of figure 7 It is not possible however to ensure at thesame time that λh remains positive for large field values of h In other words the electroweakvacuum is always metastable in the low-mass region (for the preferred SM parameters)

In the high-mass region we found that the coupling λhs is bounded from below by thedesire to stabilise the electroweak vacuum and from above by constraints from Planck andthe requirement of perturbativity In the low-mass region on the other hand we obtain anupper bound on λhs from LUX and a lower bound on λhs from the relic density requirementCompared to the high-mass region the allowed range of couplings in the low-mass region ismuch larger and therefore much harder to probe in direct detection experiments If indeedms is very close to mh2 and λs λhs lt 10minus3 it will be a great challenge to test the modelpredictions with cosmological or particle physics measurements

ndash 18 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplingsLUXexcluded

(95

CL)

ns - nstree

0

0002

0004

0006

0008

0010

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

log10 ξS

20

25

30

35

40

45

50

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

LUXexcluded

(95

CL)

ln Λuvϕinf

0

1

2

3

4

5

52 54 56 58 60 62-5

-4

-3

-2

-1

0

m s [GeV]

log 10λS

log10 λhs

Non-perturbative couplings

ξh(mt) gt 0

ξh(mt) lt 0

LUXexcluded

(95

CL)

log10 |ξH(m t)|

-10

-05

0

05

10

15

20

Figure 7 Top row predictions for inflation in the low-mass region (525GeV lt ms lt 625GeV) as afunction of the couplings λhs and λs for fixed ξh(mP) = 1 Shown are the deviations from the tree-levelpredictions ntrees = 0965 (left) and the value of ξs at the beginning of inflation (right) Bottom rowthe scale of unitarity-violation Λ compared to the field value at the beginning of inflation sinf (left)and the value of ξh at the electroweak scale (right) In all panels the grey shaded region indicatesthe parameter region where couplings become non-perturbative below the scale of inflation and thelight blue shaded region represents the 95 CL bound from LUX We do not show the metastabilitybound since it covers the entire low-mass region

7 Conclusions

The origin of inflation and the nature of DM are two of the fundamental questions of cos-mology In the present work we have revisited the possibility that both issues are unifiedby having a common explanation in terms of a real gauge singlet scalar which is one of thesimplest possible extensions of the SM Considering the most recent experimental constraintsfor this model from direct detection experiments the LHC and Planck we have shown thatlarge regions of parameter space remain viable Furthermore we find that in parts of theparameter space the scalar singlet can stabilise the electroweak vacuum all the way up to thePlanck scale while at the same time avoiding the problem of unitarity-violation present inconventional models of Higgs inflation

The scalar singlet can efficiently pair-annihilate into SM particles via the Higgs portalso that it is straight-forward in this model to reproduce the observed DM relic abundance

ndash 19 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

via thermal freeze-out We find two distinct mass regions where the model is consistent withexperimental constraints from LUX LHC searches for invisible Higgs decays and Fermi-LATthe low-mass region 53 GeV ms 624 GeV where DM annihilation via Higgs exchangereceives a resonant enhancement and the high-mass region ms amp 93 GeV where a largenumber of annihilation channels are allowed

In both mass regions it is possible without problems to fix the non-minimal couplingsξs and ξh in such a way that inflation proceeds in agreement with all present constraintsIn particular the tensor-to-scalar ratio and the running of the spectral index are expectedto be unobservably small On the other hand radiative corrections to the spectral indextypically lead to a value of ns slightly larger than the classical estimate ie ns gt 0965 Thiseffect is largest for large values of ms and λhs and current Planck constraints already requirems 2 TeV The entire high-mass region compatible with Planck constraints will thereforebe tested by XENON1T which can constrain gauge singlet scalar DM up to ms sim 4 TeV

In the high-mass region the value of ξs required to obtain a sufficiently flat potentialduring inflation is typically ξs sim 104ndash105 In spite of such a large non-minimal couplingit is possible to have unitarity-conservation during inflation in the sense that the scale ofunitarity-violation can be much larger than the inflaton field The reason is that only ξsneeds to be large in order to reproduce the observed density perturbation while the Higgsnon-minimal coupling ξh can be arbitrarily small In the limit ξh rarr 0 there will be only onenon-minimally coupled scalar field and therefore no unitarity-violation provided that theinflaton and so the DM particle is a real scalar

We find that at large singlet field values s the scale of unitarity-violation is given byΛ sim s

radicξsξh If the non-minimal couplings satisfy ξs(mP) ξh(mP) at the Planck scale it

is possible for the unitarity-violation scale during inflation to be orders of magnitude largerthan s Such a hierarchy of couplings is stable under radiative corrections and consistent withthe assumption that inflation proceeds along the s-direction Furthermore in the low-massregion λhs and λs can be so small that ξs sim 102ndash103 is sufficient to obtain a flat enoughpotential

We conclude that it is possible for the inflaton potential in S-inflation to be safe fromnew physics or strong-coupling effects associated with the unitarity-violation scale Thiscontrasts with the case of Higgs Inflation where unitarity is always violated at the scale ofthe inflaton

Another interesting observation is that if the singlet mass and the coupling λhs aresufficiently large (roughly ms amp 1 TeV and λhs amp 03) the presence of the additional scalarsinglet stabilises the electroweak vacuum because the additional contribution to βλh pre-vents the quartic Higgs coupling from running to negative values This observation becomesimportant if a metastable electroweak vacuum is physically disfavoured for example if thepotential energy relative to the absolute minimum defines an observable vacuum energy

Given how tightly many models for DM are constrained by direct detection and LHCsearches and how strong recent bounds on models for inflation have become it is quiteremarkable that one of the simplest models addressing both problems still has a large allowedparameter space Nevertheless the model is highly predictive In particular if the DM scalaris also the inflaton and unitarity is conserved during inflation then DM is predicted to be areal scalar Direct detection experiments will soon reach the sensitivity necessary to probethe entire parameter space relevant for phenomenology with the exception of a small windowin ms close to the Higgs resonance The next few years will therefore likely tell us whetherindeed a singlet scalar extension of the SM can solve two of the central problems of particlephysics and cosmology

ndash 20 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

Acknowledgments

We would like to thank Rose Lerner for her contribution during the early stages of thisproject We are grateful to Kyle Allison Ido Ben-Dayan Andreas Goudelis Huayong HanThomas Konstandin Kai Schmidt-Hoberg Pat Scott and Christoph Weniger for helpfuldiscussions and to Guillermo Ballesteros for carefully reading the manuscript and providinga number of useful comments The work of FK was supported by the German ScienceFoundation (DFG) under the Collaborative Research Center (SFB) 676 Particles Strings andthe Early Universe The work of JMcD was partly supported by the Lancaster-Manchester-Sheffield Consortium for Fundamental Physics under STFC grant STJ0004181 FK wouldlike to thank the Instituto de Fisica Teorica (IFT UAM-CSIC) in Madrid for its support viathe Centro de Excelencia Severo Ochoa Program under Grant SEV-2012-0249 during theProgram ldquoIdentification of dark matter with a Cross-Disciplinary Approachrdquo where part ofthis work was carried out

A RG equations

The RG equations for the scalar couplings can be obtained using the techniques detailedin [66ndash68] as in [8] The one-loop β-functions for the scalar couplings are

16π2 β(1)λh

= minus6 y4t +3

8

(2 g4 +

(g2 + gprime2

)2)+(minus9 g2 minus 3 gprime2 + 12 y2t

)λh

+(18 c2h + 6

)λ2h +

1

2c2s λ

2hs (A1)

16π2 β(1)λhs

= 4 ch cs λ2hs + 6

(c2h + 1

)λh λhs minus

3

2

(3 g2 + gprime2

)λhs

+ 6 y2t λhs + 6 c2s λs λhs (A2)

16π2 β(1)λs

=1

2(c2h + 3)λ2hs + 18 c2s λ

2s (A3)

The propagator suppression factors are given by

cφ =1 +

ξφ φ2

m2P

1 + (6 ξφ + 1)ξφ φ2

m2P

(A4)

where φ is s or hThe RG equations for the non-minimal coupling can be derived following [69] as in [8]

(see also [10]) One obtains

16π2dξsdt

= (3 + ch)λhs

(ξh +

1

6

)+

(ξs +

1

6

)6 cs λs (A5)

16π2dξhdt

=

((6 + 6 ch)λh + 6 y2t minus

3

2(3 g2 + gprime2)

)(ξh +

1

6

)+

(ξs +

1

6

)cs λhs (A6)

References

[1] Planck collaboration PAR Ade et al Planck 2015 results XX Constraints on inflationarXiv150202114 [INSPIRE]

[2] V Silveira and A Zee Scalar Phantoms Phys Lett B 161 (1985) 136 [INSPIRE]

ndash 21 ndash

JCAP11(2015)015

[3] J McDonald Gauge singlet scalars as cold dark matter Phys Rev D 50 (1994) 3637[hep-ph0702143] [INSPIRE]

[4] CP Burgess M Pospelov and T ter Veldhuis The Minimal model of nonbaryonic darkmatter A Singlet scalar Nucl Phys B 619 (2001) 709 [hep-ph0011335] [INSPIRE]

[5] Planck collaboration PAR Ade et al Planck 2015 results XVII Constraints on primordialnon-Gaussianity arXiv150201592 [INSPIRE]

[6] FL Bezrukov and M Shaposhnikov The Standard Model Higgs boson as the inflaton PhysLett B 659 (2008) 703 [arXiv07103755] [INSPIRE]

[7] DS Salopek JR Bond and JM Bardeen Designing Density Fluctuation Spectra in InflationPhys Rev D 40 (1989) 1753 [INSPIRE]

[8] RN Lerner and J McDonald Gauge singlet scalar as inflaton and thermal relic dark matterPhys Rev D 80 (2009) 123507 [arXiv09090520] [INSPIRE]

[9] N Okada and Q Shafi WIMP Dark Matter Inflation with Observable Gravity Waves PhysRev D 84 (2011) 043533 [arXiv10071672] [INSPIRE]

[10] TE Clark B Liu ST Love and T ter Veldhuis The Standard Model Higgs Boson-Inflatonand Dark Matter Phys Rev D 80 (2009) 075019 [arXiv09065595] [INSPIRE]

[11] FL Bezrukov A Magnin and M Shaposhnikov Standard Model Higgs boson mass frominflation Phys Lett B 675 (2009) 88 [arXiv08124950] [INSPIRE]

[12] A De Simone MP Hertzberg and F Wilczek Running Inflation in the Standard Model PhysLett B 678 (2009) 1 [arXiv08124946] [INSPIRE]

[13] AO Barvinsky AY Kamenshchik and AA Starobinsky Inflation scenario via the StandardModel Higgs boson and LHC JCAP 11 (2008) 021 [arXiv08092104] [INSPIRE]

[14] F Bezrukov and M Shaposhnikov Standard Model Higgs boson mass from inflation Two loopanalysis JHEP 07 (2009) 089 [arXiv09041537] [INSPIRE]

[15] AO Barvinsky AY Kamenshchik C Kiefer AA Starobinsky and C SteinwachsAsymptotic freedom in inflationary cosmology with a non-minimally coupled Higgs field JCAP12 (2009) 003 [arXiv09041698] [INSPIRE]

[16] RN Lerner and J McDonald Distinguishing Higgs inflation and its variants Phys Rev D 83(2011) 123522 [arXiv11042468] [INSPIRE]

[17] ATLAS and CMS collaborations Combined Measurement of the Higgs Boson Mass in ppCollisions at

radics = 7 and 8 TeV with the ATLAS and CMS Experiments Phys Rev Lett 114

(2015) 191803 [arXiv150307589] [INSPIRE]

[18] LUX collaboration DS Akerib et al First results from the LUX dark matter experiment atthe Sanford Underground Research Facility Phys Rev Lett 112 (2014) 091303[arXiv13108214] [INSPIRE]

[19] JM Cline K Kainulainen P Scott and C Weniger Update on scalar singlet dark matterPhys Rev D 88 (2013) 055025 [arXiv13064710] [INSPIRE]

[20] Y Mambrini Higgs searches and singlet scalar dark matter Combined constraints fromXENON 100 and the LHC Phys Rev D 84 (2011) 115017 [arXiv11080671] [INSPIRE]

[21] A Djouadi O Lebedev Y Mambrini and J Quevillon Implications of LHC searches forHiggs-portal dark matter Phys Lett B 709 (2012) 65 [arXiv11123299] [INSPIRE]

[22] A Djouadi A Falkowski Y Mambrini and J Quevillon Direct Detection of Higgs-PortalDark Matter at the LHC Eur Phys J C 73 (2013) 2455 [arXiv12053169] [INSPIRE]

[23] A De Simone GF Giudice and A Strumia Benchmarks for Dark Matter Searches at theLHC JHEP 06 (2014) 081 [arXiv14026287] [INSPIRE]

ndash 22 ndash

JCAP11(2015)015

[24] L Feng S Profumo and L Ubaldi Closing in on singlet scalar dark matter LUX invisibleHiggs decays and gamma-ray lines JHEP 03 (2015) 045 [arXiv14121105] [INSPIRE]

[25] FS Queiroz and K Sinha The Poker Face of the Majoron Dark Matter Model LUX to keVLine Phys Lett B 735 (2014) 69 [arXiv14041400] [INSPIRE]

[26] JLF Barbon and JR Espinosa On the Naturalness of Higgs Inflation Phys Rev D 79(2009) 081302 [arXiv09030355] [INSPIRE]

[27] CP Burgess HM Lee and M Trott Power-counting and the Validity of the ClassicalApproximation During Inflation JHEP 09 (2009) 103 [arXiv09024465] [INSPIRE]

[28] T Han and S Willenbrock Scale of quantum gravity Phys Lett B 616 (2005) 215[hep-ph0404182] [INSPIRE]

[29] RN Lerner and J McDonald A Unitarity-Conserving Higgs Inflation Model Phys Rev D 82(2010) 103525 [arXiv10052978] [INSPIRE]

[30] RN Lerner and J McDonald Unitarity-Violation in Generalized Higgs Inflation ModelsJCAP 11 (2012) 019 [arXiv11120954] [INSPIRE]

[31] U Aydemir MM Anber and JF Donoghue Self-healing of unitarity in effective field theoriesand the onset of new physics Phys Rev D 86 (2012) 014025 [arXiv12035153] [INSPIRE]

[32] F Bezrukov A Magnin M Shaposhnikov and S Sibiryakov Higgs inflation consistency andgeneralisations JHEP 01 (2011) 016 [arXiv10085157] [INSPIRE]

[33] RN Lerner and J McDonald Higgs Inflation and Naturalness JCAP 04 (2010) 015[arXiv09125463] [INSPIRE]

[34] DP George S Mooij and M Postma Quantum corrections in Higgs inflation the real scalarcase JCAP 02 (2014) 024 [arXiv13102157] [INSPIRE]

[35] V Barger P Langacker M McCaskey MJ Ramsey-Musolf and G Shaughnessy LHCPhenomenology of an Extended Standard Model with a Real Scalar Singlet Phys Rev D 77(2008) 035005 [arXiv07064311] [INSPIRE]

[36] VV Khoze Inflation and Dark Matter in the Higgs Portal of Classically Scale InvariantStandard Model JHEP 11 (2013) 215 [arXiv13086338] [INSPIRE]

[37] G Belanger F Boudjema A Pukhov and A Semenov MicrOMEGAs 3 A program forcalculating dark matter observables Comput Phys Commun 185 (2014) 960[arXiv13050237] [INSPIRE]

[38] Planck collaboration PAR Ade et al Planck 2015 results XIII Cosmological parametersarXiv150201589 [INSPIRE]

[39] ATLAS collaboration Measurements of the Higgs boson production and decay rates andcoupling strengths using pp collision data at

radics = 7 and 8 TeV in the ATLAS experiment

arXiv150704548 [INSPIRE]

[40] CMS collaboration Precise determination of the mass of the Higgs boson and tests ofcompatibility of its couplings with the standard model predictions using proton collisions at 7and 8 TeV Eur Phys J C 75 (2015) 212 [arXiv14128662] [INSPIRE]

[41] Fermi-LAT collaboration M Ackermann et al Search for gamma-ray spectral lines with theFermi large area telescope and dark matter implications Phys Rev D 88 (2013) 082002[arXiv13055597] [INSPIRE]

[42] N Craig HK Lou M McCullough and A Thalapillil The Higgs Portal Above ThresholdarXiv14120258 [INSPIRE]

[43] XENON1T collaboration E Aprile The XENON1T Dark Matter Search ExperimentSpringer Proc Phys 148 (2013) 93 [arXiv12066288] [INSPIRE]

ndash 23 ndash

JCAP11(2015)015

[44] JR Espinosa GF Giudice and A Riotto Cosmological implications of the Higgs massmeasurement JCAP 05 (2008) 002 [arXiv07102484] [INSPIRE]

[45] J Elias-Miro JR Espinosa GF Giudice G Isidori A Riotto and A Strumia Higgs massimplications on the stability of the electroweak vacuum Phys Lett B 709 (2012) 222[arXiv11123022] [INSPIRE]

[46] G Degrassi et al Higgs mass and vacuum stability in the Standard Model at NNLO JHEP 08(2012) 098 [arXiv12056497] [INSPIRE]

[47] K Allison Higgs ξ-inflation for the 125ndash126 GeV Higgs a two-loop analysis JHEP 02 (2014)040 [arXiv13066931] [INSPIRE]

[48] LN Mihaila J Salomon and M Steinhauser Gauge Coupling β-functions in the StandardModel to Three Loops Phys Rev Lett 108 (2012) 151602 [arXiv12015868] [INSPIRE]

[49] KG Chetyrkin and MF Zoller Three-loop β-functions for top-Yukawa and the Higgsself-interaction in the Standard Model JHEP 06 (2012) 033 [arXiv12052892] [INSPIRE]

[50] Particle Data Group collaboration KA Olive et al Review of Particle Physics ChinPhys C 38 (2014) 090001 [INSPIRE]

[51] K Kannike RGErun 20 Solving Renormalization Group Equations in Effective FieldTheories (2011) httpkoduuteesimkkannikeenglishsciencephysicsRGE run EFT

[52] D Buttazzo et al Investigating the near-criticality of the Higgs boson JHEP 12 (2013) 089[arXiv13073536] [INSPIRE]

[53] V Branchina and E Messina Stability Higgs Boson Mass and New Physics Phys Rev Lett111 (2013) 241801 [arXiv13075193] [INSPIRE]

[54] V Branchina E Messina and A Platania Top mass determination Higgs inflation andvacuum stability JHEP 09 (2014) 182 [arXiv14074112] [INSPIRE]

[55] V Branchina E Messina and M Sher Lifetime of the electroweak vacuum and sensitivity toPlanck scale physics Phys Rev D 91 (2015) 013003 [arXiv14085302] [INSPIRE]

[56] M Gonderinger Y Li H Patel and MJ Ramsey-Musolf Vacuum Stability Perturbativityand Scalar Singlet Dark Matter JHEP 01 (2010) 053 [arXiv09103167] [INSPIRE]

[57] S Profumo L Ubaldi and C Wainwright Singlet Scalar Dark Matter monochromatic gammarays and metastable vacua Phys Rev D 82 (2010) 123514 [arXiv10095377] [INSPIRE]

[58] N Khan and S Rakshit Study of electroweak vacuum metastability with a singlet scalar darkmatter Phys Rev D 90 (2014) 113008 [arXiv14076015] [INSPIRE]

[59] J Elias-Miro JR Espinosa GF Giudice HM Lee and A Strumia Stabilization of theElectroweak Vacuum by a Scalar Threshold Effect JHEP 06 (2012) 031 [arXiv12030237][INSPIRE]

[60] O Lebedev On Stability of the Electroweak Vacuum and the Higgs Portal Eur Phys J C 72(2012) 2058 [arXiv12030156] [INSPIRE]

[61] G Ballesteros and C Tamarit Higgs portal valleys stability and inflation JHEP 09 (2015)210 [arXiv150507476] [INSPIRE]

[62] G Cynolter E Lendvai and G Pocsik Note on unitarity constraints in a model for a singletscalar dark matter candidate Acta Phys Polon B 36 (2005) 827 [hep-ph0410102] [INSPIRE]

[63] MP Hertzberg On Inflation with Non-minimal Coupling JHEP 11 (2010) 023[arXiv10022995] [INSPIRE]

[64] A Kogut et al The Primordial Inflation Explorer (PIXIE) A Nulling Polarimeter for CosmicMicrowave Background Observations JCAP 07 (2011) 025 [arXiv11052044] [INSPIRE]

ndash 24 ndash

JCAP11(2015)015

[65] T Matsumura et al Mission design of LiteBIRD J Low Temp Phys 176 (2014) 733[arXiv13112847] [INSPIRE]

[66] ME Machacek and MT Vaughn Two Loop Renormalization Group Equations in a GeneralQuantum Field Theory 1 Wave Function Renormalization Nucl Phys B 222 (1983) 83[INSPIRE]

[67] ME Machacek and MT Vaughn Two Loop Renormalization Group Equations in a GeneralQuantum Field Theory 2 Yukawa Couplings Nucl Phys B 236 (1984) 221 [INSPIRE]

[68] ME Machacek and MT Vaughn Two Loop Renormalization Group Equations in a GeneralQuantum Field Theory 3 Scalar Quartic Couplings Nucl Phys B 249 (1985) 70 [INSPIRE]

[69] I Buchbinder S Odintsov and I Shapiro Effective action in quantum gravity Institute ofPhysics Bristol UK (1992)

ndash 25 ndash