Higgs Sector of the Left-Right Symmetric Theory

Alessio Maiezza,1, ∗ Goran Senjanović,1, 2, † and Juan Carlos Vasquez3, 1, 41International Centre for Theoretical Physics, Trieste, Italy

2Gran Sasso Science Institute, L’Aquila, Italy3Centro Científico Tecnológico de Valparaíso-CCTVal,

Universidad Técnica Federico Santa María, Valparaíso, Chile‡4International School for Advanced Studies/INFN, Trieste, Italy

We perform an in-depth analysis of the Higgs sector in the Minimal Left-Right Symmetric Model andcompute the scalar mass spectrum and associated mixings, offering simple physical and symmetryarguments in support of our findings. We identify the tree-level quartic and cubic potential couplingsin terms of the physical states and compute the quantum corrections for the latter ones. Thedeviations from the Standard Model prediction of the cubic Higgs doublet coupling are considered.Moreover we discuss the possible implications concerning the stability of the potential under therenormalization-group-equations evolution. In particular we examine three possible energy scalesof parity restoration: LHC reach, next hadronic collider and very high energy relevant for grandunification.

PACS numbers: 12.60.Cn, 12.60.Fr, 12.10.Kt

I. INTRODUCTION

There has been a great deal of interest in the Left-Right symmetric electro-weak gauge theory [1, 2] in re-cent years due its potential accessibility at the LHC. Af-ter more than four decades since its birth, there is fi-nally hope that experiment could confirm it. Moreover,it has emerged [3] that the minimal such model is a self-contained and predictive theory of neutrino mass in fullanalogy with the standard model (SM) for the Higgs ori-gin of charged fermions masses. We can say that whatseemed originally its curse, the prediction of massive neu-trino, over the years turned into a great blessing. Inthis, the crucial role was played by the seesaw mecha-nism [4–6] which not only suggestively accounts for smallneutrino mass, but moreover makes it be of Majorana na-ture. This implies Lepton Number Violation (LNV) bothat low energies through the neutrinoless double beta de-cay [7] and at high energies through a production of samesign charged lepton pairs at hadronic colliders [8]. In theminimal Left-Right symmetric model (LRSM) there is adeep connection between these processes [9].

There has recently been another important advance-ment in the minimal LR model, the analytic expressionfor the right-handed quark mixing matrix, in all of theparameter space [10]. It showed that the left and right-handed mixing angles are remarkably close to each otherin spite of near maximal parity violation in low energyweak interactions.

The LR symmetric theory is the simplest realization ofthe idea of the restoration of parity at the fundamentallevel. LR symmetry is broken spontaneously, and parityviolation is supposed to be a low energy accident. Since

∗ alessiomaiezza@gmail.com† goran@ictp.it‡ juan.vasquezcar@usm.cl

it was known fairly early that the right-handed (RH)charged gauge boson WR had to be very heavy due toits impact on the KL−KS mass difference, on the orderof few TeV [11–13], one had to wait for the advent of LHCin order to study it experimentally. This limit has beenrevisited in recent years [14] and definitively estimatedto lie in the full LHC reach [15, 16], which ranges up to∼ 6 TeV for the WR mass [17]. This value would makeneutrinoless double decay likely to be seen, even if it werenot due to neutrino mass. The LHC is slowly but surelygetting there [18], with the limitMWR

& 3 TeV in a largeportion of the parameter space of RH neutrino masses.

It is then important to study carefully the LRSM in itsfull glory, including the Higgs sector. The original anal-ysis of the Higgs sector goes back almost forty years [2],and it had cleared some essential features of the LR the-ory, such as the issue of flavor violation in the neutralscalar sector. It was quite comprehensive, but it had todo with the outdated version of the theory with Diracneutrinos. The changes are not dramatic, basically theyreduce to the existence of doubly charged scalars. Theyare important though to be taken into account and werediscussed first in [19–22] and most recently in [23–25].

The previous studies lacked the computation of themasses and mixings of scalar particles in the whole (phe-nomenologically relevant) parameter space. It is not theonly reason that drove us to go through this not very in-spiring task plagued by computational tedium, althoughwe believe that this by itself ought to suffice. The mainissue for the low scale LRSM, the one accessible at theLHC, is the issue of stability and perturbativity of thepotential at higher energies. Namely, the low energy con-straints from meson mixing, the same that drive WR tobe heavy, imply a stringent limit [13, 15, 26] on the massof the additional Higgs doublet necessarily present in theminimal model on the order of 20 TeV [16], which leadsto a worry of possibly too large couplings in the Higgspotential. This was recently studied in [24], where it was

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deduced that the theory can be perturbative for the LHCscale of symmetry breaking, but that it lives dangerously.We discuss this issue further, and in particular addressthe question of the cut-off scale where the theory ceasesto work. We show that the closeness of the cut-off tothe LR symmetry breaking scale brings important con-sequences on the parametric space of the model.

There is more to it. After all, the LR scale is not pre-dicted by theory and strictly speaking it can be anywherebetween TeV and the Planck scale. Obviously, the LHCreach is of great importance but one should be gettingready for future hadronic colliders, now being planned.There have already been studies devoted to this possi-bility, such as [25], with a hope of reaching the LR scalearound 20 TeV. We find that in this case the theory isperfectly perturbative and the cut-off can be far fromthe mass of WR, allowing for a natural suppression ofultra-violet (UV) non-renormalizable operators.

Another important scale is the one suggested by theminimal SO(10) grand unified theory (GUT), around1010 GeV. We run the whole parametric space of themodel in order to check if the scalar sector remains per-turbative preserving the picture of the two step symmetrybreaking. As a result, we get the generic constraint thatthe quartic couplings have to be of order of few percent,favoring marginally light scalars.

The rest of the paper is organized as follows. In thenext section we review briefly the essential features of theminimal Left-Right symmetric model. In the section IIIwe give the scalar mass spectrum and the relevant mix-ings. We offer simple symmetry arguments behind ourresults in order to facilitate the reading of the paper andas a check of our computations. We also give the phys-ical quartic and cubic couplings and discuss the devia-tions from the SM results. In the section IV we applyour results to the question of stability and perturbativityof the potential. We pay special attention to the issueof the cut-off which signals the breakdown of perturba-tivity at higher energies. In section V we consider thevertex renormalization, explicitly showing where the rel-evant vertices vary from the tree-level ones. Finally, insection VI we offer a summary and outlook of our results.The paper is completed by three Appendices where wegive some of the more technical details.

II. SALIENT FEATURES OF THE MINIMALLEFT-RIGHT SYMMETRIC MODEL

Gauge group and field content. The minimal LR sym-metric theory is based on the GLR ≡ SU(2)L×SU(2)R×U(1)B−L gauge group (suppressing color) and a symme-try between the left and the right sectors. Quarks andleptons come in LR symmetric representations

QL,R =

(ud

)L,R

, `L,R =

(νe

)L,R

. (1)

The formula for the electromagnetic charge becomes [27]Qem = I3L + I3R + B−L

2 which trades the hard to recallhyper-charge of the SM for B−L, the physical anomaly-free global symmetry of the SM, now gauged. Both LRsymmetry and the gauged B − L require the presence ofRH neutrinos.

The Higgs sector consists of the following multiplets [4,5, 28]: the bi-doublet Φ ∈ (2L, 2R, 0) and the SU(2)L,Rtriplets ∆L ∈ (3L, 1R, 2) and ∆R ∈ (1L, 3R, 2)

Φ =

[φ0

1 φ+2

φ−1 −φ0∗2

], ∆L,R =

[δ+/√

2 δ++

δ0 −δ+/√

2

]L,R

.

(2)We denote the bi-doublet as two SM model Y = −1doublets φi as in [23]

Φ = [φ1, εφ∗2] ; φi =

(φ0i

φ−i

), i = 1, 2 , (3)

with ε = iσ2. This manifest SM notation allows one tomake direct comparison between the LR theory and theSM with two Higgs doublets.Symmetry breaking. The symmetry breaking pro-

ceeds through two steps. First, at high scale with thebreaking of SU(2)R × U(1)B−L → U(1)Y through thevacuum expectation values (vev) [28]

〈δ0R〉 ≡ vR, 〈δ0

L〉 ≡ vL = 0 , (4)

which is responsible for the masses of new gauge bosonsWR, ZR

M2WR' g2

Rv2R, M2

ZR ' (2g2R + g2

B−L)v2R, (5)

where gR, gB−L are the gauge couplings of theSU(2)R, U(1)B−L groups. Moreover, vR gives largemasses to RH neutrinos νR, denoted N hereafter.

Next, at low scale with the usual SM symmetry break-ing through (from here on we use the notation sin γ =sγ , cos γ = cγ , tan γ = tγ for any angle γ)

〈Φ〉 = v diag(cβ ,−e−iasβ) , (6)

which gives the mass to the LH charged gauge bosonM2W = g2

L/2 v2. In turn, this in general produces a small

vev for the left-handed triplet vL, with vL ∝ v2/vR [28],ensuring the usual dominant doublet symmetry breakingof the SM symmetry. The oblique parameters impose abound vL . 5 GeV, however in the see-saw picture thatwe follow, this bound becomes much more stringent sincevL directly contributes to neutrino mass.Parity restoration: P or C. The discrete LR symme-

try can be shown to be either a generalized parity P or ageneralized charge conjugation C [15]. Under these, thefields transform as follows

P :

fL ↔ fRΦ↔ Φ†

∆L ↔ ∆R

C :

fL ↔ (fR)c

Φ↔ ΦT

∆L ↔ ∆∗R

(7)

where (fR)c = Cγ0f∗R is the usual charge-conjugate

spinor. These symmetries imply gL = gR ≡ g andstrongly characterize the form of the scalar potential thatwe are going to discuss.

3

III. THE HIGGS SCALAR SECTOR: MASSES,MIXINGS AND COUPLINGS

A. The Higgs potential.

The most general potential consistent with the GLRgauge group, without assuming any discrete LR symme-try, is given in [29]. It is too messy to be presented here.After all, if one does not believe in LR symmetry, whyassume the existence of ∆L if ∆R suffices by itself? Letus focus instead on the the part of the potential contain-ing only the bi-doublet, since it is quite instructive andwill ease the reader’s pain in facing the full potential. Itsgeneral form is given by

VΦ = −µ2ΦTr(Φ

†Φ)− µ2Φ[Tr(Φ†Φ) + h.c.]

+ λ1[Tr(Φ†Φ)]2 + λ2[eid2Tr2(ΦΦ†) + h.c.]

+ λ3[Tr(ΦΦ†)Tr(Φ†Φ)] + λ4Tr(Φ†Φ)[eid4Tr(ΦΦ†) + h.c.] ,

(8)

where Φ = εΦ∗ε = [φ2, εφ∗1] simply amounts for the in-

terchange of the two SU(2)L doublets φ1 and φ2, yet

another advantage of using the notation used in (3). Wehave used the phase freedom of Φ to make the mass termµΦ real. The potential has two real mass parameters andsix real quartic couplings. It is instructive to compare itwith the two Higgs doublet model case in SU(2)L×U(1),where one has three real mass terms and ten real quarticcouplings [30]. In spite of being much more restricted,the above potential still allows for a spontaneous viola-tion of CP as shown originally in [2], however the gener-ated phase would be too small due to the large mass ofthe second doublet, to be discussed below.

Clearly, the SU(2)R gauge symmetry plays an impor-tant role in restricting the number of parameters. Wewill see that the generalized charge conjugation as LRdiscrete symmetry makes no further restriction whatso-ever on this part of the potential, as opposed to gener-alized parity that makes the couplings real. Of course,both of these LR symmetries connect the couplings ofthe LH and RH triplets ∆L,R and simplify the potentialconsiderably.Case C. We start with case of the generalized charge

conjugation C as the LR symmetry, since the case P issimply obtained in the limit of some vanishing phases(see below). This further restricts the numbers of theparameters in the potential which now reads as [23, 31]

VC = −µ2ΦTr(Φ

†Φ)− µ2Φ[Tr(Φ†Φ) + h.c.]− µ2

∆[Tr(∆L∆†L) + Tr(∆R∆†R)] + λ1[Tr(Φ†Φ)]2

+ λ2[eid2Tr2(ΦΦ†) + h.c.] + λ3[Tr(ΦΦ†)Tr(Φ†Φ)] + λ4Tr(Φ†Φ)[eid4Tr(ΦΦ†) + h.c.] + [ρ1Tr2(∆L∆†L)

+ ρ2Tr(∆L∆L)Tr(∆†L∆†L) + L→ R] + ρ3Tr(∆L∆†L)Tr(∆R∆†R) + ρ4[eir4Tr(∆†L∆†L)Tr(∆R∆R) + h.c.]

+ α1[Tr(Φ†Φ) + α2(eicTr(ΦΦ†) + h.c.)][Tr(∆L∆†L) + Tr(∆R∆†R)] + α3[Tr(ΦΦ†∆L∆†L) + Tr(Φ†Φ∆R∆†R)]

+ [β1eib1Tr(Φ∆RΦ†∆†L) + β2e

ib2Tr(Φ∆RΦ†∆†L) + β3eib3Tr(Φ∆RΦ†∆†L) + h.c. (βi = 0 in the seesaw picture)] (9)

The potential appears messy, simply because we havemore than one same type couplings: the bi-doublet self-couplings λi, the triplet self-couplings ρi and mixed cou-plings αi and βi. It turns out that in the seesaw limitthe β terms can be safely ignored as we discuss now.

What helps is the separation of the two scales of sym-metry breaking, and the fact that for the physically in-teresting seesaw picture of neutrino mass one can safelyignore the small vL. Namely, its contribution to neutrinomass matrix has the form [28] Mν ∝ vL/vRMN , whereMN denotes the mass matrix of RH neutrinos N . Thusfor a large portion portion of RH neutrino mass param-eter space, vL must be quite small. For example, evenin the case when N are light and the lightest one pro-vides warm dark matter [32] with mN ' keV, one hasvL . 10−6 GeV which can be safely ignored in the anal-ysis of the potential. In the scenario where RH neutrinoscan be actually seen at the colliders, mN & 10 GeV, vLbecomes completely negligible.

In what follows we thus work in the limit vL = 0 (or

equivalently β = 0). The question is whether it is tech-nically natural. The positive answer was given alreadyin the original work [28] but we go through it once againfor the sake of completeness. It is easy to see that vLis generated by the β terms in the potential, and thesmallness of vL is directly controlled by the smallness ofβi couplings. It is equally easy to see that in the limitβi = 0 there is more symmetry in the potential, e.g.∆L → −∆L which guarantees its stability to all ordersin perturbation theory. This symmetry is broken only bythe Yukawa couplings of ∆L, the same ones that lead tothe seesaw picture since ∆L,R have the same couplingsbecause of the LR symmetry. In short, vL is naturallysmall in the technical sense, and in principle its effect canbe sub-dominant to the usual seesaw contribution of RHneutrinos to neutrino mass.

This said, it is fair to admit that an extremely smallβ, as does the smallness of neutrino mass itself, pointsto the possible large LR-scale, which is natural in thecontext of the SO(10) grand unified theory. Namely, in

4

the minimal model one predicts vR ' 1010 GeV [34]. Forthis reason, we also include here a section dedicated tothe SO(10) embedding of the LRSM.

Since the LR-scale on the order of TeV is still perfectlyacceptable, both theoretically and phenomenologically,one may wonder whether there is a more natural alter-native to small β. Indeed, it is sometimes claimed thatthis can be achieved by decoupling ∆L from the theoryin order to have its vev small. We disagree with this fora number of reasons that we now go through.

First, unlike small protected dimensionless couplings,large scales are not technically natural because they bringin the usual hierarchy problem. Second, in order to de-couple ∆L in the context of the spontaneous symmetrybreaking one needs to break the discrete LR symmetryat a large scale by the GLR gauge singlet vev [33]. Noticethat keeping ∆R light while decoupling ∆L requires theusual fine-tuning between the original symmetric massterms and the corrections induced by the singlet vev. Un-like in the case of small β, there is no protective symmetryhere. Moreover, a decoupled ad-hoc singlet is physicallyequivalent to the soft, non-spontaneous, breaking, andthus not well motivated.

If the LRSM is embedded in the SO(10) theory how-ever, the GLR parity odd singlets are often automaticallypresent [33], but then, as mentioned above, MWR

is pre-dicted to be huge, around 1010 GeV [34], and one is leftbasically with the SM at low energies (and massive neu-trinos). One may find ways to lower MWR

, but in thatcase one loses all the predictivity of grand unification.

Imagine for a moment that in any case one does invokethe GUT fields to argue in favor of a parity odd GLRgauge singlet field. In this case the LR theory has toremain perturbative and consistent all the way to theGUT scale. We will show in the following section thatfor the LR scale accessible at the LHC, the theory breaksdown quite quickly. It helps to raise the LR scale to theone reachable at the future colliders, but it is still notenough, the quartic couplings become large well belowthe GUT scale.

Still, one can opt for the parity odd GLR gauge sin-glet and claim that this helps the domain wall problemsince the domain walls can be washed by the subsequentinflation. However, the domain wall problem is not so se-rious, for it may be solved by tiny Planck scale inducedgravitational effects [35] or through [36] the symmetrynon-restoration at high temperature [37]. All this said,it is perfectly legitimate to decouple ∆L, but the natu-ralness argument is not the right one to use.

Bottom line: in the LR-symmetric seesaw picture thatwe employ, it is natural, both physically and technically,to work in the limit of vanishing vL and the βi couplings.Case P. We do not write down explicitly the poten-

tial in the case of P. It is enough to say that this case,being more constrained, is obtained from that of C byrequiring most of the couplings in the potential in (9) tobe real. More precisely, a number of phases must vanishand the potential can be obtained from the one in the

case C

VP = VC(d2 = d4 = r4 = b1 = b2 = b3 = 0)−2iα2scTr(ΦΦ† − Φ†Φ)Tr(∆L∆†L) (10)

We should add that in this case the mass term µΦ isautomatically real, unlike in the case of C which requireda phase redefinition of Φ.

B. Scalar spectrum.

Before we go into the gory detail, it is instructive toanticipate the results on physical grounds, at least in thedecoupling limit of large MWR

when the spectrum re-duces to the SM Higgs boson h with the usual relationfor its mass, the heavy triplets ∆L,R withm2

∆ ∝ ρv2R cou-

plings (where ρ stands for the appropriate combinationthe couplings ρi) and of the heavy flavor violating doubletfrom the bi-doublet with the mass-squared proportionalto α3v

2R. These essential features get complicated by the

possible mixings in the case of accessible scale MWR, but

most of them can be understood by symmetry argumentswhich we present below.

The only relevant relation coming from the first-derivative minimization conditions is for a generic tβ [22]

t2βsa ' −4α2

α3sc , (11)

which holds for both the LR symmetries P and C. Thereis an important distinction though. In the case of P, onehas t2βsa ≤ 2mb/mt [10], which from (11) implies

|2α2/α3sc| ≤ mb/mt . (12)

In the case of C the parameter t2βsa is unconstrained andno further restriction emerges from (11). In both cases,as seen from (11), there is no possibility for spontaneousCP violation as opposed to the generic two Higgs doubletsituation; the phase a vanishes in the limit of explicit CPconservation (c = 0). The reason for this is phenomeno-logical, not structural, as we can explain below.

Let us define the following couplings that are useful forthe discussion below

λΦ ≡ λ1 + s22β(2λ2cd2+2a + λ3) + 2s2βλ4cd4+a ,

α ≡ α1 + 2α2s2βca+c + α3s2β ,

α ≡ α2s2βsasc ' −4α3c2β(t2βsa)2 , (13)

where λΦ is the quartic coupling of the SM Higgs if themixing with ∆R fields is neglected, α is the quartic cou-pling that mixes the SM Higgs with the new Higgs bosonin ∆R and finally α is the effective quartic responsibleof the electroweak corrections to the masses of the ∆L

multiplets. Notice that α is negative since β is limiteddue to the perturbativity of Yukawa couplings [38], andis controlled by the physical parameters as we discussbelow.

5

As usual, the next step is to write down the mass ma-trix through the Hessian of the potential. It is useful todiagonalize it in two steps: in the first one, we neglect themixing of the Φ with ∆R; in the second one, we considerthe whole matrix. Thus we first introduce

φSM = (cβφ1 + sβe−iaφ2) =

(hSM + iG

G−

), (14)

and

φFV = (−sβeiaφ1 + cβφ2) =

(H + iAH−

), (15)

where φSM is the SM doublet and FV stands for thetree-level flavor violating interactions in which the heavyscalar doublet φFV takes part (φSM is the doublet witha non-vanishing vev, while φFV has a zero vev). In thegeneric two SM Higgs doublet case these doublets wouldmix, but here they are eigenstates to a great precision,since φFV has to be extremely heavy, on the order of 20TeV. This allows to ignore the electro-weak contributionof order v to the mass of φFV . Moreover, this scalardoublet is basically decoupled, which is why there canbe no observable spontaneous CP violation, which as is

well known, requires two Higgs doublets with masses atthe electro-weak scale [39]. Since m2

φFV∝ α3v

2R (see the

Tab. I), in order to break CP spontaneously one wouldneed α3 ' (v/vR)2, clearly in contradiction with the limiton the φFV mass. This is made explicit in [22].

There are possible mixings though with the ∆R compo-nents (see Appendix A), in particular the mixing betweenhSM and <e(δ0

R) is approximatively given by

θ ' α

2ρ1

v

vR

[1 +O(

v2

v2R

)

], (16)

or more precisely as in appendix A. This mixing is onlyrelevant when the mass of δR in Tab. I is not far fromthe electroweak scale (small ρ1). Recent limits from elec-troweak precision tests allow a fairly large sθ as a functionof the mass of the new Higgs [40, 41], up to sθ . 0.4.

In general, the relevant mixing terms among the neu-tral scalars appearing in Tab. I can be found in Ap-pendix A. Using the constraint in (12), the expressionsin the mass matrix (A1) get somewhat simplified for thecase P, which is reflected in the results given in the Tab. I.

Physical scalars Mass2 (case C) Mass2 (case P)

h ' cθhSM − sθ<e(δ0R) 4(λΦ − α2

4ρ1)v2 The same with the restrictions in (10).

δR ' cθ<e(δ0R) + sθhSM 4ρ1v

2R + α2

ρ1v2 The same with the restrictions in (10).

φFV (FV heavy doublet) α3c2β

v2R

α3c2β

v2R

δL = <e(δ0L) ∼ =m(δ0

L) (ρ3 − 2ρ1)v2R + 4αv2 (ρ3 − 2ρ1)v2

R

δ−L (ρ3 − 2ρ1)v2R + ( 1

2α3c2β + 4α)v2 (ρ3 − 2ρ1)v2

R + 12α3c2βv

2

δ−−L (ρ3 − 2ρ1)v2R + (α3c2β + 4α)v2 (ρ3 − 2ρ1)v2

R + α3c2βv2

δ−−R 4ρ2v2R + α3c2βv

2 4ρ2v2R + α3c2βv

2

TABLE I. Physical scalar content of the LRSM and the associated mass spectrum. In the case of P we discard small terms ofO(t2βsa) for both heavy and light scalars, and in general terms of O(v2) for the heavy flavor changing doublet φFV . We alsoignore small v/vR corrections which imply that the would-be Goldstone bosons from the light and heavy sectors do not mix.The only phenomenological exception is the mixing θ which may be non-negligible for light δR, in spite of being of order v/vR.Further details on the spectrum and particle mixings are discussed in the Appendix A.

We should comment on the results presented above.What is new in Tab. I is the β dependence, ignored in theliterature by assuming tanβ ' 0. It particularly affectsthe SM Higgs mass. The β dependence enters in the restof the table mainly through the electro-weak corrections,but it can be important, especially for δR in case it islight, as discussed in subsection III C.

Notice also an interesting fact regarding the sum rulefor the masses in the LH triplet, compared to the usualsituation of the simple type II seesaw case [42]. Thearbitrary sign of the mass splitting is now fixed since α3

must be positive, being responsible for the mass of the

heavy FV doublet in the bi-doublet.Understanding the spectrum: symmetry argu-

ments. Let us try to make sense out of the above Tab. Iby offering simple symmetry considerations; we believethey ease the reader’s pain.

• Notice that in the limit ρ2 = ρ4 = c = α3 = 0,ρ3 = 2ρ1 the masses of the ∆ states vanish, ex-cept for Reδ0

R. It is easy to understand why thisis so, since in this limit the potential exhibits anaccidental global SO(12) symmetry which involves12 real fields in ∆L,R multiplets. The SO(12) isbroken down to SO(11) through the vR (assuming

6

vL =0). Hence 11 Goldstone bosons, the three ofthem eaten by the heavy gauge fields ZR and WR.In the α3 = 0 limit the mass of the heavy doubletvanishes too, but it is not due to the SO(12) sym-metry arguments. Simply, it is only the α3 that cansplit the doublets in the bi-doublet since the termsof type Tr(∆†R∆R) do not affect the Φ sector. Thisis what makes the heavy doublet live at the MWR

scale [2] and what cures the usual problem of flavorviolation in two-Higgs doublet models [38].

• In the ∆R sector there is a global SO(6) symme-try when ρ2 = α3 = 0 and once again the SO(6)symmetry is broken down to SO(5) by vR 6= 0.There are then 5 Goldstone bosons, three of themare eaten by the gauge fields ZR and W±R and theother two correspond to δ++

R , which is manifestlymassless in that limit. Notice that this is indepen-dent of the quartic coupling α2 which explains whyα2 contribution is absent in the masses of the ∆R

triplet whereas it appears as a common contribu-tion in all the fields that belongs to ∆L.

• It is also instructive to consider the limit vR = 0in which case only v gives mass to the scalars. Itgives mass also to both W and WR (as well theneutral gauge bosons), thus one expects doublingof the Goldstone bosons compared to the SM situ-ation, and it is confirmed by looking at the Tab. Isince only the real components of the neutral fieldsin the bi-doublet pick up masses. There is an in-teresting exception: tβ = 1 (only a mathematicallimit, physically not reachable). In this case onlyone linear combination of the two W ’s get massiveand thus we expect halving the number of Gold-stone bosons in the bi-doublet. An explicit com-putation confirms it, with φFV becoming massive.The limit must be studied apart, it is not smooth.

C. The Higgs self-couplings: a window to newphysics.

In the SM the Higgs mass is given in terms of thequartic coupling appearing in the Higgs potential andtherefore its determination is a crucial test of the Higgsmechanism. Several studies have been proposed in orderto probe the Higgs self-couplings at the LHC and futurecolliders [43–50]. In particular in [45, 48] the LHC reachis studied in the context of the scalar singlet extension ofthe SM, which is effectively the situation encountered inthe LRSM for the light δR Higgs scalar.

In Tab. II we give relations between the physical andthe original quartic couplings that enter in the scalarmasses in Tab. I. We drop the ∼ θ2 corrections in thefirst two lines, since the forthcoming experiments willnot be very sensitive to these interactions.

Physical couplings Quartic couplingsλhhhh λΦ/4λδRδRδRδR ρ1/4λδ++R

δ++R

δ−−R

δ−−R

ρ1

λδ+Lδ−Lδ+Lδ−L− λ

δ++R

δ−−R

δ++R

δ−−R

ρ2

λδ++R

δ++R

δ−−L

δ−−L

ρ3

4λφ†FV

φFV δ∗LδL− λ

φ†FV

φFV δ++R

δ−−R

c2βα3

TABLE II. Relations among the quartic couplings in the po-tential and the physical quartic couplings. Small terms oforder θ2 are ignored in the first two lines.

The LHC is more sensitive to the triple coupling λhhh,since it can be probed in Higgs pair production at theLHC, the reason being that the gluon fusion pair produc-tion is the dominant channel (the order of 30 fb at

√s =

14 TeV [43]). The other channels, such as vector-bosonfusion and associated production with gauge bosons andheavy quarks are generically a factor 10 - 30 smaller. Intable III we show the expressions for the relevant trilinearcouplings in term of the scalar masses using the relationspresented in the Appendix A. A detailed study of theLHC sensitivity to the trilinear coupling [43] concludedthat the LHC with an integrated luminosity of 3000 fb−1

could see the Higgs pair production through the scalarcouplings at significant rates. In contrast to the trilinearcoupling, the quartic one needs the production of threeHiggs bosons in the final state; it is therefore suppressedand probably it cannot be determined at the LHC.

Tri-linear couplings Expression

λhhhm2h

2√

2

c3θv

λδRδRδRm2δR

2√

2

(s3θv

+c3θvR

)λhhδR

s2θcθ(m2δR

+2m2h)

4√

2v

λhδRδRs2θ(2m2

δR+m2

h)

4√

2( sθv− cθ

vR)

TABLE III. Triple scalar couplings in the LR model. Weused the relations in the Appendix A to express the tri-linearcouplings in terms of the scalar masses. Due to the LHCsensitivity of these couplings, we do not ignore leading v/vRterms.

Using the expressions in table III for a quite light δR,the trilinear coupling λhhh can be written as [45]

λhhh 'm2h

2√

2

c3θv. (17)

It is instructive to compare it to the SM trilinear cou-pling, which is λSMhhh = 1

2√

2

m2h

v and it gives a deviationwith respect the standard model expectation of the form

∆λhhh ≡λSMhhh − λhhh

λSMhhh' 3/2θ2. (18)

7

Therefore a deviation of around 20% can result for a fairlylarge θ (order ∼ 0.4 [40]). We shall see in section V thatthis deviation may be much larger once quantum cor-rections are included1. This is encouraging, since at theLHC program with 3000 fb−1 of integrated luminosity,the trilinear coupling is expected to be measured with±30%

20% of accuracy [49].The prospects for future hadron colliders are even bet-

ter. For instance, in [48] it is found that a deviation of13% can be measured in a 100 TeV collider for 3 ab−1 ofintegrated luminosity, so it is clear that a deviation withrespect to the SM values can be found in the present andthe next generation of hadron colliders. Notice that thisis complementary to the LNV Higgs decays first consid-ered in [52] and phenomenologically investigated withinan effective approach in [53]. Recently an in-depth col-lider study, including displaced vertices, has been pro-vided in [54] within the LRSM, where this decay is ex-plicitly linkable to the SM deviation of the Higgs bosonself-coupling through θ. Furthermore, even if δR is tooheavy to be seen at the LHC, for mδR & 2MW [55], theabove deviation may be still present for δR below TeV.Scalar masses and naturalness. As discussed

above, one can relate directly the scalar masses to therelevant interaction couplings, a general feature of spon-taneous broken gauge theories. Trouble occurs for verylow scale LRSM though. A WR in the reach of LHCwould require an effective potential beyond tree-level be-cause of the large α3. The issue is analyzed in [24] andfurther discussed in the next sections.

What about the mass scales of various scalar states?First of all, as repeatedly stated the second SM doubletφFV is rather heavy, above 20 TeV or so, due to its flavorviolating couplings in the quark sector. The left-handedtriplet ∆L could be light, but for WR accessible at theLHC it ends up being too heavy to be observed (just asδ++R ), as we discuss in the next section. Ironically, byincreasing MWR

the constraints on ∆L and δ++R masses

go away, allowing them to be light, close to the electro-weak scale. Of course, this become less and less naturalas theWR mass keeps growing. The last remaining state,the RH neutral δR scalar can be as light as one wishes,although again its lightness certainly violates naturalnessexpectations.

A very light δR, with a mass close to the electro-weakscale (and thus decoupled from the RH scale) becomeseffectively a SM singlet. This implies a tiny ρ1, so thatone looses a direct relation between masses and associ-ated vertices because the latter would be dominated bythe Coleman-Weinberg potential [56]. In order to have apredictive theory, one would need the full effective poten-tial, beyond the one in [24] that is focused on the leading

1 A complete analysis on the one-loop corrections to the cubicHiggs couplings in the singlet extension of the SM can be foundin [51].

quantum corrections related to α3. This is explicitly dis-cussed for the relevant trilinear couplings above in sec-tion V.The decoupling limit. It is worth noticing that

in the limit of θ → 0 (mδR → ∞), the expression forthe quartic coupling λhhhh in Tab. II does not coincidewith the effective quartic coupling appearing in the Higgsmass, with an apparent mismatch with the expected de-coupling. The well-known reason is that one must in-clude the reducible diagram hh → hh with an interme-diate δR. Since the relevant trilinear coupling can beexpanded from Tab. III as λhhδR ' 1/

√2αvR, in the de-

coupling limit one obtains for the Higgs effective quarticinteraction

1

4(λΦ −

α2

4ρ1)h4 ≡ 1

4λhh

4. (19)

This is precisely the same effective quartic entering in theHiggs mass m2

h = 4λhv2 in Tab. I, where λΦ and α are

defined in (13).

IV. THE SCALAR POTENTIAL AT WORK

In this section we examine the behavior of the poten-tial under the running in the whole parametric spaceof the model for three different LR scales: LHC reach,next collider reach (i.e.∼ 20 TeV) and very high en-ergy (109 − 1011 GeV). The complete renormalizationgroup equations (RGE) for the quartics were first pro-vided by [57] and recently revisited in [58], where someconstraints on the parametric space are derived.

For low LR scale one has to deal with a large α3 re-quired by the heaviness of the doublet φFV . The the fi-nite one-loop contributions to the generation of the quar-tics at vR due to the large α3 coupling were taken sys-tematically into account in [24]. Here we consider thedivergent loop contributions and the running of the cou-plings by choosing randomly the initial quartics (consis-tently with the expression for the masses). Moreover,we allow the possibility of tβ 6= 0 since it enters directlyin the RGE’s and more important, it changes drasticallythe matching conditions of the starting quartics with thescalar masses in Tab. I. It is not justified to set the ini-tial values of λ2,3,4 to zero as in the present literaturesince these couplings contribute to the Higgs mass mh asclear from Tab. I and (13) and furthermore they are notself-renormalizable.

We extend the analysis for the LR scale at next collidergeneration, where the LRSM is less constrained, showingthat the theory becomes completely natural and remainsperturbative all the way to high scales. Finally, we con-sider the case of very high energy RH scale, relevant forthe two step symmetry breaking of the SO(10) GUT. Itis crucial to make sure that the theory remains pertur-bative in the energy regime between the intermediate LRbreaking scale and the scale of grand unification. As we

8

Α3

Λ1

MWR=6 TeV

Landau

pole

3.8 4.0 4.2 4.4 4.6 4.8 5.0 5.20

2

4

6

8

10

12

14

LogHEGeVL

cut-off>13M

WR

cut-off>3M

WR

Instability

MWR=6 TeV

3.8 4.0 4.2 4.4 4.6 4.8 5.0-0.2

0.0

0.2

0.4

0.6

LogHEGeVL

Ρ3-

2Ρ

1

cut-off>10 M

WR

cut-off>16.4 M

WR

Instability

MWR=6 TeV

3.8 4.0 4.2 4.4 4.6 4.8 5.0-1.0

-0.5

0.0

0.5

1.0

1.5

2.0

LogHEGeVL

4Ρ

2

FIG. 1. Left. Running of λ1, α3 (the other λ and α couplings exhibit a similar behavior), they become non-perturbative around105 GeV. Center. Running of ρ3 − 2ρ1 which provides the leading masses for the ∆L multiplets. The values for the cut-off areread off from the point where ρ3 − 2ρ1 goes to zero. Right. The same for 4ρ2 which provides the leading mass term for δ++

R .In all plots the bands denote the dependence on the random initial choices consistent with the mass spectrum.

shall see, this can be satisfied as long as the scalar statestend to live below the LR scale.

A. Left-Right symmetry at LHC

Let us start our discussion on RGE’s in the phe-nomenologically most relevant case of low RH scale. Asalready remarked above, in this case the scalar poten-tial is strongly affected by the large α3 and its inducedquantum effects. The evaluation of the self-induced α3 atone-loop and at vR scale yields [24] α(1)

3 /α3 = 3α3/(8π2),

which means a perturbativity of∼ 10% forMWR' 6 TeV

(the value for which the perturbativity issue is maximallyalleviated, while WR is still detectable at LHC [17]).Therefore we focus exactly on this portion of parameterspace of the model, which means α3 . 5 [16, 24]. Tak-ing the lower limit as an input and choosing the otherquartics randomly within the range2 (0,0.1) but consis-tently with the spectrum, several couplings become non-perturbative above 105 GeV.

The running of λ1, α3 is shown in Fig 1(Left). The re-sult depends on the random choice of the initial quarticswhile being quite insensitive to tβ . Increasing the rangeto be (0,1), the situation worsens and the Landau pole ofthe theory gets too low. The cutoff from Fig. 1(Left) islower than the one shown in [58], due to the larger initialα3. In the running, the threshold effects are taken intoaccount, the light scalars start to run below vR at theirown mass values.

Other important results of the RGE’s of the scalarsector with the RH scale at LHC is represented by inFigs. 1(Center,Right). The combination ρ3 − 2ρ1 pro-vides the leading mass term of the ∆L components (seeTab. I). The parameter ρ3 − 2ρ1 can become negative asin Fig. 1(Center), destabilizing the potential below the

2 The analysis is done even by choosing randomly negative valuesfor those quartics not responsible for the leading mass terms ofthe scalars. No significative differences emerge.

limits from perturbativity in Fig. 1(Left). In order toget the cut-off (defined as the point where this parame-ter vanishes) as far as possible above MWR

, one has tochoose those configuration where the initial ρ3 − 2ρ1 islarge enough, without worsening significantly the pertur-bative limit (Landau pole).Theoretical limits on the masses of the triplet

components. In terms of the masses of ∆L triplet, forthe chosen value MWR

= 6 TeV, this arguments readsfrom the Fig. 1(Center) as

cutoff & 10MWR⇒ mδL,δ

+L ,δ

++L

& 9TeV. (20)

This is not the actual limit on the masses of ∆L. It onlyapplies to the WR accessible at LHC, while for a WR

mass above roughly 20 TeV, it goes away completely, aswe discuss in the next subsection. Physically, it says thatif the WR were to be discovered at the LHC, ∆L shouldnot be seen.

Exactly the same discussion applies for Fig. 1(Right)that shows the running of the quartic ρ2, related to theleading mass term of δ++

R . One has to choose the initialρ2 & 0.35 consistently with the cut-off in Fig. 1(Right)and without spoiling significantly the Landau pole inFig. 1(Left), thus

mδ++R

& 12TeV. (21)

These LHC constraints are stronger than the phenomeno-logical ones from the oblique parameters in [24], andlarger than the benchmark values considered in [59].

We believe that a cut-off as in (20) is the small-est value for living safe, just enough to suppress non-renormalizable operators from a new physics scale, atleast in those configurations with cutoff & 10MWR

. Themodel requires though a UV completion already in thereach of the next collider generation, which can be seenas a challenge. Still, the conclusion is that the entirescalar content of the LRSM has to be heavy, except forδR that remains unconstrained. This is crucial in rela-tion with the discussions encountered in section III C. Weshould stress that by lowering the WR mass, the cut-offgoes down, below the order of magnitude limit we used

9

cut-off>1.8´105MWR

cut-off>8.9´10 3M

WR

Instability

MWR=20 TeV

5 6 7 8 9 10

-0.1

0.0

0.1

0.2

0.3

0.4

LogHEGeVL

ΛH

iggs

cut-off>1.2´103MWR

cut-off>250MWR

Instability

MWR=20 TeV

4.5 5.0 5.5 6.0 6.5 7.0 7.5

0.0

0.5

1.0

1.5

LogHEGeVL

ΛH

iggs

FIG. 2. Left. Running of λHiggs ≡ 4λh defined in (19) for tβ=0. Right. The same for tβ=0.3 giving a lower cut-off. Thecut-offs are defined in the same manner as in the Fig. 1

as a definition of a sensible renormalizable theory. Thisimplies MWR

& 6TeV, whereas, as remarked before, theLHC reach requires [17] MWR

. 6TeV - at the LHC thetheory lives at the edge.

A final comment is in order. What is exhibitedin Fig. 1(Center, Right) represents proper instabilities(not meta-stabilities), since the estimated decay timefrom [60] is very short with respect to the age of theuniverse. The same holds for the instabilities discussedbelow.

B. Left-Right symmetry at next hadronic collider

The proper machine for the LRSM would be a 100TeV collider in any case, since the FV scalar doublet isfar away from the LHC reach. Therefore we choose tofocus in this section on the LRSM with MWR

= 20 TeV,consistent with next generation colliders. This choice,besides eliminating any tension in the parametric spaceof the model, represents a scale for which the LRSM of-fers an insight on the strong CP problem. Namely, therestoration of parity makes θ computable [61] leading toMWR

≥ 20 TeV [62]. This also fits well with the poten-tially strong limit due to ε′ [63].

The general setup of the RGE analysis is the same ofthe one discussed in the previous subsection, except thatnow α3 can be fairly small. From the FV constraints onehas α3 & 0.38 [16, 24], being the lower value our inputparameter.

The most stringent limits are obtained by the runningof λHiggs ≡ 4λh defined in (19), and they depend ontβ . In the left panel of Fig. 2 we choose tβ = 0, lead-ing to a destabilization of the potential around 109 GeV.The result is seen to depend on the random choices ofthe initial values, and we conservatively quote the worstconfiguration.

A non-vanishing tβ enters directly in the RGE’sthrough the Yukawa couplings and the cut-off gets low-

ered. For tβ = 0.3, chosen for the sake of illustration3,the potential is destabilized around 107 GeV, as shownin the right side of Fig 2.

As a result, we believe that it is not well motivated tofocus on versions of the theory in which parity is brokenat very high energy, while the gauge symmetry GLR ispreserved up to 10-100 TeV - at least, not by appealingto grand unification. The quartic couplings become non-perturbative well below the GUT scale and this holdseven for the truncated potential [25, 33] consistent withthe high scale parity breaking picture.

In short, a RH scale in the range 10-100 TeV leadsto a well-defined perturbative model, with a high scalecutoff. Moreover, the theoretical limits on the masses of∆L states and δ++

R are now gone away and one is leftonly with the experimental bounds on the order of a fewhundred GeV.

C. High scale Left-Right symmetry and SO(10)GUT

The LRSM can be naturally embedded in the SO(10)GUT with the generalized charge conjugation C a discreteSO(10) gauge symmetry. With the minimal fine-tuninghypothesis, the LR and GUT scales are predicted to be∼ 1010 GeV and ∼ 1016 GeV respectively [34]. A questionarises naturally: are there any conditions on the scalarpotential needed to ensure the consistence of this picture?After all, the quartics of the potential have to remainperturbative up to the scale of grand unification.

In the Fig. 3 we illustrate once again the cases of tβ =0; 0.3 for two different ranges of the quartics. For thesake of completeness we plot also the gauge couplings asa benchmark. By varying randomly the initial quartics,one sees that the two step SO(10) symmetry breaking can

3 Larger values imply a less perturbative interaction of the FVscalars with the quarks [38].

10

g3

gB-L

g

tΒ=0

ΛHiggsHΛi,Αi,ΡiÎ@0, 1DL

ΛHiggsHΛi,Αi,ΡiÎ@0, 0.1DL

10 12 14 16 180.0

0.2

0.4

0.6

0.8

1.0

LogHEGeVL

g3´5

gB-L´5

g´5

ΛHiggsHΛi,Αi,ΡiÎ@0, 1DL

ΛHiggsHΛi,Αi,ΡiÎ@0, 0.1DL

tΒ=0.3

10 11 12 13 14 15 160

1

2

3

4

5

6

LogHEGeVLFIG. 3. Left. Running of λHiggs ≡ 4λh defined in (19) for tβ=0. Right. The same for tβ=0.3 which shows a lower cut-off andλHiggs can become slightly large at GUT scale. The cut-offs are defined as in the previous figures.

be preserved with |λi, αi, ρi| < 0.1, albeit non-trivially.The case of non-null tβ is slightly disfavored, as clearfrom the right side of Fig. 3. In fact, although the cut-offis still around GUT scale, λHiggs can become fairly largebelow the destabilization point of the potential.

In any case, keeping the quartics of order of few per-cent is sufficient to preserve the standard SO(10) picture.This implies that the scalar masses tend to be lower thanvR, in reasonable accord with the extended survival prin-ciple (equivalent to minimal fine-tuning) needed in orderto make predictions on the mass scales in grand unifica-tion [64]. In short, all is well with the naive picture, aslong as the scalars live somewhat below the correspond-ing symmetry breaking scale.

Higher order effects. Before closing this section, adiscussion is needed regarding higher order effects. Theone-loop RGE’s for the LRSM show fairly large coeffi-cients in the pure quartics part [57] due to the rich scalarfield content. One has to wonder whether at higher or-ders even larger coefficients appear, breaking down theperturbative expansion and spoiling the one-loop results.A complete two-loops analysis is beyond the scope of thiswork. Still, it is important to check the impact on therunning from this main part of β2−loop related to thequartics only. In the Appendix B, as generic example,we show the β-function for λ1 at the two-loop order.

As can be seen from (B1), no relevant impact on therunning is expected in the cases of very high energy RHscale and next collider reach, since there the quartics canbe fairly small (we verify this by explicit calculation).In the case of LR symmetry at LHC, α3 is large andso most of the other couplings grow quickly during therunning. A direct evaluation shows that the two-loopcorrection reduces a bit the already low destabilizationpoint. However, the Landau pole appears still slightlyabove the cut-off shown in Figs. 1(Center, Right), whichin turn is not drastically modified. In conclusion, theresults presented in this section are quite stable.

V. TRILINEAR VERTEX CORRECTIONS

Here we discuss the one-loop renormalization vertexfor the cubic couplings; similar results hold also for thequartic couplings. Of particular importance is the limitof mδR MWR

, since a phenomenologically apprecia-ble impact on the Higgs physics requires a light δR, par-tially decoupled from RH scale. This, in turn, impliesdomination of the quantum corrections for the trilinearand quartic couplings involving δR in the effective po-tential. Moreover, a WR in the reach of LHC requiresa large α3 and therefore its related loop effects may bethe dominant ones. In this case the leading quantumcorrection can be read off from the effective potentialin [24], where in particular one sees the trilinear δ3

R cou-pling (re-scaling δR → δR/

√2 in usual normalization)

'(√

2ρ1 + 2√

2α23/(48π2)

)vR.

Clearly, for δR sufficiently light (small ρ1), the loopeffect becomes dominant. One should not confuse thiswith the perturbativity issue in the LRSM discussedin [24]; simply the perturbation theory starts at the one-loop level when the tree level is made artificially small,as known from the classic work of Coleman and Wein-berg [56].

We consider here the quantum corrections to the tree-level exact expressions in the Table III and the Ap-pendix A by including the whole scalar spectrum. Thelatter is especially important in the case of the RH scalein the LHC reach, where one has to consider even theconstraints in (20) and (21). The complete expressionsof the effective trilinear couplings are too long to be re-ported here, thus we show the leading corrections to the

11

10%

50%

100%

hhh

MWR=6 TeV

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

500

1000

1500

2000

2500

3000

Θ

m∆

Rin

GeV

10%

50%

100%

hh∆R

MWR=6 TeV

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

500

1000

1500

2000

2500

3000

Θ

m∆

Rin

GeV

10%

50%100%

h∆R∆R

MWR=6 TeV

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

500

1000

1500

2000

2500

3000

Θ

m∆

Rin

GeV

10%50%

100%

∆R∆R∆R

MWR=6 TeV

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

500

1000

1500

2000

2500

3000

Θ

m∆

Rin

GeV

10%

50%

hhh

MWR=20 TeV

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

500

1000

1500

2000

2500

3000

Θ

m∆

Rin

GeV

10%

50%hh∆R

MWR=20 TeV

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

500

1000

1500

2000

2500

3000

Θ

m∆

Rin

GeV

10%

50%

100%h∆R∆R

MWR=20 TeV

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

500

1000

1500

2000

2500

3000

Θ

m∆

Rin

GeV

10%50%

100%

∆R∆R∆R

MWR=20 TeV

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

500

1000

1500

2000

2500

3000

Θ

m∆

Rin

GeV

FIG. 4. Plots for the quantities shown in (26) (in %) for MWR = 6 TeV (Top) and MWR = 20 TeV (Bottom). For the sake ofclearness the plots run up to θ ' 0.7, although some regions are ruled out phenomenologically [40].

expressions in Table III in the limit θ, ρ1 → 0:

λapproxhhh = λhhh+

1

π2

[v3

v2R

(√2λ3

Φ

3α3+

α33

96√

2ρ2

+3α3

3

64√

2ρ3

)+

9λ2Φv

8√

2

],

(22)

λapproxhhδR= λhhδR +

v2(9α2

3 + 32λΦ2)

32√

2π2vR, (23)

λapproxhδRδR= λhδRδR +

α3v (8 (λΦ + ρ2) + 3ρ3)

16√

2π2, (24)

λapproxδRδRδR= λδRδRδR +

(2α2

3 + 16ρ22 + 3ρ2

3

)vR

24√

2π2. (25)

In Fig. 4 we show the deviation from the expressions (22)-(25) of the the full quantum corrections due to the entirescalar sector. More precisely we plot

|(λtotali,j,k )− λapprox

i,j,k )/λapproxi,j,k |, (26)

in the (mδR , θ) plane, where λtotali,j,k are the trilinear cou-

plings with the full quantum corrections included and theindices i, j, k range on h and δR.

Notice that λapproxhhh would be affected by the furtherquantum corrections in the presence of non-vanishingmixing, as shown in Fig. 4. Therefore a larger SM de-viation than the one in (18) may result in some por-tions of the parameter space. This can be understood bynoticing that for non zero mixing, λΦ has to be largerthan its SM value (see first line in Tab. I), thus affect-ing the tree level values for the couplings entering di-rectly in the loops. Furthermore, there are contributionsdepending on both α3 and θ. This is particularly true

for the effective λhδRδR , as clear from Fig. 4, which re-ceives contributions ∼ α2

3θ. Nevertheless, the approxi-mations in (22)-(25) work quite well for wide regions inthe (mδR , θ) plane.

In the natural case with mδR ' vR and θ negligible,in accord with the perturbativity constraints [24], theeffective vertices discussed above assume a simple formgiven in the Appendix C. In the Fig. 4, for the caseMWR

= 6 TeV we have used the bounds in (20) and (21)on ρ2, ρ3, while for MWR

= 20 TeV the experimentalconstraints [65] on m++

δR,m++

δLof a few hundred GeV. As-

suming larger values, especially in the latter case, changesonly slightly the effective vertices and the explicit checkshows that the Fig. 4 remains quite stable.

VI. CONCLUSIONS AND OUTLOOK

The LR symmetric theory has gone through a revivalof interest in recent years, and for good reasons. Due tothe theoretical limits on its scale, obtained in the earlyeighties, one had to wait for the LHC in order to hope forits verification. The possible LHC signatures are remark-able: lepton number violation through the production ofsame sign charged lepton pairs and the way of directlytesting the Majorana nature of RH neutrinos [8]. Thisis connected with the low energy lepton number and lep-ton flavor violating processes [9]. Moreover, the theoryallows for a direct probe of the origin of neutrino massand a disentangling of the seesaw mechanism [3], as longas one can measure the masses and mixings of the RHneutrinos [66, 67] and their Majorana nature [68]. In re-cent years, one has also finally computed analytically theRH quark mixing matrix [10].

12

One cannot overemphasize the importance of the studyof the Higgs sector of the theory, especially today whenit appears that the SM Higgs boson has been found. Therich scalar sector of the LRSM merges two milestones ofthe present day phenomenology, the Higgs boson physicsand the origin of neutrino mass. An example of the re-lated literature can be found in [54], where a LNV Higgsdecay is analyzed in the light of LHC. Nevertheless, acomplete analysis of the whole phenomenologically rel-evant parametric space of the scalar potential was stillmissing, and in the present work we have attempted to fillthe gap. In particular we have discussed the scalar spec-trum with a generic tβ and the spontaneous CP-phase.This configuration, moreover, would be the one neededfor a RH scale in the reach of LHC [15] if the LR symme-try were P. In any case the full knowledge of the scalarmasses is fundamental for the matching of the parame-ters of the model with the relevant physical observables.Such an example is the evolution of the quartics underthe RGE’s, which requires a direct match with the an-alytical expression of the masses, in order to ensure thestability of the potential.

We have examined the behavior of the model in threedifferent regime: LHC energy reach, next 100 TeVhadronic collider and very high energy, in accordancewith the SO(10) GUT constraints. In the first regime,our analysis shows that the model lives dangerously.While it is not ruled out from LHC reach, new physics be-yond the LRSM is already required at energy ∼ 10MWR

.This cut-off implies stringent bounds for the entire scalarspectrum, and except that for mδR that might be light asan effective SM singlet, all other states end up too heavyto be seen at the LHC. A light δR could have direct im-plications for the standard-like Higgs physics, with fairlylarge deviations of the Higgs self-couplings from the SMpredictions, measurable in the near future.

The second energy regime considered is the one of nexthadronic collider. Here the model becomes more natural.The cut-off appears far away from MWR

, although wellbelow GUT scale.

In the last energy regime we discussed the SO(10) em-bedding, within the scenario of two-step symmetry break-ing. We have shown that the usual picture fits well withthe RGE evolution of the whole parametric space of theLRSM, as long as the quartics are fairly small of order of10−1.

We conclude with the vertex renormalization for thephenomenologically important system h, δR, showing theanatomy of the quantum corrections that may be domi-nant in some regions of the parametric space.

ACKNOWLEDGMENTS

We thanks Fabrizio Nesti and Vladimir Tello for use-ful discussions. JV was funded by Conicyt PIA/BasalFB0821 and Fondecyt project N. 3170154.

Appendix A: Neutral scalar masses

Here we discuss the neutral mass matrix for the scalarpotential in (9). What in principle could be a compli-cated 4 × 4 matrix, reduces effectively to the hSM , δsystem, since the flavor violating neutral components Hand A decouple and form a part of the super-heavy dou-blet φFV with the mass m2

φFV= α3

c2βv2R.

Some comments are in order. First of all, the mass ofthe heavy doublet φFV receives corrections of the orderv2 that we discard because of the strong limit on its massof around 20 TeV [16] and the φFV components (scalarand pseudo-scalar) are degenerate for any phenomenolog-ical purpose. For the same reason we neglect in m2

δRin

Tab. I those terms suppressed as 1/m2FV and moreover,

we neglect the small mixing between δ and φFV states,which can be relevant in the case of their quasi degen-eracy, of little phenomenological interest, in which caseone could trust the tree-level anyway. It is worth noticingthat a very light δR, well below the electro-weak scale,requires some more care because of potential FCNC ef-fects. This subject has been recently studied in [69], inwhich a strong constraint on θ is obtained. However, thisdoes not affect our results, since we consider mδR ≥ mh.In such a case, this mixing is suppressed by the electro-weak scale, completely negligible due to the huge mass ofφFV field. The only mixing to consider is between hSMand δ, and only if δ is relatively light.

The mass matrix for the hSM , δ system is then foundto be

M20 '

m2hSM

m2δh

m2δh m2

δ

, (A1)

where

m2hSM = 4λΦv

2, (A2)

m2δ = 4 ρ1v

2R, (A3)

m2δh = 2αvvR, (A4)

and λΦ and α are given by (13).This matrix has the following eigenvalues (it is effec-

tively the SM augmented by a real scalar singlet studiedin [45])

m2h,δR =2

[ρ1v

2R + v2λΦ ∓

(ρ1v

2R − v2λΦ

)× (A5)√

α2v2v2R

(ρ1v2R − λΦv2)

2 + 1

]

where h = cθhSM − sθδ, δR = sθhSM + cθδ, with themixing given by

t2θ =αvvR

ρ1v2R − λΦv2

. (A6)

Finally we quote the expressions of λΦ, ρ1, α in terms of

13

the masses mh,mδR and the mixing [45]

λΦ =c2θ(m2h −m2

δR

)+m2

δR+m2

h

8v2, (A7)

ρ1 =c2θ(m2δR−m2

h

)+m2

δR+m2

h

8v2R

, (A8)

α =s2θ

(m2δR−m2

h

)4vvR

. (A9)

Appendix B: A look at RGE’s at higher order

In this appendix, we estimate the impact of the two-loop corrections to the running of the quartic couplingsin the potential. Since the complete two-loop correctionsis out of the scope of this work, we consider the correc-tions due to the scalar self-couplings only. We expectthat the leading contribution from the two-loop is due tothe self-quartics part, in full analogy with the one-loopresult [57] where the full expressions are provided. Also,the gauge couplings remain always smaller than unity,even for larger quartics (this is precisely the case in whichtwo-loop might be relevant), and for this reason they playa secondary role. With this in mind and as an illustra-tive example, we show the partial two-loop and one-loopβ-function of λ1 for a direct comparison, including onlythe contributions of the scalar quartics since only thesemay become dangerously large

(4π)2βλ1

(1−loop) ⊃ 6α21 + 6α3α1 + 2.5α2

3 + 32λ21 + 64λ2

2 + 16λ23 + 48λ2

4 + 16λ1λ3 ; (B1)

(4π)2βλ1

(2−loop) ⊃1

384π2

−36α2

1 (α3 − 30λ1)− 2α1 [α3 (19α3 − 540λ1) + 48α2 (α2 + 3λ4)] + 826α23λ1

− 48α22 (α3 − 94λ1 + 8λ2 + 4λ3)− 144α2α3λ4 − 24α3

1 − 13α33 + 2304λ1ρ

21 + 3456λ1ρ

22

+ 432λ1ρ23 + 2304λ1λ

24 + 3456λ1ρ

24 + 2304λ1ρ1ρ2 + 1424λ3

1 − 384λ33 + 14592λ1λ

22

+ 2304λ1λ23 − 3328λ2λ

24 − 1792λ3λ

24 + 1152λ2

1λ3 − 5632λ22λ3

.

Let us emphasize once again that in section IV the com-plete one-loop RGE’s were used. The expressions in (B1)can be worked out from the general formalism in [70] andare both normalized with (4π)2 for a direct comparison.A drastic gap between the size of the coefficients of one-loop and two-loops is evident, although the number ofthe contributions clearly increases for the latter. Similar

expressions hold for the other quartics.

Appendix C: Effective trilinear vertices

Here we report the expressions of the trilinear verticeswith negligible mixing θ

λeff.hhh =√

2λΦv +1

384π2α3ρ2 (2ρ1 − ρ3) v2R

(432√

2α3λ2Φρ2ρ1vv

2R − 216

√2α3λ

2Φρ2ρ3vv

2R − 9

√2α4

3ρ2v3 (C1)

−2√

2α43ρ3v

3 + 4√

2α43ρ1v

3 + 256√

2λ3Φρ2ρ1v

3 − 128√

2λ3Φρ2ρ3v

3),

λeff.hhδR=v2(α2

3

(8ρ2ρ1 − 2ρ3ρ1 − 9ρ2ρ3 + 4ρ2

1

)+ 32λ2

Φρ2 (2ρ1 − ρ3))

32√

2π2ρ2 (2ρ1 − ρ3) vR, (C2)

λeff.hδRδR=α3v

(−2ρ3

(4λΦρ2 + (2ρ2 + ρ1) 2

)+ 4ρ1

(4λΦρ2 + (2ρ2 + ρ1) 2

)− 3ρ2ρ

23

)16√

2π2ρ2 (2ρ1 − ρ3), (C3)

λeff.δRδRδR=√

2ρ1vR +1

48π2ρ2 (2ρ1 − ρ3)

(4√

2α23ρ2ρ1vR − 2

√2α2

3ρ2ρ3vR + 78√

2ρ2ρ31vR − 2

√2ρ3ρ

31vR (C4)

+48√

2ρ22ρ

21vR − 39

√2ρ2ρ3ρ

21vR + 32

√2ρ3

2ρ1vR − 24√

2ρ22ρ3ρ1vR − 3

√2ρ2ρ

33vR − 16

√2ρ3

2ρ3vR + 4√

2ρ41vR

).

14

[1] J. C. Pati and A. Salam, “Lepton Number As The FourthColor,” Phys. Rev. D 10 (1974) 275.R. N. Mohapatra and J. C. Pati, “A ’Natural’ Left-RightSymmetry,” Phys. Rev. D 11 (1975) 2558.G. Senjanović and R. N. Mohapatra, “Exact Left-RightSymmetry And Spontaneous Violation Of Parity,” Phys.Rev. D 12 (1975) 1502.

[2] G. Senjanović, “Spontaneous Breakdown of Parity in aClass of Gauge Theories,” Nucl. Phys. B 153, 334 (1979).doi:10.1016/0550-3213(79)90604-7

[3] M. Nemevšek, G. Senjanović and V. Tello, “Con-necting Dirac and Majorana Neutrino Mass Matri-ces in the Minimal Left-Right Symmetric Model,”Phys. Rev. Lett. 110, no. 15, 151802 (2013)doi:10.1103/PhysRevLett.110.151802 [arXiv:1211.2837[hep-ph]].G. Senjanović and V. Tello, “Probing Seesaw with ParityRestoration,” [arXiv:1612.05503 [hep-ph]].

[4] P. Minkowski, “µ → eγ at a Rate of One Out of109 Muon Decays?,” Phys. Lett. B 67, 421 (1977).doi:10.1016/0370-2693(77)90435-X

[5] R. N. Mohapatra and G. Senjanović, “Neutrino Mass andSpontaneous Parity Violation,” Phys. Rev. Lett. 44, 912(1980). doi:10.1103/PhysRevLett.44.912

[6] S. L. Glashow, “The Future of Elementary ParticlePhysics,” NATO Sci. Ser. B 61, 687 (1980).M. Gell-Mann, P. Ramond and R. Slansky, “ComplexSpinors and Unified Theories,” Conf. Proc. C 790927(1979) 315 [arXiv:1306.4669 [hep-th]].T. Yanagida, “Horizontal Symmetry And Masses Of Neu-trinos,” Conf. Proc. C 7902131, 95 (1979).

[7] G. Racah, “On the symmetry of particle and antiparti-cle,” Nuovo Cim. 14, 322 (1937)W. H. Furry, “On transition probabilities in double beta-disintegration,” Phys. Rev. 56, 1184 (1939).

[8] W. Y. Keung and G. Senjanović, “Majorana Neutri-nos and the Production of the Right-handed ChargedGauge Boson,” Phys. Rev. Lett. 50, 1427 (1983).doi:10.1103/PhysRevLett.50.1427

[9] V. Tello, M. Nemevšek, F. Nesti, G. Senjanović andF. Vissani, “Left-Right Symmetry: from LHC toNeutrinoless Double Beta Decay,” Phys. Rev. Lett.106, 151801 (2011) doi:10.1103/PhysRevLett.106.151801[arXiv:1011.3522 [hep-ph]].M. Nemevšek, F. Nesti, G. Senjanovi c and V. Tello,“Neutrinoless Double Beta Decay: Low Left-Right Sym-metry Scale?,” [arXiv:1112.3061 [hep-ph]].See also, P. S. Bhupal Dev, S. Goswami, M. Mi-tra and W. Rodejohann, “Constraining Neutrino Massfrom Neutrinoless Double Beta Decay,” Phys. Rev.D 88, 091301 (2013) doi:10.1103/PhysRevD.88.091301[arXiv:1305.0056 [hep-ph]].W. C. Huang and J. Lopez-Pavon, “On neutrinolessdouble beta decay in the minimal left-right sym-metric model,” Eur. Phys. J. C 74, 2853 (2014)doi:10.1140/epjc/s10052-014-2853-z [arXiv:1310.0265[hep-ph]].R. L. Awasthi, A. Dasgupta and M. Mitra, “Limit-ing the effective mass and new physics parametersfrom 0νββ,” Phys. Rev. D 94, no. 7, 073003 (2016)doi:10.1103/PhysRevD.94.073003 [arXiv:1607.03835

[hep-ph]].[10] G. Senjanović and V. Tello, “Right Handed

Quark Mixing in Left-Right Symmetric The-ory,” Phys. Rev. Lett. 114, no. 7, 071801 (2015)doi:10.1103/PhysRevLett.114.071801 [arXiv:1408.3835[hep-ph]].G. Senjanović and V. Tello, “Restoration of Parityand the Right-Handed Analog of the CKM Matrix,”[arXiv:1502.05704 [hep-ph]].For a previous work on the analytical form of the RHquark mixing matrix in a portion of the parameter space,see Y. Zhang, H. An, X. Ji and R. N. Mohapatra,“Right-handed quark mixings in minimal left-right sym-metric model with general CP violation,” Phys. Rev.D 76, 091301 (2007) doi:10.1103/PhysRevD.76.091301[arXiv:0704.1662 [hep-ph]].

[11] G. Beall, M. Bander and A. Soni, “Constraint on theMass Scale of a Left-Right Symmetric Electroweak The-ory from the K(L) K(S) Mass Difference,” Phys. Rev.Lett. 48, 848 (1982). doi:10.1103/PhysRevLett.48.848.

[12] R. N. Mohapatra, G. Senjanović and M. D. Tran,“Strangeness Changing Processes and the Limit on theRight-handed Gauge Boson Mass,” Phys. Rev. D 28, 546(1983). doi:10.1103/PhysRevD.28.546

[13] G. Ecker and W. Grimus, “CP violation and left-rightsymmetry”, Nucl. Phys. B 258, 328 (1985).

[14] Y. Zhang, H. An, X. Ji and R. N. Mohapatra, “GeneralCP Violation in Minimal Left-Right Symmetric Modeland Constraints on the Right-Handed Scale,” Nucl. Phys.B 802, 247 (2008) [arXiv:0712.4218 [hep-ph]].

[15] A. Maiezza, M. Nemevšek, F. Nesti and G. Sen-janović, “Left-Right Symmetry at LHC,” Phys. Rev.D 82, 055022 (2010) doi:10.1103/PhysRevD.82.055022[arXiv:1005.5160 [hep-ph]].

[16] S. Bertolini, A. Maiezza and F. Nesti, “Presentand Future K and B Meson Mixing Constraints onTeV Scale Left-Right Symmetry,” Phys. Rev. D 89,no. 9, 095028 (2014) doi:10.1103/PhysRevD.89.095028[arXiv:1403.7112 [hep-ph]].

[17] A. Ferrari et al., “Sensitivity study for new gauge bosonsand right-handed Majorana neutrinos in p p collisions ats = 14-TeV,” Phys. Rev. D 62, 013001 (2000).S. N. Gninenko, M. M. Kirsanov, N. V. Krasnikov andV. A. Matveev, “Detection of heavy Majorana neutrinosand right-handed bosons,” Phys. Atom. Nucl. 70, 441(2007). doi:10.1134/S1063778807030039

[18] V. Khachatryan et al. [CMS Collaboration], “Searchfor heavy neutrinos and W bosons with right-handed couplings in proton-proton collisions at√s = 8TeV,” Eur. Phys. J. C 74, no. 11, 3149 (2014)

doi:10.1140/epjc/s10052-014-3149-z [arXiv:1407.3683[hep-ex]].

[19] N. G. Deshpande, J. F. Gunion, B. Kayser andF. I. Olness, “Left-right symmetric electroweak mod-els with triplet Higgs,” Phys. Rev. D 44 (1991) 837.doi:10.1103/PhysRevD.44.837

[20] P. Duka, J. Gluza and M. Zralek, “Quantizationand renormalization of the manifest left-right sym-metric model of electroweak interactions,” AnnalsPhys. 280, 336 (2000) doi:10.1006/aphy.1999.5988 [hep-ph/9910279].

15

[21] G. Barenboim, M. Gorbahn, U. Nierste and M. Raidal,“Higgs sector of the minimal left-right symmet-ric model,” Phys. Rev. D 65, 095003 (2002)doi:10.1103/PhysRevD.65.095003 [hep-ph/0107121].

[22] K. Kiers, M. Assis and A. A. Petrov, “Higgs sector of theleft-right model with explicit CP violation,” Phys. Rev.D 71, 115015 (2005) doi:10.1103/PhysRevD.71.115015[hep-ph/0503115].

[23] V. Tello, PhD Thesis, SISSA (2012) “Connections be-tween the high and low energy violation of Leptonand Flavor numbers in the minimal left-right symmet-ric model,”

[24] A. Maiezza, M. Nemevšek and F. Nesti, “Perturbativityand mass scales in the minimal left-right symmet-ric model,” Phys. Rev. D 94, no. 3, 035008 (2016)doi:10.1103/PhysRevD.94.035008 [arXiv:1603.00360[hep-ph]].

[25] P. S. B. Dev, R. N. Mohapatra and Y. Zhang, “Prob-ing the Higgs Sector of the Minimal Left-Right Symmet-ric Model at Future Hadron Colliders,” JHEP 1605, 174(2016) doi:10.1007/JHEP05(2016)174 [arXiv:1602.05947[hep-ph]].

[26] M. Blanke, A. J. Buras, K. Gemmler and T. Heidsieck,“∆F = 2 observables and B → Xq gamma decays inthe Left-Right Model: Higgs particles striking back,”JHEP 1203, 024 (2012) doi:10.1007/JHEP03(2012)024[arXiv:1111.5014 [hep-ph]].

[27] R. N. Mohapatra and R. E. Marshak, “Local B-L Sym-metry of Electroweak Interactions, Majorana Neutri-nos and Neutron Oscillations,” Phys. Rev. Lett. 44(1980) 1316 Erratum: [Phys. Rev. Lett. 44 (1980) 1643].doi:10.1103/PhysRevLett.44.1316

[28] R. N. Mohapatra and G. Senjanović, “Neutrino Massesand Mixings in Gauge Models with SpontaneousParity Violation,” Phys. Rev. D 23 (1981) 165.doi:10.1103/PhysRevD.23.165

[29] W. Dekens, “Testing left-right symmetric models,”[arXiv:1505.06599 [hep-ph]].

[30] See for example J. F. Gunion and H. E. Haber, “The CPconserving two Higgs doublet model: The Approach tothe decoupling limit,” Phys. Rev. D 67, 075019 (2003)doi:10.1103/PhysRevD.67.075019 [hep-ph/0207010].

[31] W. Dekens and D. Boer, “Viability of minimal left-right models with discrete symmetries,” Nucl. Phys.B 889, 727 (2014) doi:10.1016/j.nuclphysb.2014.10.025[arXiv:1409.4052 [hep-ph]].

[32] F. Bezrukov, H. Hettmansperger and M. Lindner, “keVsterile neutrino Dark Matter in gauge extensions ofthe Standard Model,” Phys. Rev. D 81, 085032 (2010)doi:10.1103/PhysRevD.81.085032 [arXiv:0912.4415 [hep-ph]].M. Nemevšek, G. Senjanović and Y. Zhang, “WarmDark Matter in Low Scale Left-Right Theory,” JCAP1207, 006 (2012) doi:10.1088/1475-7516/2012/07/006[arXiv:1205.0844 [hep-ph]].

[33] D. Chang, R. N. Mohapatra and M. K. Parida, “De-coupling Parity and SU(2)-R Breaking Scales: A NewApproach to Left-Right Symmetric Models,” Phys. Rev.Lett. 52, 1072 (1984). doi:10.1103/PhysRevLett.52.1072D. Chang, R. N. Mohapatra and M. K. Parida, “ANew Approach to Left-Right Symmetry Breaking in Uni-fied Gauge Theories,” Phys. Rev. D 30, 1052 (1984).doi:10.1103/PhysRevD.30.1052

[34] F. del Aguila and L. E. Ibanez, “Higgs Bosons in SO(10)

and Partial Unification,” Nucl. Phys. B 177, 60 (1981).doi:10.1016/0550-3213(81)90266-2T. G. Rizzo and G. Senjanović, “Can There BeLow Intermediate Mass Scales in Grand Uni-fied Theories?,” Phys. Rev. Lett. 46, 1315 (1981).doi:10.1103/PhysRevLett.46.1315T. G. Rizzo and G. Senjanović, “Grand Unificationand Parity Restoration at Low-energies. 2. Unifica-tion Constraints,” Phys. Rev. D 25, 235 (1982).doi:10.1103/PhysRevD.25.235W. E. Caswell, J. Milutinović and G. Senjanović,“Predictions of Left-right Symmetric Grand Uni-fied Theories,” Phys. Rev. D 26, 161 (1982).doi:10.1103/PhysRevD.26.161D. Chang, R. N. Mohapatra, J. Gipson, R. E. Marshakand M. K. Parida, “Experimental Tests of New SO(10)Grand Unification,” Phys. Rev. D 31, 1718 (1985).doi:10.1103/PhysRevD.31.1718

[35] B. Rai and G. Senjanović, “Gravity and domainwall problem,” Phys. Rev. D 49, 2729 (1994)doi:10.1103/PhysRevD.49.2729 [hep-ph/9301240].

[36] G. R. Dvali and G. Senjanović, “Is there a domainwall problem?,” Phys. Rev. Lett. 74, 5178 (1995)doi:10.1103/PhysRevLett.74.5178 [hep-ph/9501387].G. R. Dvali, A. Melfo and G. Senjanović, “Non-restoration of spontaneously broken P and CP athigh temperature,” Phys. Rev. D 54, 7857 (1996)doi:10.1103/PhysRevD.54.7857 [hep-ph/9601376].

[37] S. Weinberg, “Gauge and Global Symmetries atHigh Temperature,” Phys. Rev. D 9, 3357 (1974).doi:10.1103/PhysRevD.9.3357R. N. Mohapatra and G. Senjanović, “Broken Symmetriesat High Temperature,” Phys. Rev. D 20 (1979) 3390.doi:10.1103/PhysRevD.20.3390R. N. Mohapatra and G. Senjanović, “High Tempera-ture Behavior of Gauge Theories,” Phys. Lett. B 89, 57(1979). doi:10.1016/0370-2693(79)90075-3For a discussion the higher order effects that casts ashadow on this phenomenon in gauge theories, see G. Bi-monte and G. Lozano, “On Symmetry nonrestorationat high temperature,” Phys. Lett. B 366, 248 (1996)doi:10.1016/0370-2693(95)01395-4 [hep-th/9507079].

[38] G. Senjanović and P. Senjanović, “Suppression of HiggsStrangeness Changing Neutral Currents in a Classof Gauge Theories,” Phys. Rev. D 21, 3253 (1980).doi:10.1103/PhysRevD.21.3253

[39] T. D. Lee, “A Theory of Spontaneous T Violation,” Phys.Rev. D 8, 1226 (1973). doi:10.1103/PhysRevD.8.1226

[40] A. Falkowski, C. Gross and O. Lebedev, “A secondHiggs from the Higgs portal,” JHEP 1505, 057 (2015)doi:10.1007/JHEP05(2015)057 [arXiv:1502.01361 [hep-ph]].

[41] S. I. Godunov, A. N. Rozanov, M. I. Vysot-sky and E. V. Zhemchugov, “Extending the Higgssector: an extra singlet,” Eur. Phys. J. C 76,no. 1, 1 (2016) doi:10.1140/epjc/s10052-015-3826-6[arXiv:1503.01618 [hep-ph]].

[42] A. Melfo, M. Nemevsek, F. Nesti, G. Senjanovićand Y. Zhang, “Type II Seesaw at LHC: TheRoadmap,” Phys. Rev. D 85 (2012) 055018doi:10.1103/PhysRevD.85.055018 [arXiv:1108.4416[hep-ph]].

[43] J. Baglio, A. Djouadi, R. GrÃČÂűber, M. M. MÃČÅŠh-lleitner, J. Quevillon and M. Spira, “The measurement of

16

the Higgs self-coupling at the LHC: theoretical status,”JHEP 1304, 151 (2013) doi:10.1007/JHEP04(2013)151[arXiv:1212.5581 [hep-ph]].

[44] M. J. Dolan, C. Englert and M. Spannowsky, “Higgs self-coupling measurements at the LHC,” JHEP 1210, 112(2012) doi:10.1007/JHEP10(2012)112 [arXiv:1206.5001[hep-ph]].

[45] R. S. Gupta, H. Rzehak and J. D. Wells, “Howwell do we need to measure the Higgs boson massand self-coupling?,” Phys. Rev. D 88, 055024 (2013)doi:10.1103/PhysRevD.88.055024 [arXiv:1305.6397 [hep-ph]].

[46] A. Efrati and Y. Nir, “What if λhhh 6= 3m2h/v,”

[arXiv:1401.0935 [hep-ph]].[47] B. Fuks, J. H. Kim and S. J. Lee, “Probing Higgs

self-interactions in proton-proton collisions at a center-of-mass energy of 100 TeV,” Phys. Rev. D 93,no. 3, 035026 (2016) doi:10.1103/PhysRevD.93.035026[arXiv:1510.07697 [hep-ph]].

[48] H. J. He, J. and W. Yao, “Probing new physicsof cubic Higgs boson interaction via Higgs pair pro-duction at hadron colliders,” Phys. Rev. D 93,no. 1, 015003 (2016) doi:10.1103/PhysRevD.93.015003[arXiv:1506.03302 [hep-ph]].

[49] F. Goertz, A. Papaefstathiou, L. L. Yang and J. Zu-rita, “Higgs Boson self-coupling measurements usingratios of cross sections,” JHEP 1306, 016 (2013)doi:10.1007/JHEP06(2013)016 [arXiv:1301.3492 [hep-ph]].

[50] R. Contino et al., “Physics at a 100 TeV pp collider: Higgsand EW symmetry breaking studies,” [arXiv:1606.09408[hep-ph]].

[51] J. E. Camargo-Molina, A. P. Morais, R. Pasechnik,M. O. P. Sampaio and J. WessÃľn, “All one-loopscalar vertices in the effective potential approach,”JHEP 1608, 073 (2016) doi:10.1007/JHEP08(2016)073[arXiv:1606.07069 [hep-ph]].

[52] J.F. Gunion, B. Kayser, R.N. Mohapatra, N.G. Desh-pande, J. Grifols, A. Mendez, F.I. Olness and P.B. Pal,“Production and detection at ssc of higgs bosons in left-right symmetric theories,” PRINT-86-1324 (UC,DAVIS).

[53] A. Pilaftsis, “Radiatively induced neutrino masses andlarge Higgs neutrino couplings in the standard modelwith Majorana fields,” Z. Phys. C 55, 275 (1992)doi:10.1007/BF01482590 [hep-ph/9901206].M. L. Graesser, “Broadening the Higgs boson with right-handed neutrinos and a higher dimension operator atthe electroweak scale,” Phys. Rev. D 76, 075006 (2007)doi:10.1103/PhysRevD.76.075006 [arXiv:0704.0438 [hep-ph]].M. L. Graesser, “Experimental Constraints on HiggsBoson Decays to TeV-scale Right-Handed Neutrinos,”[arXiv:0705.2190 [hep-ph]].

[54] A. Maiezza, M. Nemevšek and F. Nesti, “Lepton Num-ber Violation in Higgs Decay at LHC,” Phys. Rev. Lett.115, 081802 (2015) doi:10.1103/PhysRevLett.115.081802[arXiv:1503.06834 [hep-ph]].A. Maiezza, M. Nemevšek and F. Nesti, “LNV Hig-gses at LHC,” AIP Conf. Proc. 1743, 030008 (2016).doi:10.1063/1.4953289

[55] M. Nemevšek, F. Nesti and J. C. Vasquez,[arXiv:1612.06840 [hep-ph]].

[56] S. R. Coleman and E. J. Weinberg,“Radiative Corrections as the Origin of Spontaneous

Symmetry Breaking,” Phys. Rev. D 7, 1888 (1973).doi:10.1103/PhysRevD.7.1888

[57] I. Z. Rothstein, “Renormalization group analysis of theminimal left-right symmetric model,” Nucl. Phys. B 358,181 (1991). doi:10.1016/0550-3213(91)90536-7See also, R. Kuchimanchi, “Leptonic CP problemin left-right symmetric model,” Phys. Rev. D 91,no. 7, 071901 (2015) doi:10.1103/PhysRevD.91.071901[arXiv:1408.6382 [hep-ph]].

[58] J. Chakrabortty, J. Gluza, T. Jelinski and T. Srivas-tava, “Theoretical constraints on masses of heavy par-ticles in Left-Right Symmetric Models,” Phys. Lett.B 759, 361 (2016) doi:10.1016/j.physletb.2016.05.092[arXiv:1604.06987 [hep-ph]].

[59] G. Bambhaniya, J. Chakrabortty, J. Gluza, M. Kor-diaczynska and R. Szafron, “Left-Right Symmetry andthe Charged Higgs Bosons at the LHC,” JHEP 1405, 033(2014) doi:10.1007/JHEP05(2014)033 [arXiv:1311.4144[hep-ph]].G. Bambhaniya, J. Chakrabortty, J. Gluza, T. Jelin-ski and M. Kordiaczynska, “Lowest limits on the dou-bly charged Higgs boson masses in the minimal left-rightsymmetric model,” Phys. Rev. D 90, no. 9, 095003 (2014)doi:10.1103/PhysRevD.90.095003 [arXiv:1408.0774 [hep-ph]].

[60] K. M. Lee and E. J. Weinberg, “Tunneling Without Bar-riers,” Nucl. Phys. B 267, 181 (1986). doi:10.1016/0550-3213(86)90150-1

[61] R. N. Mohapatra and G. Senjanović, “Natural Suppres-sion of Strong p and t Noninvariance,” Phys. Lett. B 79(1978) 283;M. A. B. Beg and H.-S. Tsao, “Strong P, T Noninvari-ances in a Superweak Theory,” Phys. Rev. Lett. 41, 278(1978). doi:10.1103/PhysRevLett.41.278See also,K. S. Babu and R. N. Mohapatra, “A Solution to theStrong CP Problem Without an Axion,” Phys. Rev. D41 (1990) 1286.S. M. Barr, D. Chang and G. Senjanović, “Strong CPproblem and parity,” Phys. Rev. Lett. 67 (1991) 2765.

[62] A. Maiezza and M. Nemevšek, “Strong P invariance, neu-tron electric dipole moment, and minimal left-right par-ity at LHC,” Phys. Rev. D 90, no. 9, 095002 (2014)doi:10.1103/PhysRevD.90.095002 [arXiv:1407.3678 [hep-ph]].

[63] S. Bertolini, J. O. Eeg, A. Maiezza and F. Nesti, “Newphysics in ε′ from gluomagnetic contributions and lim-its on Left-Right symmetry,” Phys. Rev. D 86, 095013(2012) Erratum: [Phys. Rev. D 93, no. 7, 079903(2016)] doi:10.1103/PhysRevD.86.095013, 10.1103/Phys-RevD.93.079903 [arXiv:1206.0668 [hep-ph]].S. Bertolini, A. Maiezza and F. Nesti, “K -> pphadronic matrix elements of left-right current-currentoperators,” Phys. Rev. D 88, no. 3, 034014 (2013)doi:10.1103/PhysRevD.88.034014 [arXiv:1305.5739 [hep-ph]].

[64] F. del Aguila and L. E. Ibanez, “Higgs Bosons in SO(10)and Partial Unification,” Nucl. Phys. B 177, 60 (1981).doi:10.1016/0550-3213(81)90266-2R. N. Mohapatra and G. Senjanović, “Higgs Boson Ef-fects in Grand Unified Theories,” Phys. Rev. D 27, 1601(1983). doi:10.1103/PhysRevD.27.1601

[65] G. Aad et al. [ATLAS Collaboration], JHEP 1503, 041(2015) doi:10.1007/JHEP03(2015)041 [arXiv:1412.0237

17

[hep-ex]].[66] S. P. Das, F. F. Deppisch, O. Kittel and J. W. F. Valle,

“Heavy Neutrinos and Lepton Flavour Violation in Left-Right Symmetric Models at the LHC,” Phys. Rev.D 86, 055006 (2012) doi:10.1103/PhysRevD.86.055006[arXiv:1206.0256 [hep-ph]].

[67] J. A. Aguilar-Saavedra and F. R. Joaquim, “Measur-ing heavy neutrino couplings at the LHC,” Phys. Rev.D 86, 073005 (2012) doi:10.1103/PhysRevD.86.073005[arXiv:1207.4193 [hep-ph]].J. C. Vasquez, “Right-handed lepton mix-ings at the LHC,” JHEP 1605, 176 (2016)doi:10.1007/JHEP05(2016)176 [arXiv:1411.5824 [hep-ph]].T. Han, I. Lewis, R. Ruiz and Z. g. Si, Phys. Rev. D87, no. 3, 035011 (2013) Erratum: [Phys. Rev. D 87,no. 3, 039906 (2013)] doi:10.1103/PhysRevD.87.035011,10.1103/PhysRevD.87.039906 [arXiv:1211.6447 [hep-ph]].C. Y. Chen, P. S. B. Dev and R. N. Mo-hapatra, Phys. Rev. D 88, 033014 (2013)

doi:10.1103/PhysRevD.88.033014 [arXiv:1306.2342[hep-ph]].P. S. B. Dev, D. Kim and R. N. Mohapatra,JHEP 1601, 118 (2016) doi:10.1007/JHEP01(2016)118[arXiv:1510.04328 [hep-ph]].

[68] J. Gluza, T. Jelinski and R. Szafron, “Lepton num-ber violation and "Diracness" of massive neutrinoscomposed of Majorana states,” Phys. Rev. D 93,no. 11, 113017 (2016) doi:10.1103/PhysRevD.93.113017[arXiv:1604.01388 [hep-ph]].G. Anamiati, M. Hirsch and E. Nardi, JHEP 1610, 010(2016) doi:10.1007/JHEP10(2016)010 [arXiv:1607.05641[hep-ph]].

[69] P. S. B. Dev, R. N. Mohapatra and Y. Zhang,[arXiv:1612.09587 [hep-ph]]. P. S. B. Dev, R. N. Mohap-atra and Y. Zhang, [arXiv:1703.02471 [hep-ph]].

[70] M. E. Machacek and M. T. Vaughn, “Two Loop Renor-malization Group Equations in a General Quantum FieldTheory. 3. Scalar Quartic Couplings,” Nucl. Phys. B 249,70 (1985). doi:10.1016/0550-3213(85)90040-9