Eur. Phys. J. C (2014) 74:2994DOI 10.1140/epjc/s10052-014-2994-0

Regular Article - Theoretical Physics

Momentum-dependent two-loop QCD corrections to the neutralHiggs-boson masses in the MSSM

S. Borowka1,a, T. Hahn1,b, S. Heinemeyer2,c, G. Heinrich1,d, W. Hollik1,e

1 Max-Planck-Institut für Physik (Werner-Heisenberg-Institut), Föhringer Ring 6, 80805 Munich, Germany2 Instituto de Física de Cantabria (CSIC-UC), Santander, Spain

Received: 16 June 2014 / Accepted: 19 July 2014 / Published online: 12 August 2014© The Author(s) 2014. This article is published with open access at Springerlink.com

Abstract Results are presented for the momentum-depe-ndent two-loop contributions of O(αtαs) to the masses andmixing effects in the Higgs sector of the MSSM. Theyare obtained in the Feynman-diagrammatic approach usinga mixed on-shell/DR renormalization that can directly bematched onto the higher-order corrections included in thecode FeynHiggs. The new two-loop diagrams are eval-uated with the program SecDec. The combination of thenew momentum-dependent two-loop contribution with theexisting one- and two-loop corrections in the on-shell/DRscheme leads to an improved prediction of the light MSSMHiggs boson mass and a correspondingly reduced theoreti-cal uncertainty. We find that the corresponding shifts in thelightest Higgs-boson mass Mh are below 1 GeV in all sce-narios considered, but they can extend up to the level of thecurrent experimental uncertainty. The results are included inthe code FeynHiggs.

1 Introduction

The ATLAS and CMS experiments at CERN have recentlydiscovered a new boson with a mass around 125.6 GeV [1,2].Within the present experimental uncertainties this new bosonbehaves like the Higgs boson of the Standard Model (SM) [3–6]. However, the newly discovered particle can also be inter-preted as the Higgs boson of extended models. The Higgs sec-tor of the Minimal Supersymmetric Standard Model (MSSM)[7–9] with two scalar doublets accommodates five physi-

a e-mail: sborowka@mpp.mpg.deb e-mail: hahn@mpp.mpg.dec e-mail: Sven.Heinemeyer@cern.chd e-mail: gudrun@mpp.mpg.dee e-mail: hollik@mpp.mpg.de

cal Higgs bosons. In lowest order these are the light andheavy CP-even h and H , the CP-odd A, and the chargedHiggs bosons H±. The measured mass value, having alreadyreached the level of a precision observable with an experi-mental accuracy of about 500 MeV, plays an important rolein this context. In the MSSM the mass of the light CP-evenHiggs boson, Mh , can directly be predicted from the otherparameters of the model. The accuracy of this predictionshould at least match the one of the experimental result.

The Higgs sector of the MSSM can be expressed at low-est order in terms of the gauge couplings, the mass of theCP-odd Higgs boson, MA, and tan β ≡ v2/v1, the ratio ofthe two vacuum expectation values. All other masses andmixing angles can therefore be predicted. Higher-order con-tributions can give large corrections to the tree-level relations[10–12]. An upper bound for the mass of the lightest MSSMHiggs boson of Mh � 135 GeV was obtained [13], and theremaining theoretical uncertainty in the calculation of Mh ,from unknown higher-order corrections, was estimated to beup to 3 GeV, depending on the parameter region. Recentimprovements have lead to a somewhat smaller estimate ofup to ∼2 GeV [14,15] (see below).

Experimental searches for the neutral MSSM Higgsbosons have been performed at LEP [16,17], placing impor-tant restrictions on the parameter space. At Run II of theTevatron the search was continued but is now supersededby the LHC Higgs searches. Besides the discovery of a SMHiggs-like boson the LHC searches place stringent bounds,in particular in the regions of small MA and large tan β [18].At a future linear collider (ILC) a precise determination ofthe Higgs boson properties (either of the light Higgs bosonat ∼125.6 GeV or heavier MSSM Higgs bosons within thekinematic reach) will be possible [19]. In particular a massmeasurement of the light Higgs boson with an accuracybelow ∼0.05 GeV is anticipated [20]. The interplay of theLHC and the ILC in the neutral MSSM Higgs sector has beendiscussed in Refs. [21–23].

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For the MSSM1 the status of higher-order corrections tothe masses and mixing angles in the neutral Higgs sectoris quite advanced. The complete one-loop result within theMSSM is known [28–35]. The by far dominant one-loopcontribution is the O(αt ) term due to top and stop loops(αt ≡ h2

t /(4π), ht being the top-quark Yukawa coupling).The computation of the two-loop corrections has meanwhilereached a stage where all the presumably dominant contribu-tions are available [36–53]. In particular, the O(αtαs) con-tributions to the self-energies—evaluated in the Feynman-diagrammatic (FD) as well as in the effective potential(EP) method—as well as the O(α2

t ), O(αbαs), O(αtαb) andO(α2

b) contributions—evaluated in the EP approach—areknown for vanishing external momenta. An evaluation ofthe momentum dependence at the two-loop level in a pureDR calculation was presented in Ref. [54]. A (nearly) fulltwo-loop EP calculation, including even the leading three-loop corrections, has also been published [55–62]. How-ever, within the EP method all contributions are evaluatedat zero external momentum, in contrast to the FD method,which in principle allows non-vanishing external momen-tum. Further, the calculation presented in Refs. [55–62] isnot publicly available as a computer code for Higgs-masscalculations. Subsequently, another leading three-loop cal-culation of O(αtα

2s ), depending on the various SUSY mass

hierarchies, has been performed [63–65], resulting in thecode H3m (which adds the three-loop corrections to theFeynHiggs result). Most recently, a combination of thefull one-loop result, supplemented with leading and sub-leading two-loop corrections evaluated in the Feynman-diagrammatic/effective potential method and a resummationof the leading and subleading logarithmic corrections fromthe scalar-top sector has been published [14] in the latest ver-sion of the code FeynHiggs [13,14,24,38,66,67]. Whileprevious to this combination the remaining theoretical uncer-tainty on the lightest CP-even Higgs boson mass had beenestimated to be about 3 GeV [12,13], the combined resultwas roughly estimated to yield an uncertainty of about 2 GeV[14,15]; however, more detailed analyses will be necessaryto yield a more solid result.

In the present paper we calculate the two-loop O(αtαs)

corrections to the Higgs boson masses in a mixed on-shell/DR scheme. Compared to previously known results[36–38,44] we evaluate here corrections that are proportionalto the external momentum of the relevant Higgs boson self-energies. These corrections can directly be added to the cor-rections included in FeynHiggs. An overview of the rele-vant sectors and the calculation is given in Sect. 2, whereas inSect. 3 we discuss the size and relevance of the new two-loopcorrections. Our conclusions are given in Sect. 4.

1 We concentrate here on the case with real parameters. For the case ofcomplex parameters see Refs. [24–27] and references therein.

2 Calculation

2.1 The Higgs-boson sector of the MSSM

The MSSM requires two scalar doublets, which are conven-tionally written in terms of their components as follows:

H1 =( H0

1H−

1

)=(

v1 + 1√2(φ0

1 − iχ01 )

−φ−1

),

H2 =(H+

2H0

2

)=(

φ+2

v2 + 1√2(φ0

2 + iχ02 )

). (1)

The Higgs boson sector can be described with the help of twoindependent parameters (besides the SM gauge couplings),conventionally chosen as tan β = v2/v1, the ratio of the twovacuum expectation values, and M2

A, the mass of the CP-oddHiggs boson A. The bilinear part of the Higgs potential leadsto the tree-level mass matrix for the neutral CP-even Higgsboson,

M2,treeHiggs =

⎛⎝m2

φ1m2

φ1φ2

m2φ1φ2

m2φ2

⎞⎠

=(

M2A sin2β + M2

Z cos2β − (M2A + M2

Z ) sin β cos β

−(M2A + M2

Z ) sin β cos β M2A cos2β + M2

Z sin2β

),

(2)

in the (φ1, φ2) basis and being expressed in terms of theparameters MZ , MA and the angle β. Diagonalization yieldsthe tree-level masses mh,tree, m H,tree.

The higher-order corrected CP-even Higgs boson massesin the MSSM are obtained from the corresponding propa-gators dressed by their self-energies. The inverse propagatormatrix in the (φ1, φ2) basis is given by

(�Higgs)−1

= −i

(p2 − m2

φ1+ �̂φ1(p2) −m2

φ1φ2+ �̂φ1φ2(p2)

−m2φ1φ2

+ �̂φ1φ2(p2) p2 − m2φ2

+ �̂φ2(p2)

),

(3)

where the �̂(p2) denote the renormalized Higgs-boson self-energies, p being the external momentum. The renormalizedself-energies can be expressed through the unrenormalizedself-energies, �(p2), and counterterms involving renormal-ization constants δm2 and δZ from parameter and field renor-malization. With the self-energies expanded up to two-looporder, �̂ = �̂(1) + �̂(2), one has for the CP-even part at thei-loop level (i = 1, 2),

�̂(i)φ1

(p2) = �(i)φ1

(p2) + δZ (i)φ1

(p2 − m2φ1

) − δm2(i)φ1

, (4a)

�̂(i)φ2

(p2) = �(i)φ2

(p2) + δZ (i)φ2

(p2 − m2φ2

) − δm2(i)φ2

, (4b)

�̂(i)φ1φ2

(p2) = �(i)φ1φ2

(p2) − δZ (i)φ1φ2

m2φ1φ2

− δm2(i)φ1φ2

. (4c)

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The counterterms are determined by appropriate renormal-ization conditions and are given in the appendix.

The renormalized self-energies in the (φ1, φ2) basis canbe rotated into the physical (h, H) basis where the tree-levelpropagator matrix is diagonal, via(

�̂H H �̂h H

�̂h H �̂hh

)= D(α)

(�̂φ1 �̂φ1φ2

�̂φ1φ2 �̂φ2

)DT (α) (5)

with the matrix

D(α) =(

cos α sin α

− sin α cos α

), (6)

which diagonalizes the tree-level mass matrix (2). The CP-even Higgs boson masses are determined by the poles of the(h, H)-propagator matrix. This is equivalent to solving theequation[

p2 − m2h,tree + �̂hh(p2)

] [p2 − m2

H,tree + �̂H H (p2)]

−[�̂h H (p2)

]2 = 0, (7)

yielding the loop-corrected pole masses, Mh and MH .Here we use the implementation in the code FeynHiggs[13,14,24,38,66,67], supplemented by the new momentum-dependent O(αtαs) corrections, as described in Sect. 2.4.

Our calculation is performed in the Feynman-diagra-mmatic (FD) approach. To arrive at expressions for theunrenormalized self-energies and tadpoles at O(αtαs), theevaluation of genuine two-loop diagrams and one-loopgraphs with counterterm insertions is required. Example dia-grams for the neutral Higgs-boson self-energies are shownin Fig. 1, and for the tadpoles in Fig. 2. For the countert-erm insertions, described in Sect. 2.2, one-loop diagramswith external top quarks/squarks have to be evaluated aswell, as displayed in Fig. 3. The complete set of contribut-ing Feynman diagrams has been generated with the programFeynArts [68–71] (using the model file including countert-erms from Ref. [72]), tensor reduction and the evaluation oftraces was done with support from the programs FormCalc[73] and TwoCalc [74,75], yielding algebraic expressionsin terms of the scalar one-loop functions A0, B0 [76], themassive vacuum two-loop functions [77], and two-loop inte-grals which depend on the external momentum. These inte-grals have been evaluated with the programSecDec [78,79];see Sect. 2.3.

2.2 The scalar-top sector of the MSSM

The bilinear part of the top-squark Lagrangian,

Lt̃,mass = −(

t̃†L , t̃†

R

)Mt̃

(t̃L

t̃R

), (8)

contains the stop mass matrix Mt̃ , given by

Mt̃ =⎛⎝M2

t̃L+ m2

t + M2Z cos 2β (T 3

t − Qt s2w) mt Xt

mt Xt M2t̃R

+ m2t + M2

Z cos 2β Qt s2w

⎞⎠ ,

(9)

with

Xt = At − μ cot β. (10)

Qt and T 3t denote the charge and isospin of the top quark,

At is the trilinear coupling between the Higgs bosons andthe scalar tops, and μ is the Higgsino mass parameter. Belowwe use MSUSY := Mt̃L

= Mt̃Rfor our numerical evalua-

tion. However, the analytical calculation has been performedfor arbitrary Mt̃L

and Mt̃R. Mt̃ can be diagonalized with the

help of a unitary transformation matrix Ut̃ , parametrized bya mixing angle θt̃ , to provide the eigenvalues m2

t̃1and m2

t̃2as

the squares of the two on-shell top-squark masses.For the evaluation of the O(αtαs) two-loop contributions

to the self-energies and tadpoles of the Higgs sector, renor-malization of the top/stop sector at O(αs) is required, givingrise to the counterterms for one-loop subrenormalization (seeFigs. 1, 2). We follow the renormalization at the one-looplevel given in Refs. [40,80–82], where details can be found.In the context of this paper, we only want to emphasize thaton-shell (OS) renormalization is performed for the top-quarkmass as well as for the scalar-top masses. This is differ-ent from the approach pursued, for example, in Ref. [54],where a DR renormalization has been employed. Using theOS scheme allows us to consistently combine our new correc-tion terms with the hitherto available self-energies includedin FeynHiggs.

Finally, at O(αtαs), gluinos appear as virtual particlesonly at the two-loop level (hence, no renormalization for thegluinos is needed). The corresponding soft-breaking gluinomass parameter M3 determines the gluino mass, Mg̃ = M3.

2.3 The program SecDec

The calculation of the momentum-dependent two-loop cor-rections to the Higgs-boson masses at order O(αtαs) involvestwo-loop two-point functions with up to four differentmasses, in addition to the mass scale given by the exter-nal momentum p2. For two-loop diagrams of propagatortype, analytical results in four space-time dimensions areknown only sparsely if different masses are occurring in theloops [77,83–90]. The integrals which are lacking analyt-ical results can be classified into four different topologies,shown in Fig. 4. We have calculated these integrals numeri-cally using the program SecDec [78,79], where up to fourdifferent masses in 34 different mass configurations neededto be considered, with differences in the kinematic invariantsof several orders of magnitude.

The program SecDec is a publicly available tool [91]to calculate multi-loop integrals numerically. Dimensionally

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2994 Page 4 of 15 Eur. Phys. J. C (2014) 74:2994

Fig. 1 Generic two-loopdiagrams and diagrams withcounterterm insertions for theHiggs-boson self-energies(φ = h, H, A)

t̃i

t̃j

φφ

(a)

t

t φφ

(b)

t

t̃i φφg̃

t̃j

t

(c)

t̃j

φφg

t̃i

(d)

t

φφg

(e)

t̃k

φφg̃

tt̃i t̃j

(f)

t̃i

t

φφg̃

(g)

φφ

t̃i

g

(h)

t̃i φφ

t

g̃

t̃j

(i)

φ φ

t̃k

t̃i t̃j

(j)

φφ

t̃j t̃l

t̃i t̃k

(k)

φ φ

t̃l

t̃k

t̃i t̃j

(l)

t

t

φφt

(m)

t

φφ

t

(n)

t̃i

t̃j

φφt̃i

(o)

t̃j

φφ

t̃i

(p)

t̃i

φφ

(q)

t̃i

φφ

t̃i

(r)

regulated poles are factorized by sector decomposition asdescribed in Refs. [92,93], while kinematic thresholds arehandled by a deformation of the integration contour into thecomplex plane, as described e.g. in Ref. [79]. The numericalintegration is done using the Cuba library [94].

The program has also been extended to be able to calculatetensor integrals of any rank [95], and to process efficiently

the evaluation of large ranges of kinematic points using the“multinumerics” feature of the program, which is of par-ticular importance for the calculation presented here. Thisfeature allows one to produce input files for large sets ofkinematic points automatically, and to process the evalua-tion of these points in parallel if several cores or a cluster areavailable, without repeating the algebraic part of the sector

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Fig. 2 Generic two-loopdiagrams and diagrams withcounterterm insertions for theHiggs-boson tadpoles(φ = h, H ; i, j, k = 1, 2)

t̃k

t̃j

φt̃i

(a)

t̃j

tφ

g̃

t̃i

(b)

φt̃i

t̃k

t̃j

(c)

t

t

φt

(d)

t

t̃iφ

g̃

t

(e)

t̃iφ

(f)

t̃j

φ

t̃i

(g)

tφ

(h)

t

φ

t

(i)

Fig. 3 Generic one-loopdiagrams for subrenormalizationcounterterms, involving topquarks t , top squarks t̃ , gluons gand gluinos g̃ (i, j, k = 1, 2)

t̃i t̃i

g

t̃i

(a)

t t

g

t

(b)

t̃i t̃j

g̃

t

(c)

t t

t̃i

g̃

(d)

t̃i t̃j

t̃k

(e)

T234 T1234 T11234 T12345

Fig. 4 Topologies which have been calculated numerically using SecDec

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decomposition, which can be done once and for all. The eval-uation of a single phase space point for the most complicatedtopology, to reach a relative accuracy of at least 10−5, rangesbetween 0.01 and 100 s on an Intel core i7 processor, wherethe larger timings are for points very close to a kinematicthreshold.

2.4 Evaluation and implementation in the programFeynHiggs

The resulting new contributions to the neutral CP-evenHiggs-boson self-energies, containing all momentum-depe-ndent and additional constant terms, are assigned to the dif-ferences

��̂(2)ab (p2)=�̂

(2)ab (p2)−�̃

(2)ab (0), ab={H H, h H, hh}.

(11)

Note the tilde (not hat) on �̃(2)(0) which signifies thatnot only the self-energies are evaluated at zero externalmomentum but also the corresponding counterterms, follow-ing Refs. [36–38]. A finite shift ��̂(2)(0) therefore remainsin the limit p2 → 0 due to δM2(2)

A = Re �(2)AA(M2

A) beingcomputed at p2 = M2

A in �̂(2), but at p2 = 0 in �̃(2); fordetails see Eqs. (22) and (24) in the appendix.

The numerical evaluation to derive the physical massesfor h, H as the poles (real parts) of the dressed propagatorsproceeds on the basis of Eq. (7) in an iterative way.

• In a first step, the squared masses M2h,0, M2

H,0 are deter-mined by solving Eq. (7) excluding the new terms��̂

(2)ab (p2) from the self-energies.

• In a second step, the shifts ��̂(2)ab (M2

h,0) ≡ chab and

��̂(2)ab (M2

H,0) ≡ cHab are calculated and added as con-

stants to the self-energies in Eq. (7), �̂ab(p2) →�̂ab(p2) + ch(H)

ab .• In the third step, Eq. (7) is solved again, now including

the constant shifts ch(H)ab in the self-energies, to deliver

the refined masses Mh (with chab) and MH (with cH

ab).

This procedure can be repeated for improving the accuracy;numerically it turns out that going beyond the first iterationyields only marginal changes.

The corrections of Eq. (11) are incorporated inFeynHiggs by the following recipe, which is more generaland in principle applicable also to the case of the complexMSSM with CP violation.

1. Determine Higgs masses Mhi ,0 without the momentum-dependent terms of Eq. (11); the index i = 1, . . . , 4 enu-merates the masses of h, H, A, H± in the real MSSM.This is done by invoking the FeynHiggs mass-finder.

2. Compute the shifts chkab = ��̂

(2)ab (M2

hk ,0) with a, b, hk =

h, H .3. Run FeynHiggs’ mass-finder again including the chk

abas constant shifts in the self-energies to determine therefined Higgs masses Mh and MH .

This procedure could conceivably be iterated until full self-consistency is reached; yet the resulting mass improvementsturn out to be too small to justify extra CPU time.

On the technical side we added an interface for anexternal program to FeynHiggs which exports relevantmodel parameters to the external program’s environment,currently:

FHscalefactor ren. scale multiplicator, FHTB tan β,FHAlfasMT αs(mt ), FHGF G F ,FHMHiggs2i M2

hi ,0, i = 1 . . . 4, FHMSti mt̃,i , i = 1, 2,

FH{Re,Im}USt1i Ut̃,1i , i = 1, 2, FHMGl mg̃ ,FH{Re,Im}MUE μ, FHMA0 MA ,

where the Ut̃,1i denote the elements of the stop mixing matrix,αs(mt ) the running strong coupling at the scale mt , andG F the Fermi constant. The renormalization scale is definedwithin FeynHiggs as μR = mt ·FHscalefactor. Invo-cation of the external program is switched on by providingits path in the environment variable FHEXTSE. The programis executed from inside a temporary directory which is after-wards removed.

The output (stdout) is scanned for lines of the form‘se@m cr ci ’ which specify the correction cr + ici [withcr = Re(chk

ab), ci = Im(chkab)] to self-energy se in the com-

putation of mass m, where m is one of Mh0, MHH, MA0, MHp,and se is one of h0h0, HHHH, A0A0, HmHp, h0HH, h0A0,HHA0, G0G0, h0G0, HHG0, A0G0, GmGp, HmGp, F1F1,F2F2, F1F2. The latter three, if given, substitute

HHHH = cos2 α F1F1+ sin2 α F2F2+ sin 2α F1F2 ,

(12a)

h0h0 = sin2 α F1F1+ cos2 α F2F2− sin 2α F1F2,

(12b)

h0HH = cos 2α F1F2+ 12 sin 2α (F2F2− F1F1), (12c)

where α is the tree-level 2 × 2 neutral-Higgs mixing anglein Eq. (6). Self-energies not given are assumed zero.

The zero-momentum contributions �̃(2)ab (0), ab = {H H,

h H, hh}, are subtracted if the output of the external pro-gram contains one or more of ‘sub asat’, ‘sub atat’,‘sub asab’, ‘sub atab’ for the αsαt , α2

t , αsαb, and αtαb

contributions, respectively. All other lines in the output areignored.

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-5500

-5000

-4500

-4000

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-3000

-2500

-2000

-1500

-1000

-500

0

500

0 100 200 300 400 500 600 700 800 900 1000

Re[

Δ^ Σ hh(

p2 )] (

GeV

2 )

p (GeV)

TB= 5TB=20

0

5

10

15

25 50 75 100 125

-4500

-4000

-3500

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-1000

-500

0

500

1000

0 100 200 300 400 500 600 700 800 900 1000

Im[Δ

^ Σ hh(

p2 )] (

GeV

2 )

p (GeV)

TB= 5TB=20

-200

0

200

400

600

800

1000

1200

1400

0 100 200 300 400 500 600 700 800 900 1000

Re[

Δ^ Σ hH

(p2 )]

(G

eV2 )

p (GeV)

TB= 5TB=20

-2

0

2

25 50 75 100 125

-400

-200

0

200

400

600

800

1000

1200

0 100 200 300 400 500 600 700 800 900 1000

Im[Δ

^ Σ hH

(p2 )]

(G

eV2 )

p (GeV)

TB= 5TB=20

-400

-350

-300

-250

-200

-150

-100

-50

0

50

0 100 200 300 400 500 600 700 800 900 1000

Re[

Δ^ Σ HH

(p2 )]

(G

eV2 )

p (GeV)

TB= 5TB=20

0 1 2 3 4 5

25 50 75 100 125

-350

-300

-250

-200

-150

-100

-50

0

50

0 100 200 300 400 500 600 700 800 900 1000

Im[ Δ

^ Σ HH

(p2 )]

(G

eV2 )

p (GeV)

TB= 5TB=20

Fig. 5 Momentum dependence of the real (left column) andimaginary (right column) parts of the two-loop self-energies��̂hh ,��̂h H ,��̂H H , within scenario 1, for tan β = 5 (red squares)

and tan β = 20 (blue crosses) and MA = 250 GeV. One can see that theself-energies change substantially beyond the threshold at p2 = (2mt )

2

3 Numerical results

We show results for the subtracted two-loop self-energies��̂

(2)ab (p2) given in Eq. (11), as well as for the mass shifts

�Mh = Mh − Mh,0, �MH = MH − MH,0 (13)

i.e. the difference in the physical Higgs-boson masses evalu-ated including and excluding the newly obtained momentum-dependent two-loop corrections. This quantity, in particular�Mh for the light CP-even Higgs boson, can directly becompared with the current experimental uncertainty as well

as with the anticipated future ILC accuracy of [20],

δMexp,ILCh � 0.05 GeV. (14)

The results are obtained for two different scenarios, vary-ing parameters like tan β, MA, Mg̃ , and illustrate the impactof these parameters via the new two-loop corrections on theneutral CP-even Higgs boson masses, Mh and MH . The cor-responding renormalization scale, μDR, is set to μDR = mt

in all numerical evaluations. The scale uncertainties areexpected to be much smaller than the parametric uncertain-ties due to variations of parameters like tan β, MA, Mg̃, mt̃ .

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0

500

0 100 200 300 400 500 600 700 800 900 1000

Re[

Δ^ Σ hh(

p2 )] (

GeV

2 )

p (GeV)

MA=100.9,TB= 5, ReMA=900.1,TB= 5, ReMA=100.9,TB=20, ReMA=900.1,TB=20, Re

0 5

10 15 20 25

80 100 120 140 160

-500

0

500

1000

1500

2000

2500

3000

0 100 200 300 400 500 600 700 800 900 1000

Re[

Δ^ Σ hH

(p2 )]

(G

eV2 )

p (GeV)

MA=100.9,TB= 5, ReMA=900.1,TB= 5, ReMA=100.9,TB=20, ReMA=900.1,TB=20, Re

-10

-5

0

80 100 120 140 160

-2250

-2000

-1750

-1500

-1250

-1000

-750

-500

-250

0

250

500

0 100 200 300 400 500 600 700 800 900 1000

Re[

Δ^ Σ HH

(p2 )]

(G

eV2 )

p (GeV)

MA=100.9,TB= 5, ReMA=250.0,TB= 5, ReMA=600.0,TB= 5, ReMA=900.1,TB= 5, ReMA=100.9,TB=20, ReMA=250.0,TB=20, ReMA=900.1,TB=20, Re

Fig. 6 Momentum dependence of the real part of the two-loop self-energies ��̂hh,��̂h H ,��̂H H , within scenario 1, for two differentvalues of tan β and a range of MA values

3.1 Scenario 1: mmaxh

Scenario 1 is oriented at the mmaxh scenario described in

Ref. [96]. We use the following parameters:

mt = 173.2 GeV, MSUSY = 1 TeV, Xt = 2 MSUSY,

Mg̃ = 1500 GeV, μ = 200 GeV, (15)

-60

-50

-40

-30

-20

-10

0

0 100 200 300 400 500 600 700 800 900 1000

Δ M

h (M

eV)

MA (GeV)

TB=20TB= 5

-70

-60

-50

-40

-30

-20

-10

0

10

20

30

40

0 200 400 600 800 1000

Δ M

H (

MeV

)

MA (GeV)

TB=20TB= 5

Fig. 7 Variation of the mass shifts �Mh ,�MH with the A-boson massMA within scenario 1, for tan β = 5 (red) and tan β = 20 (blue). Thepeak in �MH originates from a threshold at 2 mt

leading to stop mass values of

mt̃1 = 826.8 GeV, mt̃2 = 1173.2 GeV.

With the introduction of the momentum dependence, thresh-olds occur in the self-energy diagrams when the externalmomentum p = √

p2, in the time-like region, is such that acut of the diagram would correspond to on-shell productionof the massive particles of the cut propagators. The resultingimaginary parts enter in the search for the complex poles ofthe inverse propagator matrix of the Higgs bosons. There-fore it is interesting to study the behavior of the real andimaginary parts of the self-energies. In Fig. 5 we show themomentum-dependent parts of the renormalized two-loopself-energies in the physical basis, Eq. (11) for two differ-ent values of tan β, tan β = 5 and tan β = 20, at a fixedA-boson mass MA = 250 GeV. The data points are notconnected by a line in order to show that each numericalpoint is obtained from a calculation of the 34 analyticallyunknown integrals with the program SecDec. The inlays inFig. 5 magnify the region p2 ≤ (125 GeV)2, where one canobserve that for p2 → 0, the subtracted self-energies are notexactly zero. As mentioned in Sect. 2.4, this is due to the fact

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-450

-400

-350

-300

-250

-200

-150

-100

-50

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Δ M

h (M

eV)

M~g (GeV)

TB= 5TB=20

-40

-35

-30

-25

-20

-15

-10

-5

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Δ M

H (

MeV

)

M~g (GeV)

TB= 5TB=20

Fig. 8 Variation of the mass shifts �Mh,�MH with the gluino mass,within scenario 1, for two different values of tan β = 5, 20 and MA =250 GeV

that the on-shell renormalization condition for the A-bosonself-energy is defined differently with regard to the calcula-tion without momentum dependence. The resulting constantcontributions are additionally suppressed by factors sin2β,sin β cos β and cos2β appearing in the counter terms δm2(2)

φ1,

δm2(2)φ1φ2

and δm2(2)φ2

, respectively, according to Eqs. (24) inthe appendix.

The imaginary part is independent of the A-boson mass, asthis mass parameter solely appears in the counterterms of DRrenormalized quantities and the δM2(2)

A counterterm, whereonly the real part contributes. Therefore, the imaginary partsdisplayed in Fig. 5 do not contain additional constant terms.As to be expected, the imaginary parts are zero below thet t̄ production threshold at p = 2 mt , which results from thefact that the top mass is the smallest mass appearing in theloops. Beyond this threshold, the imaginary parts are growingsubstantially with increasing p2. From these observations,the mass shifts in the region below the first threshold at p =2 mt are expected not to be large.

Similar results, now including a variation of MA are shownin Fig. 6. In the upper plot for ��̂hh and in the middle plotfor ��̂h H the solid lines depict MA ∼ 100 GeV, while the

118

120

122

124

126

128

130

132

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Mh,

0 an

d M

h (G

eV)

M~g (GeV)

TB= 5, Mh,0TB= 5, MhTB=20, Mh,0TB=20, Mh

Fig. 9 Variation of Mh and Mh,0 as a function of Mg̃ within scenario1, for tan β = 5, 20 and MA = 250 GeV

dashed lines are for MA ∼ 900 GeV. In these plots the lightshading covers the range for tan β = 5, while the dark shad-ing for tan β = 20. In the lower plot for ��̂H H we showresults for MA ∼ 100, 250, 600, 900 GeV as solid, dotted,dot-dashed, dashed lines, respectively (and shading has beenomitted). For ��̂hh at low p values only a small variationwith MA can be observed. For p and MA large, the contri-butions to the self-energy are bigger. In ��̂h H larger effectsare observed at smaller MA for both small and large p val-ues. For ��̂H H , on the other hand, at low p values, largeeffects can be observed for large MA due to the aforemen-tioned counterterm contribution ∼δM2(2)

A = Re �(2)AA(M2

A).At large p, as before, small MA values give a more sizablecontribution.

We now turn to the effects of our newly computedmomentum-dependent two-loop corrections on the Higgs-boson masses Mh,H via the mass shifts �Mh and �MH . InFig. 7 we show �Mh (upper plot) and �MH (lower plot)as a function of MA for tan β = 5 (blue) and tan β = 20(red). In the mmax

h scenario for MA � 200 GeV we find�Mh ∼ −60 MeV, i.e. of the size of the future experimen-tal precision; see Eq. (14). The contribution to the heavyCP-even Higgs-boson is suppressed with tan β. While thesize of �MH becomes negligible for MA � 150 GeV fortan β = 20, its variation is more pronounced for tan β = 5.�MH can reach about −60 MeV for very small or intermedi-ate values of MA and steadily decreases for MA � 500 GeV.The peak in �MH for tan β = 5 originates from a thresholdat 2 mt .

Finally, in scenario 1, we analyze the dependence of Mh

and MH on the gluino mass, Mg̃ . The results are shownin Fig. 8 for �Mh (upper plot) and �MH (lower plot) forMA = 250 GeV, with the same color coding as in Fig. 7.In the upper plot one can observe that the effects are par-ticularly small for the default value of Mg̃ in scenario 1.More sizeable shifts occur for larger gluino masses, by more

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-9000

-7500

-6000

-4500

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-1500

0

1500

3000

0 100 200 300 400 500 600 700 800 900 1000

Re[

Δ^ Σ hh(

p2 )] (

GeV

2 )

p (GeV)

TB= 5TB=20

0

5

10

15

25 50 75 100 125

-1500

-1000

-500

0

500

1000

1500

2000

2500

3000

3500

0 100 200 300 400 500 600 700 800 900 1000

Im[Δ

^ Σ hh(

p2 )] (

GeV

2 )

p (GeV)

TB= 5TB=20

-2000

-1500

-1000

-500

0

500

1000

1500

2000

2500

0 100 200 300 400 500 600 700 800 900 1000

Re[

Δ^ Σ hH

(p2 )]

(G

eV2 )

p (GeV)

TB= 5TB=20

-5-4-3-2-1 0 1

25 50 75 100 125

-1600

-1400

-1200

-1000

-800

-600

-400

-200

0

200

0 100 200 300 400 500 600 700 800 900 1000

Im[Δ

^ Σ hH

(p2 )]

(G

eV2 )

p (GeV)

TB= 5TB=20

-1000

-800

-600

-400

-200

0

200

400

600

800

1000

1200

0 100 200 300 400 500 600 700 800 900 1000

Re[

Δ^ Σ HH

(p2 )]

(G

eV2 )

p (GeV)

TB= 5TB=20

0 2 4 6 8

25 50 75 100 125

-100

0

100

200

300

400

500

600

700

800

900

0 100 200 300 400 500 600 700 800 900 1000

Im[ Δ

^ Σ HH

(p2 )]

(G

eV2 )

p (GeV)

TB= 5TB=20

Fig. 10 Momentum dependence of the real and imaginary parts of the two-loop self-energies ��̂hh,��̂h H ,��̂H H within scenario 2, withtan β = 5, 20 and MA = 250 GeV with the same color coding as in Fig. 5

than −400 MeV for Mg̃ � 4 TeV, reaching thus the level ofthe current experimental accuracy in the Higgs-boson massdetermination. The corrections to MH , for the given value ofMA = 250 GeV do not exceed −50 MeV in the consideredMg̃ range.

The dependence of the light CP-even Higgs-boson masson Mg̃ is analyzed in Fig. 9 for tan β = 5, 20 and MA =250 GeV. Here we show as dashed lines the results for Mh,0

(i.e. without the newly obtained momentum-dependent two-loop corrections) and as solid lines the results for Mh (i.e.including the new corrections). While a maximum of theHiggs-boson mass can be observed around Mg̃ ∼ 800 GeV,

in agreement with the original definition of the mmaxh sce-

nario [97], a downward shift by more than 4 GeV is foundfor Mg̃ ∼ 5 TeV. Such a strong effect is due to a (squared)logarithmic dependence of theO(αtαs) corrections evaluatedat p2 = 0, as given in Eq. (73) of Ref. [38]. In Fig. 9 it canbe seen that the size of the momentum-dependent two-loopcorrections similarly grows with Mg̃ , reaching ∼400 MeV,as was shown above in Fig. 8. Consequently, the logarithmicdependence of the light CP-even Higgs-boson mass on thegluino mass that was found analytically for the O(αtαs) cor-rections at p2 = 0, is now also found numerically for themomentum-dependent two-loop corrections.

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-8000

-7000

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-5000

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-3000

-2000

-1000

0

1000

2000

3000

0 100 200 300 400 500 600 700 800 900 1000

Re[

Δ^ Σ hh(

p2 )] (

GeV

2 )

p (GeV)

MA=100.9,TB= 5MA=900.1,TB= 5MA=100.9,TB=20MA=900.1,TB=20

0 5

10 15 20 25

80 100 120 140 160

-2000

-1000

0

1000

2000

3000

4000

0 100 200 300 400 500 600 700 800 900 1000

Re[

Δ^ Σ hH

(p2 )]

(G

eV2 )

p (GeV)

MA=100.9,TB= 5MA=900.1,TB= 5MA=100.9,TB=20MA=900.1,TB=20

-12-10

-8-6-4-2 0

80 100 120 140 160

-4000

-3000

-2000

-1000

0

1000

2000

3000

0 100 200 300 400 500 600 700 800 900 1000

Re[

Δ^ Σ HH

(p2 )]

(G

eV2 )

p (GeV)

MA=100.9,TB= 5MA=250.8,TB= 5MA=600.4,TB= 5MA=900.1,TB= 5MA=100.9,TB=20MA=250.8,TB=20MA=900.1,TB=20

Fig. 11 Momentum dependence of the real parts of the two-loop self-energies ��̂hh,��̂h H ,��̂H H in scenario 2 for two different valuesof tan β and various values of MA (see text)

3.2 Scenario 2: light stops

Scenario 2 is oriented at the “light-stop scenario” ofRef. [96].2 We use the following parameters:

2 While the original scenario in Ref. [96] is challenged by recent scalar-top searches at ATLAS and CMS, a small modification in the gaugino-mass parameters (which play no or only a very minor role here) to

-60

-50

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-30

-20

-10

0

0 100 200 300 400 500 600 700 800 900 1000

Δ M

h (M

eV)

MA (GeV)

TB= 5TB=20

-1000

-800

-600

-400

-200

0

200

400

600

800

0 100 200 300 400 500 600 700 800 900 1000

Δ M

H (

MeV

)

MA (GeV)

TB= 5TB=20

-80-60-40-20

0

0 100 200 300

Fig. 12 Variation of the mass shifts �Mh,�MH with the A-bosonmass MA within scenario 2, for two different values of tan β = 5, 20

mt = 173.2 GeV, MSUSY = 0.5 TeV, Xt = 2 MSUSY,

Mg̃ = 1600 GeV, μ = 200 GeV, (16)

leading to stop mass values of

mt̃1 = 326.8 GeV, mt̃2 = 673.2 GeV.

Scenario 2 is analyzed with the same set of plots shown forscenario 1 in the previous subsection. The effects of the newmomentum-dependent two-loop contributions on the renor-malized Higgs-boson self-energies, ��̂ab(p2), are shownin Fig. 10. As before, we show the results separately forthe real and imaginary parts of the self-energies. An addi-tional threshold beyond the top-mass threshold appears atp = 2 mt̃1 , where the discontinuity stems from the deriva-tive of the imaginary part of the B0 function(s). Analogouslyto scenario 1, the largest contributions in the region below200 GeV arise in the real part of ��̂hh amounting to about15 GeV2 at p = 125 GeV, where the dependence on thevalue of tan β is rather weak.

Footnote 2 continuedM1 = 340 GeV, M2 = μ = 400 GeV leads to a SUSY spectrum thatis very difficult to test at the LHC.

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-700

-600

-500

-400

-300

-200

-100

0

100

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Δ M

h (M

eV)

M~g (GeV)

TB= 5TB=20

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10

0

500 1000 1500 2000 2500 3000 3500 4000 4500 5000

Δ M

H (

MeV

)

M~g (GeV)

TB= 5TB=20

Fig. 13 Variation of the mass shifts �Mh,�MH with the gluino mass,within scenario 2, for two different values of tan β = 5, 20 and MA =250 GeV

The dependence of ��̂ab(p2) on MA is shown in Fig. 11,using the same line styles as in Fig. 6. The curves show thesame qualitative behavior as in Fig. 10, exhibiting again thenew threshold at p = 2 mt̃1 . In general, outside the thresholdregion the effects in scenario 2 are slightly smaller than inscenario 1.

We now turn to the effects on the physical neutral CP-evenHiggs boson masses. In Fig. 12 we show the results for �Mh

(upper plot) and �MH (lower plot) as a function of MA (withthe same line styles as in Fig. 7). As can be expected fromthe previous figures, the effects on Mh and MH are in generalslightly smaller in scenario 2 than in scenario 1, where �Mh

still reaches the anticipated ILC accuracy; see Eq. (14). For�MH around the threshold p = 2 mt̃1 the largest shift of∼−1 GeV can be observed. However, this shift is still belowthe anticipated mass resolution at the LHC [98].

Finally we analyze the dependence on Mg̃ in Fig. 13. Inthe upper plot we show �Mh for tan β = 5 and tan β = 20,where both values yield very similar results. As in sce-nario 1, “accidentally” small values of �Mh are found aroundMg̃ ∼ 1600 GeV. For larger gluino mass values the shiftsinduced by the new momentum-dependent two-loop correc-tions exceed −500 MeV and are thus larger than the current

experimental uncertainty. The results for �MH are shown inthe lower plot. While they are roughly twice as large as inscenario 1, they do not exceed −100 MeV.

4 Conclusions

We have presented results for the leading momentum-dependent O(αtαs) contributions to the masses of neutralCP-even Higgs bosons in the MSSM. They are obtained bycalculating the corresponding contributions to the dressedHiggs-boson propagators obtained in the Feynman-diagra-mmatic approach using a mixed on-shell/DR renormaliza-tion scheme. In the Higgs sector a two-loop renormaliza-tion has to be carried out for the mass of the neutral Higgsbosons and the tadpole contributions. Furthermore, renor-malization of the top/stop sector at O(αs) is needed enteringat the two-loop level via one-loop subrenormalization. Thediagrams were generated with FeynArts and reduced toa set of basic integrals with the help of FormCalc andTwoCalc. The two-loop integrals which are analyticallyunknown have been calculated numerically with the programSecDec.

We have analyzed numerically the effect of the newmomentum-dependent two-loop corrections on the predic-tions for the CP-even Higgs boson masses. This is partic-ularly important for the interpretation of the scalar bosondiscovered at the LHC as the light CP-even Higgs state ofthe MSSM. While currently a precision below the level of∼500 MeV is reached, a reduction by about an order of mag-nitude can be expected at the future e+e− International LinearCollider (ILC).

In our numerical analysis we found that the effects on thelight CP-even Higgs boson mass, Mh , depend strongly on thevalue of the gluino mass, Mg̃ . For values of Mg̃ ∼ 1.5 TeVcorrections to Mh of about −50 MeV are found, at the levelof the anticipated future ILC accuracy. For very large gluinomasses, Mg̃ � 4 TeV, on the other hand, substantially largercorrections are found, at the level of the current experimentalaccuracy. Consequently, this type of momentum-dependenttwo-loop corrections should be taken into account in preci-sion analyses interpreting the discovered Higgs boson in theMSSM.

For the heavy CP-even Higgs boson mass, MH , the effectsare mostly below current and future anticipated accuracies.Only close to thresholds, e.g. around p = 2 mt̃1 , larger cor-rections to MH around ∼1 GeV are found.

The new results of O(αtαs) have been implemented intothe program FeynHiggs. A detailed description of our cal-culation will be presented in a forthcoming publication [99].

Acknowledgments We thank S. Paßehr for help with the interfaces toTwoCalc and FormCalc, S. di Vita for numerical comparisons andG. Weiglein for helpful discussions. The work of S.H. was supported by

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the Spanish MICINN’s Consolider-Ingenio 2010 Program under GrantMultiDark No. CSD2009-00064.

Open Access This article is distributed under the terms of the CreativeCommons Attribution License which permits any use, distribution, andreproduction in any medium, provided the original author(s) and thesource are credited.Funded by SCOAP3 / License Version CC BY 4.0.

Appendix: Renormalization and counterterms

Renormalization and calculation of the renormalized self-energies is performed in the (φ1, φ2) basis, which has theadvantage that the mixing angle α does not appear andexpressions are in general simpler.

Field renormalization is performed by assigning onerenormalization constant for each doublet,

H1 → (1 + 12δZH1)H1, H2 → (1 + 1

2δZH2)H2, (17)

which can be expanded to one- and two-loop order accordingto

δZH1 = δZ (1)

H1+ δZ (2)

H1, δZH2 = δZ (1)

H2+ δZ (2)

H2. (18)

The field renormalization constants appearing in (4) are thengiven by

δZ (i)φ1

= δZ (i)H1

, δZ (i)φ2

= δZ (i)H2

,

δZ (i)φ1φ2

= 12 (δZ (i)

H1+ δZ (i)

H2).

(19)

The mass counterterms δm2(i)ab in (4) are derived from

the Higgs potential, including the tadpoles, by the follow-ing parameter renormalization:

M2A → M2

A + δM2(1)A + δM2(2)

A ,

T1 → T1 + δT (1)1 + δT (2)

1 , (20)

M2Z → M2

Z + δM2(1)Z + δM2(2)

Z ,

T2 → T2 + δT (1)2 + δT (2)

2 ,

tan β → tan β(

1 + δ tan β(1) + δ tan β(2))

.

The parameters T1 and T2 are the terms linear in φ1 and φ2

in the Higgs potential. The renormalization of the Z massMZ does not contribute to the O(αsαt ) corrections we arepursuing here; it is listed, however, for completeness.

The basic renormalization constants for parameters andfields have to be fixed by renormalization conditions accord-ing to a renormalization scheme. Here we choose the on-shellscheme for the parameters and the DR scheme for field renor-malization and give the expressions for the two-loop part.

The tadpole coefficients are chosen to vanish at all orders;hence their two-loop counterterms follow from

T (2)1,2 +δT (2)

1,2 =0, i.e. δT (2)1 =−T (2)

1 , δT (2)2 =−T (2)

2 ,

(21)

where T (2)1 , T (2)

2 are obtained from the two-loop tapole dia-grams. The two-loop renormalization constant of the A-boson mass reads

δM2(2)A = Re �

(2)AA(M2

A), (22)

in terms of the A-boson unrenormalized self-energy �AA.The appearance of a non-zero momentum in the self-energygoes beyond the O(αtαs) corrections evaluated in Refs. [36–38,44].

For the renormalization constants δZH1 , δZH2 and δ tan β

several choices are possible; see the discussion in [100–102].As shown there, the most convenient choice is a DR renor-malization of δ tan β, δZH1 and δZH2 , which reads at thetwo-loop level

δZ (2)

H1= δZ (2)DR

H1= −

[Re �

′(2)φ1

]div

|p2=0, (23a)

δZ (2)

H2= δZ (2)DR

H2= −

[Re �

′(2)φ2

]div

|p2=0, (23b)

δ tan β(2) = δ tan β(2)DR = 12

(δZ (2)

H2− δZ (2)

H1

). (23c)

The term in Eq. (23c) is in general not the proper expres-sion beyond one-loop order even in the DR scheme. For ourapproximation, however, with only the top Yukawa couplingat the two-loop level, it is the correct DR form [103,104].

The two-loop mass counterterms in the self-energies (4)are now expressed in terms of the parameter renormalizationconstants determined above as follows:

δm2(2)φ1

= δM2(2)Z cos2β + δM2(2)

A sin2β

−δT (2)1

e

2MW swcos β(1 + sin2β)

+δT (2)2

e

2MW swcos2β sin β

+2 δ tan β(2) cos2β sin2β (M2A − M2

Z ), (24a)

δm2(2)φ1φ2

= −(δM2(2)Z + δM2(2)

A ) sin β cos β

−δT (2)1

e

2MW swsin3β − δT (2)

2e

2MW swcos3β

−δ tan β(2) cos β sin β cos 2β(M2A+M2

Z ), (24b)

δm2(2)φ2

= δM2(2)Z sin2β + δM2(2)

A cos2β

+δT (2)1

e

2MW swsin2β cos β

−δT (2)2

e

2MW swsin β(1 + cos2β)

−2 δ tan β(2) cos2β sin2β (M2A − M2

Z ). (24c)

Note that the Z -mass counterterm is kept for completeness;it does not contribute in the approximation of O(αsαt ) con-sidered here.

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