Reconciling EFT and hybrid calculations of the light MSSM Higgs-boson mass
Henning Bahl, Sven Heinemeyer, Wolfgang Hollik, Georg Weiglein
DDESY 17-072IFT-UAM/CSIC-17-047MPP-2017-108
Reconciling EFT and hybrid calculationsof the light MSSM Higgs-boson mass
Henning Bahl a ∗ , Sven Heinemeyer b,c,d † , Wolfgang Hollik a ‡ , Georg Weiglein e § a Max-Planck Institut f¨ur Physik, F¨ohringer Ring 6, D-80805 M¨unchen, Germany b Campus of International Excellence UAM+CSIC, Cantoblanco, E-28049 Madrid, Spain c Instituto de F´ısica Te´orica, (UAM/CSIC), Universidad Aut´onoma de Madrid, Cantoblanco, E-28049Madrid, Spain d Instituto de F´ısica Cantabria (CSIC-UC), E-39005 Santander, Spain e Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany
Abstract
Various methods are used in the literature for predicting the lightest CP -even Higgs boson mass inthe Minimal Supersymmetric Standard Model (MSSM). Fixed-order diagrammatic calculations captureall effects at a given order and yield accurate results for scales of supersymmetric (SUSY) particles thatare not separated too much from the weak scale. Effective field theory calculations allow a resummationof large logarithmic contributions up to all orders and therefore yield accurate results for a high SUSYscale. A hybrid approach, where both methods have been combined, is implemented in the computer code FeynHiggs . So far, however, at large scales sizeable differences have been observed between
FeynHiggs and other pure EFT codes. In this work, the various approaches are analytically compared with eachother in a simple scenario in which all SUSY mass scales are chosen to be equal to each other. Threemain sources are identified that account for the major part of the observed differences. Firstly, it isshown that the scheme conversion of the input parameters that is commonly used for the comparisonof fixed-order results is not adequate for the comparison of results containing a series of higher-orderlogarithms. Secondly, the treatment of higher-order terms arising from the determination of the Higgspropagator pole is addressed. Thirdly, the effect of different parametrizations in particular of the topYukawa coupling in the non-logarithmic terms is investigated. Taking into account all of these effects, inthe considered simple scenario very good agreement is found for scales above 1 TeV between the resultsobtained using the EFT approach and the hybrid approach of
FeynHiggs . ∗ email: [email protected] † email: [email protected] ‡ email: [email protected] § email: [email protected] a r X i v : . [ h e p - ph ] J a n ontents FeynHiggs to other codes 136 Numerical results 147 Conclusions 19A Fixed-order conversion: additional two-loop terms 20B Logarithms arising from the determination of the propagator pole 21 Introduction
The properties of the Higgs boson that has been discovered by the ATLAS and CMS collaborations at theCERN Large Hadron Collider [1, 2] are compatible with those predicted for the Higgs boson of the StandardModel (SM) at the present level of accuracy. Despite this apparent success of the SM, there are severalopen questions that cannot be answered by the SM and ask for extended or alternative theoretical concepts.Supersymmetry is one of best motivated frameworks for physics beyond the Standard Model (BSM), and inparticular the Minimal Supersymmetric Standard Model (MSSM) is the most intensively studied scenarioproviding precise predictions for experimental phenomena in the LHC era.Apart from associating a superpartner to each SM degree of freedom, the MSSM extends the Higgssector of the SM by a second complex doublet. Consequently, the MSSM employs two Higgs-boson doublets,denoted by H and H , with hypercharges − H and H acquire vacuum expectation values (vevs), v and v . Without loss ofgenerality, one can assume that the vevs are real and non-negative, yielding v ≡ v + v , tan β ≡ v /v . (1)The two Higgs doublets in the MSSM accommodate five physical Higgs bosons. In lowest order these are thelight and heavy CP -even Higgs bosons, h and H , the CP -odd Higgs boson, A , and two charged Higgs bosons, H ± . Two parameters are required to describe the Higgs sector at the tree level (conventionally chosen astan β and the mass M A of the CP -odd Higgs particle); masses and couplings, however, are substantiallyaffected by higher-order contributions.Until now, experiments have not found direct evidence for supersymmetric (SUSY) particles. On the otherhand, precision observables provide an indirect access to the MSSM parameter space from which significantconstraints on the allowed parameter regions can be obtained. On top of the classical set of electroweakprecision observables, the mass of the detected Higgs boson constitutes an additional important precisionobservable, M exp h = 125 . ± .
24 GeV [3]. If the measured value is associated with the mass M h of thelightest CP -even Higgs boson within the MSSM (for a recent discussion of the viability of the interpretationin terms of the heavy CP -even Higgs boson H , see [4]), the comparison of the predicted value with themeasurement constitutes an important test of the model with high sensitivity to the SUSY mass scales(see e.g. [5–7] for reviews). In order to fully exploit the high precision of the experimental measurement forconstraining the SUSY parameter space the accuracy of the theoretical prediction for M h has to be improvedvery significantly.So far, the full one-loop corrections [8–11], dominant two-loop corrections [12–35] and partial three-loop results [36–38] for the light MSSM Higgs-boson mass have been calculated diagrammatically. Besidesfixed-order calculations, effective field theory (EFT) methods have been used to resum large logarithmiccontributions in case of a large mass hierarchy between the electroweak and the SUSY scale [39–43]. TheseEFT calculations, however, are less accurate for relatively low SUSY mass scales owing to terms suppressedby the SUSY scale(s) which correspond to higher-dimensional operators in the EFT framework (see [44] forrecent work in this direction).In order to profit from the advantages of both methods – high accuracy for relatively low SUSY scalesin the case of the diagrammatic approach versus high accuracy for a high SUSY scale in the case of theEFT approach – a hybrid method combining both approaches has been developed [45, 46], see also [47, 48]for different implementations. The method introduced in [45, 46] has been implemented into the publiclyavailable code FeynHiggs [11, 17, 49–51] such that the fixed-order result is supplemented with higher-orderlogarithmic contributions.Comparisons between
FeynHiggs and pure EFT codes in the literature [43, 47, 48] have revealed non-negligible differences between the predicted values for M h . In particular, deviations have been observed forlarge SUSY scales, where terms not captured in the EFT framework are supposed to be negligible. At firstglance, such differences appear to be unexpected since the resummation of logarithms included in FeynHiggs is at the same level of accuracy as in pure EFT calculations.In order to clarify the situation, it is the purpose of this work to perform an in-depth comparison of thevarious approaches to explain the origin of the observed differences. For simplicity, we choose a single-scalescenario, M soft = µ = M A ≡ M SUSY , (2)2here M soft are the soft SUSY-breaking masses and µ is the Higgsino mass parameter. Furthermore, allparameters are assumed to be real, i.e. we work in the CP -conserving MSSM with real parameters. Whilethe chosen single-scale scenario is particularly suitable for the EFT approach, it should be noted that inrealistic cases the actual task is to provide the most accurate prediction (together with a reliable estimateof the remaining theoretical uncertainties) for the Higgs-boson masses of the model for a given SUSY massspectrum which may contain a variety of SUSY scales. We leave an investigation of such multi-scale scenariosfor future work.We shall explain that there are essentially three sources of the observed differences. In a first step, weshow that the usual scheme conversion of input parameters is not suitable for the comparison of results con-taining a series of higher-order logarithms. Such a scheme conversion can lead to large shifts correspondingto formally uncontrolled higher-order terms. Secondly, we analytically identify specific terms arising throughthe determination of the Higgs propagator pole which cancel with subloop renormalization contributions inthe irreducible self-energies of the diagrammatic approach for a large SUSY scale. We develop an improvedtreatment where unwanted effects from incomplete cancellations are avoided. Thirdly, we show how dif-ferent parametrizations of non-logarithmic terms can explain remaining differences between the results of
FeynHiggs and pure EFT codes for high scales.The paper is organized as follows. In Section 2, we review the different approaches with a particular focuson how the Higgs pole mass is extracted. In Section 3, we compare the results of the various approachesfor the Higgs pole mass to each other. In Section 4, we discuss the issue of using DR input parameters asinput of an OS calculation. In Section 5, we give a brief overview about the levels of accuracy of the M h evaluation implemented in various codes. In Section 6, we present a numerical analysis showing the impactof the effects discussed in the previous Sections and numerically compare FeynHiggs to other codes. Theconclusions can be found in Section 7. Two appendices provide additional details.
In this Section, we shortly review how the pole mass of the lightest CP -even Higgs boson of the MSSM iscalculated in a pure diagrammatic calculation, in a pure EFT calculation, and in the hybrid approach of FeynHiggs . A well-established way to calculate corrections to the mass of the SM-like Higgs of the MSSM, as well as tothe mass of the heavier CP -even neutral Higgs boson and the charged Higgs boson, is a fixed-order Feynmandiagrammatic (FD) calculation. The prediction is based on the calculation of Higgs self-energies involvingcontributions from SM particles, extra Higgs bosons, as well as their corresponding superpartners. In thisapproach the contributions from all sectors of the model and of all particles in the loop can be incorporatedat a given order. The mass effects of all particles in the loop can be taken into account for any pattern of themass spectrum. If there is however a large splitting between the relevant scales, in particular a large masshierarchy between the electroweak and the scale of some or all of the SUSY particles, the fixed-order resultwill contain numerically large logarithms that can spoil the convergence of the perturbative expansion.In the MSSM with real parameters, after calculating the renormalized Higgs-boson self-energies, thephysical masses of the CP -even Higgs bosons h, H can be obtained by finding the poles of their propagatormatrix, whose inverse is given by∆ − hH = i (cid:18) p − m h + ˆΣ MSSM hh ( p ) ˆΣ MSSM hH ( p )ˆΣ MSSM hH ( p ) p − m H + ˆΣ MSSM HH ( p ) (cid:19) , (3)where m h ( m H ) denotes the tree-level mass of the h ( H ) boson and ˆΣ hh,hH,HH are the corresponding self-energies. We introduced the label “MSSM” to indicate that the corresponding self-energy contains SM-typecontributions as well as non-SM contributions.Concerning the renormalization, we follow here the approach used in the program FeynHiggs . Accord-ingly, the circumflex ˆ indicates that the self-energies have been renormalized using the mixed on-shell (OS) Note that
FeynHiggs works also with complex parameters including an interpolation of the resummation routines. A -boson mass is renormalized on-shell, whereas the Higgs fieldrenormalization and the renormalization of tan β is performed using the DR scheme.The masses of the weak gauge bosons ( M Z , M W ) and the electromagnetic charge e are renormalizedon-shell, and the tadpole renormalization is carried out such that the tadpole contributions are cancelled bytheir respective counterterms. The OS vev is a dependent quantity, which is given in terms of the OS valuesof the observables M W , s w and e by v = 2 s M W e , (4)where s w denotes the sine of the weak mixing angle. The renormalization of this quantity at the one-looplevel is therefore given in terms of the OS counterterms of M W , s w and e ,2 s M W e → s M W e (cid:26) δM W M W + c s (cid:18) δM Z M Z − δM W M W (cid:19) − δe e (cid:27) , (5)where δM W,Z are the mass counterterms of the W and Z bosons, respectively, and δe is the counterterm ofthe electromagnetic charge ( c = 1 − s ). Motivated by the fact that the renormalization of the vev receivesa contribution from the field renormalization of the Higgs doublet, we identify the counterterm given inEq. (5) with δv /v + δZ hh , where δZ hh is the field renormalization counterterm of the SM-like Higgsfield fixed in the DR scheme. Accordingly, the OS counterterm of the vev defined in this way reads δv v = δM W M W + c s (cid:18) δM Z M Z − δM W M W (cid:19) − δe e − δZ hh . (6)The results for the self-energies in FeynHiggs have been reparametrized in terms of the Fermi constant G F instead of the electric charge e . The corresponding vev v G F is related to v OS via v = v G F (1 + ∆ r ) with v G F = 12 √ G F . (7)MSSM predictions for the quantity ∆ r can be found in [52–55]. The effect of this reparametrization in theone-loop self-energies is formally of two-loop order.Furthermore (in the default choice), the stop sector is renormalized using the OS scheme, which is definedby applying on-shell conditions for the respective masses: the top-quark mass M t , and the top-squark masses M ˜ t and M ˜ t . A fourth renormalization condition fixes the mixing of the stops and can be identified with acondition for the top-squark mixing angle.Employing this scheme, in FeynHiggs the full one-loop corrections to the Higgs self-energies as well as two-loop corrections of O ( α t α s , α b α s , α t , α t α b , α b ) are implemented [11, 17, 22, 23, 25, 27, 30, 33, 34, 49–51]. Whilethose two-loop corrections in the gauge-less limit have been obtained for vanishing external momentum, thereis futhermore an option to incorporate the momentum dependence of the corrections at O ( α t α s ) [56,57] (seealso [58]). Finding the (complex) poles for the case where CP conservation is assumed corresponds to solvingthe equation (cid:16) p − m h + ˆΣ MSSM hh ( p ) (cid:17) (cid:16) p − m H + ˆΣ MSSM HH ( p ) (cid:17) − (cid:16) ˆΣ MSSM hH ( p ) (cid:17) = 0 . (8)In the decoupling limit, M A (cid:29) M Z , the physical mass of the lightest Higgs boson can approximately beobtained as solution of the simpler equation p − m h + ˆΣ MSSM hh ( p ) = 0 (9)up to corrections from the hH and HH self-energies, which are suppressed by powers of M A . In thefollowing discussion we will for simplicity use Eq. (9) for determining the pole of the propagator and wewill furthermore neglect the imaginary parts of the self-energies. In FeynHiggs the complex poles of the Here, we already implictly assume the decoupling limit ( M A (cid:29) M Z ) in the sense that we identify the h boson as theSM-like Higgs. M h ) FD = m h − ˆΣ MSSM hh ( m h ) + ˆΣ MSSM (cid:48) hh ( m h ) ˆΣ MSSM hh ( m h ) + . . . , (10)where the prime denotes the derivative of the self-energy with respect to the momentum squared. The ellipsisstands for terms involving higher-order derivatives and products of differentiated self-energies. In App. B weprovide a formula from which these terms can be derived recursively. The Higgs pole mass at a given orderis obtained from Eq. (10) via a loop expansion to the appropriate order. Another approach to calculate the mass of the SM-like Higgs boson in the MSSM is using effective fieldtheory (EFT) methods. These allow the resummation of large logarithmic contributions, so that higher-order contributions beyond the order of fixed-order diagrammatic calculations can be incorporated. Withoutincluding higher dimensional operators in the effective Lagrangian, contributions suppressed by a heavy scaleare however not captured.In the simplest EFT framework, all SUSY particles are integrated out from the full theory at a commonmass scale M SUSY . Below M SUSY the SM remains as the low-energy EFT. The couplings of the EFT aredetermined by matching to the MSSM at the scale M SUSY . In the case of the SM as the EFT below M SUSY this concerns only the effective Higgs self-coupling λ , all the other couplings are fixed by matching themto observables at the low-energy scale. Renormalization group equations (RGEs) are used to correlate thecouplings at the high scale M SUSY and the low scale, typically chosen to be the OS top mass M t (or M Z ).The effective Higgs self coupling λ ( M t ) obtained from the matched λ ( M SUSY ) determines the MS massof the SM Higgs boson at the scale M t via( m MS , SM h ) = 2 λ ( M t ) v , (11)with the MS vev (at the scale M t ). The MS vev can be related to the on-shell vev via the finite part of δv defined in Eq. (6), v = v + δv (cid:12)(cid:12)(cid:12) fin . (12)It should be noted that since the quantity in Eq. (11) is the SM MS vev, in Eq. (12) only SM-type contri-butions have to be considered in δv .Getting from the running mass (11) to the physical Higgs mass one has to solve the pole equation forthe Higgs-boson propagator, p − ( m MS , SM h ) + ˜Σ SM hh ( p ) = 0 , (13)involving the renormalized SM Higgs boson self-energy (denoted by a tilde)˜Σ SM hh ( p ) = Σ SM hh ( p ) (cid:12)(cid:12)(cid:12) fin − √ v MS T SM h (cid:12)(cid:12)(cid:12) fin , (14)which is renormalized accordingly in the MS scheme at the scale M t but with the Higgs tadpoles renormalizedto zero, i.e. the tadpole counterterm is chosen to cancel the sum of the tadpole diagrams, T SM h , for theHiggs field, δT SM h = − T SM h . (15)With all these ingredients, the Higgs pole mass is now obtained as the solution of the equation M h = 2 λ ( M t ) v − ˜Σ SM hh ( M h ) . (16) In case of M A ∼ M t the effective theory is a Two-Higgs-Doublet model and not the SM, see [42]. m h of the MSSM yields( M h ) EFT = 2 v λ ( M t ) − ˜Σ SM hh ( m h ) − ˜Σ SM (cid:48) hh ( m h ) · (cid:104) v λ ( M t ) − ˜Σ SM hh ( m h ) − m h (cid:105) + · · · , (17)where the ellipsis indicates higher-order terms in the expansion.We discuss the current status of EFT calculations in Section 5. In FeynHiggs , the fixed-order approach is combined with the EFT approach in order to supplement thefull diagrammatic result with leading higher-order contributions [45, 46]. The logarithmic contributionsresummed using the EFT approach are incorporated into Eq. (9), p − m h + ˆΣ MSSM hh ( p ) + ∆ ˆΣ hh = 0 . (18)The quantity ∆ ˆΣ hh contains all logarithmic contributions obtained via the EFT approach as well as sub-traction terms compensating the logarithmic terms already present in the diagrammatic fixed-order resultfor ˆΣ MSSM hh , ∆ ˆΣ hh = − (cid:2) v λ ( M t ) (cid:3) log − (cid:2) ˆΣ MSSM hh ( m h ) (cid:3) log . (19)The subscript ‘log’ indicates that we take only logarithmic contributions into account. Note that inˆΣ MSSM hh ( m h ) the logarithms appear only explicitly when expanding in v/M SUSY . For more details on thecombination of the fixed-order and the EFT result, we refer to [45, 46].Plugging the expression for ∆ ˆΣ hh into Eq. (18), we obtain for the physical Higgs mass( M h ) FH = m h − ˆΣ MSSM hh ( M h ) + (cid:2) v λ ( M t ) (cid:3) log + (cid:2) ˆΣ MSSM hh ( m h ) (cid:3) log == m h + (cid:2) v λ ( M t ) (cid:3) log − (cid:2) ˆΣ MSSM hh ( m h ) (cid:3) nolog − ˆΣ MSSM (cid:48) hh ( m h ) (cid:16)(cid:2) v λ ( M t ) (cid:3) log − (cid:2) ˆΣ MSSM hh ( m h ) (cid:3) nolog (cid:17) + . . . . (20)We use the label ‘nolog’ to indicate that we take only terms not involving large logarithms into accountfor the labelled quantity. We again would like to stress that the large logarithms (and thereby the meantnon-logarithmic terms) appear only explicitly in ˆΣ MSSM hh ( m h ) when expanding in v/M SUSY .Before comparing the various approaches in depth, we also shortly comment on the renormalizationscheme conversion needed for the combination of the fixed-order and the EFT calculation. As mentionedbefore, in
FeynHiggs (in the default choice) the stop sector is renormalized using the OS scheme. In contrast,in the EFT calculation, i.e. the calculation of λ ( M t ), all SUSY parameters enter in DR-renormalized form.As argued in [46], it is sufficient to convert only the stop mixing parameter X t using only the one-loop largelogarithmic terms, X DR , EFT t = X OS t (cid:20) (cid:18) α s π − α t π (1 − X t /M S ) (cid:19) ln M S M t (cid:21) , (21)where M S = M ˜ t M ˜ t , α s = g / (4 π ) (with g being the strong gauge coupling) and α t = y t / (4 π ) (with y t being the top Yukawa coupling). In the following we will discuss the differences between the various approaches. It is obvious from thediscussion of the previous section that the diagrammatic fixed-order result and the pure EFT result differ byhigher-order logarithmic terms that are contained in the EFT result but not in the diagrammatic fixed-orderresult as well as by non-logarihmic terms that are contained in the diagrammatic fixed-order result but not6n the pure EFT result. In the hybrid approach the diagrammatic fixed-order result is supplemented by thehigher-order logarithmic terms obtained by the EFT approach. We focus in the following on the comparisonbetween the hybrid approach and the pure EFT result. In the present section we leave aside issues relatedto the used renormalization schemes, which will be addressed in Section 4.While the hybrid approach and the pure EFT approach both incorporate the higher-order logarithmicterms obtained by the EFT approach, this does not necessarily imply that all logarithmic terms in the tworesults are the same. This is due to the fact that the determination of the Higgs-boson mass from the poleof the progagator within the hybrid approach is performed in the full model (in the example consideredhere the MSSM, incorporating loop contributions from all SUSY particles), while in the EFT approach it isdetermined in the effective low-scale model (in the considered example the SM). We will demonstrate belowthat the determination of the propagator pole in the hybrid approach generates logarithmic terms beyondthe ones contained in the EFT approach at the two-loop level and beyond which actually cancel in the limitof a heavy SUSY scale with contributions from the subloop renormalization. This cancellation is explicitlydemonstrated at the two-loop level. We will furthermore discuss the difference in non-logarithmic termsbetween the results of the hybrid and the EFT approach.
In the EFT approach where the Higgs boson mass is determined as the pole of the propagator in the SM asthe effective low-scale model, while the SUSY particles have been integrated out, the logarithmic terms aregiven by (see Eq. (17))( M h ) logEFT = (cid:2) v λ ( M t ) (cid:3) log − ˜Σ SM (cid:48) hh ( m h ) (cid:2) v λ ( M t ) (cid:3) log + . . . . (22)The logarithmic terms contained in the result of the hybrid approach implemented in FeynHiggs are givenby (see Eq. (20)) ( M h ) logFH = (cid:2) v λ ( M t ) (cid:3) log + (cid:104) ˆΣ MSSM (cid:48) hh ( m h ) (cid:105) log (cid:104) ˆΣ MSSM hh ( m h ) (cid:105) nolog − ˆΣ MSSM (cid:48) hh ( m h ) (cid:2) v λ ( M t ) (cid:3) log + . . . . (23)In the decoupling limit ( M SUSY = M A (cid:29) M t , where in particular the light CP -even Higgs boson has SM-likecouplings), we can split up the MSSM Higgs self-energy into a SM part and a non-SM part,ˆΣ MSSM hh ( m h ) = ˆΣ SM hh ( m h ) + ˆΣ nonSM hh ( m h ) . (24)In the mixed OS/DR scheme of the full diagrammatic calculation, the Higgs field renormalization constantsare fixed in the DR scheme. For scalar propagators, there is no difference between the DR and the MSscheme at the one-loop level. Consequently,ˆΣ SM (cid:48) hh ( m h ) = ˜Σ SM (cid:48) hh ( m h ) (25)holds.Using this relation, we obtain for the difference between the higher-order logarithmic terms from thedetermination of the pole of the propagator obtained in the EFT and the hybrid approach∆ log ≡ ( M h ) logFH − ( M h ) logEFT = (cid:104) ˆΣ nonSM (cid:48) hh ( m h ) (cid:105) log (cid:104) ˆΣ MSSM hh ( m h ) (cid:105) nolog − ˆΣ nonSM (cid:48) hh ( m h ) (cid:2) v λ ( M t ) (cid:3) log + . . . (26)= : ∆ log p . Since this difference, which is of two-loop order and beyond, results only from the momentum dependenceof the non-SM contributions to the Higgs self-energy, we call it ∆ log p in the following. We give analyticexpressions for ∆ log p in App. B.In Section 3.3 we will demonstrate at the two-loop level that in the limit of a heavy SUSY scale thequantity ∆ log p consisting of “momentum-dependent non-SM contributions” as given in Eq. (26) cancels outwith contributions of the Higgs self-energy’s subloop renormalization. Before we address this issue we firstcompare the non-logarithmic terms in the two approaches.7 .2 Non-logarithmic terms In the EFT approach, the non-logarithmic terms are given by (see Eq. (17))( M h ) nologEFT = (cid:2) v λ ( M t ) (cid:3) nolog − ˜Σ SM hh ( m h ) − ˜Σ SM (cid:48) hh ( m h ) (cid:16) (cid:2) v λ ( M t ) (cid:3) nolog − ˜Σ SM hh ( m h ) − m h (cid:17) + . . . . (27)By construction, all non-logarithmic terms contained in the result of the hybrid approach originate fromthe fixed-order diagrammatic calculation (see Eq. (20)),( M h ) nologFH = m h − (cid:104) ˆΣ MSSM hh ( m h ) (cid:105) nolog + (cid:104) ˆΣ MSSM (cid:48) hh ( m h ) (cid:105) nolog (cid:104) ˆΣ MSSM hh ( m h ) (cid:105) nolog + . . . . (28)In this way one- and two-loop terms that are suppressed by the SUSY scale, ∆ nolog v/M SUSY , are included inthe result of the hybrid approach. Terms of this kind would result from higher-dimensional operators inthe EFT approach. Those terms that are included in the hybrid result as implemented in
FeynHiggs butnot in the publicly available pure EFT results constitute an important source of difference between thecorresponding results, which is expected to be sizeable if some or all SUSY particles are relatively light (seealso [44] for a recent discussion of contributions of this kind in the EFT approach). It should be noted thatin general terms of O ( v/M SUSY ) also originate from solving the full pole mass equation, Eq. (8), rather thanthe approximated one, Eq. (9).At zeroth order in v/M
SUSY , the non-logarithmic terms of the EFT approach contained in λ ( M t ) inEq. (27) agree with the non-SM contributions in Eq. (28). They result from the threshold corrections atthe matching scale M SUSY . These threshold corrections are so far only known fully at the one-loop order.At the two-loop order only the O ( α s α t , α t ) corrections are implemented in publicly available codes so far. Thus, those terms in (cid:104) ˆΣ nonSM (cid:48) hh ( m h ) (cid:105) nolog (cid:104) ˆΣ MSSM hh ( m h ) (cid:105) nolog being not of O ( α t ) are not present in ( M h ) EFT .At higher orders, all terms involving a derivative of ˆΣ nonSM hh are affected. As we will demonstrate in thefollowing section, also the non-logarithmic non-SM contributions arising from the determination of the poleof the propagator cancel out with contributions of the subloop renormalization in the limit of a high SUSYscale.Apart from these terms and from the non-logarithmic terms of O ( v/M SUSY ) discussed above, ∆ nolog v/M
SUSY ,a further difference between the hybrid approach and the EFT aproach is due to the parametrization ofthe non-logarithmic terms. In the EFT approach all low-scale parameters are MS quantities. The resultsof
FeynHiggs , on the other hand, are expressed in terms of physical, i.e. on-shell, parameters. For thetop-quark mass both the results expressed in terms of the pole mass, M t , and the running mass at the scale M t , m t ( M t ) (see [59] for details on the involved reparametrization) have been implemented (the appliedrenormalization schemes for SUSY parameters will be discussed below). The Higgs vev is a dependentquantity in FeynHiggs which is expressed in terms of the physical observables M W , s w and e according toEq. (4) (where e is furthermore reparametrized in terms of the Fermi constant, see Eq. (7)). Accordingly, ifchoosing low-energy SM parameters to express the EFT result, the non-logarithmic terms in this result areparametrized in terms of the MS quantities m t ( M t ) and v MS ( M t ), while depending on the option chosen forthe top-quark mass the non-logarithmic terms in FeynHiggs are expressed in terms of either m t ( M t ) and v G F or M t and v G F . Those parametrizations differ from each other by higher-order terms. The observeddifferences are therefore related to the remaining uncertainties of unknown higher-order corrections.It should be noted that also within the EFT approach there is a certain freedom for choosing differentparametrizations. For instance, the threshold corrections at the matching scale can be expressed in terms ofthe SM MS top Yukawa coupling or in terms of the MSSM DR top Yukawa coupling.As a result, the deviations ∆ nolog between the non-logarithmic terms in the hybrid approach and theEFT approach arise from the following sources,∆ nolog ≡ ( M h ) nologFH − ( M h ) nologEFT == ∆ nolog v/M SUSY + ∆ nologpara + ∆ nolog p . (29) Two-loop corrections controlled by the bottom and tau Yukawa couplings have recently been derived in [44]. nolog v/M
SUSY are terms present in the hybrid approach that would correspond to higher-dimensionaloperators in the EFT approach. The term ∆ nologpara indicates the differences in the parametrization of thenon-logarithmic terms, and∆ nolog p := (cid:104) ˆΣ nonSM (cid:48) hh ( m h ) (cid:105) nolog (cid:104) ˆΣ MSSM hh ( m h ) (cid:105) nolog − (cid:104) ˆΣ nonSM (cid:48) hh ( m h ) (cid:105) O ( α t )nolog (cid:104) ˆΣ MSSM hh ( m h ) (cid:105) O ( α t )nolog + (cid:104) higher order terms involving( ∂/∂p ) n ˆΣ nonSM hh , n ≥ (cid:105) (30)are terms arising from the different determination of the propagator poles, as discussed above. We saw in Section 3.1 and Section 3.2 that the different determination of the propagator pole in the hybridapproach and the EFT approach gives rise to both logarithmic and non-logarithmic contributions in whichthe expressions given for the two approaches in the previous sections differ from each other. We will nowexplicitly demonstrate at the two-loop level that those differences in fact cancel out in the limit of a heavySUSY scale if all the relevant terms at this order are taken into account.As a first step, we write down the correction to M h , derived by an explicit diagrammatic calculation. Atstrict two-loop order, we obtain( M h ) FD = m h − ˆΣ MSSM , (1) hh ( m h ) − ˆΣ MSSM , (2) hh ( m h )+ (cid:16) ˆΣ nonSM , (1) (cid:48) hh ( m h ) + ˆΣ SM , (1) (cid:48) hh ( m h ) (cid:17) ˆΣ MSSM , (1) hh ( m h ) . (31)The superscripts indicate the loop-order of the corresponding self-energy. We obtain the renormalized two-loop self-energy from the unrenormalized one viaˆΣ
MSSM , (2) hh ( m h ) = Σ MSSM , (2) hh ( m h ) + (two-loop counterterms) + (subloop-ren.) . (32)The subloop-renormalization can be derived from the one-loop self-energy via a counterterm-expansion.Expressing all couplings appearing in the one-loop self-energy through masses divided by v G F (for theremainder of this section we drop the subscript “ G F ”, i.e. we use the shorthand v ≡ v G F ), we can write(subloop-ren.) == ( δv ) MSSM ∂∂v ˆΣ MSSM , (1) hh ( m h ) + (cid:88) i ( δm i ) MSSM ∂∂m i ˆΣ MSSM , (1) hh ( m h ) + (field ren.) . == − ( δv ) MSSM v ˆΣ MSSM , (1) hh ( m h ) + (cid:88) i ( δm i ) MSSM ∂∂m i ˆΣ MSSM , (1) hh ( m h ) + (field ren.) , (33)where we used in the last line that ˆΣ MSSM , (1) hh ∝ /v if all couplings are expressed by the respective massdivided by v .We are interested in terms involving the finite parts of the derivative of the Higgs self-energy, i.e. termswhich could potentially cancel the term proportional to ˆΣ nonSM , (1) (cid:48) hh ( m h ) in Eq. (31). At first sight it wouldseem that terms of this kind could arise from an on-shell field renormalization of the Higgs field. It is well-known, however, that those field renormalization constants drop out of the prediction of the mass parameterorder by order in perturbation theory (in FeynHiggs , a DR renormalization is employed for the Higgs fields).Also the mass counterterms as well as the genuine two-loop counterterms do not contribute terms that are In our discussion here we treat the two-loop self-energy as the full result containg all contributions that appear at thisorder. The specific approximations that have been made at the two-loop level in
FeynHiggs will be discussed below. nonSM , (1) (cid:48) hh ( m h ). The only remaining term is the vev counterterm. According to Eq. (6) andEq. (7) it is given at the one-loop level by, having the same form in the SM and the MSSM, δv v = δM W M W + c s (cid:18) δM Z M Z − δM W M W (cid:19) − δe e + ∆ r − δZ hh . (34)The renormalization constant δZ hh represents within the MSSM the DR field renormalization constant ofthe SM-like Higgs field, while in the SM it is understood to be the MS field renormalization constant of theHiggs field.We verified by explicit calculation that in the limit of a large SUSY scale the following relation holds( δv ) MSSM v = ( δv ) SM v − ˆΣ nonSM , (1) (cid:48) hh ( m h ) + O ( v/M SUSY ) . (35)Using this relation, we can rewrite the two-loop self-energy (omitting terms of O ( v/M SUSY )),ˆΣ
MSSM , (2) hh ( m h ) = ˆΣ MSSM , (2) hh ( m h ) (cid:12)(cid:12)(cid:12) ( δv ) MSSM → ( δv ) SM + ˆΣ nonSM , (1) (cid:48) hh ( m h ) ˆΣ MSSM , (1) hh ( m h ) , (36)where the subscript ‘( δv ) MSSM → ( δv ) SM ’ is used to indicate that the MSSM vev counterterm, appearingin the subloop renormalization, is replaced by its SM counterpart.Plugging this expression back into Eq. (33) and Eq. (31), we obtain( M h ) FD = m h − ˆΣ MSSM , (1) hh ( m h ) − (cid:18) ˆΣ MSSM , (2) hh ( m h ) (cid:12)(cid:12)(cid:12) ( δv ) MSSM → ( δv ) SM + ˆΣ nonSM , (1) (cid:48) hh ( m h ) ˆΣ MSSM , (1) hh ( m h ) (cid:19) + (cid:16) ˆΣ nonSM , (1) (cid:48) hh ( m h ) + ˆΣ SM , (1) (cid:48) hh ( m h ) (cid:17) ˆΣ MSSM , (1) hh ( m h ) == m h − ˆΣ MSSM , (1) hh ( m h ) − ˆΣ MSSM , (2) hh ( m h ) (cid:12)(cid:12)(cid:12) ( δv ) MSSM → ( δv ) SM + ˆΣ SM , (1) (cid:48) hh ( m h ) ˆΣ MSSM , (1) hh ( m h ) . (37)We observe that the corresponding subloop renormalization term cancels in Eq. (31) the term ˆΣ nonSM , (1) (cid:48) hh ( m h )involving the non-SM contributions to the Higgs self-energy by which the determination of the propagatorpole in the hybrid approach differs from the EFT approach.The origin of Eq. (35) is the different normalization of the SM-like MSSM Higgs doublet Φ MSSM and theSM Higgs doublet Φ SM . Comparing the derivative of the two-point function, appearing in the LSZ factor ofamplitudes with external Higgs fields, we obtain in the limit of a heavy SUSY scaleΦ MSSM (cid:18)
MSSM , (1) (cid:48) hh ( m h ) (cid:19) = Φ SM (cid:18) SM , (1) (cid:48) hh ( m h ) (cid:19) , (38)or equivalently Φ MSSM = Φ SM (cid:18) −
12 ˆΣ nonSM , (1) (cid:48) hh ( m h ) (cid:19) . (39)Expressed in terms of a relation between the counterterms of the vevs, this implies Eq. (35).While as mentioned above the Higgs field renormalization constant drops out in the Higgs mass predictionorder by order, it is nevertheless noteworthy that the introduction of an OS field renormalization constantwould lead to ˆΣ MSSM (cid:48) hh ( m h ) (cid:12)(cid:12) δZ OS h = 0 (40)and ( δv ) MSSM (cid:12)(cid:12) δZ OS h = ( δv ) SM (cid:12)(cid:12) δZ OS h , (41) It should be noted that such an LSZ factor enters in the EFT approach via the matching condition at the high scale. nonSM (cid:48) hh appear in the subloop renormalization at the two-loop level.While we have demonstrated this cancellation at the two-loop level, it is to be expected that it wouldalso occur at higher orders. Explicit formulas for higher-order terms of this kind are given in App. B. Whilethe described cancellation occurs at the full two-loop level, only partial cancellations occur between the fullone-loop self-energy times its derivative and the two-loop self-energy if for the latter certain approximationsare made.In FeynHiggs , the two-loop self-energies are derived in the gaugeless limit (i.e., two-loop corrections of O ( α t α s , α b α s , α t , α t α b , α b ) are incorporated [22,23,25,27,33,34]), and by default the external momentum ofthe two-loop graphs is neglected. There is, however, an option to include momentum dependence at O ( α t α s )(see [56, 57]). Accordingly, all O ( α t , α t α b , α b ) non-SM terms arising through the determination of thepropagator pole at the two-loop level are cancelled in the limit of a large SUSY scale by corresponding sublooprenormalization contributions within the diagrammatic calculation (the determination of the propagatorpole obviously does not give rise to terms of O ( α t α s , α b α s )). In previous versions of FeynHiggs , we havealready taken care when constructing the subtraction terms according to Eq. (19) that we do not subtractlogarithmic contributions that are needed for the cancellation with the corresponding terms arising from thedetermination of the propagator poles. For terms arising through the determination of the propagator polebeyond O ( α t , α t α b , α b ), however, so far the cancellation in the limit of a large SUSY scale did not occurbecause the corresponding contributions in the irreducible self-energies at the two-loop level and beyond arenot incorporated. In order to avoid unwanted effects from an incomplete cancellation, we have removed theuncompensated terms arising from the determination of the propagator pole in FeynHiggs . In this section we discuss issues related to the conversion between parameters of OS and DR renormalizationschemes. While the discussion will focus on the case where DR input parameters are converted into OSones that are then inserted into a result in the OS scheme, it should be stressed that the related problemsare not intrinsic to the OS approach. The same problems would occur if a DR result were used with OSinput parameters. The discussed problems are also not specific to Higgs mass predictions in SUSY models,but would appear whenever there are numerically large higher-order logarithms arising from a large splittingbetween the relevant scales of the considered quantity. In predictions for the mass of the SM-like Higgs bosonwithin the MSSM, the result is however particularly sensitive to higher-order effects of this kind through thepronounced dependence on the stop mixing parameter X t , which receives large corrections when convertingfrom the DR to the OS scheme or vice versa.In the case where fixed-order results at the n -loop level obtained in two different renormalization schemesare compared with each other, and higher-order logarithms are unknown and not expected to be particularlyenhanced, it is well known that the results based on the same type of corrections in two schemes differ byterms that are of O ( n + 1). The same is true for different options regarding how to perform the parameterconversion that differ from each other by higher-order contributions. The numerical differences observedin such a comparison can therefore be used as an indication of the possible size of unknown higher-ordercorrections.The situation is different, however, in the case that we are considering here, since the comparison is notperformed between fixed-order results but between results incorporating a series of (resummed) higher-orderlogarithms. It is crucial in such a case that the correct form of the higher-order logarithms that can be derivedvia EFT methods, which in our case arise from the large splitting between the assumed SUSY scale andthe weak scale, is maintained in the parameter conversion. We will demonstrate below that the parameterconversion that is usually applied for a comparison of renormalization schemes in fixed-order results does notmaintain the correct form of the higher-order logarithms. Since those higher-order logarithms are numericallyimportant, a conversion carried out in the described way leads to very large numerical discrepancies for largevalues of the SUSY scale. The recent results of [35] for the O ( α t α b , α b ) corrections in the general case of complex parameters will be implementedinto FeynHiggs . .1 Conversion between DR and OS parameters applicable to fixed-order res-ults The most straightforward method used for the conversion of DR input parameters to OS parameters in fixed-order results is to derive the shift between a parameter p in the two schemes according to p OS = p DR + ∆ p at the considered loop order, see e.g. [60]. Accordingly, at the full one-loop level, including logarithmic aswell as non-logarithmic terms, the conversion from DR to OS parameters for the stop mixing parameter andthe stop masses, which are particularly relevant in the context of MSSM Higgs mass predictions, reads (forexplicit formulas see [22, 23, 25, 59]) X OS t = X DR t + ∆ X t , (42) M ˜ t = m DR˜ t + ∆ m ˜ t , (43) M ˜ t = m DR˜ t + ∆ m ˜ t . (44)Here ∆ m ˜ t , is given by the corresponding difference of the DR and the OS counterterm. In FeynHiggs , theshift of X t is obtained by first calculating the OS stop masses and the OS stop mixing angle θ OS˜ t . These arethen used to obtain X OS t via M t X OS t = ( M t − M t ) sin θ OS˜ t cos θ OS˜ t . (45)Relating this prescription for X OS t to the DR input parameters X DR t , m DR˜ t , m DR˜ t , one can see that Eq. (45)contains products of one-loop contributions and therefore involves higher-order terms. Alternatively onecould have used an expression for the conversion that is truncated at the one-loop level. The differencebetween the two prescriptions would be of the order of unknown higher-order corrections in a fixed-ordercomparison. The on-shell parameters obtained as described above are then used as input of the fixed-orderOS renormalized calculation. This means in particular that the knowledge of the initial DR parameters isnot used any further once the conversion to OS parameters has been carried out. While this procedure issuitable for fixed-order results, it leads to problems if results containing a series of higher-order logarithmsare meant to be converted.Indeed, applying the described parameter conversion to the case of a DR result that incorporates higher-order logarithms generates additional higher-order terms causing a deviation in the logarithmic contributions.This can be seen by investigating the Higgs self-energy up to the two-loop level where the parameter X OS t obtained from the conversion has been inserted,ˆΣ OS hh ( X OS t ) = ˆΣ (1) , OS hh ( X OS t ) + ˆΣ (2) , OS hh ( X OS t ) . (46)Using instead Eq. (42) to write X OS t in terms of X DR t ,ˆΣ OS hh ( X OS t ) = ˆΣ (1) , OS hh ( X DR t + ∆ X t ) + ˆΣ (2) , OS hh ( X DR t + ∆ X t ) , (47)and performing an expansion in ∆ X t yieldsˆΣ OS hh ( X OS t ) = ˆΣ (1) , OS hh ( X DR t ) + (cid:20) ∂∂X t ˆΣ (1) , OS hh ( X DR t ) (cid:21) ∆ X t + ˆΣ (2) , OS hh ( X DR t ) + (cid:20) ∂∂X t ˆΣ (2) , OS hh ( X DR t ) (cid:21) ∆ X t + O (∆ X t ) = (48)= ˆΣ DR hh ( X DR t ) + (cid:20) ∂∂X t ˆΣ (2) , OS hh ( X DR t ) (cid:21) ∆ X t + O (∆ X t ) . (49)Thus, the obtained expression obviously differs from the original DR result by terms of 3-loop order andbeyond. One would furthermore need to convert also all other parameters entering the self-energy to theDR scheme in order to exactly recover the DR renormalized self-energy.12 .2 The case of large higher-order logarithms The higher-order terms in Eq. (49) that are not present in the original DR result contain in general logarithmiccontributions which for a result containing a series of higher-order logarithms cause a deviation from thelogarithmic corrections determined via the RGE. In our numerical discussion in Section 6 below we willdemonstrate that those higher-order contributions that are induced by the parameter conversion are indeednumerically sizeable.Another issue that is relevant in a hybrid approach, as pursued in
FeynHiggs , where a fixed-order resultin the OS scheme is combined with higher-order logarithmic expressions that are expressed in the DR schemeconcerns the DR value of X t that is used in the EFT part of the calculation. Only logarithmic terms are keptin the relation between X DR , EFT t and X OS t , see Eq. (21). If instead an input value for X DR t were convertedto X OS t using the full one-loop contributions according to Eq. (42), the stop mixing parameter used in theEFT calculation of FeynHiggs , X DR , EFT t , would differ from the input parameter X DR t .In order to properly address the case where DR parameters associated with a result containing a seriesof higher-order logarithms are used as input for FeynHiggs , we follow the strategy to perform the parameterconversion in the fixed-order result rather than in the infinite series of higher-order logarithms. For thispurpose we have extended
FeynHiggs such that the incorporated fixed-order result is given in terms of theDR parameters X DR t , m DR˜ t , m DR˜ t (the actual input parameters are the soft-breaking parameters of the stopsector). This new result complements the existing result that is given in terms of the on-shell parameters X OS t , M ˜ t ≡ m OS˜ t , M ˜ t ≡ m OS˜ t . The reparametrisation on which the new result is based can be viewed asthe parameter conversion described in the example of the previous section, but truncated at the two-looplevel, ˆΣ OS hh ( X OS t ) → ˆΣ OS hh ( X DR t ) + (cid:20) ∂∂X t ˆΣ (1) , OS hh ( X DR t ) (cid:21) ∆ X t = ˆΣ DR hh ( X DR t ) . (50)We have used the same procedure as the one described here for the stop mixing parameter also for the stopmasses. The two-loop terms that are induced by the conversion at the one-loop level have been added tothe two-loop result derived in the on-shell scheme in order to arrive at the corresponding expression in theDR scheme. Compact expressions for these additional terms valid in the case M SUSY (cid:29) M t and degenerate M L = M ˜ t R = M SUSY can be found in App. A. It should be noted that we would have obtained the same resultif we had performed the diagrammatic calculation with a DR renormalization of the respective parametersinstead of reparametrizing the final result. Using the above result given in terms of DR parameters, the valueof X t that is used in the EFT part of the calculation equals the DR input parameter, X DR , EFT t = X DR t . Forthis setting in FeynHiggs with DR input parameters the subtraction terms have been adjusted such that thelogarithms already contained in the fixed-order result for the DR renormalized self-energy are subtracted(rather than the ones contained in the OS renormalized self-energy, as it is the case for OS input parameters).Accordingly, depending on the provided input parameters the evaluation of the prediction for the massof the SM-like Higgs boson in
FeynHiggs proceeds in the following ways: • For on-shell input parameters the on-shell fixed-order result is combined with the higher-order logsobtained in the EFT approach, where X DR , EFT t is related to X OS t as specified in Eq. (21). • For DR input parameters in the stop sector associated with a result containing a series of higher-orderlogarithms the DR fixed-order result is combined with the higher-order logs obtained in the EFTapproach, where X DR , EFT t = X DR t . • For DR input parameters in a low-scale SUSY scenario where the impact of higher-order logarithmsis expected to be small, both the fixed-order DR result and the fixed-order on-shell result can beemployed, where for the latter the parameter conversion described in the previous section is used.
FeynHiggs to other codes
In the previous sections, we investigated methodical differences between the different approaches for pre-dicting the lightest CP -even Higgs boson mass in the MSSM, focusing in particular on the comparison of13he hybrid approach implemented in FeynHiggs with a pure EFT calculation. In the following, we compare
FeynHiggs numerically to other codes.Publicly available codes based on diagrammatic fixed-order results or effective potential methods include
CPSuperH [61–63],
SoftSUSY [64],
SPheno [65, 66] and
SUSPECT [67]. Publicly available pure EFT calcula-tions are
SUSYHD [43] and
MhEFT [40, 42, 68].
FlexibleSUSY [69], based on
SARAH [70–73], includes both adiagrammatic and an EFT result. Furthermore, it also has the option to use a hybrid method different fromthe one pursued in
FeynHiggs , called
FlexibleEFTHiggs [47]. Its basic idea is to include terms suppressedby the SUSY scale into the matching conditions in order to obtain accurate results for both low and highscales. Recently, the same approach has been included into
SPheno [48].The different levels of higher-order corrections implemented in the various diagrammatic codes are listedin [74]. A detailed numerical comparison between various diagrammatic and EFT codes can be found in [47].In there, it is also discussed in detail how
FlexibleEFTHiggs compares to other codes. We therefore focusin this work on a comparison of
FeynHiggs to SUSYHD as an exemplary EFT calculation.Before we can investigate the impact of the effects discussed in the previous Sections on the comparisonof
FeynHiggs and
SUSYHD , we have to ensure that the RGE results, i.e. the results for λ ( M t ), of both codesare compatible with each other. Both codes implement full leading and next-to-leading resummation and O ( α s α t , α t ) next-to-next-to-leading resummation of large logarithms. So the levels of accuracy are basicallyidentical. There are however several differences which are listed below. • SUSYHD by default uses the top-Yukawa coupling extracted at the NNNLO level.
FeynHiggs insteaduses the NNLO value by default, which is formally the appropriate setting for the resummation of NNLLcontributions. For all numerical results shown in this work, we deactivate the NNNLO corrections tothe top-Yukawa coupling in
SUSYHD . • SUSYHD includes the bottom- and tau-Yukawa couplings in the renormalization group running and alsoincludes corresponding one-loop threshold corrections. In
FeynHiggs , the bottom and tau Yukawacouplings are set to zero in the EFT calculation. In the fixed-order diagrammatic calculation, however,terms proportional to the bottom Yukawa coupling are included at the one- and two-loop level (at theone-loop level for the case of the tau Yukawa coupling). • SUSYHD includes the electroweak gauge couplings in the running up to the three-loop level.
FeynHiggs takes them into account up to the two-loop level. At the three-loop level, they are set to zero. • FeynHiggs includes a one-loop running of tan β to relate tan β ( M t ), which is used as input of FeynHiggs ,to tan β ( M SUSY ), which enters through the matching at the SUSY scale. In contrast,
SUSYHD usestan β ( M SUSY ) as input. • Similarly,
FeynHiggs uses a DR renormalized Higgsino mass parameter µ at the scale M t . The runningto the scale M SUSY , at which it enters the EFT calculation via the matching conditions at the SUSYscale, is neglected.
SUSYHD uses µ ( M SUSY ) as input.More details on the implemented EFT calculations are given in [43, 46].Despite the listed differences including the different treatment of the renormalization scales of tan β and µ , we find excellent agreement between the results of the RGE running of both codes. The numericaldifference of the quantity v λ ( M t ) calculated using the two codes is always (cid:46)
50 GeV for the single scalescenario defined in Eq. (2) and tan β ∼ O (10). This translates into a difference in M h of (cid:46) . In this Section, we present a numerical investigation of the effects discussed in the previous Sections andcompare the result obtained by
FeynHiggs to SUSYHD as an exemplary pure EFT code. We restrict ourselvesto the single scale scenario defined in Eq. (2). Apart from the parameters of the stop sector, we neglect allrenormalization scheme conversions necessary to relate the parameters of Eq. (2) as defined in
FeynHiggs to the parameters as defined in
SUSYHD . We furthermore settan β = 10 , (51)14 -
404 110115120125130135140 - - - - Figure 1: Left: M h as a function of M SUSY for X DR t /M SUSY = 0 (solid) and X DR t /M SUSY = 2 (dashed).The results of
FeynHiggs2.13.0 with a DR to OS conversion of the input parameters (blue) and a DRrenormalization of the fixed-order result (red) are compared. Right: Same as left plot, apart that M h isshown in dependence of X DR t /M SUSY for M SUSY = 1 TeV (solid), M SUSY = 5 TeV (dashed) and M SUSY = 20TeV (dot-dashed). In the bottom panels, the difference between the blue and red curves is shown (∆ M h = M h (FH 2.13.0 param conv) − M h (FH 2.13.0 DR)).i.e. we use tan β ( M t ) = 10 as input for FeynHiggs and tan β ( M SUSY ) = 10 as input for
SUSYHD . Asmentioned in Section 5, the difference in the renormalization scales is negligible for the considered scenario.All soft-breaking trilinear couplings except the one of the scalar top quarks are choosen to be A e,µ,τ,u,d,c,s,b = 0 . (52)For all soft-breaking parameters (i.e. those of the stop sector), we use the DR scheme with the renormalizationscale being M SUSY . If not stated otherwise, we use a parametrization of the non-logarithmic contributions interms of the SM MS NNLO top mass and v G F (see Section 3.2), corresponding to choosing runningMT = 1 as FeynHiggs flag.We first look at the numerical difference between employing the type of conversion from DR to OS inputparameters which is suitable for the comparison of fixed-order results (“FH 2.13.0 param conv”) and using aDR renormalized fixed-order result (“FH 2.13.0 DR”), see the discussion in Section 4. The left plot of Fig. 1shows the corresponding results for X DR t /M SUSY = 0 (2) as solid (dashed) lines as a function of M SUSY . Onecan see that for M SUSY (cid:46) M h . For vanishing mixing the prediction obtained by using a DR renormalizedfixed-order result is ∼ . X t /M SUSY = 2, the shift is larger. The result obtained using a DR fixed-order result is ∼ − . M SUSY (cid:39) . | X DR t /M SUSY | ∼ M SUSY (cid:38) | X DR t /M SUSY | ∼ M SUSY ∼
20 TeV.This behavior is mainly due to the fact that the parameter conversion that is used for the comparison of fixed-order results induces higher-order logarithmic contributions that are not compatible with the implementedresummation of logarithms to all orders (see the discussion in Section 4.1). For high SUSY scales, where the15 - - - Figure 2: Comparison of the M h predictions of FeynHiggs2.13.0 DR with
FeynHiggsnew DR , where in thenew version terms arising from the determination of the propagator pole are omitted that go beyond the levelof the corrections implemented in the irreducible self-energies. Left: Prediction for M h as function of M SUSY for vanishing stop mixing and X DR t /M SUSY = 2. Right: Prediction for M h as function of of X DR t /M SUSY for M SUSY = 1 TeV (solid), M SUSY = 5 TeV (dashed) and M SUSY = 20 TeV (dot-dashed). In the bottom panels,the difference between the blue and red curves is shown (∆ M h = M h (FH 2.13.0 DR) − M h (FH new DR)).higher-order logarithmic contributions become numerically large, this mismatch leads to the observed largedeviations. To a lesser extent, also the deviation between the input X DR t and the X DR , EFT t used in the EFTcalculation plays a role in this context, see Section 4.2.In the right plot of Fig. 1 the two results are compared as a function of X DR t /M SUSY for M SUSY =1 , ,
20 TeV, shown as solid, dashed and dot-dashed lines, respectively. For M SUSY = 1 TeV and M SUSY =5 TeV the deviations stay relatively small except for the highest values of | X DR t /M SUSY | . In contrast,for M SUSY = 20 TeV the uncontrolled higher-order contributions induced by the naive conversion of theinput parameters are seen to have a huge effect which even reverts the usual pattern of the dependence on | X DR t /M SUSY | , giving rise to local minima at | X DR t /M SUSY | (cid:39) ± .
3. We emphasize again that the samekind of uncontrolled higher-order effects would occur if a naive conversion of OS to DR parameters wouldbe used as input for a DR result containing a series of numerically large higher-order logarithms. Fig. 1shows that numerical instabilities noticed in comparisons of EFT results with
FeynHiggs carried out in theliterature are a consequence of an inappropriate application of the conversion of input parameters betweenthe OS and the DR schemes. The higher-order contributions implemented in
FeynHiggs are seen to benumerically stable up to very high SUSY scales in the considered scenario.For the further
FeynHiggs results shown below we use the DR renormalization of the parameters in thestop sector. As a next step we investigate the impact of the terms arising from the determination of thepropagator pole. As explained in Section 3, there occurs a cancellation in the limit of a large SUSY scalebetween non-SM terms arising through the determination of the propagator pole and contributions from thesubloop renormalization of the irreducible self-energy diagrams. While up to the version
FeynHiggs2.13.0 this cancellation was incomplete for terms beyond O ( α t , α t α b , α b ) (see Eq. (20)), we have modified thedetermination of the propagator poles in the new version of FeynHiggs such that terms are omitted thatwould not cancel because their counterpart in the irreducible self-energies is not incorporated at present. InFig. 2
FeynHiggs2.13.0 DR is compared with the new version, which is labelled as
FeynHiggsnew DR . Thedifference between the two results corresponds essentially to the terms ∆ logs p and ∆ nolog p given in Eqs. (26) and(29). In the left plot of Fig. 2, we show the results as a function of M SUSY for X DR t = 0 and X DR t /M SUSY = 2.One observes that the difference grows nearly logarithmically with M SUSY . This is expected since the16 -
101 110115120125130135 - - - - - Figure 3: Comparison of the M h predictions of FeynHiggsnew DR with
SUSYHD . Left: M h as function of M SUSY for X DR t /M SUSY = 0 (solid) and X DR t /M SUSY = 2 (dashed). Right: M h as function of X DR t /M DRSUSY for M SUSY = 1 TeV (solid), M SUSY = 5 TeV (dashed) and M SUSY = 20 TeV (dot-dashed). In the bottompanels, the difference between the blue and red curves is shown (∆ M h = M h (FH new DR) − M h (SUSYHD)).largest terms in ∆ logs p + ∆ nolog p are in fact logarithms of the SUSY scale over M t . Consequently, for smallscales ( M SUSY (cid:46) (cid:46) . M SUSY (cid:38) . X DR t /M SUSY = 2. In the right plot of Fig. 2 the difference is depicted as a function of X DR t /M SUSY for M SUSY = 1 , ,
20 TeV, shown as solid, dashed and dot-dashed lines, respectively. One can see that thedifference between the two results is approximately quadratically depependent on X DR t /M SUSY . This reflectsthe X DR t dependence of the derivative of the Higgs-boson self-energy (see Eq. (77) below).Having investigated the numerical impact of the scheme conversion of the input parameters as well asof the terms arising from the determination of the propagator pole, we now turn to a direct comparisonof FeynHiggs with
SUSYHD . The
FeynHiggs results in this comparison are the ones of the new version,
FeynHiggsnew DR , where the stop sector is renormalized in the DR scheme and terms arising from thedetermination of the propagator pole are omitted that go beyond the level of the corrections implementedin the irreducible self-energies, as described above.The left plot of Fig. 3 shows M h as a function of M SUSY for X DR t /M SUSY = 0 (2) as solid (dashed) lines.For vanishing stop mixing and M SUSY (cid:38)
FeynHiggs curvewith the
SUSYHD result. Even for very large scales M SUSY (cid:39)
20 TeV, we find agreement within ∼ . M SUSY (cid:46)
FeynHiggs result is higher by up to ∼ . SUSYHD result. The origin of this difference are terms suppressed by the SUSY scale, whichare included in
FeynHiggs but not in
SUSYHD , as will be discussed below. For X DR t /M SUSY = 2, we basicallyobserve the same behavior as in case of vanishing stop mixing. The overall agreement in the simple numericalscenario is very good (within ∼ . M SUSY (cid:38) . M SUSY (cid:46) . FeynHiggs result is lower compared to the
SUSYHD result by up to ∼ M h prediction of the new FeynHiggs versionand
SUSYHD is shown as a function of X DR t /M SUSY for M SUSY = 1 , ,
20 TeV, shown as solid, dashed and We remind the reader that we use
SUSYHD with the top Yukawa coupling evaluated at the NNLO level. Using instead theNNNLO value would shift the results of
SUSYHD shown here downwards by ∼ .
000 10 000 15 000 20 0000100200 - - - Figure 4: Left: Difference of the M h predictions of FeynHiggsnew DR and
SUSYHD as a function of M SUSY for X DR t /M SUSY = 0 (solid) and X DR t /M SUSY = 2 (dashed). For the parametrization of the diagrammaticresult of
FeynHiggs the SM NNLO MS top-quark mass is chosen. Right: Differences due to the differentparametrization of the top-quark mass and the vev in a fixed-order O ( α t α s , α t ) calculation, taking intoaccount only non-logarithmic terms, as a function of X DR t /M SUSY . The difference between the result para-metrized in terms of the MS NNLO top-quark mass and v G F and the one parametrized in terms of the MSNNLO top-quark mass and v MS is shown.dot-dashed lines, respectively. Again one can see an overall very good agreement between both codes for M SUSY (cid:38) | X DR t /M SUSY | , but the deviations stay below 1 GeV also for increasing mixing in the stopsector except for the highest values of | X DR t /M SUSY | in the case of M SUSY = 1 TeV. The larger deviationsof up to ∼ | X DR t /M SUSY | (cid:38) . M SUSY = 1 TeV are due to terms suppressed by M SUSY which become large for increasing | X DR t /M SUSY | .In Fig. 4, we further investigate these remaining differences between FeynHiggs and
SUSYHD observed inFig. 3. In the left plot we show the difference between the results of
FeynHiggs and
SUSYHD for M h (not for M h ). Since in both codes actually M h is calculated, taking the square root of these results can obscure thetrue dependences of the difference. As an example, if the difference in M h is constant when varying M SUSY ,we would not observe a constant difference when comparing the difference in M h . We show in the plot thedifference in M h for the case where the fixed-order result of FeynHiggs is parametrized in terms of the SMNNLO MS top mass. For M SUSY (cid:46) M SUSY (cid:46) X DR t /M SUSY = 2 we observe large gradients. For larger scales ( M SUSY (cid:38) M SUSY . For vanishing stop mixing, the difference is growing by ∼
50 GeV when raising M SUSY from 3 TeV to 20 TeV. For X DR t /M SUSY = 2, similarly a growth of ∼
50 GeV is recognizable. This behavior is mostly due to the differences in the EFT calculations implemented in FeynHiggs and
SUSYHD discussed in Section 5. In addition, however, we observe an offset relative to the zeroaxis for M SUSY (cid:38) ∼
50 GeV ), whereas for X DR t /M SUSY = 2,the shift is more significant ( ∼
150 GeV ). The nearly constant offset between the two codes can be tracedback to the different parametrization of the non-logarithmic terms discussed in Section 3.2.We further analyse the influence of the different ways to parametrize the non-logarithmic terms in theright plot of Fig. 4. It shows the difference in M h obtained from a diagrammatic calculation of O ( α t α s , α t )using different parametrizations of the vev for the non-logarithmic one- and two-loop terms (see Section 3.2for more details). Note that these non-logarithmic terms, apart of O ( v/M SUSY ) contributions, stay constantwhen varying M SUSY . For X DR t /M SUSY ∼ v G F and v MS (both using the SM NNLO MS top-quark mass) amounts to ∼
170 GeV . Such a shift accounts forthe main part of the nearly constant offset observed in the left plot of Fig. 4. For X DR t /M SUSY ∼ v G F and v MS is seen to become very small. The nearlyconstant offset for vanishing stop mixing observed in the left plot of Fig. 4 can be explained in a similar way18y different parameterization of terms that are not of O ( α t α s , α t ).Finally, we briefly comment on the differences between FeynHiggs and other codes that have beenreported in the literature. In [43] it was claimed that differences between
FeynHiggs and
SUSYHD of up to ∼ M SUSY = 2 TeV and X DR t /M SUSY ∼ √
6. As already noted in [43], this differencewas somewhat reduced if the NNLO MS top mass was employed in the calculation of
FeynHiggs . While atthe time of the comparison carried out in [43] the EFT calculation of
FeynHiggs was not yet at the samelevel of accuracy as the one of
SUSYHD , the differences claimed in [43] were in fact primarily caused by aninappropriate application of the conversion of input parameters between the DR and the OS scheme. Theinappropriate parameter conversion, for which the authors of [43] used their own routine, caused a deviationof 3–4 GeV for M SUSY = 2 TeV and X DR t /M SUSY ∼ √
FeynHiggs curve with X DR t /M SUSY = 0 shown in [43]. The numericaleffect of this deviation was larger than the shift caused by employing the NNLO or NNNLO MS top-quarkmass in
FeynHiggs , in contrast to the claim made in [43].Also the comparison figures shown in [47, 48] are plagued by deficiencies arising from an inappropriateapplication of the parameter conversion between the DR and the OS scheme. We stress again that such aparameter conversion would give rise to the same kind of problems when starting from OS parameters andconverting to DR ones.
We have presented a detailed comparison between various approaches used to predict the mass of the SM-likeHiggs boson in the MSSM in a scenario in which all SUSY mass scales are chosen equal to each other. Inparticular we have compared pure EFT calculations with the hybrid approach, in which an explicit Feynman-diagrammatic fixed-order result is combined with the leading higher-order contributions obtained from EFTmethods. In the literature significant deviations between the results obtained via the two approaches havebeen reported especially at large SUSY scales. In this work, we have identified three sources of the observeddifferences.We could show that a large part of the reported discrepancies can be traced back to parameter conver-sions between different renormalization schemes. In EFT calculations typically the DR scheme is used for therenormalization of SUSY breaking parameters, e.g. the stop mixing parameter. In the diagrammatic calcula-tion of
FeynHiggs (in the default case) however, the OS scheme is employed in the scalar top sector. We havedemonstrated that the usual scheme conversion of input parameters used for the comparison of fixed-orderresults is not suitable for the comparison of results containing a series of higher-order logarithms. This kindof parameter conversion would induce higher-order logarithmic contributions that are not compatible withthe implemented resummation of logarithms to all orders. We have shown that the form of the higher-orderlogarithms obtained in one scheme can manifestly be maintained if the fixed-order part of the calculationis consistently reparametrized to this scheme. In order to enable this approach for DR input parameters,we have extended
FeynHiggs such that the results are provided both in terms of the on-shell parameters X OS t , M ˜ t ≡ m OS˜ t , M ˜ t ≡ m OS˜ t (as before) and the DR parameters X DR t , m DR˜ t , m DR˜ t . In practice, this wasachieved by reparametrizing the existing OS fixed-order result. We have demonstrated that many of theapparent discrepancies reported in the literature have mainly been caused by an inappropriate applicationof the conversion of input parameters between the OS and the DR schemes. It should be emphasized thatthis issue is not a problem of the OS renormalization, but analogously appears if OS parameters are used asinput for codes employing the DR scheme.Another difference between pure EFT calculations and the hybrid approach arises from the determinationof the poles of the Higgs propagator matrix. We have shown explicitly at the two-loop level that there occursa cancellation in the limit of a large SUSY scale between non-SM terms arising through the determinationof the propagator pole and contributions from the subloop renormalization of the irreducible self-energydiagrams. Since we expect that similar cancellations will happen at higher loops, we have modified thedetermination of the propagator poles in the new version of FeynHiggs such that terms are omitted that In the
FeynHiggs version employed in the comparison by default the NLO MS top mass was used. This was formally correctfor the resummation of the LL and NLL contributions that was implemented in
FeynHiggs at that time. Numerically, the shiftin the top-quark mass from NLO to NNLO generated the main effect when going to NNLL resummation [46].
FeynHiggs . Numerically, we found that the terms beyond O ( α t , α t α b ) for which in previousversions of FeynHiggs the cancellation was incomplete are negligible for low scales ( M SUSY (cid:46) . ∼ . M SUSY ∼
20 TeV).Furthermore, we investigated the impact of different parametrizations of the non-logarithmic one- andtwo-loop terms. In this context, we found the top-quark quark mass and the vev to be especially relevant.Despite the results being formally identical at the strict two-loop level, using e.g. a SM NNLO MS top quarkmass instead of the OS top quark mass induces changes in the higher-order non-logarithmic contributions.In our numerical comparison, we focused on a single scale scenario with a moderate tan β , which isparticularly suited for an EFT calculation. We specifically compared the results of FeynHiggs and the EFTcode
SUSYHD . Using the NNLO value of the MS top Yukawa coupling in
SUSYHD (by default the NNNLOvalue is used in
SUSYHD , which leads to a downward shift by ∼ . M h ), we find very good agreementbetween the new version of FeynHiggs and
SUSYHD for scales M SUSY (cid:38)
FeynHiggs incorporates essentially the samelogarithmic contributions as pure EFT calculations. For M SUSY (cid:46)
FeynHiggs and
SUSYHD due to terms suppressed by the SUSY scale that are not incorporated in theEFT calculation of
SUSYHD . The observed differences stay relatively small for the considered simple scenariowith a single SUSY scale, reaching ∼ M SUSY ∼
300 GeV. Larger deviations can be expected inSUSY scenarios with non-negligible mass splittings between the various SUSY particles. Such kind of masspatterns are accounted for in the diagrammatic fixed-order part of the hybrid approach.The new version of
FeynHiggs described in this paper, comprising an improvement in the determinationof the propagator poles and an option for using the DR scheme for the renormalization of the stop sector,will be made public soon.The results obtained in this paper provide important input for an improved estimate of the remainingtheoretical uncertainties from unknown higher-order corrections. In this context, we would like to stress oncemore that for the numerical evaluations in this paper we have used a rather simple scenario where all SUSYmasses have been set to be equal to each other. Having reconciled the hybrid approach of
FeynHiggs withpure EFT calculations for this simple single scale scenario, we are now in a position to assess the accuracyof the theoretical predictions also for more general scenarios with different hierarchies of scales. This will beanalysed in a fothcoming publication.
Acknowledgments
We thank Pietro Slavich for useful discussions. H.B. is thankful to Thomas Hahn for his invaluable helpconcerning all issues related to
FeynHiggs . H.B. and W.H. gratefully acknowledge support by the DeutscheForschungsgemeinschaft (DFG) under Grant No. EXC-153 (Excellence Cluster “Structure and Origin of theUniverse”). The work of S.H. is supported in part by CICYT (Grant FPA 2013-40715-P), in part by theMEINCOP Spain under contract FPA2016-78022-P, in part by the “Spanish Agencia Estatal de Investigacin”(AEI) and the EU “Fondo Europeo de Desarrollo Regional” (FEDER) through the project FPA2016-78645-P, and by the Spanish MICINN’s Consolider-Ingenio 2010 Program under Grant MultiDark CSD2009-00064.The work of G.W. is supported in part by the DFG through the SFB 676 “Particles, Strings and the EarlyUniverse” and by the European Commission through the “HiggsTools” Initial Training Network PITN-GA-2012-316704.
A Fixed-order conversion: additional two-loop terms
In the limit M SUSY (cid:29) M t and degenerate M L = M ˜ t R = M SUSY , the one-loop contributions from thestop/top sector to the neutral Higgs self-energies at O ( α t ) are given by (for the remainder of this section we20rop the subscript “ G F ”, i.e. we use the shorthand v ≡ v G F )ˆΣ = 116 π s β m t v µ X t M S , (53)ˆΣ = 116 π s β m t v µX t M S (cid:20) − X t M S − t β µX t M S (cid:21) , (54)ˆΣ = 116 π s β m t v (cid:34) −
12 ln M S m t − X t M S + X t M S − t β µX t M S (cid:18) − X t M S (cid:19) + 1 t β µ X t M S (cid:35) , (55)where M S = M ˜ t M ˜ t , and m t is either the OS top mass or the MS SM top mass. We furthermore introducedthe abbreviations s x ≡ sin x, c x ≡ cos x, t x ≡ tan x. (56)If we convert the stop masses and the stop mixing parameter from the OS to the DR scheme using theshifts defined in Eqs. (42)-(44), the following two-loop terms are generated (see Eq. (50)),∆ ˆΣ = 18 π s β m t v (cid:20) ∆ X t M S µ X t M S − M S M S µ X t M S (cid:21) , (57)∆ ˆΣ = 116 π s β m t v (cid:20) ∆ X t M S (cid:18) − µX t M S − t β µ X t M S + 6 µM S (cid:19) + ∆ M S M S (cid:18) µX t M S + 4 t β µ X t M S − µX t M S (cid:19)(cid:21) , (58)∆ ˆΣ = 18 π s β m t v (cid:34) ∆ X t M S (cid:32) − X t M S (cid:18) − X t M S (cid:19) − t β µM S (cid:18) − X t M S (cid:19) + 1 t β µ X t M S (cid:33) − M S M S (cid:32) − X t M S + X t M S − t β µX t M S (cid:18) − X t M S (cid:19) + 1 t β µ X t M S (cid:33)(cid:35) . (59)The quantity ∆ M S is given by ∆ M S = 12 (cid:18) ∆ m ˜ t M ˜ t + ∆ m ˜ t M ˜ t (cid:19) M S , (60)where ∆ X t and ∆ m ˜ t , are defined in Eqs. (42)-(44).Note that for all numerical results presented in this work, we used expressions valid also for low M SUSY ( M SUSY ∼ M t ) and general SUSY breaking. Note also that the shifts are performed for all self-energies andnot only for the hh self-energy as shown exemplary in Section 4. Therefore, the procedure remains also validin non-decoupling scenarios ( M A ∼ M Z ).As described in Section 4, these two-loop terms are finally added to the respective self-energies, i.e.,the ∆ ˆΣ’s are added to the two-loop self-energies obtained from the diagrammatic calculation. Higher-orderterms which would be generated by a scheme conversion of the input parameters are omitted. In thisway, the renormalization of the stop sector is changed from the OS to the DR scheme. This alternativerenormalization scheme will be available as an option in the next FeynHiggs version.
B Logarithms arising from the determination of the propagatorpole
In this Appendix, we give explicit expressions, valid in the decoupling limit, for the logarithms induced bythe momentum dependence of the non-SM contributions to the MSSM Higgs self-energy, i.e. for the quantity∆ logs p defined in Eq. (26).In order to derive the ( n + 1)th order iterative solution to the Higgs pole mass equation (see Eq. (9)) interms of lower order solutions, F`aa di Bruno’s formula (extended chain rule for derivatives) is used,( M h ) ( n +1) = − (cid:88) ( a ,...,a n ) ∈ T n a ! · ... · a n ! · (cid:34)(cid:18) ∂∂p (cid:19) ( a + ... + a n ) ˆΣ MSSM hh ( p ) (cid:35) p = m h · n (cid:89) m =1 ( M h ) ( m ) , (61)21here an n -tuple of non negative integers ( a , ..., a n ) is an element of T n if 1 · a + 2 · a + ... + n · a n = n .The zeroth order correction ( M h ) (0) = m h (62)serves as starting point of the recursion.We split ∆ logs p into a leading, a next-to-leading and a next-to-next-to-leading logarithm piece,∆ logs p = ∆ LL p + ∆ NLL p + ∆ NNLL p + . . . . (63)In FeynHiggs , the full momentum dependence by default is taken into account only at the one-loop level.At the two-loop level, the external momentum is set to zero (see [56, 57] for a discussion of the momentumdependence at the two-loop level). We can therefore split up the non-SM contributions to the Higgs self-energy into a one- and a two-loop piece,ˆΣ nonSM hh ( p ) = ˆΣ nonSM , (1) hh ( p ) + ˆΣ nonSM , (2) hh (0) . (64)To shorten the expressions for the individual contributions, we first introduce abbreviations. We writethe non-SM contributions to the Higgs self-energy asˆΣ nonSM , (1) hh ( m h ) = k (cid:16) c χ , L χ + c A , L A + c ˜ f , L S + c , (cid:17) , (65)ˆΣ nonSM , (2) hh (0) = k (cid:0) c , L S + c , L S + c , (cid:1) , (66)where k ≡ (4 π ) − is used to keep track of the loop order and L χ ≡ ln M χ m t , L A ≡ ln M A m t , L S ≡ ln M m t . (67)Here it should be noted that in this work we set M χ ≡ M = M = µ and M χ = M A = M SUSY . (68)In this Appendix, however, we keep them independent to be able to use the expressions also for more generalcases.The subscript of a coefficient c a,b indicates that it is the prefactor of the term k a L b ( L = L χ , L A , L S ).The corresponding superscript marks the origin of the respective term (from EWinos χ , from heavy Higgses A or from sfermions ˜ f ). These superscripts are used only at the one-loop level to be able to differentiatebetween the different types of appearing logarithms ( L χ , L A and L S ). In the DR scheme, the appearingcoefficients up to O ( v /M ) ( M heavy = M χ , M A , M SUSY ) are given by (for the remainder of this section22e drop the subscript “ G F ”, i.e. we use the shorthand v ≡ v G F ) c ˜ f , = − v (cid:20) y t + 32 y t ( g + g (cid:48) ) c β + 12 g + 56 g (cid:48) + 16 (3 g + 5 g (cid:48) ) c β (cid:21) , (69) c χ , = − v (cid:20) g (cid:48) ( −
11 + c β ) − g (5 + c β ) − g g (cid:48) s β (cid:21) , (70) c A , = − v (cid:20) g (53 − c β − c β ) + 1192 g (cid:48) (29 − c β − c β ) + 18 g g (cid:48) (5 + 3 c β ) s β (cid:21) , (71) c , = − v (cid:40) y t (cid:20)(cid:18) y t + 18 ( g + g (cid:48) ) c β (cid:19) ˆ X t − y t ˆ X t (cid:21) − y t ( g + g (cid:48) ) ˆ X t c β −
316 ( g + g (cid:48) ) s β − (cid:20)(cid:18) − c β (cid:19) g + 12 g g (cid:48) + 14 g (cid:48) (cid:21) + 124 ( s β + c β ) ·· (cid:20) − g − g g (cid:48) − g (cid:48) + (cid:0) ( g + g (cid:48) ) c β + 2( g − g (cid:48) ) s β (cid:1) (3 g + g (cid:48) ) (cid:21)(cid:41) , (72) c , = − v y t (cid:0) − g + 9 y t (cid:1) , (73) c , = − v y t (cid:104) g (cid:16) −
12 ˆ X t + ˆ X t (cid:17) − y t (cid:16) −
12 ˆ X t + ˆ X t (cid:17) (cid:105) , (74)where all appearing couplings are SM MS couplings evaluated at Q = M t ( g , g (cid:48) are the electroweak gaugecouplings, and ˆ X t ≡ X t /M SUSY ). We write the derivative of the non-SM contributions to the Higgs self-energy as ˆΣ nonSM , (1) (cid:48) hh ( m h ) = k (cid:0) c (cid:48) , L χ + c (cid:48) , (cid:1) , (75)with the primes denoting that the corresponding coefficient appears in the derivative of the self-energy. Weagain drop contributions of O ( v /M ). The coefficient multiplying L χ originates purely from EWinographs and reads c (cid:48) , = −
12 (3 g + g (cid:48) ) . (76)The non-logarithmic coefficient has contributions from EWinos as well as from stops (neglecting all otherYukawa couplings), c (cid:48) , = 12 y t ˆ X t (cid:124) (cid:123)(cid:122) (cid:125) stop contr. −
16 (3 g + g (cid:48) )( s β + c β ) (cid:124) (cid:123)(cid:122) (cid:125) EWino contr. . (77)All higher derivatives of ˆΣ nonSM hh ( p ) are suppressed, i.e. of O ( p /M ).The SM contributions are written in a similar way, (cid:18) ∂∂p (cid:19) n ˆΣ SM , (1) hh ( p ) (cid:12)(cid:12)(cid:12)(cid:12) p = m h = k ˜ c n , (78)where the superscript ’ n ’ denotes the n th derivative of ˆΣ SM , (1) hh . Here, we only give explicit expressions for23he pure top Yukawa contributions to the first five derivatives of ˆΣ SM , (1) hh ,˜ c (1)1 = − y t v , (79)˜ c (2)1 = 35 y t v − , (80)˜ c (3)1 = 970 y − t v − , (81)˜ c (4)1 = 235 y − t v − , (82)˜ c (5)1 = 477 y − t v − . (83)Eq. (61) allows now to successively derive all corrections induced by the momentum dependence of thenon-SM contributions to the hh self-energy. The generated leading logarithms can be resummed easily, sincehigher derivatives of ˆΣ nonSM hh are always suppressed, as noted before. The resummed expression is given interms of the c coefficients by∆ LL p = k c (cid:48) , L χ kc (cid:48) , L χ (cid:104) c χ , L χ + c A , L A + c ˜ f , L S + kc , L S (cid:105) . (84)A similar expression can be derived at the NLL level. We obtain∆ NLL p = k kc (cid:48) , L χ ) ·· (cid:104) c χ , c (cid:48) , L χ + c A , c (cid:48) , L A + c ˜ f , c (cid:48) , L S + c , c (cid:48) , L χ + k (cid:0) c , ( c (cid:48) , ) L χ + c , c (cid:48) , L χ L S + c , c (cid:48) , L S (cid:1) + k c , ( c (cid:48) , ) L χ L S (cid:105) . (85)At the NLL level however, additional terms proportional to derivatives of the light self-energy exist. Sincethese derivatives are not suppressed by a heavy mass, it seems not to be possible to resum the correspondinglogarithms. Nevertheless, including terms up to the 7-loop order we find a good convergence behavior andan induced shift of O ( ± ) to M h in the parameter region M t < M heavy (cid:46)
20 TeV. The respectiveshift in M h is of O (50 MeV). We therefore neglect this contribution completely.At the NNLL level, we take into account only terms proportional to the strong gauge coupling and thetop-Yukawa coupling (terms proportional to electroweak gauge couplings are negligible). We find that atthis level all terms include derivatives of the SM self-energy. We also find that this contribution to M h isnot negligible, O (20 GeV ). Therefore, we include terms up to the 7-loop order, which are given by∆ NNLL p = k L S c (cid:48) , (cid:104) c , − c ˜ f , ˜ c (cid:48) (cid:105) − k L S c (cid:48) , (cid:20) c , c (cid:48) , + c , ˜ c (1)1 − (cid:16) c ˜ f , (cid:17) ˜ c (1)1 (cid:21) + k L S c (cid:48) , (cid:20) c ˜ f , c , ˜ c (2)1 − (cid:16) c ˜ f , (cid:17) ˜ c (3)1 (cid:21) + 12 k L S c (cid:48) , (cid:20) ( c , ) ˜ c (2)1 − c , (cid:16) c ˜ f , (cid:17) ˜ c (3)1 + 112 (cid:16) c ˜ f , (cid:17) ˜ c (4)1 (cid:21) − k L S c (cid:48) , (cid:20) ( c , ) c ˜ f , ˜ c (3)1 − c , (cid:16) c ˜ f , (cid:17) ˜ c (4)1 + 160 (cid:16) c ˜ f , (cid:17) ˜ c (5)1 (cid:21) + O ( k ) , (86)where all terms in the c coefficients proportional to g or g (cid:48) are set to zero. Correspondingly, the derivativesof the light self-energy only include terms proportional to y t . These are listed in Eqs. (79)-(83). This loopexpansion quickly converges such that we can safely drop higher-order contributions (8-loop and beyond).24e find the electroweak contributions at the NNLL level and even higher-order logarithms (N n L with n >
2) to be completely negligible. Similar expressions can easily be obtained for the non-logarithmic termsof the same origin (see Eq. (30)). 25 eferences [1]
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