Precise prediction for the light MSSM Higgsboson mass combining effective field theory and fixedorder calculations
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Abstract
In the Minimal Supersymmetric Standard Model heavy superparticles introduce large logarithms in the calculation of the lightest \(\mathscr {CP}\)even Higgsboson mass. These logarithmic contributions can be resummed using effective field theory techniques. For light superparticles, however, fixedorder calculations are expected to be more accurate. To gain a precise prediction also for intermediate mass scales, the two approaches have to be combined. Here, we report on an improvement of this method in various steps: the inclusion of electroweak contributions, of separate electroweakino and gluino thresholds, as well as resummation at the NNLL level. These improvements can lead to significant numerical effects. In most cases, the lightest \(\mathscr {CP}\)even Higgsboson mass is shifted downwards by about 1 GeV. This is mainly caused by higherorder corrections to the \({\overline{\text {MS}}}\) topquark mass. We also describe the implementation of the new contributions in the code FeynHiggs.
Keywords
Minimal Supersymmetric Standard Model Effective Field Theory Subtraction Term Lead Logarithm SUSY Scale1 Introduction
With the discovery of the Higgs boson by the ATLAS [1] and CMS [2] experiments at the CERN Large Hadron Collider the Standard Model (SM) has been completed; there is, however, still ample room for Beyond Standard Model (BSM) physics. One of the best motivated and studied BSM models is the Minimal Supersymmetric Standard Model (MSSM) realizing the concept of supersymmetry (SUSY). It extends the Higgs sector of the SM by a second complex doublet leading to five physical Higgs particles (h, H, A, and \(H^\pm \)) and three (wouldbe) Goldstone bosons. The light \(\mathscr {CP}\)even state h can be identified with the discovered boson. At the tree level, the Higgs sector can be conveniently parametrized by the mass of the A boson, \(M_A\), and the ratio of the vacuum expectation values of the two doublets, \(\tan \beta = v_2/v_1\).
So far, no direct hints for SUSY particles have been found. Still, the SUSY parameter space can be constrained indirectly by precision observables, with the Higgsboson mass constituting an important precision observable on its own. Since the Higgs mass \(M_\mathrm{h}\) is very sensitive to quantum effects via loop contributions, much work has been dedicated to their calculation within the MSSM. The full oneloop result [3, 4, 5], the dominant twoloop corrections [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20] as well as partial threeloop results [21, 22, 23] are known. For heavy SUSY particles, fixedorder calculations suffer from large logarithms originating in a potentially huge hierarchy between the electroweak scale and the SUSY scale. Therefore, effective field theory (EFT) calculations have been developed to resum these logarithmic contributions [17, 24, 25]. Recent works have refined these methods to include gaugino/higgsino thresholds [26, 27, 28] and to allow for light nonstandard Higgs particles [29]. Furthermore, resummation at the nexttonexttoleading logarithm (NNLL) level has been adressed in [28, 30, 31].
These computations, however, do not capture the effect of terms that would be suppressed only in the case of a heavy SUSY scale. Thus, fixedorder calculations are expected to be more accurate for low SUSY scales. To gain the most accurate prediction for intermediate SUSY scales, both approaches have to be combined. This allows also to profit from the other advantages of the diagrammatic approach: the easy inclusion of many different SUSY scales, and the full control over the Higgsboson selfenergies, which are needed for other observables (e.g. production and decay rates). The authors of [32] first realized the idea of combining the diagrammatic and the EFT approach and implemented the method into the publicly available program FeynHiggs [8, 32, 33, 34, 35, 36], which also contains the complete fixedorder oneloop result as well as dominant twoloop results; NLL resummation was done for the strong and top Yukawa coupling enhanced logarithmic terms beyond the twoloop order. Here, we report on an extension of this work in a threefold respect: the inclusion of the electroweak contributions, the inclusion of separate electroweakino and gluino thresholds, and resummation of logarithms proportional to the top Yukawa coupling and the strong gauge coupling at the NNLL level.
In Sect. 2, we outline the EFT calculation, focusing on the ingredients needed to include electroweak contributions, gaugino/higgsino thresholds and NNLL resummation. In Sect. 3, we describe how the result of the EFT calculation is consistently combined with the fixedorder diagrammatic result. In Sect. 4, we discuss the implementation of the improvements in FeynHiggs. In Sect. 5, we present a numerical analysis showing the impact of the improved version on the calculation of \(M_\mathrm{h}\), with conclusions in Sect. 6.
2 Effective field theory calculation
In the case of heavy SUSY particles, large logarithms appear in explicit diagrammatic calculations making a fixedorder calculation an unreliable tool. The origin of this problem is the large hierarchy between the electroweak scale and the SUSY scale. Effective field theory techniques allow one to resum these large logarithms to all orders and thus get stable predictions.
The effective couplings of the EFT are fixed by matching to the full MSSM at the matching scale \(M_\mathrm{S}\) (in the simplest case of an effective SM below \(M_\mathrm{S}\), this concerns only the Higgs selfcoupling \(\lambda \)). All of the other couplings of the EFT are fixed by matching them to physical observables at the electroweak scale [37], e.g. the top Yukawa coupling is extracted from the topquark pole mass.
For the resummation of leading logarithms (LL), oneloop RGEs and treelevel matching conditions are needed; for the resummation of leading and nexttoleading logarithms (NLL), twoloop RGEs and oneloop matching conditions, and, accordingly, for the resummation of leading, nexttoleading and nexttonexttoleading logarithms, threeloop RGEs and twoloop matching conditions.
2.1 Electroweak contributions
As a first improvement with respect to [32], we include electroweak contributions in the resummation procedure at the NLL level. Correspondingly, we use the full twoloop RGEs of the SM (see [37] and references therein), including terms proportional to the electroweak gauge couplings g and \(g'\) (for SU(2) and U(1)), to evolve the SM couplings.
2.2 Gaugino–higgsino thresholds
At the scale \(M_\chi \), all electroweakinos are integrated out, and the remaining EFT below \(M_\chi \) is the SM. We match the SM to the split model using the threshold corrections given in [27, 28], i.e. the term \(\Delta _{\text {EWino}}\lambda \) in Eq. (3) is now part of the matching condition of \(\lambda \) at \(M_\chi \). Also the top Yukawa coupling receives a threshold correction at the electroweakino scale. Below \(M_\chi \) the SM RGEs are used for evolving the couplings.
In addition to allowing for light charginos and neutralinos, we also consider the case of a light gluino. This case is implemented by introducing an additional threshold marked by the gluino mass \(M_{\tilde{g}}\), below which the gluino is integrated out. The gluino is also assumed to be much heavier than the electroweak scale such that eventually the SM is recovered as the EFT close to the electroweak scale. However, no assumption as regards the ordering of \(M_{\tilde{g}}\) and \(M_\chi \) is made, i.e. \(M_{\tilde{g}}\le M_\chi \) as well as \(M_{\tilde{g}}> M_\chi \) is allowed. Since the gluino does not couple directly to the Higgs boson, no additional oneloop matching condition for \(\lambda \) has to be considered. The same argument applies for the electroweak gauge couplings, the Yukawa couplings (in the absence of sfermions) and the effective Higgs–Higgsino–gaugino couplings of the split model. An explicit calculation also shows that the strong gauge coupling does not receive a threshold correction. However, the presence of the gluino in the EFT above \(M_{\tilde{g}}\) modifies the RGEs (see Appendix A).
2.3 NNLL resummation
Also the matching conditions for the SM gauge and Yukawa couplings at \(M_\mathrm{t}\) have to be extended to include the \(\mathscr {O}(\alpha _\mathrm{s}^2,\alpha _\mathrm{s}\alpha _\mathrm{t},\alpha _\mathrm{t}^2)\) corrections. These are taken from [37]. The matching condition for the top Yukawa coupling involves the \({\overline{\text {MS}}}\) topquark mass which for NNLL resummation is obtained from the pole mass by means of the standard QCD and top Yukawa corrections at the twoloop level [37].
Furthermore, threeloop RGEs are needed for the coupling constant evolution. Since only NNL logarithms of \(\mathscr {O}(\alpha _\mathrm{s},\alpha _\mathrm{t})\) are resummed in this step, we neglect the electroweak gauge couplings at the threeloop level of the needed RGEs. All couplings of electroweakinos, being present below \(M_\mathrm{S}\) for \(M_\chi <M_\mathrm{S}\), are proportional to the electroweak gauge couplings when their matching conditions at \(M_\mathrm{S}\) are plugged in. In consequence, their presence has no influence on the form of the threeloop RGEs at this level of approximation. Hence for all considered hierarchies at all scales below \(M_\mathrm{S}\), the needed threeloop RGEs are just the corresponding SM RGEs, which are well known [39, 40, 41, 42, 43, 44, 45]. The same argument implies that the twoloop matching conditions of \(\lambda \) do not have to be modified for \(M_\chi \) lower than \(M_\mathrm{S}\).
For NNLL resummation, we have to restrict ourselves to the case of \(M_{\tilde{g}}\) equal to \(M_\mathrm{S}\) in the resummation procedure, since threeloop RGEs for the SM with added gluino are not known. Nevertheless, the numerical effect of a gluino threshold is so small that it can be safely neglected, as will be seen in the numerical results.
3 Combining fixedorder and EFT calculations
The first subtraction term \((\Delta M_\mathrm{h}^2)^{{1L,2L \mathrm{logs}}}\) ensures that the one and twoloop logarithms in the OS scheme, already contained in the Feynmandiagrammatic result, are not counted twice. We extracted these logarithms in the EFT framework by solving the system of RGEs iteratively and converting to the OS scheme afterwards. As a crosscheck, we also identified the oneloop logarithms within the Feynmandiagrammatic result finding agreement (see Appendix B for explicit expressions). It should be noted that FeynHiggs also allows one to choose a \({\overline{\text {MS}}}\) topquark mass [8]. If this option is switched on, we have to subtract the one and twoloop logarithms as contained in the Feynmandiagrammatic result, i.e. as obtained with a \({\overline{\text {MS}}}\) topquark mass.
The second subtraction term \((\Delta M_\mathrm{h}^2)^{\text {EFT, nonlog}}\) is introduced to cancel all nonlogarithmic terms contained in the EFT result. They originate from the matching conditions of the Higgs selfcoupling and have to be subtracted when only higherorder logarithmic contributions are added to the Feynmandiagrammatic result.
A particular issue to be taken care of when combining the diagrammatic result with the EFT calculation, is the choice of the renormalization scheme. The EFT calculation uses minimal subtraction schemes (\({\overline{\text {DR}}}\) for scales above \(M_\mathrm{S}\), \({\overline{\text {MS}}}\) for scales below \(M_\mathrm{S}\)) for renormalization. In contrast, in the diagrammatic calculation a mixed OS/\({\overline{\text {DR}}}\) scheme is employed (see [36] for a detailed description). Consequently, the input parameters of the EFT calculation are \({\overline{\text {MS}}}\)/\({\overline{\text {DR}}}\) parameters, whereas they are OS parameters in the diagrammatic calculation [38], as indicated in Eq. (7). The logarithmic subtraction term takes OS parameters as input, because we want to avoid doublecounting of the one and twoloop logarithms in the OS scheme. Also the nonlogarithmic subtraction term takes OS parameters as input, although the nonlogarithmic terms contained in the EFT result are parametrized with \({\overline{\text {DR}}}\) parameters. This is owing to the fact that nonlogarithmic terms in the \({\overline{\text {DR}}}\) scheme lead to logarithmic terms in the OS scheme; consequently, the OS twoloop logarithms of the Feynmandiagrammatic result would not be reproduced when \({\overline{\text {DR}}}\) parameters were used as input.
We choose to work with OS parameters as principal input. This means that OS input parameters are converted to \({\overline{\text {DR}}}\) parameters when used as input for the EFT calculation. We restrict ourselves to a oneloop conversion involving only terms proportional to large logarithms. This conversion is sufficient to reproduce all large logarithms already contained in the diagrammatic twoloop result of FeynHiggs. In contrast, nonlogarithmic terms and higher looporder terms would lead to terms in the EFT result which correspond to unknown higherorder corrections in an OS renormalized diagrammatic result. We, however, intend to add the resummed logarithms as obtained in the \({\overline{\text {MS}}}\)/\({\overline{\text {DR}}}\) scheme to the diagrammatic result. In consequence, all terms beyond oneloop logarithms have to be omitted.
4 Implementation in FeynHiggs
As explained in [32], the shift \(\Delta M_\mathrm{h}^2\) is implemented in FeynHiggs by adding it with a factor \(1/\sin ^2\beta \) to the \(\phi _2\phi _2\) selfenergy (\(\phi _2\) is the \(\mathscr {CP}\)even neutral component of the second Higgs doublet). In this way, the result of the resummation procedure enters also the calculation of other observables that are available from FeynHiggs.

loglevel=0: no resummation;

loglevel=1: \(\mathscr {O}(\alpha _\mathrm{s},\alpha _\mathrm{t})\) LL and NLL resummation (corresponds to former looplevel=3);

loglevel=2: full LL and NLL resummation;

loglevel=3: full LL, NLL and \(\mathscr {O}(\alpha _\mathrm{s},\alpha _\mathrm{t})\) NNLL resummation.
So far, all matching conditions are only implemented for degenerate softbreaking masses, meaning that all softbreaking masses are set equal to their corresponding threshold scale. The diagrammatic part of the calculation, however, captures the effects of nondegeneracy in an exact way at the one and twoloop level. The matching condition will be extended to the nondegenerate case in a future update to FeynHiggs.
5 Numerical analysis
To analyze the numerical impact of the improved resummations, we first compare the results of the previous FeynHiggs version 2.11.3 with the new version 2.12.0. As an example case, we look at a scenario where all softbreaking masses as well as the Higgsino mass parameter are chosen to be equal to \(M_\mathrm{S}\), together with \(t_\beta \equiv \tan \beta =10\). The results of FeynHiggs2.11.3 are obtained with switched on \(\mathscr {O}(\alpha _\mathrm{s},\alpha _\mathrm{t})\) LL and NLL resummation. Also the twoloop QCD correction to the \({\overline{\text {MS}}}\) topquark mass are enabled, although no NNLL resummation is performed. This is done because of the large numerical impact of this twoloop correction on the Higgs mass calculation (see the discussion at the end of this section). For the results of FeynHiggs2.12.0, all improvements discussed above are activated (loglevel=3).
To explore the origin of these shifts, we examine first the contribution of the resummation of logarithms proportional to the electroweak gauge couplings. The upper panel of Fig. 2 shows \(M_\mathrm{h}\) as a function of \(M_\mathrm{S}\) for \(X_\mathrm{t}/M_\mathrm{S}=0\) and \(X_\mathrm{t}/M_\mathrm{S}=2\). The results with a resummation of logarithms proportional to the electroweak gauge couplings (loglevel=2) and without such a resummation are compared (loglevel=1). The latter corresponds, apart from some minor fixes, to the result of FeynHiggs2.11.3. Furthermore, the result without resummation of logarithms proportional to the electroweak gauge couplings but with electroweak NLO corrections to the \({\overline{\text {MS}}}\) top mass is shown. For vanishing stop mixing, we observe a downwards shift of \({\sim }1.2\) GeV for \(M_\mathrm{S} = 1\) TeV. This shift is almost completely caused by the electroweak NLO corrections to the \({\overline{\text {MS}}}\) top mass yielding a reduction of the \({\overline{\text {MS}}}\) top mass by 1.1 GeV. This translates directly to a downwards shift of \(M_\mathrm{h}\) [48]. For rising \(M_\mathrm{S}\), the downwards shift caused by the corrections to the \({\overline{\text {MS}}}\) top mass is more and more compensated by the upwards shift caused by the resummed logarithms proportional to the electroweak gauge couplings. For \(X_\mathrm{t}/M_\mathrm{S}=2\), the behavior is very similar. For \(M_\mathrm{S}=1\) TeV, the downwards shift is larger (\(\sim \)1.7 GeV) owing to the increased dependence on the \({\overline{\text {MS}}}\) top mass for nearly maximal stop mixing. For rising \(M_\mathrm{S}\), this downwards shift is again more and more compensated by the positive contributions of the resummed electroweak logarithms.
The effect of the electroweakino threshold is investigated in the upper panel of Fig. 3, which displays \(M_\mathrm{h}\) as a function of \(M_\mathrm{S}\) for \(X_\mathrm{t}/M_\mathrm{S}=0\) and \(X_\mathrm{t}/M_\mathrm{S}=2\). In contrast to the previous figures, the electroweakino mass scale \(M_\chi \) is not chosen to be equal to \(M_\mathrm{S}\), but is fixed to 1 TeV. To disentangle the effect of the electroweakino threshold in the EFT calculation from the fixedorder oneloop corrections due to neutralinos and charginos, we compare the results with an electroweakino threshold to the results without a separate electroweakino threshold. To get the results without a separate electroweakino threshold, we set \(M_\chi =M_\mathrm{S}\) in the EFT calculation (namely in \(\Delta M_\mathrm{h}^2\)), but keep \(M_\chi =1\) TeV in the Feynmandiagrammatic calculation. The plot clearly shows that the implementation of a separate electroweakino threshold becomes only relevant for \(M_\mathrm{S}\gtrsim 5\) TeV. This behavior does not depend on the size of the stop mixing.
The effect of a separate gluino threshold is found to be negligible. For \(M_\mathrm{S}\) between 1 TeV and 20 TeV, its inclusion shifts \(M_\mathrm{h}\) downwards by at most 0.2 GeV for \(X_\mathrm{t}/M_\mathrm{S}\le 2\). The diagrammatic twoloop corrections capture almost the entire effect of varying \(M_{\tilde{g}}\), which can be sizable (\(\sim \)2 GeV) for maximal mixing. This justifies to set \(M_{\tilde{g}}=M_\mathrm{S}\) in the resummation procedure in the case of NNLL resummation, as explained in Sect. 2.
In the lower panel of Fig. 3, the difference between the results without (loglevel=2) and with (loglevel=3) NNLL resummation as a function of \(X_\mathrm{t}/M_\mathrm{S}\) is shown for \(M_\mathrm{S}=1\) TeV, \(M_\mathrm{S}=2\) TeV and \(M_\mathrm{S}=5\) TeV. Between \(X_\mathrm{t}/M_\mathrm{S} \sim 1\) and \(X_\mathrm{t}/M_\mathrm{S} \sim 1.5\), we observe only small shifts (\(\lesssim \)0.3 GeV). For \(X_\mathrm{t}/M_\mathrm{S}\sim 2\), \(M_\mathrm{h}\) is shifted upwards by the inclusion of NNLL resummation by up to 1 GeV, whereas \(M_\mathrm{h}\) is shifted downwards by up to 0.5 GeV for \(X_\mathrm{t}/M_\mathrm{S} =2\). This behavior is mainly caused by the \(\mathscr {O}(\alpha _\mathrm{s}\alpha _\mathrm{t})\) matching condition of \(\lambda \), which exhibits a similar dependence on \(X_\mathrm{t}/M_\mathrm{S}\). The large positive shift for negative \(X_\mathrm{t}\) compensates the downwards shift originating from the electroweak NLO correction to the \({\overline{\text {MS}}}\) topquark mass. This downwards shift is, however, enhanced by the negative shift for positive \(X_\mathrm{t}\). This is the reason for the asymmetric behavior observed in the lower panel of Fig. 1.
Note that the comparison made in the bottom panel of Fig. 3 does not exhibit the effect of the twoloop corrections to the \({\overline{\text {MS}}}\) top mass, since also for the curve without NNLL resummation the twoloop QCD corrections in the \({\overline{\text {MS}}}\)mass–polemass relation are employed. We have kept them because they constitute the by far dominant part of the twoloop corrections to the \({\overline{\text {MS}}}\) top mass, shifting the \({\overline{\text {MS}}}\) top mass down by 1.9 GeV. This downwards shift causes a downwards shift in \(M_\mathrm{h}\) of about the same size, as discussed before in the context of the electroweak NLO corrections to the \({\overline{\text {MS}}}\) top mass. Twoloop corrections to the \({\overline{\text {MS}}}\) top mass are formally not needed in the case of LL and NLL resummation. This means actually that the main effect of going from NLL to NNLL resummation is caused by the higherorder matching condition of the \({\overline{\text {MS}}}\) top mass, as in the case of including electroweak corrections into the resummation procedure.
6 Conclusions
We have presented and discussed the inclusion of electroweak contributions, electroweakino and gluino thresholds, and NNLL resummation in the EFT resummation of logarithmically enhanced terms in the calculation of the lightest Higgsboson mass \(M_\mathrm{h}\), on top of the fixedorder one and twoloop computation as currently available in the code FeynHiggs. Special attention is payed to a consistent combination of fixedorder diagrammatic and EFT methods taking care of scheme conversion and proper subtractions to avoid double counting. These improvements have become part of FeynHiggs. They shift the prediction for \(M_\mathrm{h}\), especially pronounced for positive values of the stopmixing parameter \(X_\mathrm{t}\) with downwards shifts in \(M_\mathrm{h}\) of about 1.7 GeV.
We found that this is mainly caused by the electroweak NLO corrections to the \({\overline{\text {MS}}}\) topquark mass. The genuine effect of resumming electroweak contributions shifts the Higgs mass upwards compensating the downwards shift induced by the smaller \({\overline{\text {MS}}}\) topquark mass. This effect becomes only relevant for SUSY scales larger than a few TeV. Furthermore, electroweak NLL contributions are found to be much smaller than electroweak LL contributions.
We also investigated the effect of various intermediate thresholds. In our framework, an electroweakino threshold yields significant contributions only for SUSY scales above 5 TeV. We found that a gluino threshold is completely negligible, since the main contributions sensitive to the gluino mass are already captured by the twoloop Feynman diagrammatic result.
Furthermore, we found NNLL resummation of \(\mathscr {O}(\alpha _\mathrm{s},\alpha _\mathrm{t})\) to shift the lightest Higgs mass downwards for positive stop mixing, whereas it leads to a larger upwards shift for negative values of \(X_\mathrm{t}\).
We aim to compare the results thoroughly to other publicly available codes [31, 49] in an upcoming publication. We also plan to extend the resummation procedure to scenarios with light nonSM Higgs bosons [29, 50].
Footnotes
Notes
Acknowledgments
We are thankful to Thomas Hahn for his invaluable help concerning all issues related to FeynHiggs, and to Sven Heinemeyer and Georg Weiglein for useful discussions.
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