Uncertainties in the lightest \(CP\) even Higgs boson mass prediction in the minimal supersymmetric standard model: fixed order versus effective field theory prediction
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Abstract
We quantify and examine the uncertainties in predictions of the lightest \(CP\) even Higgs boson pole mass \(M_h\) in the Minimal Supersymmetric Standard Model (\({\text {MSSM}}\)), utilising current spectrum generators and including some threeloop corrections. There are two broadly different approximations being used: effective field theory (EFT) where an effective Standard Model (\(\text {SM}\)) is used below a supersymmetric mass scale, and a fixed order calculation, where the \({\text {MSSM}}\) is matched to \(\text {QCD}\times \text {QED}\) at the electroweak scale. The uncertainties on the \(M_h\) prediction in each approach are broken down into logarithmic and finite pieces. The inferred values of the stop mass parameters are sensitively dependent upon the precision of the prediction for \(M_h\). The fixed order calculation appears to be more accurate below a supersymmetry (SUSY) mass scale of \(M_S\approx 1.2~\text {TeV}\), whereas above this scale, the EFT calculation is more accurate. We also revisit the range of the lightest stop mass across finetuned parameter space that has an appropriate stable vacuum and is compatible with the lightest \(CP\) even Higgs boson h being identified with the one discovered at the ATLAS and CMS experiments in 2012; we achieve a maximum value of \(\sim 10^{11}\) GeV.
1 Introduction
The truncation of perturbation theory at a finite order generates a theoretical uncertainty on the prediction of \(M_{h}\). This then leads to an associated uncertainty in the inferred masses and mixings of stops that agree with the experimentally inferred value of \(M_{h}\). The allowed range of stop parameter space depends very sensitively on the accuracy of the \(M_{h}\) prediction. Equation (2) shows that the stop mass scale depends roughly exponentially upon \(M_h\) in the high \(m_{{\tilde{t}}_i}\) limit.^{1} Achieving the most precise prediction for \(M_h\) is then of paramount importance. In order to predict \(M_{h}\) with higher accuracy and greater precision, higherorder contributions and large log resummation are required. To date, terms up to twoloop order have been computed in the onshell scheme [8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19] and up to threeloop order in the \(\overline{\text {DR}}\)/\(\overline{\text {DR}}'\) scheme [4, 13, 14, 15, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32].
There are several current publicly available MSSM spectrum calculator computer programs on the market. These calculate the spectrum consistent with weakscale data on the gauge couplings and the masses of SM states. Each employs an approximation scheme. The two approximation schemes examined here are called the fixed order \(\overline{\text {DR}}'\) scheme and the EFT scheme, depicted in Fig. 1. The fixed order \(\overline{\text {DR}}'\) scheme matches effective QED\(\times \)QCD to the \({\text {MSSM}}\) at a single scale \({Q_\text {match}}\). The values \({Q_\text {match}}=M_Z\) or \({Q_\text {match}}=M_t\) are commonly taken, and data on gauge couplings and quark, lepton and electroweak boson masses are input at this scale \({Q_\text {match}}\) (see Ref. [43] for a more detailed description). These couplings are then run using \({\text {MSSM}}\) renormalisation group equations (RGEs) to M, where the Higgs potential minimisation conditions are imposed and supersymmetric physical observables including \(M_h\) are calculated. \(M_h\) is calculated using the known higher order diagrammatic corrections, up to three loops, of the order \(o \in \{{\alpha _{\text {s}}}{\alpha _t}\), \({\alpha _{\text {s}}}{\alpha _b}\), \({\alpha _t}^2\), \({\alpha _t}{\alpha _b}\), \({\alpha _b}^2\), \({\alpha _\tau }^2\), \({\alpha _{\text {s}}}^2{\alpha _t}\}\), where \({\alpha _{\text {s}}}= {g_{3}}^{2}/(4\pi )\), \(\alpha _{t,b,\tau } = y_{t,b,\tau }^2/(4\pi )\) and \(y_t\), \(y_b\), \(y_\tau \) are the top, bottom and tau Yukawa couplings, respectively, and \(g_3\) is the QCD gauge coupling. These fixedorder corrections include twoloop terms which are proportional to \(o \ln ^2 (M_S/M_Z)/(4 \pi )^2\) as well as terms of order \(o M_Z^2 / M_S^2 / (4 \pi )^2\). However, some threeloop terms, for example of order \(\{{\alpha _t}^{2}{\alpha _{\text {s}}}, {\alpha _t}^{3}\}\) \(\times \) \(\ln ^3 (M_S/M_Z) / (4 \pi )^3\), are missed. As \(M_S\) becomes larger (for example as motivated by the negative results of sparticle searches), such missing logarithmic higher order terms become numerically more important, and missing them will imply a larger uncertainty in the fixed order \(\overline{\text {DR}}'\) prediction of \(M_h\). This has motivated the approximation scheme which we call the EFT scheme, where the heavy SUSY particles are decoupled at the SUSY scale \(M_S\) and the RGEs are used to resum the large logarithmic corrections. However, the EFT scheme neglects terms of order \({M_{Z}}^{2}/M_S^2\) at the tree level and therefore is less accurate the closer \(M_S\) is to \(M_Z\). Which scheme is the most accurate for various different physical predictions is not obvious beforehand and depends on the MSSM parameters. It is one of our goals to determine in which domain of \(M_S\) the fixedorder scheme becomes less accurate than the EFT scheme.
The preceding paragraph has been greatly simplified for clarity of discussion. In the \({\text {MSSM}}\) there are many gauge and Yukawa couplings and oneloop corrections from all of these are included in the fixed order \(\overline{\text {DR}}'\) calculations. Also, we have used \(M_S\) as a catchall supersymmetric scale, but really the individual sparticles contribute to the logarithms and finite terms with their own masses, not with some universal value of \(M_S\).
The programs used for our \(M_{h}\) predictions are the fixed order \(\overline{\text {DR}}'\) spectrum generators SOFTSUSY 4.1.1 [43, 44], FlexibleSUSY 2.1.0 [45, 46] and HSSUSY 2.1.0 [46], which uses the EFT approach. We include the threeloop corrections that are available in Himalaya 1.0.1 [47].
In Ref. [48], the hybrid fixed order \(\overline{\text {DR}}'\)EFT calculation of FeynHiggs [49, 50] was compared to the purely EFT calculation of SUSYHD [42]. The observed numerical differences between the (mostly) onshell hybrid calculation of FeynHiggs and the \(\overline{\text {DR}}'\) calculation of SUSYHD were found to be mainly caused by renormalisation scheme conversion terms, the treatment of higherorder terms in the determination of the Higgs boson pole mass and the parametrisation of the top Yukawa coupling. When these differences are taken into consideration, excellent agreement was found between the two programs for SUSY scales above \(1~\text {TeV}\). This finding confirms that above this scale the terms neglected in the EFT calculation are in fact negligible and the EFT calculation yields an accurate prediction of the Higgs boson mass. Similarly, in Ref. [51] the \(\overline{\text {DR}}'\) hybrid fixed order/EFT calculation implemented in FlexibleSUSY (denoted as FlexibleEFTHiggs) was compared to the \(\overline{\text {DR}}'\) fixed order calculation available in FlexibleSUSY. A prescription for an uncertainty estimation of both calculations was given and it was found that (based on that uncertainty estimate) above a few TeV the hybrid and the pure EFT calculations are more precise than the fixed order \(\overline{\text {DR}}'\) calculation.
Our work differs from Refs. [48, 51] in that we perform a comparison between the \(\overline{\text {DR}}'\) fixed order and the pure EFT predictions. Our \(\overline{\text {DR}}'\) fixed order calculation is also a loop higher in order than the previous work. We shall give a prescription for the estimation of the theoretical uncertainties of the two \(M_h\) predictions in the \(\overline{\text {DR}}'\) scheme based on the procedures described in Refs. [40, 42, 51]. Based on our uncertainty estimates we derive an \(M_S\) region in that scheme, above which the EFT prediction becomes more precise than the fixed order one.
In Sect. 2, we estimate and dissect theoretical uncertainties in stateofthe art predictions of the lightest \(CP\) even Higgs boson pole mass in the \(\overline{\text {DR}}'\) scheme. Then, in Sect. 3, we update the upper bounds on the lightest stop mass from the experimental determination of the Higgs boson mass and from the stability of an appropriate vacuum by our detailed quantification of the theoretical uncertainties and stateoftheart calculation of \(M_h\). We summarise in Sect. 4.
2 Higgs boson mass prediction uncertainties

Missing higher order contributions to the Higgs self energy and to the electroweak symmetry breaking (EWSB) conditions.

Missing higher order corrections in the determination of the running \(\overline{\text {DR}}'\) gauge and Yukawa couplings and the VEVs from experimental quantities.
where \(M_S\) is the SUSY scale, usually set to \(M_S= \sqrt{m_{\tilde{t}_1}m_{\tilde{t}_2}}\). In Fig. 2 we show this uncertainty as the blue dashed line for a scenario with degenerate \(\overline{\text {DR}}'\) mass parameters (aside from the Higgs mass parameters, which are fixed in order to achieve successful EWSB), \(\tan \beta = 20\) and maximal stop mixing, \(X_t = \sqrt{6}M_S\). For this scenario \(\varDelta M_h^{(Q_\text {pole})}\) varies between 0.5–\(1~\text {GeV}\), depending on the SUSY scale. In Ref. [51] this uncertainty is larger, because the threeloop contribution to the Higgs boson mass was not included. The kink at \(M_S\approx 700~\text {GeV}\) is due to a switch in the approximation scheme being used in the calculation of the threeloop contribution of Himalaya: the integrands of the threeloop integrals were expanded in different sparticle “mass hierarchies” where different sparticles were approximated as being massless [31]. As \(M_S\) changes, Himalaya switches from one mass hierarchy to another one that fits better to the given parameter point, resulting in the kink. We note that \(\varDelta M_h^{(Q_\text {pole})}\) is approximately independent of \(M_S\) as \(M_S\) becomes large. The dominant \(Q_\text {pole}\) dependence comes from the first term on the righthand side of Eq. (2): from the running Z mass, at oneloop order. This will be cancelled by to leading order in \(\log (Q_\text {pole})\) by the oneloop electroweak corrections that are added to \(M_h\) by the spectrum generators that we employ. However, higher order logarithms (formally at the twoloop order) in the electroweak gauge couplings remain. These remaining pieces have no explicit dependence at leading order on \(M=M_S\). In the limit of large \(M_S\), the first term in the square brackets of Eq. (2) contains both a dependence on a large \(M_S\) and \(Q_\text {pole}\) through renormalisation of the running top Yukawa coupling \(y_t = \sqrt{2} m_t / (v \sin \beta )\). The \(Q_\text {pole}\) dependence of leading logarithm terms due to this are cancelled by the explicit twoloop terms of order \(\alpha _t^2/(4 \pi )^2\) in the \(M_h\) calculation that the spectrum generators employ, but higher powers of the logarithms do not cancel. The \(Q_\text {pole}\) dependence from this term is then formally of threeloop order, but is boosted somewhat by the large value of \(y_t\). For \(\tan \beta =20\) and large \(M_S\), the \(Q_\text {pole}\) dependence is small, partly aided by cancellations in the beta function of \(y_t\). However, for \(\tan \beta =5\), as is the case in Ref. [51], for example, one can see an increase in scale uncertainty with a larger \(M_S\) due to a larger value of \(y_t\) (and consequently a larger beta function/scale dependence).
In the following we compare the fixedorder Higgs boson mass prediction for this scenario to the pure EFT calculation of HSSUSY [46]. HSSUSY is a spectrum generator from the FlexibleSUSY package, which implements the highscale SUSY scenario, where the quartic \(\text {SM}\) Higgs coupling \(\lambda (M_S)\) is predicted from matching to the \({\text {MSSM}}\) at a high SUSY scale \(M_S\). It provides a prediction of the Higgs pole mass in the \({\text {MSSM}}\) in the pure EFT limit, \(v^2 \ll M_S^2\), up to the twoloop level \(\mathcal {O}({\alpha _{\text {s}}}({\alpha _t}+{\alpha _b}) + ({\alpha _t}+{\alpha _b})^2 + {\alpha _\tau }{\alpha _b}+ {\alpha _\tau }^2)\) [40, 55, 56, 57], including nexttonexttoleadinglog (NNLL) resummation [58, 59]. Additional pure \(\text {SM}\) three and fourloop corrections [60, 61, 62, 63, 64, 65] can be taken into account on demand.

\(\text {SM}\) uncertainty from missing higher order corrections in the determination of the running \(\text {SM}\) \(\overline{\text {MS}}\) parameters

EFT uncertainty from neglecting terms of order \(\mathcal {O}(v^2/M_S^2)\)

SUSY uncertainty from missing higher order contributions from SUSY particles
In Fig. 4 the \(M_h\) prediction in the fixedorder and the EFT approximation schemes are shown, together with their uncertainties.^{4} We see from the figure that the allowed \(M_S\) range depends sensitively on the approximation scheme: 1.3–\(3.0~\text {TeV}\) for fixedorder and 2.5–\(4.6~\text {TeV}\) for EFT. The Higgs mass increases as a function of the SUSY scale due to the logarithmic enhancement from heavy SUSY particles. As discussed above, the combined uncertainty of the fixedorder calculations (red band) tends to increase with increasing \(M_S\), while the uncertainty of the EFT calculation (grey band) decreases. The point where the fixedorder and the EFT calculation have the same estimated uncertainty is \(M_S^\text {equal}=1.2~\text {TeV}\). To improve the prediction near this point, a “hybrid” calculation should be used, where the large logarithms are resummed and \(\mathcal {O}(v^2/M_S^2)\) terms are included [46, 48, 50, 51, 57, 68, 69].
3 Upper bound on the lightest stop mass
4 Summary
We also compared the precision of the Higgs boson mass predictions of the stateoftheart \(\overline{\text {DR}}'\) fixedorder and EFT spectrum generators SOFTSUSY, FlexibleSUSY and HSSUSY in the \({\text {MSSM}}\). We estimated the uncertainties of the Higgs boson mass of the fixedorder and the EFT calculation by considering unknown logarithmic and nonlogarithmic higherorder corrections. As part of our work, we have provided a scheme to estimate the theoretical uncertainties in fixedorder \(\overline{\text {DR}}'\) calculations, based on the prescription used in Ref. [51]. Our prescription is an extension of Ref. [51], which takes further sources of uncertainty into account. By comparing the precision of the predictions of the two methods, we concluded that for SUSY masses below \(M_S^\text {equal}=1.2~\text {TeV}\), the fixedorder calculation is more precise, while above that scale the EFT method is more precise. To estimate this scale, we took the maximal mixing case where all soft supersymmetry breaking masses are set to be degenerate at \(M_S\) (except for \(m_{H_i}\), which are fixed in order to achieve successful EWSB) and where \(\tan \beta =20\). The precise value of \(M_S^\text {equal}\) will change depending upon the scenario and can vary between \(M_S= 1.0~\text {TeV}\) and \(1.3~\text {TeV}\) for minimal/maximal stop mixing and small/large values of \(\tan \beta \). However, once one imposes the experimental measurement upon \(M_h\), \(M_S\ge 1.3~\text {TeV}\) according to the fixedorder calculation^{7} and \(2 ~\text {TeV}\) according to the EFT calculation, as Fig. 4 shows. For \(M_S\ge 1.3~\text {TeV}\), the EFT has smaller uncertainties and so one is likely to be in a régime where \(M_h\) is better approximated by EFT methods. It is unclear as yet, however, whether details of the \({\text {MSSM}}\) spectrum other than \(M_h\) are better approximated by EFT methods. One question which we have not addressed is: which approximation scheme (fixed order \(\overline{\text {DR}}'\) or EFT) is more accurate when there is a hierarchical sparticle spectrum? It is quite possible, for example, that the stops are heavy but several of the other \({\text {MSSM}}\) sparticles are significantly lighter. For such scenarios the precision of the fixed order \(\overline{\text {DR}}'\) calculation would have to be compared with the precision of an appropriate EFT that contains the light sparticles. We leave such a study for future work.
Footnotes
 1.
More precisely, \(M_{h}^2\) has a logarithmic dependence on M in the large M limit.
 2.
Large stop masses make the running softbreaking squared Higgs mass parameters very large, requiring a huge cancellation in the minimisation of the Higgs potential to achieve \(v\sim 246~\text {GeV}\).
 3.
 4.
There are small differences in the calculations of SOFTSUSY and of FlexibleSUSY producing their \(M_h\) predictions: for example, the twoloop calculations of the electroweak corrections differ.
 5.
Around \(X_t \approx 0\) the Higgsinos and electroweak gauginos give the dominant negative contribution to \(\lambda (M_S)\) for \(\tan \beta = 1\). For slightly larger values of \(X_t\) the stop contributions become dominant, leading to a positive \(\lambda (M_S)\). For large stop mixing, the stop contribution becomes negative as well, driving \(\lambda (M_S) < 0\) again.
 6.
For slightly larger values of \(\tan \beta \) the region around \(X_t \approx 0\) becomes allowed. However, with larger \(\tan \beta \) the treelevel Higgs boson mass rapidly increases, which leads to a significantly lower bound on the lightest stop mass.
 7.
For a positive \(X_t\) and varying \(\tan \beta \), this may be reduced slightly to 1.1\(~\text {TeV}\).
Notes
Acknowledgements
We would like to thank the KUTS series of workshops, and LPTHE Paris for hospitality extended during the commencement of this work. A.V. would like to thank Emanuele Bagnaschi, Pietro Slavich and Georg Weiglein for many helpful discussions on the Higgs boson mass uncertainty estimate and Emanuele Bagnaschi for providing the two and threeloop shifts to reparametrise the quartic Higgs coupling in terms of the MSSM top Yukawa coupling. We thank Pietro Slavich for detailed comments on the manuscript. B.C.A. would like to thank the Cambridge SUSY Working Group for helpful discussions.
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