High Precision Prediction for M h in the MSSM

The new particle discovered at the LHC can be interpreted as the lightest CP -even Higgs boson of the Minimal Supersymmetric Standard Model (MSSM), where M h denotes its mass. The experimental precision of this mass has reached the level below ∼ 500 MeV. In the MSSM M h can directly be predicted from the other parameters of the model. The accuracy of this prediction should at least match the one of the experimental result. The relatively high mass value of about 125 GeV has led to many investigations where the scalar top quarks are in the multi-TeV range. We review the recent improvement in the prediction for M h in the MSSM that has been reached by combining the existing ﬁxed-order result, comprising the full one-loop and leading and subleading two-loop corrections, with a resummation of the leading and subleading logarithmic contributions from the scalar top sector to all orders.


Introduction
After the spectacular discovery of a signal in the Higgs-boson searches at the LHC by ATLAS and CMS [1,2], now the exploration of the properties of the observed particle is in the main focus. In particular, the observation in the γγ and the ZZ ( * ) → 4 channels made it possible to determine its mass with already a remarkable precision. Currently, the combined mass measurement from ATLAS is 125.36 ± 0.37 ± 0.18 GeV [3], and the one from CMS is 125.03 +0.26 At the (planned) future International e + e − Linear Collider (ILC), using the Z-recoil method a precision of [5] δM ILC is currently anticipated. The other properties that have been determined so far, in particular the coupling strength modifiers [6,7], as well as spin, are compatible with the minimal realisation of the Higgs sector within the Standard Model (SM) [8]. However, the discovery can also be interpreted as a Higgs boson in a model beyond the SM, where the Minimal Supersymmetric Standard Model (MSSM) [9] is one of the leading candidates.
In the MSSM the Higgs sector consists of two scalar doublets accommodating five physical Higgs bosons. In lowest order these are the light and heavy CPeven h and H, the CP-odd A, and the charged Higgs bosons H ± . The parameters characterising the MSSM Higgs sector at lowest order are the gauge couplings, the mass of the CP-odd Higgs boson, M A , and tan β ≡ v 2 /v 1 , the ratio of the two vacuum expectation values. Accordingly, all other masses and mixing angles can be predicted in terms of those parameters, leading to the famous tree-level upper bound for the mass of the light CP-even Higgs boson, M h ≤ M Z , determined by the mass of the Z boson, M Z . This tree-level upper bound, which arises from the gauge sector, receives large corrections from the Yukawa sector of the theory, which can be of O(50%) (depending on the model parameters) upon incorporating the full one-loop and the dominant two-loop contributions [10,11,12].
The prediction for the light CP-even Higgs-boson mass in the MSSM is affected by two kinds of theoretical uncertainties. First, the parametric uncertainties induced by the experimental errors of the input parameters. Here the dominant source of parametric uncertainty is the experimental error on the top-quark mass, m t . Very roughly, the impact of the experimental error on m t on the prediction for M h scales like [13] δM As a consequence, high-precision top-physics providing an accuracy on m t much below the GeV-level is a crucial ingredient for precision physics in the Higgs sector [13]. The second type of uncertainties are the intrinsic theoretical uncertainties that are due to unknown higherorder corrections. An overall estimate for the lightest CP-even Higgs mass of δM intr h ∼ 3 GeV had been given in Refs. [10,12] (the more recent inclusion of the leading O(α t α 2 s ) 3-loop corrections [14], see below, has slightly reduced this estimated uncertainty by few times O(100 MeV)). It was pointed out that a more detailed estimate needs to take into account the dependence on the considered parameter region of the model. In particular, the uncertainty of this fixed-order prediction is somewhat larger for scalar top masses in the multi-TeV range.
The MSSM parameter space with scalar top masses in the multi-TeV range has received considerable attention recently, partly because of the relatively high value of M h ≈ 125 GeV, which generically requires either large stop masses or large mixing in the scalar top sector, and partly because of the limits from searches for supersymmetric (SUSY) particles at the LHC. While within the general MSSM the lighter scalar superpartner of the top quark is allowed to be relatively light (down to values even as low as m t ), both with respect to the direct searches and with respect to the prediction for M h (see e.g. Ref. [15]), the situation is different in more constrained models. For instance, global fits in the Constrained MSSM (CMSSM) prefer scalar top masses in the multi-TeV range [16,17,18,19].
Here we review the significantly improved prediction for M h in the MSSM from resumming large logarithms with large scalar top masses [20], which has an important impact on the phenomenology in the region of heavy squarks and on its confrontation with the experimental results. We briefly review the relevant sectors and the new, improved prediction for M h . The numerical analysis focuses on the effects of heavy scalar top masses in the general MSSM and in the CMSSM. The feasability of reaching the anticipated ILC precision will be briefly discussed.

The Higgs and scalar top sectors of the MSSM
In the MSSM with real parameters (we restrict to this case for simplicity; for the treatment of complex parameters see Refs. [21,22,23] and references therein), using the Feynman diagrammatic (FD) approach, the higher-order corrected CP-even Higgs-boson masses are derived by finding the poles of the (h, H)-propagator matrix. The inverse of this matrix is given by where m h,H,tree denote the tree-level masses, and Σ hh,HH,hH (p 2 ) are the renormalized Higgs boson selfenergies evaluated at the squared external momentum p 2 . For the computation of the leading contributions to those self-energies it is convenient to use the basis of the fields φ 1 , φ 2 , which are related to h, H via the (tree-level) mixing angle α: The new higher-order corrections reviewed here originate in the top/stop sector of the MSSM. The bilinear part of the top-squark Lagrangian, contains the stop mass matrix, Mt, given by with Z denote numerically small D-terms, A t is the trilinear coupling between the Higgs bosons and the scalar top quarks, and μ is the Higgs mixing parameter. The mass matrix can be diagonalized with the help of a unitary transformation Ut, yielding the two stop mass eigenvalues, mt 1 and mt 2 .
For the MSSM with real parameters the status of higher-order corrections to the masses and mixing angles in the neutral Higgs sector is quite advanced.
The complete one-loop result within the MSSM is known [24,25,26,27]. The by far dominant one-loop contribution is the O(α t ) term due to top and stop loops (α t ≡ h 2 t /(4π), h t being the top-quark Yukawa coupling). The computation of the two-loop corrections is already quite advanced and has meanwhile reached a stage where all the presumably dominant contributions are available [23,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43]. In particular, the O(α t α s ), O(α 2 t ) contributions to the self-energies -evaluated in the Feynman-diagrammatic (FD) and in the effective potential (EP) method -as well as the contributions -evaluated in the EP approach -are known. (For latest corrections to the charged Higgs boson mass, see Ref. [44].) The public code FeynHiggs [45,29,10,21,20] includes all of the above corrections 1 , where the on-shell (OS) scheme for the renormalization of the scalar quark sector has been used (another public code, based on the Renormalization Group (RG) improved Effective Potential, is CPsuperH [47]). A full 2-loop effective potential calculation (supplemented by the momentum dependence for the leading pieces and the leading 3loop corrections) has been published [48,49]. However, no computer code is publicly available (see, however, Ref. [50]). Most recently another leading 3-loop calculation at O(α t α 2 s ) became available (based on a DR or a "hybrid" renormalisation scheme for the scalar top sector), where the numerical evaluation depends on the various SUSY mass hierarchies [14], resulting in the code H3m (which adds the 3-loop corrections to the FeynHiggs result).

Improved calculation of M h
We review here the improved prediction for M h where we combine the fixed-order result obtained in the OS scheme with an all-order resummation of the leading and subleading contributions from the scalar top sector. We have obtained the latter from an analysis of the RG Equations (RGEs) at the two-loop level [51]. Assuming a common mass scale M S = √ mt 1 mt 2 (M S M Z ) for all relevant SUSY mass parameters, the quartic Higgs coupling λ can be evolved via SM RGEs from M S to the scale Q (we choose Q = m t in the following) where M 2 h is to be evaluated (see, for instance, Ref. [32] and references therein), Here v ∼ 174 GeV denotes the vacuum expectation value of the SM. Three coupled RGEs, the ones for λ, h t , g s (10) are relevant for this evolution, with the strong coupling constant given as α s = g 2 s /(4 π). Since SM RGEs are used, the relevant parameters are given in the MS scheme. We incorporate the one-loop threshold corrections to λ(M S ) as given in Ref. [32], where as mentioned above X t is an MS parameter. Furthermore, in Eq. (11) we have set the SM gauge couplings to g = g = 0, ensuring that Eq. (9) consists of the "pure loop correction" and will be denoted (ΔM 2 h ) RGE below. Using RGEs at two-loop order [51], including fermionic contributions from the top sector only, leads to a prediction for the corrections to M 2 h including leading and subleading logarithmic contributions at n-loop order, originating from the top/stop sector of the MSSM. We have obtained both analytic solutions of the RGEs up to the 7-loop level as well as a numerical solution incorporating the leading and subleading logarithmic contributions up to all orders. In a similar way in Ref. [49] the leading logarithms at 3-and 4-loop order have been evaluated analytically. Most recently a calculation using 3-loop SM RGEs appeared in Ref. [52].
A particular complication arises in the combination of the higher-order logarithmic contributions obtained from solving the RGEs with the fixed-order FD result implemented in FeynHiggs comprising corrections up to the two-loop level in the OS scheme. We have used the parametrisation of the FD result in terms of the running top-quark mass at the scale m t , where m pole t denotes the top-quark pole mass. Avoiding double counting of the logarithmic contributions up to the two-loop level and consistently taking into account the different schemes employed in the FD and the RGE approach, the correction ΔM 2 h takes the form Here (M 2 h ) FD denotes the fixed-order FD result, (ΔM 2 h ) FD,LL1,LL2 are the logarithmic contributions up to the two-loop level obtained with the FD approach in the OS scheme, while (ΔM 2 h ) RGE are the leading and subleading logarithmic contributions (either up to a certain loop order or summed to all orders) obtained in the RGE approach, as evaluated via Eq. (9). In all terms of Eq. (14) the top-quark mass is parametrised in terms of m t ; the relation between X MS t and X OS t is given by up to non-logarithmic terms, and there are no logarithmic contributions in the relation between M MS S and M OS S . Since our higher-order contributions beyond 2-loop have been derived under the assumption M A M Z , to a good approximation these corrections can be incorporated as a shift in the prediction for the φ 2 φ 2 selfenergy (where ΔM 2 h enters with a coefficient 1/ sin 2 β). In this way the new higher-order contributions enter not only the prediction for M h , but also consistently all other Higgs sector observables that are evaluated in FeynHiggs, such as the effective mixing angle α eff or the finite field renormalization constant matrix Z n [21].
The latest version of the code, FeynHiggs 2.10.0, which is available at feynhiggs.de, contains those improved predictions as well as a refined estimate of the theoretical uncertainties from unknown higher-order corrections. Taking into account the leading and subleading logarithmic contributions in higher orders reduces the uncertainty of the remaining unknown higherorder corrections. Accordingly, the estimate of the uncertainties arising from corrections beyond two-loop order in the top/stop sector is adjusted such that the impact of replacing the running top-quark mass by the pole mass (see Ref. [10]) is evaluated only for the nonlogarithmic corrections rather than for the full two-loop contributions implemented in FeynHiggs. First investigations using this new uncertainty estimate can be found in Refs. [17,53]. Some details in these directions can be found in Ref. [52]. We leave those refinements for future work.

Numerical analysis
In this section we review the analysis of the phenomenological implications of the improved M h prediction for large stop mass scales, as evaluated with FeynHiggs 2.10.0. In Ref. [20] it was shown that TeV. The plot furthermore shows that for M S ≈ 7 TeV (and the value of tan β = 10 chosen here) a predicted value of M h of about 125 GeV is obtained even for the case of vanishing mixing in the scalar top sector (X t = 0). Since the predicted value of M h grows further with increasing M S it becomes apparent that the measured mass of the observed signal, when interpreted as M h , can be used (within the current experimental and theoretical uncertaintes) to derive an upper bound on the mass scale M S in the scalar top sector, see also Ref. [54]. However, more robust statements in this direction will require a careful analysis of still present intrinsic as well as the parametric uncertainties.  In Fig. 2 we compare our result with the prediction obtained from the code H3m [14]. The comparison was performed in the CMSSM with the parameters set to m 0 = m 1/2 = 200 GeV . . . 15000 GeV, A 0 = 0, tan β = 10 and μ > 0. The spectra were generated with SoftSusy 3.3.10 [55]. The H3m result shown as blue line, containing the terms in O(α t α 2 s )×O(L 3 , L 2 , L 1 , L 0 ), can be compared with the FeynHiggs 3-loop result, O(L 3 , L 2 ), but restricted to O(α t α 2 s ) (green dashed). We find that the latter result agrees rather well with H3m, with maximal deviation of O(1 GeV) for M S < ∼ 10 TeV. The observed deviations can be attributed to the terms of O(L 1 , L 0 ) included in H3m, to the SUSY mass hierarchies taken into account in H3m, and to the use of different scalar top renormalization schemes employed in the two codes (where the latter effect is already expected to be at the GeV-level). Further investigations will be needed to explore the source of the main differences. However, adding also the 3-loop O(α 2 t α s , α 3 t )×O(L 3 , L 2 ) terms (solid green), as included in the FeynHiggs result, leads to a strong reduction of M h by ∼ 5 GeV for M S = 10 TeV (see also Ref. [49]). Going to the full resummed FeynHiggs result (red) exhibits a further, but smaller reduction of M h of about 2 GeV for M S = 10 TeV, even larger changes are found for M S > 10 TeV. Consequently, 3-loop corrections at O(α 2 t α s , α 3 t ) as well as corrections beyond 3-loop are clearly important for a precise M h prediction. Finally, in Fig. 3 we analyze the effects of the new M h evaluation in the CMSSM [53]. In the upper (lower) plot we show the m 1/2 -m 0 plane for tan β = 10(30), A 0 = 2.5 m 0 and μ > 0. Regions where the lightest SUSY particle (LSP) is charged are shaded brown, those where there is no consistent electroweak vacuum are shaded mauve, regions excluded by BR(b → sγ) measurements at the 2 σ level are shaded green, those favored by the SUSY interpretation of (g − 2) μ are shaded pink (with lines indicating the 1 σ (dashed) and 2 σ ranges (solid)), and strips with an LSP density appropriate to make up all the cold dark matter are shaded dark blue. The 95% CL limit from the ATLAS MET search is shown as a continuous purple contour, and the 68 and 95% CL limits from the CMS and LHCb measurements of BR(B s → μ + μ − ) are shown as continuous green contours, see Ref. [53]    In view of the anticipated future accuracy at the ILC, as given in Eq. (2), the remaining theory uncertainties in the current status of the M h evaluations will have to be re-analyzed carefully. It can be expected, see also Ref. [53], that for scalar top mass scales below the few-TeV level the intrinsic uncertainty is now, i.e. including the resummed contributions, at or below the level of ∼ 2 GeV. However, still substantial further refinements will be needed to reach the sub-GeV level. On the other hand, no investigation of the size of the intrinsic uncer-tainties has been performed for scalar top masses in the multi-TeV range, as explored here. Consequently, the prospects of reaching the sub-GeV level in δM intr h are so far unclear.

Conclusions
We have reviewed the improved prediction for the light CP-even Higgs-boson mass in the MSSM, obtained by combining the FD result at the one-and two-loop level with an all-order resummation of the leading and subleading logarithmic contributions from the top/stop sector obtained from solving the two-loop RGEs. Particular care has been taken to consistently match these two different types of corrections. The result, providing the most precise prediction for M h in the presence of large masses of the scalar partners of the top quark, has been implemented into the public code FeynHiggs and can be obtained at feynhiggs.de . We have found a sizable effect of the higher-order logarithmic contributions to M h in the MSSM and in particular in the CMSSM, where M h ∼ 125 GeV can be reached for lower values of m 1/2 and m 0 .