Supersymmetric Models in Light of Improved Higgs Mass Calculations

We discuss the parameter spaces of supersymmetry (SUSY) scenarios taking into account the improved Higgs-mass prediction provided by FeynHiggs 2.14.1. Among other improvements, this prediction incorporates three-loop renormalization-group effects and two-loop threshold corrections, and can accommodate three separate mass scales: m_{\tilde q} (for squarks), m_{\tilde g} (for gluinos) and m_{\tilde\chi} (for electroweakinos). Furthermore, it contains an improved treatment of the DRbar scalar top parameters avoiding problems with the conversion to on-shell parameters, that yields more accurate results for large SUSY-breaking scales. We first consider the CMSSM, in which the soft SUSY-breaking parameters m_0 and m_{1/2} are universal at the GUT scale, and then sub-GUT models in which universality is imposed at some lower scale. In both cases, we consider the constraints from the Higgs-boson mass M_h in the bulk of the (m_0, m_{1/2}) plane and also along stop coannihilation strips where sparticle masses may extend into the multi-TeV range. We then consider the minimal anomaly-mediated SUSY-breaking (mAMSB) scenario, in which large sparticle masses are generic. In all these scenarios the substantial improvements between the calculations of M_h in FeynHiggs 2.14.1 and FeynHiggs 2.10.0, which was used in an earlier study, change significantly the preferred portions of the models' parameter spaces. Finally, we consider the pMSSM11, in which sparticle masses may be significantly smaller and we find only small changes in the preferred regions of parameter space.


Introduction
These are the squark and gluino masses, mq, mg, and a scale mχ characterizing the overall electroweakino mass scale, thus making the connection to DM, assuming it to be given by the lightest neutralino,χ 0 1 [13,14]. In addition, problems that occur when combining an infinite tower of resummed logarithms with a fixed-order result where DR input parameters of the scalar top sector have been converted into the corresponding parameters of the on-shell (OS) renormalization scheme can now be avoided by performing the calculation directly in the DR scheme. Finally a new, improved procedure for determining the poles of the Higgs-boson propagator matrix has been introduced. Section 2.1 contains a review of FeynHiggs 2.14.1 and its relations to other codes for calculating M h in the MSSM, and Section 2.2 makes a specific comparison of FeynHiggs 2.14.1 with FeynHiggs 2.10.0.
In Section 3 of this paper we explore the significance of these advances for a number of MSSM scenarios with different phenomenological features that are sensitive to different aspects of FeynHiggs 2.14.1. 1 The first of these is the CMSSM [16][17][18][19][20], in which the soft SUSYbreaking scalar mass parameter m 0 and gaugino mass parameter m 1/2 are each assumed to be universal at the GUT scale M GUT .
The second example is provided by 'sub-GUT' models in which this universality is imposed at some scale M in ≤ M GUT [17,18,[20][21][22]. The LHC searches impose severe constraints on these models, favoring parameter sets along the stop coannihilation [23][24][25][26][27] and focus-point strips [28]. These extend out to multi-TeV sparticle masses with stop masses mt 1,2 that are strongly nondegenerate in general. Moreover, in the focus-point case mχ0 1 mt 1 , whereas these masses are very similar along the stop coannihilation strip.
Thirdly, we consider the minimal anomaly-mediated SUSY-breaking (mAMSB) model [29,30], in which sfermion masses are typically several tens of TeV, whereas values of mχ0 1 1 TeV or 3 TeV are preferred by the DM density constraint. For a recent global analysis of this model taking into account the constraints from Run 1 of the LHC, see [31].
Finally, we consider a phenomenological MSSM scenario [32,33] with 11 free parameters specified at the electroweak scale, as has recently been analyzed including LHC Run 2 data in [34]. A priori, this scenario would allow many possible mass hierarchies, as well as many near-degeneracies between sparticle masses that could dilute the classic missing-transverseenergy (/ E T ) signatures at the LHC and permit lighter sparticles than are allowed in the CMSSM and sub-GUT models.
In each of these scenarios, our primary concern is the implications of improvements in the FeynHiggs 2.14.1 calculation of M h (compared to previous, less sophisticated calculations) for the model parameter space.

Higgs Mass Calculations
The experimental accuracy of the measured mass of the observed Higgs boson has already reached the level of a precision observable, with an uncertainty of less than 300 MeV [9]. This precision should ideally be matched by the theoretical uncertainty in the prediction of the with mg/mq 1, where mg denotes the gluino mass and mq the scalar top mass scale, are not yet included in any code: the corrections by log(mg/mq) in this hierarchy can presently not yet be resummed. These logarithms could lead to large effects for mg/mq 4, a possibility that we comment on later in our numerical analysis.
In order to provide a reliable prediction for the Higgs-boson masses in both low-and high-scale MSSM scenarios, the resummation of the leading and subleading logarithms can be combined with the fixed-order results in the MSSM in the so-called "hybrid approach", thereby keeping track of the power-suppressed terms that are neglected in a simple EFT approach in which the low-energy EFT does not include higher-dimensional operators 3 . The hybrid approach was first implemented into the code FeynHiggs [12,15,41,73,[84][85][86][87][88][89]. In the first version that adopted this method, FeynHiggs 2.10.0, one light Higgs doublet at the low scale was assumed, and the logarithms originating in the top/scalar top sector were resummed [12]. Further refinements have been presented more recently in Refs. [84,85] 4 . More recently, the hybrid approach has been extended to support such spectra where a full Two-Higgs-Doublet-Model (2HDM) is required as the low-energy EFT [90]. However the latter are not implemented in the current public release of FeynHiggs and therefore they are not used in the current paper, see the discussion in Sect. 2.2 for more details.
For completeness, we also mention here some further corrections that are available in the literature. The full O(αα s ) corrections, including the complete momentum dependence at the two-loop level, became recently available in Ref. [64]. A (nearly) full two-loop effective potential calculation, including also the leading three-loop corrections up to next-to-leadinglogarithm (NLL) level, has also been published [61,74,91,92], but is not publicly available as a computer code. Another leading three-loop calculation of O(α t α 2 s ), depending on various SUSY mass hierarchies, has been performed in [93,94], and is included in the code H3m that is now available as a stand-alone code, Himalaya [95]. Another approach to the combination of logarithmic resummation with fixed-order results has been presented in Ref. [96] and included in FlexibleSUSY. Subsequently it was also implemented in the SARAH+SPheno [97] framework. We also note that Ref. [98] has studied the issue of the comparison of the theoretical uncertainties in SoftSUSY vs. HSSUSY. Finally, there is a recent calculation [99] that resums terms of leading order in the top Yukawa coupling and NNLO in the strong coupling α s , including the three-loop matching coefficient for the quartic Higgs coupling of the SM to the MSSM between the EFT and the fixed-order expression for the Higgs mass, which is available in an updated version of the Himalaya code [95]. However, a detailed numerical comparison of FeynHiggs 2.14.1 with other codes to calculate M h is beyond the scope of this paper. The main advances in FeynHiggs 2.14.1 in comparison to FeynHiggs 2.10.0 are related to the EFT part of the calculation. The resummation of large logarithmic contributions in FeynHiggs 2.10.0 was restricted to O(α s , α t ) leading-logarithmic (LL) and NLL contributions.

Comparison between
Since then, electroweak LL and NLL contributions as well as O(α s , α t ) next-to NLL (NNLL) contributions have been included. This means, in particular, that the full SM two-loop RGEs and partial three-loop RGEs 5 are used for evolving the couplings between the electroweak scale and the SUSY scale M SUSY . At the SUSY scale, full one-loop threshold corrections and (nondegenerate) threshold corrections of O(α s α t , α 2 t ) are used for the matching of the effective SM to the full MSSM, taken from Ref. [76] and from Refs. [77,78], respectively. Numerically, the electroweak LL and NLL contributions amount to an upward shift of M h of ∼ 1 GeV for a SUSY scale of a few TeV. The NNLL contributions are numerically relevant only for large stop mixing, shifting M h downwards by ∼ 1 GeV for positive X t and upwards by ∼ 1 GeV for negative X t (where the off-diagonal entry in the stop mass matrix for real parameters is m t X t ).
For consistency with this logarithmic precision, one must choose appropriate matching conditions with physical observables at the electroweak scale. This is relevant, in particular, for the MS top quark mass in the SM. In FeynHiggs 2.10.0, the corrections of O(α s , α t ) in the mass were used. The inclusion of electroweak LL and NLL resummation as well as NNLL of O(α s , α t ) implies the need to use instead the NNLO MS top quark mass of the SM, as done in FeynHiggs 2.14.1. This modification not only implies changes for large SUSY scales but also impacts significantly the prediction of M h for low SUSY scales, as the shift in the top quark mass affects the non-logarithmic terms that are relevant in this regime. The combined electroweak one-loop as well as the two-loop corrections amount to a downwards shift of the MS top mass of the SM by ∼ 3 GeV. The effect on M h is of similar size.
The EFT calculation in the new FeynHiggs version allows one to take into account three different relevant scales. In addition to the SUSY scale mq-which was the only scale in FeynHiggs 2.10.0-an electroweakino scale mχ and a gluino scale mg are available. They allow one to investigate scenarios with light electroweakinos and/or gluinos. This corresponds to a tower of up to three EFTs (SM, SM with electroweakinos, SM with gluinos, SM with electroweakinos and gluinos). Besides the limitation that mg/mq should not be too large (see the discussion above, all scales can be chosen independently from each other, though the gluino threshold has a negligible numerical influence in this case. Also, the electroweakino threshold becomes relevant only for a large hierarchy between the electroweakino scale and the SUSY scale (mχ/M SUSY 1/10), leading to upward shifts of M h of ∼ 1 GeV.
The second main advance is a better handling of DR input parameters. The fixed-order calculation of FeynHiggs by default employs a mixed OS/DR scheme for renormalization, in which the parameters of the stop sector are fixed employing the OS scheme. In FeynHiggs 2.10.0, this was the only available renormalization scheme. Therefore, a one-loop conversion between the DR and the OS scheme was employed in the case of DR input parameters. Whilst, for a fixed-order result, such a conversion leads to shifts that are beyond the calculated order, this is no longer the case if a fixed-order result is supplemented by a resummation of large logarithms. As shown in [85], the parameter conversion in this case induces additional logarithmic higher-order terms that can spoil the resummation. As a solution for this issue, an optional DR renormalization of the stop sector is implemented in FeynHiggs 2.14.1. This renders a conversion of the stop parameters unnecessary. Note, however, that the DR sbottom input parameters are still converted to the OS scheme. In particular for large SUSY scales, employing directly the 5 The electroweak gauge couplings are neglected at the three-loop level.

6
DR scheme for the stop sector parameters and avoiding the conversion to the OS scheme affects the results significantly: e. g., for SUSY scales of ∼ 20 TeV, shifts in M h of ∼ 10 GeV were observed compared to the result based on the parameter conversion with the sign of the shift depending on the size of the stop mixing. Also, for low SUSY scales of ∼ 1 TeV, the prediction using the DR scheme of the stop sector parameters differs from that employing the conversion to the OS scheme by a downward shift in M h of ∼ 1 GeV in the case of large stop mixing. For SUSY scales below 1 TeV, where the impact of higher-order logarithmic contributions is relatively small, the observed shift can be interpreted to a large extent as an indication of the possible size of unknown higher-order corrections.
In addition to these improvements, also the Higgs pole determination has been reworked. It was noted in [85] that there is a cancellation between two-loop contributions from subloop renormalization and terms arising through the pole determination. In the fixed-order calculation, these terms are of higher order, which are not controlled. In FeynHiggs 2.10.0, the pole determination was performed numerically employing the DR scheme for the Higgs field renormalization. As a consequence of this procedure, the two-loop contributions from sub-loop renormalization were not included at the same order as the terms arising through the pole determination, resulting in an incomplete cancellation. In FeynHiggs 2.14.1 the pole determination has been adapted in order to ensure a complete cancellation. 6 The numerical impact of this improved pole determination procedure increases with rising M SUSY . For M SUSY in the multi-TeV range, it amounts to a downward shift of M h of ∼ 1 GeV.
Finally, the handling of complex input parameters in FeynHiggs was improved. In the fixedorder calculation, the corrections of O(α 2 t ) with full dependence on the phases of complex parameters were implemented [51,52,100,101] (see also [53]). In addition, an interpolation of the EFT calculation in the case of non-zero phases was introduced. Numerically, this can lead to shifts of M h of up to 3 GeV. As we do not discuss here the effects of the phases of complex parameters, we do not provide further details that can be found in Ref. [15].
Summing up this discussion, we generally expect the prediction of M h of FeynHiggs 2.14.1 to be lower than that of FeynHiggs 2.10.0. In the case of DR input parameters, the large shifts compared to the previous result that employed a conversion to the OS scheme for the renormalization of the stop sector can, however, outweigh the other effects and lead to an overall upward shift of M h .

Calculations in Specific MSSM Scenarios
In this Section, we illustrate the implications of the improved prediction for M h implemented in FeynHiggs 2.14.1 in the context of several specific MSSM scenarios. The first of these is the CMSSM [16][17][18][19][20], in which the soft supersymmetry-breaking scalar masses m 0 , the gaugino 6 In FeynHiggs 2.14.1, the Higgs poles are determined by expanding the Higgs propagator matrix around the one-loop solutions for the Higgs masses. Due to instabilities in this method close to crossing points, where two of the Higgs bosons change their role, in the most recent FeynHiggs version 2.14.3 [15] the Higgs poles are again determined numerically. In order to avoid inducing higher-order terms that would cancel in a more complete calculation, the Higgs field renormalization is used to absorb these. Since no crossing points appear in the scenarios investigated in this work, using FeynHiggs 2.14.3 instead of FeynHiggs 2.14.1 would not lead to significant numerical differences. masses m 1/2 and the trilinear parameters are all constrained to be universal at the GUT scale. The second scenario we study is a class of sub-GUT models [17,18,[20][21][22], in which these universality relations hold at some renormalization scale M in < M GUT , as occurs, e. g., in miragemediation models [102]. We then discuss minimal anomaly-mediated models [29,30], in which the scalar masses are typically much greater than the gaugino masses. For all of these models, we use SSARD [103] to compute the particle mass spectrum and relic density. We note that the convention for A terms used in SSARD is opposite to that used in FeynHiggs. Finally, we study a phenomenological version of the MSSM [32,33] with 11 free parameters in the soft supersymmetry-breaking sector, the pMSSM11, allowing for many possible sparticle mass hierarchies. In all cases we assume that the lightest supersymmetric particle (LSP) is the lightest neutralinoχ 0 1 and provides the full DM density [104].

The Light Higgs-Boson Mass in the CMSSM
The four-dimensional parameter space of the CMSSM that we consider here includes a common input gaugino mass parameter, m 1/2 , a common input soft SUSY-breaking scalar mass parameter, m 0 , and a common trilinear soft SUSY-breaking parameter, A 0 , which are each assumed to be universal at the scale M GUT (defined as the renormalization scale where the two electroweak gauge couplings are equal), and the ratio of MSSM Higgs vevs, tan β. There is also a discrete ambiguity in the sign of the Higgs mixing parameter, µ. In the CMSSM, renormalization group (RG) effects typically produce hierarchies of physical sparticle masses, e. g., between gluinos and electroweakly-interacting gauginos and between squarks and sleptons. The limits from LHC searches for sparticles generally require at least the strongly-interacting sparticles to be relatively heavy. Accurate calculations of M h for MSSM spectra in the multi-TeV range require many of the improvements made in FeynHiggs 2.14.1 compared to FeynHiggs 2.10.0.
Reconciling the cosmological dark matter density [104] of the LSP with the relatively heavy spectra that LHC searches impose on the CMSSM typically requires specific relations between some of the sparticle masses. One such example is the stop coannihilation strip, and another is the focus-point region, which we discuss in the two following subsections.

Stop Coannihilation Strips in the CMSSM
We first consider in some detail examples of stop coannihilation strips. In this case the lighter stop mass mt 1 and the mass of the LSP, mχ0 1 , must be quite degenerate. The relic density constraint alone would allow them to weigh several TeV but the allowed range of mass scales is in general restricted by the measurement of M h , for more details see Ref. [20] (where FeynHiggs 2.13.1 was used). It is therefore very important that the MSSM calculation of M h along the stop coannihilation strip is optimized.
In Fig. 1 we show four examples of (m 1/2 , m 0 ) planes in the CMSSM for tan β = 5. The upper panels are for A 0 = 3 m 0 , and the lower panels are for A 0 = −4.2 m 0 , assuming that the Higgs mixing parameter µ > 0 (left panels) or µ < 0 (right panels). In each panel, the brick-red shaded regions are excluded because they feature a charged LSP, which is theτ 1 in the lower right regions and thet 1 in the upper left regions. There are very narrow dark blue strips close to these excluded regions where the LSP contribution to the dark matter density Ω χ h 2 < 0.2. This range is chosen for clarity, as the range Ω CDM h 2 = 0.1193 ± 0.0014 allowed by cosmology [104] would correspond to a much thinner strip that would be completely invisible. Even with the extended range for the relic density, the line is barely visible. 7 As we discuss in more detail below, the coannihilation strips generally have endpoints at very high masses, where the cross section becomes too small to ensure the proper relic density, even when m χ = mt 1 . The locations of these endpoints for the Planck range of Ω CDM h 2 are indicated by X marks along the strips. The panels feature contours of M h calculated using FeynHiggs 2.14.1 (red solid lines) and In the absence of a detailed uncertainty estimate that depends on the considered region of the parameter space (the update of the uncertainty estimate of FeynHiggs taking into account the latest improvements in the Higgs-mass prediction is still a work in progress), here and later we consider values of the input mass parameters as acceptable for which FeynHiggs 2.14.1 yields M h = 125 ± 3 GeV, i. e., M h ∈ [122, 128] GeV (where the additional experimental uncertainty is negligible in comparison). This range from FeynHiggs 2.14.1 is shaded light orange. We discuss this constraint in more detail below, but it is already clear from Fig. 1 that FeynHiggs 2.14.1 favors ranges of mt 1 and mχ0 1 that are quite different from those that would have been indicated by FeynHiggs 2.10.0.
For A 0 = 3 m 0 and µ > 0, the Higgs mass decreases rapidly as the stop LSP boundary is approached. In this case, the Higgs mass calculated using FeynHiggs 2.10.0 is too small all along the coannihilation strip. Furthermore, we see that FeynHiggs 2.10.0 was not able to produce a reliable result beyond m 0 13 TeV. Since the endpoint of the coannihilation strip is at much larger m 0 , FeynHiggs 2.14.1 offers a significant improvement. This version of FeynHiggs yields values of the Higgs mass that are significantly larger along the strip, rising as high as M h = 128 GeV at the endpoint which is not seen in this panel as it lies beyond the shown range in (m 1/2 , m 0 ). For A 0 = 3 m 0 and µ < 0, the Higgs mass is reduced in the newer version of FeynHiggs for most of the strip, though it is larger for m 1/2 6 TeV. While both versions of FeynHiggs provide strip segments with an acceptable Higgs mass, the location of the segment shifts upwards in the new version. In this case, the Higgs mass is M h = 135 GeV at the endpoint of the coannihilation strip, so the Higgs mass itself provides a constraint m 1/2 6 TeV, as seen more clearly in the profile plots discussed below. The endpoint is marked by an X at (m 1/2 , m 0 ) ∼ (11.3, 16.1) TeV. When A 0 = −4.2 m 0 and µ > 0, we clearly see a large difference between FeynHiggs 2.10.0 and FeynHiggs 2.14.1. In this case, the endpoint of the coannihilation strip is found at lower (m 1/2 , m 0 ). With FeynHiggs 2.10.0, we find M h < 122 GeV at the endpoint (as has also been found using FeynHiggs 2.11.3 [19]), whereas with FeynHiggs 2.14.1, we find M h = 128 GeV at the endpoint. When with FeynHiggs 2.14.1 we find M h = 128 GeV at the endpoint.
We also show in Fig. 1 as green lines contours of the lifetime for the proton decay p → K + ν of 6.6 × 10 33 yrs, the current lower limit for this decay mode. These contours have been calculated in the minimal SU(5) GUT model, neglecting possible effects due to new degrees of freedom at the GUT scale. Even though this calculation is probably inapplicable in a realistic GUT completion of the CMSSM, it does indicate that proton stability is unlikely to be a headache along the stop coannihilation strip in the CMSSM with tan β = 5 with TeV scale masses [18,19,105]. The position of this contour is similar in all four panels as the proton lifetime is mostly sensitive to tan β rather than the signs of A 0 or µ. Fig. 2 shows a similar set of plots for tan β = 20 and A 0 = 2.75 m 0 (upper panels) and for tan β = 20 and A 0 = −3.5 m 0 (lower panels), with µ > 0 (left panels) and µ < 0 (right panels). For specific values of m 1/2 and m 0 , the calculated values of M h are generally larger for tan β = 20 than for tan β = 5, as was to be expected. We see again substantial differences between the values of M h obtained from FeynHiggs 2.14.1 (red solid lines) and from FeynHiggs 2.10.0 (thin gray dashed lines), in particular along the stop coannihilation strip. Once again, we see that when A 0 > 0 and µ > 0, the contours of M h run almost parallel to the boundary of the LSP region, implying that the values of M h along the stop coannihilation strip are very sensitive to the input parameters and the level of sophistication of the M h calculation. Since the coannihilation strip extends beyond the range of the plot, both versions of FeynHiggs yield acceptable segments along the strip, albeit with different mass ranges. For A 0 < 0, the Higgs-mass contours no longer run parallel to the coannihilation strip, and FeynHiggs 2.14.1 predicts M h = 130 GeV at the endpoint, which is found at much lower m 1/2 and m 0 as marked by the X in the figure. At this higher value of tan β, there is not a large difference in the Higgs mass when the sign of µ is reversed, since the contribution to X t depends on µ/ tan β. Although the difference may appear small, when A 0 > 0, the Higgs mass is significantly larger along the strip as one approaches the endpoint at large m 1/2 and m 0 . We note that for A, µ < 0, at high m 1/2 and low m 0 there is a lack of convergence of the RGEs, due to a divergent b-quark Yukawa coupling, shown by the gray shading.
We note that the green contours where τ (p → K + ν) = 6.6 × 10 33 yrs in the minimal SU(5) GUT model are at much larger values of m 1/2 and m 0 for tan β = 20 than they were for tan β = 5, as was also to be expected. However, we emphasize that the calculation of the proton lifetime is sensitive to the details of the GUT dynamics, and that proton stability may be an issue but is not necessarily a problem for the CMSSM with tan β = 20. 8 Details of the coannihilation strips and endpoints are seen more clearly in Fig. 3, which shows the profiles of the stop coannihilation strips for tan β = 5 that were shown in Fig. 1.

Focus-Point Strips in the CMSSM
We now turn to an alternative mechanism in the CMSSM that can yield an acceptable cold dark matter density even for large values of (some) input parameters. This is the focus-point region, where the neutralino LSP acquires a significant Higgsino component that enhances (co)annihilation rates, thereby bringing the relic density down into the allowed range.
Examples of focus-point strips are visible in the (m 1/2 , m 0 ) planes shown in Fig. 5. The regions shaded pink in these plots are where the electroweak symmetry-breaking conditions cannot be satisfied, and the dark blue strips running along the boundaries of these regions (now clearly visible) are the focus-point strips. To make these strips more visible, we used the range 0.06 < Ω χ h 2 < 0.2. As before, the brick red shaded regions are where the LSP is charged. In addition to the stau-LSP regions in the lower right parts of the planes, we see in the upper panels for tan β = 10, A 0 = 0 and the two signs of µ additional brick red strips where the LSP is a chargino. In the lower right panel for tan β = 30, A 0 = 0 and µ < 0 there is a gray shaded region at large m 1/2 where the RGEs for the Yukawa coupling of the b quark break down. This region expands as tan β is increased when µ < 0. instead controlled by the value of µ, which tends towards zero as the pink region is approached. For small µ, the LSP becomes Higgsino-like, and the relic density is determined by Higgsino annihilations and coannihilations with the second Higgsino and chargino, which are nearly degenerate in mass with the LSP. While the extent of the strips is very large, as one can see in each of the panels, it is limited by the Higgs mass which differs in the two versions of FeynHiggs. The profiles of these strips are shown in Fig. 6 for tan β = 10 upper panels) and tan β = 30 (lower panels), for µ > 0 (left panels) and for µ < 0 (right panels), with A 0 = 0 in all cases. There is little difference between the calculations of M h using FeynHiggs 2.14.1 (red lines) and   For tan β = 30, the contour would lie beyond the scope of the plot. As a consequence, the proton decay limit is in conflict with the upper limit derived from the Higgs mass. However, we stress again that this should be viewed as a constraint on the GUT rather than a problem for the low-energy supersymmetric model.

Sub-GUT Models
We now extend the previous discussion to a 'sub-GUT' class of SUSY models, in which the soft SUSY-breaking parameters are universal at some input scale M in below the GUT scale M GUT but above the electroweak scale [17,18,21]. Models in this class may arise if the soft SUSYbreaking parameters in the visible sector are induced by gluino condensation or some dynamical mechanism that becomes effective below the GUT scale. Examples of sub-GUT models include those with mirage mediation [102] of soft SUSY breaking, and certain scenarios for moduli stabilization [106].
The reduced RG running below M in , relative to that below M GUT in the CMSSM and related models, leads in general to SUSY spectra that are more compressed [21]. These lead, in particular, to increased possibilities for coannihilation processes. The reduced RG running also suggests a stronger lower limit on mχ0 1 , because of a smaller hierarchy to the gluino mass, and there are also smaller hierarchies between the squark and slepton masses. For a discussion of the implications for LHC searches for sparticles in sub-GUT models, see [22].
The five-dimensional parameter space of the sub-GUT MSSM that we consider here includes, besides M in and tan β, the three soft supersymmetry-breaking parameters m 1/2 , m 0 and A 0 that are familiar from the CMSSM, but which are now assumed to be universal at the sub-GUT input mass scale M in < M GUT .  [20], the masses of the three lightest neutralinos are quite similar in these regions of parameter space, and multiple coannihilations interplay, enhanced through the heavy Higgs funnel. The chargino LSP region expands when M in = 10 9 GeV (middle left) and merges with the stau-LSP region when M in = 10 10 GeV (middle right).
Contours of M h with values determined by FeynHiggs 2.14.1 are shown by red solid curves and determined by FeynHiggs 2.10.0 with dashed gray curves. In much of the parameter space, the FeynHiggs 2.14.1 values are lower by about 4 GeV than the values produced by FeynHiggs 2.10.0. This difference shifts the viable regions of the parameter space, as is seen more clearly in Fig. 8, which shows the profiles along the dark matter strips in these sub-GUT scenarios. In the top left panel for tan β = 20, A 0 = 2.75 m 0 , M in = 10 7 GeV and µ > 0 we distinguish two groupings of lines, one extending up to m 1/2 ∼ 3 TeV, and the other from m 1/2 ∼ 3 TeV to m 1/2 ∼ 12 TeV corresponding, respectively, to the near-vertical 'peninsula' at low m 1/2 in Fig. 7 that extends up to m 0 ∼ 5 TeV and to the arc that lies close to the boundary of the   Fig. 7, and are now only viable at m 1/2 6.5 TeV. We see that mχ0 1 6.6 TeV along both the dark matter strips. At still larger M in = 10 10 GeV, the lower strip in the previous plot has now morphed into a stop-coannihilation strip reminiscent of those in the CMSSM, which runs nearly parallel with the Higgs-mass contours. In this case we find that mχ0 The two bottom panels of Fig. 7 illustrate other features of the sub-GUT parameter space. The bottom left panel has A 0 = 2.75 m 0 , M in = 10 9 GeV and µ > 0 as before, but tan β = 40. Comparing with the middle left panel for tan β = 20, we see that the stop-LSP lobe has contracted whereas the stau-LSP lobe has expanded, the stop-coannihilation band has broadened, and the chargino-LSP region has disappeared. The Higgs-mass profiles in Fig. 8 in this case correspond to the two sides of the stop-coannihilation region in Fig. 7, the upper, flatter profiles corresponding to the lower strip running parallel to the Higgs-mass contours and the steeper profiles to the upper stop-coannihilation strip. While M h is not very sensitive to the version of FeynHiggs for the former strip, FeynHiggs 2.14.1 improves the latter by lowering M h  by 3-4 GeV. Finally, the bottom right panel has tan β = 20, A 0 = 2.75 m 0 , M in = 10 9 GeV and µ < 0. It is relatively similar to the middle left panel, which has the opposite sign of µ but identical values of the other parameters. The main difference is the appearance of a 'causeway' between the chargino-LSP 'island' and the stop-LSP lobe. We see that the results for the Higgs-mass profiles are also very similar, indicating that the sign of µ is less important than the values of the other sub-GUT parameters.
As seen in Fig. 8, in general FeynHiggs 2.14.1 yields lower values of M h compared to FeynHiggs 2.10.0 along both the upper and lower sub-GUT strips we have studied. In the cases of the lower-m 0 strips (solid lines) this reduction improves consistency with the experimental value of M h over a wider range of m 1/2 . The picture is more mixed for the higherm 0 strips, where the preferred ranges of m 1/2 change, but are not necessarily more extensive when FeynHiggs 2.14.1 is used.

Minimal AMSB Models
As a contrast to the previous CMSSM and sub-GUT models, now we analyze the minimal scenario for anomaly-mediated SUSY breaking (the mAMSB) [29,30]. This has a very different spectrum, and a different composition of the LSP, giving sensitivity to different aspects of the calculation of M h . The mAMSB has three relevant continuous parameters, with the overall scale of SUSY breaking being set by the gravitino mass, m 3/2 . In pure AMSB the soft SUSY-breaking scalar masses m 0 , like the gaugino masses, are proportional to m 3/2 before renormalization. However, in this case renormalization leads to negative squared masses for sleptons. Thus, the pure AMSB is unrealistic, and some additional contributions to the scalar masses m 0 are postulated. It is simplest to assume that these are universal, as in minimal AMSB (mAMSB) models. In the mAMSB model the soft trilinear SUSY-breaking mass terms, A i , are determined by anomalies, like the gaugino masses, and hence are also proportional to m 3/2 , resulting in the following three free continuous parameters: m 3/2 , m 0 and the ratio of Higgs vevs, tan β. The µ term and the soft Higgs bilinear SUSY-breaking term, B, are determined phenomenologically via the electroweak vacuum conditions, as in the CMSSM and related models, and may have either sign.
Since the gaugino masses M 1,2,3 are induced by anomalous loop effects, they are suppressed relative to the gravitino mass, m 3/2 , which is quite heavy in this scenario: m 3/2 20 TeV. The gaugino masses have the following ratios at NLO: |M 1 | : |M 2 | : |M 3 | ≈ 2.8 : 1 : 7.1. We note that the wino-like states are lighter than the bino, which is therefore not a candidate to be the LSP. The LSP may be either a Higgsino-like or a wino-like neutralinoχ 0 1 , and is almost degenerate with a chargino partner,χ ± 1 , in both cases. If the LSP is a wino-likeχ 0 1 and it is the dominant source of the dark matter density, its mass has been shown to be 3 TeV [107,108] once Sommerfeld-enhancement effects [109] are taken into account. On the other hand, if a Higgsino-likeχ 0 1 provides the CDM density, mχ0 1 ∼ 1.1 TeV. In the mAMSB model, the Higgsino-like LSP still has a non-negligible wino component, and is therefore heavier than a pure Higgsino, with m χ 1.5 TeV.
The the input parameters, but the squark masses are typically very heavy, since they receive a contribution ∝ g 4 3 m 2 3/2 where g 2 3 /(4 π) = α s . The relatively small loop-induced values of the trilinears A i and the measured Higgs mass also favor relatively high stop masses.
We display in Fig. 9 four (m 0 , m 3/2 ) planes in the mAMSB model. They all have a pink shaded region at large m 0 and relatively small m 3/2 where the electroweak vacuum conditions cannot be satisfied. Each panel also features a prominent near-horizontal band with acceptable dark matter density where the LSP is mainly a wino with mass 3 TeV. They also feature narrower and less obvious strips close to the electroweak vacuum boundary where the LSP has a larger Higgsino fraction and a smaller mass. In this figure we use the range 0.1151 < Ω χ h 2 < 0.1235. As one can see, there is a strong preference for low tan β for the wino-dark-matter strip. At tan β > 5, most of the wino strip has Higgs masses in excess of 128 GeV. While portions of the Higgsino strip are acceptable at higher tan β, at tan β = 20 the pair of Higgsino strips is also at large M h .
The profiles of the mAMSB-dark-matter strips displayed in Fig. 9 are shown in Fig. 10. In each panel, the horizontal axis is m 0 , the left vertical axis is mχ0 1 , and the right vertical axis is M h . We can again easily distinguish between the wino and Higgsino-like strips. The wino strip spans a wide range in m 0 as seen in Fig. 9, where the Higgsino-like strip resides only at large m 0 . In the wino-like strip, the neutralino mass is shown by the blue dashed curves and mχ0 We see in the upper left panel of Fig. 10 that, for tan β = 3.5 and µ > 0, the Higgs mass M h calculated with FeynHiggs 2.14.1 (red lines) is consistent with the experimental value all along both strips, within the theoretical uncertainties. We do not show the results from FeynHiggs 2.10.0 in this case, as they were not reliable for tan β = 3.5. On the other hand, calculations of M h with FeynHiggs 2.14.1 are significantly higher than the experimental value along the wino-like strips in the other panels, which are for larger values of tan β. In contrast, FeynHiggs 2.10.0 calculations of M h were significantly lower along the wino-like strips, and compatible with experiment for tan β = 5 and µ > 0. In the cases of the Higgsino-like strips, FeynHiggs 2.14.1 calculations of M h are compatible with experiment along that for tan β = 5 and µ > 0 and most of the corresponding strip for tan β = 5 and µ < 0, though not for the Higgsino-like strip for tan β = 20 and µ > 0. FeynHiggs 2.10.0 gave generally larger values of M h along these Higgsino-like strips, which are compatible with experiment only for the strip for tan β = 5 and µ > 0 and part of the strip for tan β = 5 and µ < 0.
In the mAMSB, as seen in Fig. 10, in general FeynHiggs 2.14.1 yields values of M h along the Higgsino strips that are more consistent with the experimental measurement than the ones with FeynHiggs 2.10.0. On the other hand, the values of M h along the wino strip are generally larger for FeynHiggs 2.14.1 than for FeynHiggs 2.10.0, and in poorer agreement with experiment. Hence, in this case the improvements in FeynHiggs 2.14.1 yield a preference for a quite different region of the model parameter space.

The pMSSM11
In contrast to the above models in which soft SUSY breaking is assumed to originate from some specific theoretical mechanism, we now study a model in which the SUSY parameters are constrained by purely phenomenological considerations. In general, such phenomenological MSSM (pMSSM) [32] models contain many more parameters. Here we consider a variant of the pMSSM with 11 parameters fixed at the electroweak-scale, the pMSSM11, as analyzed in Ref. [34] using the available experimental constraints including many from the first LHC run at 13 TeV. The model parameters are three independent gaugino masses, M 1,2,3 , a common mass for the first-and second-generation squarks, mq, a mass for the third-generation squarks, m q 3 , that is allowed to be different, a common mass, m˜ , for the first-and second-generation sleptons, a mass for the stau, m˜ 3 , that is also allowed to be different, 10 a single trilinear mixing parameter, A, the Higgs mixing parameter µ, the pseudoscalar Higgs mass, M A , and the ratio of Higgs vevs, tan β. These parameters are all fixed at a renormalization scale M SUSY ≡ √ mt 1 mt 2 , where mt 1 , mt 2 are the masses of the two stop mass eigenstates. This is also the scale at which electroweak vacuum conditions are imposed. As in all the models we study, the sign of the mixing parameter µ may be either positive or negative.
The flexibility of the pMSSM11 model allows, in principle, many different mass hierarchies to be explored, and hence different aspects of the M h calculation. In particular, since the Higgs mass is most sensitive, in general, to the stop masses, we explore in Fig. 11 what stop masses and mixing are compatible with the measured Higgs mass, without being constrained by any preconceived theoretical ideas such as those arising in the models discussed in the previous sections. In each panel of Fig. 11, the regions favored at the 68% CL (1-σ), 95% CL (2-σ) and 99.7% CL (3-σ) are enclosed by red, blue and green contours, respectively, which are shown   14.1 (FeynHiggs 2.10.0) is used to calculate the χ 2 contribution 11 the LHC measurement of M h to a frequentist global analysis of the pMSSM11 parameter space [34]. 12 We see in the top left panel of Fig. 11 that the experimental value of M h can be accommodated by values of mt 1 500 GeV (1000 GeV) (1300 GeV) at the 99.7 (95) (68)% CL, whether FeynHiggs 2.14.1 or FeynHiggs 2.10.0 is used to calculate M h . The most significant difference is a tendency for FeynHiggs 2.10.0 to disfavor larger values of M h when mt 1 is large, a tendency that is absent when FeynHiggs 2.14.1 is used. We note also that the likelihood function is quite flat for mt 1 1500 GeV, and for this reason we do not quote a best-fit point. The upper right panel shows that values of mt 2 1 (1.3) (1.5) TeV are favored at the 99.7 (95) (68)% CL, again with little difference between the results with FeynHiggs 2.14.1 and FeynHiggs 2.10.0. Again, FeynHiggs 2.10.0 tends to disfavor larger values of M h when mt 2 is large, but not FeynHiggs 2.14.1.
The middle panels of Fig. 11 explore the sensitivities of the M h calculation to the stop mixing parameter X t ≡ A t − µ cot β in the two versions of FeynHiggs 13 . We see that they both favor values of |X t | 2 TeV, though X t = 0 is allowed at the 99.7% CL. However, we see in the middle right panel that this is possible only for mt 1 3 (3.5) TeV when FeynHiggs 2.14.1 (FeynHiggs 2.10.0) is used. This behavior at X t = 0 may be the origin of the often-repeated statement that the measured value of M h requires a large stop mass. In fact, as already mentioned above, the upper panels of Fig. 11 show that M h 125 GeV is quite compatible with mt 1 ∼ 1.2 TeV, and the middle right panel shows that this is possible if |X t | ∼ 2 TeV. The bottom plots of Fig. 11 show the same results as in the middle row, but with X t /M S on the vertical axes. In particular, in the lower right plot it can clearly be seen that the correct Higgs-boson mass prediction requires either large mixing in the stop sector, or large scalar top masses. Here small mixing can more easily be reached with FeynHiggs 2.14.1. Fig. 12 contains one-dimensional plots of the global χ 2 -likelihood functions for mt 1 (left panel) and X t /M S (right panel), shown as solid (dashed) lines as found using FeynHiggs 2.14.1 (FeynHiggs 2.10.0) to calculate the χ 2 contribution from the LHC measurement of M h .
Here we see again that the global minima are at mt 1 ∼ 1.5 TeV and |X t /M S | ∼ 2, with little difference between FeynHiggs 2.14.1 and FeynHiggs 2.10.0. The χ 2 function for mt 1 rises very mildly as mt 1 approaches 4 TeV, and the exact location of the minimum value cannot be regarded as significant. We note also the appearance of a secondary minimum with ∆χ 2 < 3 when mt 1 500 GeV. The χ 2 function for X t /M S exhibits no significant sign preference, but disfavors X t = 0 by ∆χ 2 4 if FeynHiggs 2.14.1 is used, compared to ∆χ 2 8 with FeynHiggs 2.10.0.
Overall, in the pMSSM11 we see no clear trend towards lower or higher values of M h when going from FeynHiggs 2.10.0 to FeynHiggs 2.14.1. Although individual parameter choices 11 We recall here that the M h contribution to the global likelihood is modeled using a Gaussian distribution with µ = 125.09 GeV and σ exp = 0.24 GeV and σ theo−SUSY = 1.5 GeV. For further details on the likelihood, including a discussion of the other constraints, we refer the reader to Ref. [34]. 12 It should be kept in mind that the set of pMSSM11 points used here was originally obtained in Ref. [34] using FeynHiggs 2.11.3. Slight shifts in the contours shown below can be expected if the full fit would be done using either FeynHiggs 2.10.0 or FeynHiggs 2.14.1. 13 Note that here we use the sign convention for A t of FeynHiggs.

Conclusions
We have investigated the physics implications of improved Higgs-boson mass predictions in the MSSM, comparing results from FeynHiggs 2.14.1 and FeynHiggs 2.10.0. The main differences, as discussed in this paper, are 3-loop RG effects and 2-loop threshold corrections that can accommodate three separate mass scales: mq, mg and an electroweakino mass scale, as well as an improved treatment of DR input parameters in the scalar top sector avoiding problems with the conversion to on-shell parameters, that yields significant improvements for large SUSY-breaking scales. These changes reflect the progress made over the last ∼ 5 years in "hybrid" Higgs-mass calculations in the MSSM.
The examples presented in this paper illustrate how the preferred ranges of the parameter space of the MSSM can change when FeynHiggs 2.14.1 is used to calculate M h , as compared to when FeynHiggs 2.10.0 is used. The first representative model is the CMSSM. As is well known, in the CMSSM reproducing the correct CDM density of neutralinos, despite the rising lower limits on sparticle masses from the LHC, tends to favor narrow strips of parameter space that extend to large m 1/2 and/or m 0 .The improvements in FeynHiggs 2.14.1 can play important roles in these parameter regions. Examples of these high-mass strips include some where stop coannihilation is important, and others where the focus-point mechanism is operative. In both these cases, using FeynHiggs 2.14.1 rather than FeynHiggs 2.10.0 changes significantly the parts of the strips that are consistent with the experimental measurement of M h . This reflects the different dependences on m 1/2 of the FeynHiggs 2.14.1 and FeynHiggs 2.10.0 calculations of M h .
We have also studied sub-GUT models, in which the soft SUSY-breaking masses are assumed to be universal at some scale M in below the conventional grand unification scale M GUT assumed in the CMSSM. Both the stop-coannihilation and focus-point mechanisms may be operational in different regions of the sub-GUT parameter space. Depending on the choice of M in , the forms of the DM strips can be very different from those allowed in the CMSSM, with the possibility of two (or more) DM strips with different values of m 0 for the same value of m 1/2 . In general, along the lower-m 0 strips the agreement between FeynHiggs 2.14.1 calculations of M h and experiment is better than that for the FeynHiggs 2.10.0 calculations.
As a third case we investigated the mAMSB, where two different classes of DM strips occur: one where the LSP may be mainly a wino, or one where it may have a large Higgsino component. Both of these types of dark-matter strips extend to relatively large values of m 0 , with an LSP mass ∼ 3 TeV or 1 TeV, respectively. Calculations of M h using FeynHiggs 2.14.1 favor the Higgsino region, whereas calculations using FeynHiggs 2.10.0 favored the wino region.
In the case of the pMSSM11, we find little change in the regions of parameter space favored by M h , which can be ascribed to the fact that there is no big mass hierarchy. The predictions from both FeynHiggs 2.14.1 and FeynHiggs 2.10.0 are consistent with M h 125 GeV and mt 1 ∼ 1.3 TeV at the 68% CL, and they both allow mt 1 ∼ 500 GeV with ∆χ 2 ∼ 3. Both versions of FeynHiggs disfavor small stop mixing, X t = 0, by ∆χ 2 ∼ 4 (8) in the case of FeynHiggs 2.14.1 (FeynHiggs 2.10.0), with |X t /M S | ∼ 2 being favored. Obtaining the correct prediction for the Higgs-boson mass requires either large mixing in the scalar top sector (with |X t /M S | ∼ 2), or large scalar top masses, though smaller values of |X t /M S | can be reached more easily with FeynHiggs 2.14.1. We find no clear preference towards lower or higher values of M h when going from FeynHiggs 2.10.0 to FeynHiggs 2.14.1 in the pMSSM11. The experimental constraints yield parameter combinations with mild effects on the light CP-even Higgs-boson mass after marginalization to minimize χ 2 .
In conclusion, we comment that in this paper we have limited ourselves to exploratory studies, and have not attempted to make global fits to the parameters of any of the SUSY models we have discussed. However, we find an overall tendency towards better compatibility with the experimental data when employing the updated Higgs-boson mass calculations. Performing new fits with updated calculations of M h would clearly be an interesting next step, and we hope that the studies described here will give some insight into the results to be expected from such more complete investigations.