1 Introduction

We perform a parameter scan of the phenomenological Minimal Supersymmetric Standard Model (pMSSM) with eight parameters taking into account the experimental Higgs boson results from Run I of the LHC and further low-energy observables. We investigate various MSSM interpretations of the Higgs signal at . First, we consider the case where the light -even Higgs boson of the MSSM is identified with the discovered Higgs boson. In this case it can impersonate the SM Higgs-like signal either in the decoupling limit, or in the limit of alignment without decoupling. In the latter case, the other states in the Higgs sector can also be light, offering good prospects for upcoming LHC searches and for searches at future colliders. Second, we demonstrate that the heavy -even Higgs boson is still a viable candidate to explain the Higgs signal — albeit only in a highly constrained parameter region, that will be probed by LHC searches for the -odd Higgs boson and the charged Higgs boson in the near future. As a guidance for such searches we provide new benchmark scenarios that can be employed to maximize the sensitivity of the experimental analysis to this interpretation.

BONN-TH-2016-06, DESY 16-151, IFT-UAM/CSIC-16-068, NIKHEF 2016-033, SCIPP 16/10

The Light and Heavy Higgs Interpretation of the MSSM

Philip Bechtle, Howard E. Haber, Sven Heinemeyer, Oscar Stål,

[.5em] Tim Stefaniak, Georg Weiglein and Lisa Zeune***Electronic addresses: bechtle@physik.uni-bonn.de, haber@scipp.ucsc.edu, Sven.Heinemeyer@cern.ch,
                                  tistefan@ucsc.edu, Georg.Weiglein@desy.de, lisa.zeune@nikhef.nl

[0.4cm] Physikalisches Institut der Universität Bonn, Nußallee 12, D-53115 Bonn, Germany

[0.1cm] Santa Cruz Institute for Particle Physics (SCIPP) and Department of Physics

University of California, Santa Cruz, 1156 High Street, Santa Cruz, CA 95060, USA

[0.1cm] Campus of International Excellence UAM+CSIC, Cantoblanco, E–28049 Madrid, Spain

[0.1cm] Instituto de Física Teórica, (UAM/CSIC), Universidad Autónoma de Madrid,

Cantoblanco, E-28049 Madrid, Spain

[0.1cm] Instituto de Física de Cantabria (CSIC-UC), E-39005 Santander, Spain

[0.1cm] The Oskar Klein Centre, Department of Physics, Stockholm University,

SE-106 91 Stockholm, Sweden (former address)

[0.1cm] Deutsches Elektronen-Synchrotron DESY, Notkestraße 85, D-22607 Hamburg, Germany

[0.1cm] Nikhef, Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands

1 Introduction

The discovery of a Higgs-like scalar boson in Run I of the Large Hadron Collider (LHC) [1, 2] marks a milestone in the exploration of electroweak symmetry breaking (EWSB). Within experimental and theoretical uncertainties, the properties of the new particle are compatible with the Higgs boson of the Standard Model (SM) [3]. However, a variety of other interpretations of the Higgs signal are possible, corresponding to very different underlying physics. Here, a prime candidate for the observed scalar boson is a -even Higgs boson of the Minimal Supersymmetric Standard Model (MSSM) [4, 5, 6], as it possesses SM Higgs-like properties over a significant part of the model parameter space with only small deviations from the SM in the Higgs production and decay rates [7].

One of the main tasks of the LHC Run II will be to determine whether the observed scalar boson forms part of the Higgs sector of an extended model. In contrast to the SM, two Higgs doublets are needed in the MSSM to give mass to up- and down-type fermions. The extended Higgs sector entails the existence of five scalar bosons, namely a light and heavy -even Higgs bosons, and , a -odd Higgs boson, , and a pair of charged Higgs bosons, . Mixing between the neutral -even and -odd states are possible in the -violating case [8, 9, 10, 11, 12, 13], which we will not considered here. At lowest order, the Higgs sector of the MSSM can be fully specified in terms of the  and  boson masses, and , the -odd Higgs boson mass, , and , the ratio of the two neutral Higgs vacuum expectation values. However, higher-order corrections are crucial for a precise prediction of the MSSM Higgs boson properties and introduce dependences on other model parameters, see e.g. Refs. [14, 15, 16] for reviews.

Many fits for the Higgs rates in various models and within the effective field theory approach have been performed over the last years, see e.g. Ref. [17, 18]. Focusing on the MSSM, recent fits have shown that the interpretation of the observed scalar as the light -even MSSM Higgs boson (“light Higgs case”) is a viable possibility, providing a very good description of all data [19, 20, 21, 22, 23, 24]; see also Refs. [25, 26, 27] for global fits including also astrophysical data. In the light-Higgs case, decoupling of the heavy Higgs bosons ([28, 29, 30, 31] naturally explains the SM-like couplings of the light MSSM Higgs boson  [7]. Another interesting possibility to explain the SM-like behavior of without decoupling in the MSSM—the so-called limit of alignment without decoupling—has been outlined in Refs. [32, 33], relying on an (accidental) cancellation of tree-level and loop contributions in the -even Higgs boson mass matrix. This led to the definition of a specific benchmark scenario [33], which has since been ruled out in the interesting low region via searches [23].

Alternatively, it was demonstrated that the heavy -even Higgs boson can also be identified with the observed signal [7, 34, 35, 36, 37, 19] (“heavy Higgs case”).111Such a situation is more common in extensions of the MSSM. In particular, in the NMSSM it occurs generically if the singlet-like -even state is lighter than the doublet-like Higgs bosons, see e.g. Refs. [38, 39, 40, 41, 42]. In this scenario all five MSSM Higgs bosons are relatively light, and in particular the lightest -even Higgs boson has a mass (substantially) smaller than with suppressed couplings to gauge bosons. This led to the development of the low- benchmark scenario [43]. This particular scenario has meanwhile been ruled out by ATLAS and CMS searches for a light charged Higgs boson [44, 45]. However the heavy Higgs interpretation in the MSSM remains viable, as we will discuss in this paper.

The questions arise, whether, and if so by how much, the MSSM can improve the theoretical description of the experimental data compared to the SM, and which parts of the MSSM parameter space are favored. In a previous analysis [19] we analyzed these questions within the MSSM. We performed a scan over the seven most relevant parameters for MSSM Higgs boson phenomenology, taking into account the data up to July 2012, which showed in particular some enhancement in the measured rate for . We found that both the light and the heavy Higgs case provided a good fit to the data. In particular, the MSSM light Higgs case gave a better fit than the SM when the data in the channel and low-energy data was included.

The situation has changed in several respects with the release of additional Higgs data by the ATLAS and CMS Collaborations [46]. In particular, the final data obtained in the LHC Run I does not show a significant enhancement over the SM prediction in the channel anymore, and the heavy Higgs case is much more restricted due to light charged Higgs boson searches. The main aim of the present paper is to study the MSSM Higgs sector in full detail taking into account the current experimental data and in particular the final LHC Run I results, and to propose paths towards a complete exploration of the heavy Higgs case at the LHC in the ongoing Run II. We incorporate the available measurements of the Higgs boson mass and signal strengths, as well as measurements of the relevant low-energy observables. Furthermore we take into account all relevant constraints from direct Higgs and supersymmetric (SUSY) particle searches. We investigate whether the MSSM can still provide a good theoretical description of the current experimental data, and which parts of the parameter space of the MSSM are favored. Within the light Higgs case we analyze the situation with very large (decoupling), as well as for small/moderate (alignment without decoupling). We also investigate the feasibility of the heavy Higgs case and define new benchmark scenarios in which this possibility is realized, in agreement with all current Higgs constraints.

The paper is organized as follows. We employ the phenomenological MSSM with 8 parameters (pMSSM 8), which is introduced in detail in Sect. 2. In this section, we also expand upon the theoretical background of the two possible limits that lead to alignment in the even Higgs sector, i.e. when one of the -even neutral MSSM Higgs bosons behaves like the SM Higgs boson. In particular, we outline how leading two-loop effects on the conditions for alignment can be assessed and present a brief quantitative discussion of these effects.222More details will be presented in a separate publication [47]. The parameter scan with sampling points, the techniques to achieve good coverage, as well as the considered experimental observables and constraints are described in Sect. 3. In Sect. 4 we present our results for the best-fit points and the preferred parameter regions for the light Higgs and the heavy Higgs interpretation. The effects of the Higgs mass and Higgs rates measurements, precision observables, and direct Higgs and SUSY searches are discussed, and the phenomenology of the other MSSM Higgs states is outlined. In particular, in Sect. 4.4 we propose new benchmark scenarios for the study of the heavy Higgs case, which can be probed at the LHC Run II. We conclude in Sect. 5. In Appendix A, we discuss the extent of the tuning associated with the regions of the MSSM parameter space that exhibit approximate Higgs alignment without decoupling. Finally in Appendix B, we provide tables listing the signal strength measurements from ATLAS, CMS and the Tevatron (DØ and CDF) that are included in our analysis.

2 Theoretical Background

2.1 The MSSM Higgs sector

In this section we briefly review the most important features of the MSSM Higgs sector and motivate the choice of the eight free pMSSM parameters in our scan. We provide a detailed description of the relevant MSSM parameter sectors and our notations, which remain unchanged compared to [19].

In the supersymmetric extension of the SM, an even number of Higgs multiplets consisting of pairs of Higgs doublets with opposite hypercharge is required to avoid anomalies due to the supersymmetric Higgsino partners. Consequently the MSSM employs two Higgs doublets, denoted by and , with hypercharges and , respectively. After minimizing the scalar potential, the neutral components of and acquire vacuum expectation values (vevs), and . Without loss of generality, we assume that the vevs are real and non-negative (this can be achieved by appropriately rephasing the Higgs doublet fields). The vevs are normalized such that


In addition, we define


Without loss of generality, we may assume that (i.e., is non-negative). This can always be achieved by a rephasing of one of the two Higgs doublet fields.

The two-doublet Higgs sector gives rise to five physical Higgs states. The mass eigenstates correspond to the neutral Higgs bosons , (with ) and , and the charged Higgs pair . Neglecting possible -violating contributions of the soft-supersymmetry-breaking terms (which can modify the neutral Higgs properties at the loop level), and are the light and heavy -even Higgs bosons, and is -odd.

At lowest order, the MSSM Higgs sector is fully described by and two MSSM parameters, often chosen as the -odd Higgs boson mass, , and . In the MSSM at the tree-level the mass of the light -even Higgs boson does not exceed . However, higher order corrections to the Higgs masses are known to be sizable and must be included, in order to be consistent with the observed Higgs signal at  [3]. Particularly important are the one- and two-loop contributions from top quarks and their scalar top (“stop”) partners. In order to shift the mass of up to , large radiative corrections are necessary, which require a large splitting in the stop sector and/or heavy stops. For large values of , the sbottom contributions to the radiative corrections also become sizable. The stop (sbottom) sector is governed by the soft SUSY-breaking mass parameter and ( and ), where SU(2) gauge invariance requires , the trilinear coupling () and the Higgsino mass parameter .

To achieve a good sampling of the full MSSM parameter space with points, we restrict ourselves to the eight MSSM parameters


most relevant for phenomenology of the Higgs sector (the scan ranges will be given in Sect. 3.1), under the assumption that the third generation squark and slepton parameters are universal. That is, we take , and . Note that the soft SUSY-breaking mass parameter in the stau sector, , can significantly impact the Higgs decays as light staus can modify the loop-induced diphoton decay. is therefore taken as an independent parameter in our scans. Even though the other slepton and gaugino parameters are generally of less importance for the Higgs phenomenology, we scan over the SU(2) gaugino mass parameter as well as over the mass parameter of the first two generation sleptons, , (assumed to be equal) as these parameters are important for the low-energy observables included in our analysis. The remaining MSSM parameters are fixed,


We choose relatively high values for the squark and gluino mass parameters, which have a minor impact on the Higgs sector, in order to be in agreement with the limits from direct SUSY searches. Finally, the U(1) gaugino mass parameter, , is fixed via the GUT relation


with and . For more details on the definition of the MSSM parameters, we refer to [19].

2.2 The Higgs alignment limit

In light of the Higgs data, which indicates that the properties of the observed Higgs boson are SM-like, we seek to explore the region of the MSSM parameter space that yields a SM-like Higgs boson. In general, a SM-like Higgs boson arises if one of the neutral Higgs mass eigenstates is approximately aligned with the direction of the Higgs vev in field space. Thus, the limit of a SM Higgs boson is called the alignment limit.

To analyze the alignment limit, it is convenient to define


where and , and there is an implicit sum over the repeated SU(2) index . For consistency of the notation, we denote the corresponding neutral Higgs vevs by and . We now define the following linear combinations of Higgs doublet fields,


such that and , which defines the so-called Higgs basis [48, 49, 50].333Since the tree-level MSSM Higgs sector is -conserving, the Higgs basis is defined up to an overall sign ambiguity, where . However, since we have adopted the convention in which is non-negative [cf. the comment below Eq. (2)], the overall sign of the Higgs basis field is now fixed. It is straightforward to express the scalar Higgs potential in terms of the Higgs basis fields and ,


where the most important terms of the scalar potential are highlighted above. The quartic couplings , and are linear combinations of the quartic couplings that appear in the MSSM Higgs potential expressed in terms of and . In particular, at tree-level,


where and are the SU(2) and U(1) gauge couplings, respectively, and . Hence, the are parameters.

One can then evaluate the squared-mass matrix of the neutral -even Higgs bosons, with respect to the neutral Higgs states,  , 


If were a Higgs mass eigenstate, then its tree-level couplings to SM particles would be precisely those of the SM Higgs boson. This would correspond to the exact alignment limit. To achieve a SM-like neutral Higgs state, it is sufficient for one of the neutral Higgs mass eigenstates to be approximately given by . In light of the form of the squared-mass matrix given in Eq. (11), we see that a SM-like neutral Higgs boson can arise in two different ways:

  1. . This is the so-called decoupling limit, where is SM-like and .

  2. . In this case is SM-like if and is SM-like if .

In particular, the -even mass eigenstates are:


where and are defined in terms of the mixing angle that diagonalizes the -even Higgs squared-mass matrix when expressed in the original basis of scalar fields, . Since the SM-like Higgs must be approximately , it follows that is SM-like if  [51] and is SM-like if  [52]. The case of a SM-like necessarily corresponds to alignment without decoupling.

In the case of exact alignment without decoupling, , the tree-level couplings of the SM-like Higgs boson are precisely those of the Higgs boson of the Standard Model. Nevertheless, deviations from SM Higgs boson properties can arise due to two possible effects. First, there might exist new particles that enter in loops and modify the loop-induced Higgs couplings to , and . For example, if is the SM-like Higgs boson, then the charged Higgs boson mass is not significantly larger than the observed Higgs mass, in which case the charged Higgs loop can shift the one-loop induced couplings of the observed Higgs boson to and  [52]. Similarly, SUSY particles can give a contribution at the loop-level to other, at the tree-level SM-like, couplings. Second, there might exist new particles with mass less that half the Higgs mass, allowing for new decay modes of the SM-like Higgs boson. An example of this possibility arises if is the SM-like Higgs boson and , in which case the decay mode is allowed. Indeed, in the exact alignment limit where , the tree-level coupling in the MSSM is given by [53]


The possibility of alignment without decoupling has been analyzed in detail in Refs. [31, 54, 32, 55, 33, 56, 51, 52] (see also the “-phobic” benchmark scenario in Ref. [57]). It was pointed out that exact alignment via can only happen through an accidental cancellation of the tree-level terms with contributions arising at the one-loop level (or higher). In this case the Higgs alignment is independent of , and . This has two phenomenological consequences. First, the remaining Higgs states can be light, which would imply good prospects for LHC searches. Second, either the light or the heavy neutral Higgs mass eigenstate can be aligned with the SM Higgs vev and thus be interpreted as the SM-like Higgs boson observed at .

The leading one-loop contributions to , and proportional to , where


is the top quark Yukawa coupling, have been obtained in Ref. [33] in the limit (using results from Ref. [58]):


where , denotes the SUSY mass scale, given by the geometric mean of the light and heavy stop masses, and


In Eqs. (15)–(18), we have assumed for simplicity that and (as well as the gaugino mass parameters that contribute subdominantly at one-loop to the ) are real parameters. That is, we are neglecting CP-violating effects that can enter the MSSM Higgs sector via radiative corrections.

Figure 1: Contours of corresponding to exact alignment, , in the plane. is adjusted to give the correct Higgs mass. The three left panels exhibit the approximate one-loop results; the three right panels exhibit the corresponding two-loop improved results. Taking the three panels on each side together, one can immediately discern the regions of zero, one, two and three values of in which exact alignment is realized. In the overlaid blue regions we have (unstable) values of .

The approximate expression for given in Eq. (17) depends only on the unknown parameters , , and . Exact alignment arises when . Note that trivially occurs when or (corresponding to the vanishing of either or ). But, this choice of parameters is not relevant for phenomenology as it leads to a massless quark or quark, respectively, at tree-level. Henceforth, we assume that is non-zero and finite. In our convention, is positive with .

We can simplify the analysis of the condition by solving Eq. (15) for and inserting this result back into Eq. (17). The resulting expression for now depends on , , and the ratios,


Using Eq. (18) to rewrite the final expression in terms of and , we obtain,


Setting , we can identify with the mass of the observed (SM-like) Higgs boson (which may be either or depending on whether is close to 1 or 0, respectively). We can then numerically solve for for given values of and . The values of the real positive solutions of obtained by using the one-loop approximate formula given in Eq. (20) are illustrated by the contour plots shown in the three left panels of Fig. 1, where each panel corresponds to a different solution of . Note that at every point in the plane, the value of has been adjusted according to Eq. (15) such that the squared-mass of the SM-like Higgs boson in the alignment limit is given by . Taking the three left panels together, one can immediately discern the regions of zero, one, two and three positive solutions of Eq. (20), and their corresponding values. A more detailed discussion of these solutions will be presented in a separate paper [47].

It is instructive to obtain an approximate analytic expression for the value of the largest real positive solution. Assuming the following approximate alignment condition, first written in Ref. [33], is obtained,


where denotes the (one-loop) mass of the SM-like Higgs boson obtained from Eq. (15), which could be either the light or heavy -even Higgs boson. It is clear from Eq. (21) that a positive solution exists if either and , or if , and is sufficiently large such that the numerator of Eq. (21) is negative. Keeping in mind that Eq. (21) was derived under the assumption that , one easily verifies that the largest of the three roots of Eq. (20) shown in Fig. 1 always satisfies the stated conditions above. Another consequence of Eq. (21) is that by increasing the value of (in the region where ), it is possible to lower the value at which alignment occurs.

If , then Eq. (21) is no longer a good approximation. Returning to Eq. (20), we set and again assume that . We can then solve approximately for ,


For example, in the parameter regime where and , we obtain .

The question of whether the light or the heavy -even Higgs boson possesses SM-like Higgs couplings in the alignment without decoupling regime depends on the relative size of and . Combining Eqs. (16) and (17), it follows that in the limit of exact alignment where , we identify as the squared mass of the observed SM-like Higgs boson and


We define a critical value of ,


where and is given by Eq. (23). Furthermore, since the squared-mass of the non-SM-like -even Higgs boson in the exact alignment limit, , must be positive, it then follows that the minimum value possible for the squared-mass of the -odd Higgs boson is


That is, if is sufficiently large and negative, then the minimal allowed value of is non-zero and positive.

We focus again on the parameter region in the ) plane, and compute from Eq. (23) using the value of obtained from setting in Eq. (20). This allows us to determine the value of for each point in the ) plane. The interpretation of is as follows. If , then can be identified as the SM-like Higgs boson with  GeV. If , then can be identified as the SM-like Higgs boson with  GeV. The corresponding contours of are exhibited in the three left panels of Fig. 2, which are in one-to-one correspondence with the three left panels of Fig. 1.

Figure 2: Critical value, , in the exact alignment, indicating the maximal value for which the mass hierarchy of the heavy Higgs interpretation is obtained, corresponding to the solutions found in Fig. 1 in the plane. The three left panels exhibit the approximate one-loop results; the three right panels exhibit the corresponding two-loop improved results. In the overlaid blue regions we have (unstable) values of .

As previously noted, the analysis above was based on approximate one-loop formulae given in Eqs. (15)–(17), where only the leading terms proportional to are included. In the exact alignment limit, we identify given by Eq. (15) as the squared-mass of the observed SM-like Higgs boson. However, it is well known that Eq. (15) overestimates the value of the radiatively corrected Higgs mass. Remarkably, one can obtain a significantly more accurate result simply by including the leading two-loop radiative corrections proportional to .

In Ref. [59], it was shown that the dominant part of these two-loop corrections can be obtained from the corresponding one-loop formulae with the following very simple two step prescription. First, we replace


where  GeV is the top quark mass [60], and the running top quark mass in the one-loop approximation is given by


In our numerical analysis, we take . Second, when multiplies that threshold corrections due to stop mixing (i.e., the one-loop terms proportional to and ), then we make the replacement,




Note that the running top-quark mass evaluated at includes a threshold correction proportional to that enters at the scale of supersymmetry breaking. Here, we only keep the leading contribution to the threshold correction under the assumption that (a more precise formula can be found in Appendix B of Ref. [59]). The above two step prescription can now be applied to Eqs. (15)–(17), which yields a more accurate expression for the radiatively corrected Higgs mass and the condition for exact alignment without decoupling. Details of this analysis will be presented in a forthcoming work [47].

The end results are summarized below. We have derived analogous expressions to Eqs. (20) and (23) that incorporate the leading two-loop effects at . It is convenient to introduce the following notation,


where is the top quark mass, and


Then, the two-loop corrected condition for the exact alignment limit corresponding to is given by,


which supersedes Eq. (20), and the correction to Eq. (23) is given by,


One can now define two-loop improved versions of and [cf. Eqs. (24) and (25)].

In the right panels of Figs. 1 and 2, we plot the two-loop improved versions of the corresponding one-loop results shown in the left panels. There are a few notable changes, which we now discuss. First, in our scan of the plane, we have observed numerically that there is a new solution to the alignment condition [cf. Eq. (32)] that is unrelated to the solutions found in the one-loop analysis. However, this solution always corresponds to a value of , which lies outside our region of interest. Henceforth, we simply discard this possibility. What remains are solutions that can be identified as the two-loop corrected versions of the one-loop results obtained above. The right panels of Fig. 1 exhibit the remaining real positive solutions of Eq. (32).

We can now see the effects of including the leading corrections. The regions where positive solutions to Eq. (32) exist shown in the right panels of Fig. 1 have shrunk somewhat as compared to the corresponding positive solutions to Eq. (20) shown in the left panels of Fig. 1. For example, only one positive solution for exists for large and in the two-loop approximation, whereas three positive solutions exist in the one-loop approximation. Using the values of found in the right panels of Fig. 1, one can now produce the corresponding two-loop corrected plots shown in the right panels of Fig. 2. The qualitative features of the one-loop and two-loop results are similar, after taking note of the slightly smaller regions in which positive solutions for exist in the two-loop approximation.

One new feature of the two-loop approximation not yet emphasized is that we must now carefully define the input parameters and . In the above formulae and plots we interpret these parameters as parameters. However, it is often more convenient to re-express these parameters in terms of on-shell parameters. In Ref. [59], the following expression was obtained for the on-shell squark mixing parameter in terms of the squark mixing parameter , where only the leading corrections are kept,


Since the on-shell and versions of are equal at this level of approximation, we also have


A more detailed examination of the above results, when expressed in terms of the on-shell parameters, will be treated in Ref. [47].

The approximations employed in the section capture some of the most important radiative corrections relevant for analyzing the alignment limit of the MSSM. However, it is important to appreciate what has been left out. First, higher-order corrections beyond are known to be relevant (see, e.g., Ref. [61]). In particular the corrections are in magnitude roughly 20% of the corrections, and enter with a different sign, thus leading to potentially non-negligible corrections to the approximate two-loop results obtained above. On more general grounds, the analysis of this section ultimately corresponds to a renormalization of , which governs the tree-level couplings of the Higgs boson and its departure from the alignment limit. However, radiative corrections also contribute other effects that modify Higgs production cross sections and branching ratios. It is well-known that for , the effective low-energy theory below the scale is a general two Higgs doublet model with the most general Higgs-fermion Yukawa couplings. These include the so-called wrong-Higgs couplings of the MSSM [62, 63], which ultimately are responsible for the and corrections that can significantly modify the coupling of the Higgs boson to bottom quarks and tau leptons.444For a review of these effects and a guide to the original literature, see Ref. [64]. The implication of these couplings will be briefly reviewed in Sect. 2.3. In addition, integrating out heavy SUSY particles at the scale can generate higher dimensional operators that can also modify Higgs production cross sections and branching ratios [65]. None of these effects are accounted for in the analysis presented in this section.

2.3 Implications of the wrong-Higgs couplings

At tree-level, the Higgs-fermion Yukawa couplings follow the Type-II pattern [66, 53] of the two-Higgs doublet model (2HDM), in which the hypercharge Higgs doublet field couples exclusively to right-handed down-type fermions and the hypercharge Higgs doublet field couples exclusively to right-handed up-type fermions. When radiative corrections are included, the so-called wrong-Higgs Yukawa couplings are induced by supersymmetry-breaking effects, in which couples to right-handed up-type fermions and couples to right-handed down-type fermions. We shall denote by a generic scale that characterizes the size of supersymmetric mass parameters. In the limit where , , the radiatively-corrected Higgs-quark Yukawa couplings can be summarized by an effective Lagrangian,555For simplicity, we ignore the couplings to first and second generation fermions. We also neglect weak isospin breaking effects that distinguish between the coupling of neutral and charged Higgs scalars.


which yields a modification of the tree-level relations between , and , as follows [67, 68, 69, 70, 71, 72, 73, 74, 64]:


The dominant contributions to are -enhanced, with . Moreover, in light of our assumption that , , it follows that is suppressed, whereas does not decouple. This non-decoupling can be explained by the fact that arises from the radiatively-generated wrong-Higgs couplings. Below the scale , the effective low energy theory is the 2HDM which contains the most general set of Higgs-fermion Yukawa couplings allowed by gauge invariance, and is no longer restricted to be of Type-II [62]. Similarly, is suppressed, whereas does not decouple. However, is not -enhanced and thus yields only small corrections to the Higgs boson couplings to fermions in the parameter regime of interest (i.e., where ).

In the parameter regime where ,  [67, 68, 69, 73, 75],


where is the gluino mass, and are the bottom and top squark masses, respectively, and smaller electroweak corrections have been ignored. The loop integral is given by


Note that


and . Thus, in the limit in which all supersymmetric parameters appearing in Eq. (39) are all very large, of , we see that does not decouple, as previously advertised.

From Eq. (36) we can obtain the couplings of the physical Higgs bosons to third generation fermions. The resulting interaction Lagrangian is of the form,


Expressions for the Higgs couplings to the third generation quarks can be found in Ref. [64]. In particular, the charged Higgs coupling to the third generation quarks is noteworthy. It is convenient to write the approximate one-loop corrected coupling in the following form,


where .

One of the important constraints on the MSSM Higgs sector is derived from the decay rate for due to the presence of one loop diagrams involving a charged Higgs boson. At large , it is important to incorporate SUSY corrections to the charged Higgs couplings to quarks666By including the radiative corrections via Eq. (43), we are effectively incorporating the leading two-loop contributions to the decay matrix element for induced by SUSY vertex corrections. in the computation of  [76]. Including the radiatively corrected and couplings using Eq. (43), suitably generalized to include intergenerational quark mixing, and taking ,


after comparing the result obtained from the contribution of the charged Higgs loop in the MSSM, including the leading SUSY radiative corrections to the charged Higgs-fermion couplings, to the corresponding results of the 2HDM with Type-II Yukawa couplings. In Eq. (44), is given by Eq. (39) and is given by [76]


where is the mass of the SUSY partner of the left-handed strange quark, is the mixing angle [77], and is defined in Eq. (40). Once again, the non-decoupling behavior of is evident in the limit in which all supersymmetric parameters appearing in Eq. (45) are of . As previously emphasized, the non-decoupling properties and arise due to the wrong-Higgs Yukawa couplings, and are responsible for the significance of the deviation from Type-II behavior of the two-Higgs doublet sector of the MSSM.

3 Parameter sampling, Observables and Constraints

3.1 Sampling of the parameter space

We sample the pMSSM 8 parameter space with uniformly distributed random values in the eight input parameters. Scans are performed separately for the light Higgs and heavy Higgs interpretation of the observed Higgs signal (see below for details) over the parameter ranges given in Table 1. Besides the scan parameters listed in Table 1, the remaining MSSM parameters are chosen as described in Sect. 2.1.

Light Higgs case Heavy Higgs case
Parameter Minimum Maximum Minimum Maximum
[GeV] 90 1000 90 200
1 60 1 20
[GeV] 200 5000 200 1500
[GeV] 200 1000 200 1000
[GeV] 200 1000 200 1000
[GeV] 5000
[GeV] 200 500 200 500
Table 1: Ranges used for the free parameters in the pMSSM 8 scan.

In both cases, we start with