Fully covering the MSSM Higgs sector at the LHC

In the context of the Minimal Supersymmetric extension of the Standard Model (MSSM), we reanalyze the search for the heavier CP-even $H$ and CP-odd $A$ neutral Higgs bosons at the LHC in their production in the gluon-fusion mechanism and their decays into gauge and lighter $h$ bosons and into top quark pairs. We show that only when considering these processes, that one can fully cover the entire parameter space of the Higgs sector of the model. Indeed, they are sensitive to the low $\tan\beta$ and high Higgs mass ranges, complementing the traditional searches for high mass resonances decaying into $\tau$-lepton pairs which are instead sensitive to the large and moderate $\tan\beta$ regions. The complementarity of the various channels in the probing of the complete $[\tan\beta, M_A]$ MSSM parameter space at the previous and upcoming phases of the LHC is illustrated in a recently proposed simple and model independent approach for the Higgs sector, the $h$MSSM, that we also refine in this paper.


Introduction
The probing of the electroweak symmetry breaking mechanism and the search for possible extensions of the Standard Model (SM) of particle physics has become the main mission of the CERN Large Hadron Collider (LHC). Among these extensions, Supersymmetry (SUSY) [1] is considered as the most appealing one as it addresses several shortcomings of the SM, including the problem of the large hierarchy between the Planck and electroweak scales. While the search for SUSY was unsuccessful at the first LHC run, the increase of the center of mass energy of the machine from 8 TeV to the 14 TeV level will significantly improve the sensitivity to the new particles that are predicted by the weak scale theory. These consists not only of the superpartners of the known fermions and gauge bosons but, also, of the additional Higgs bosons beyond the state with a mass of 125 GeV that has been observed by the ATLAS and CMS collaborations in the first LHC phase [2].
As a matter of fact, in low-energy SUSY scenarios, at least two Higgs doublet fields H u and H d are required to break the electroweak symmetry and to generate the isospin-up and down type fermion and the W/Z boson masses. In the simplest scenario, the Minimal Supersymmetric Standard Model (MSSM), the spectrum consists of five states [3][4][5]: two charged H ± , a CP-odd A and two CP-even Higgs particles h and H, with h being the state observed at the LHC while H is heavier as present LHC data is strongly indicating [6].
The phenomenology of the Higgs sector is described entirely by two input parameters, one Higgs mass that is usually taken to be that of the pseudoscalar A boson M A and the ratio tan β of the vacuum expectation values of the two doublet fields, which is generally assumed to lie in the range 1 < ∼ tan β < ∼ m t /m b ≈ 60. This is the case at tree-level where, for instance, the lightest h boson mass is an output and is predicted to be M h < ∼ M Z | cos 2β|, i.e. M h ≤ M Z at high tan β for which | cos 2β| 1. However, this relation is violated since important radiative corrections, that introduce a dependence on many SUSY parameters, occur in this context [7][8][9]. It has been recently shown that, to a good approximation, the MSSM Higgs sector can be again parametrised using the two basic inputs tan β and M A , provided that the crucial LHC information M h 125 GeV is used [10][11][12].
It is known that two efficient channels can be used to directly search for the heavier MSSM Higgs particles at the LHC and probe part of the [tan β, M A ] parameter space 1 . The first one is the search for light charged Higgs bosons that would emerge from the decays of the copiously produced top quarks and would decay almost exclusively into a τ lepton and its associated neutrino for tan β > ∼ 1. For almost all values of tan β, the latest ATLAS [14] and CMS [15] results now practically rule out the mass range M H ± < ∼ 160 GeV, which approximately corresponds to M A < ∼ 140 GeV in the MSSM. The second efficient channel is the search for high mass resonances decaying into τ -lepton pairs, which would be the signature of the production of the heavy neutral H/A states and their decay into τ leptons. The rates for this channel can be very large at high tan β values, as a consequence of the strong enhancement of the H/A couplings to bottom quarks and τ -leptons. This process is particularly favored as, for a heavy enough A boson, one has the mass degeneracy relation M H ≈ M A that in practice leads to search for a single resonance and allows to combine the rates for A and H production. The most recent ATLAS [16] and CMS [17] results with the data collected at the first LHC phase, exclude at the 95% confidence level (CL) a significant portion of the [tan β, M A ] plane for sufficiently high tan β values.
Except in the narrow mass range M Z < ∼ M A < ∼ 140 GeV, where the lower value corresponds to the exclusion limit from negative Higgs searches at the LEP collider [18,19] and the upper one is due to the present limit from charged Higgs boson searches at the LHC (which can be straightforwardly interpreted in the [tan β, M H ± ] parameter space as the H ± properties depend only on these two parameters in the low mass range), the low tan β region of the MSSM has not been considered so far by the experimental collaborations. The reason is that in the benchmark scenarios that are used to interpret the various experimental limits on the cross sections times branching ratios in the context of the MSSM [20,21], the SUSY-breaking scale is usually set to relatively low values, M S ≈ 1 TeV, that do not allow for a heavy enough h state at too low tan β. Indeed, the radiative corrections to the mass M h depend logarithmically on the scale M S and, for instance, one cannot obtain a value M h ≈ 125 GeV for tan β < ∼ 3-5 in the MSSM, even if one favorably tunes the other SUSY parameters that enter the loop radiative corrections, in particular the stop mixing parameter X t which also plays an important role in this context. This is the case of the so-called maximal mixing or M max h scenario which is defined such that the value of M h is maximized, i.e. for a stop mixing parameter X t √ 6M S in the dimensional reduction scheme [20]. The situation is even worse for different values of the X t parameter.
In fact, in most of the [tan β, M A ] parameter space, the measured value M h ≈ 125 GeV, which should be now considered as an essential information on the model, is not satisfied in the M max h benchmark scenario with M S = 1 TeV nor in the alternative benchmark scenarios that are presently used to interpret the experimental searches in the context of the MSSM. If one allows for an uncertainty of a say 3 GeV in the determination of the h mass in the MSSM, from unknown higher order contributions for instance [22], the situation is acceptable if the h mass is confined in the range 122 GeV < ∼ M h < ∼ 128 GeV. Nevertheless, it remains annoying that for each point of the [tan β, M A ] parameter space, one has a different M h value in these benchmark scenarios.
A straightforward and easily implementable solution to this problem has been proposed in Refs. [10][11][12]: if the experimental constraint M h ≈ 125 GeV is enforced, one in fact removes the dependence of the Higgs sector on the dominant radiative correction and, hence, on the additional soft SUSY-breaking parameters, in particular M S and X t . One can again parametrise the MSSM Higgs sector using only the two basic inputs tan β and M A , exactly like it was the case at tree-level. The masses of the heavier H and H ± states as well as the mixing angle α in the CP-even sector are given by very simple expressions in terms of tan β and M A with the constraint M h = 125 GeV. It was shown that this approximation is very good in most of the MSSM parameter space that is currently accessible at the LHC, even when subleading radiative corrections are also considered [10].
In this minimal and almost model independent approach, called the hMSSM in Ref. [10], one has access to the entire [tan β, M A ] parameter space without being in conflict with the LHC data, as the information M h = 125 GeV is taken into account from the very beginning (this is not always the case for the Higgs couplings which conflict with the measured ones at low M A ). In particular, the low tan β region can naturally be accessed, but at the expense of assuming a very high SUSY scale M S . The reason is that at tan β values too close to unity, the tree-level h mass becomes very small, M h ≈ M Z | cos 2β| → 0. To increase M h to ≈ 125 GeV, the radiative corrections that grow logarithmically with M S need to be maximized and hence, a very large scale, M S > ∼ O(100) TeV for tan β < ∼ 2, is required.
The low tan β region can be directly probed by the search for the heavier H/A (and eventually H ± ) states and for relatively low Higgs masses, M H ≈ M A < ∼ 350 GeV, two ways have been suggested. First, one can use the same constraint discussed above from the search of resonances decaying into τ -lepton pairs [12]. Indeed, the rates for A/H production are appreciable at low tan β as the dominant process, the gluon-fusion mechanism, is now primarily mediated by loops of top quarks that have significant couplings to the H/A bosons; at the same time, the decay of at least the A boson into τ τ pairs has a still appreciable rate. The second way is to reinterpret the existing ATLAS and CMS exclusion limits from the search for a heavy SM-like Higgs boson decaying into a pair of massive gauge bosons [23,24] in the context of the MSSM. At low tan β and not too large M H values for which we are not yet in the decoupling regime with a vanishing H coupling to massive gauge bosons, the rates for the decays H → V V with V = W, Z, as well as for gg → H production, are still significant. In addition, searches for the resonant hh [25,26] and hZ [27] topologies have been performed at the LHC with the available 25 fb −1 data at √ s = 7+8 TeV, and one can reinterpret them in the context of the MSSM where the production cross section for gg → H/A and the branching ratios for the decay modes H → hh and A → Zh below the tt threshold can be substantial; see Refs. [12,28]. The two types of searches mentioned above, with results that were preliminary and obtained with a subset of the LHC data collected at √ s = 7+8 TeV, have been used in Ref. [12] to set constraints on the [tan β, M A ] plane; excluded regions have been delineated using some approximations and extrapolations. In the present paper, we update this discussion first by using the latest ATLAS and CMS results, especially the final H/A → τ + τ − and t → bH + → bτ ν analyses [14][15][16][17] as well as heavy SM Higgs searches in the H → W W, ZZ channels [23,24], with the full set of 25 fb −1 data collected in the first LHC phase. In addition, constraints from more appropriate analyses in the A → hZ and H → hh topologies where the resonant case has now been considered [25][26][27] will be included. We will then extrapolate these results to estimate the sensitivity of the 14 TeV LHC run, with at least an order of magnitude higher integrated luminosity than the one accumulated so far. Above the tt threshold, i.e. for M A,H > ∼ 350 GeV, the previously discussed search channels will have little relevance at low tan β values as, because their couplings to b quarks and τ leptons are not enhanced anymore, the heavier H and A bosons will dominantly decay into tt pairs, the top-quark Yukawa coupling ∝ m t / tan β becoming then large. As already mentioned, the main Higgs production channel will be the gluon-fusion process gg → H/A in which the top quark loop generates the dominant contribution. We will see that the production times the decay rates in the processes gg → H/A → tt are indeed substantial in a large part of the MSSM parameter space. We perform a naive estimate of the sensitivity that can be achieved in the search for tt resonances, a sensitivity that could allow to probe a significant part of the low tan β region of the MSSM, complementing the searches for τ + τ − resonances that are instead sensitive to the high tan β region.
The main message of the present paper is that combining the searches for heavy resonances decaying into τ + τ − and tt pairs, and including the H → W W, ZZ, hh and A → hZ channels at M A < ∼ 350 GeV, one can possibly probe the entire [tan β, M A ] MSSM plane (and not only the high-tan β region as is presently the case) up to large pseudoscalar Higgs masses; M A values close to 1 TeV could be reached for any tan β with ≈ 3000 fb −1 data at the LHC with √ s = 14 TeV. This can be done in a model independent way and without relying on any additional theoretical assumption or indirect experimental constraint. The paper is organized as follows. We first summarise our simple parametrisation of the MSSM Higgs sector, further discussing and refining the hMSSM approach. In section 3, we discuss the heavier Higgs production and decay rates focusing on low tan β and summarise the possible impact of superparticles. In section 4, the probing of the [tan β, M A ] MSSM parameter space is discussed when all the search channels, including a projection for the the H/A → tt channel, are combined first at √ s = 7+8 TeV with 25 fb −1 data and then at √ s = 14 TeV and higher luminosities. A brief conclusion is given in a final section.

The hMSSM approach
In this section, we briefly describe the hMSSM introduced in Ref. [10], an approach that allows to parameterize the CP-conserving MSSM Higgs sector in a simple (as only two inputs are needed) and "model independent" (in the sense that we do not consider or fix any other SUSY parameter) way, using the information that the lightest h boson has a mass M h ≈ 125 GeV. The approach is based on several assumptions that we first summarize.
The first basic assumption of the hMSSM is that in the basis (H d , H u ) of the two MSSM Higgs doublet fields that break electroweak symmetry, the CP-even h and H mass matrix can be simply written in terms of the Z and A boson masses and the angle β as in which the radiative corrections are introduced through a 2 × 2 general matrix ∆M 2 ij . This is the usual starting point of the analyses of the neutral MSSM Higgs sector [4] and the calculation of the Higgs masses and couplings including radiative corrections and in which the SUSY scale, taken to be the geometric average of the two stop masses, M S = √ mt 1 mt 2 , can be as high as a few TeV. However, if M S is orders of magnitude higher than the TeV scale, the evolution from this high scale down to the electroweak scale might mix the quartic couplings of the MSSM Higgs sector in a non trivial way, such that the structure of the mass matrix at the low energy scale is different from the one given in eq. (2.1).
In the hMSSM, we assume that the form of the CP-even Higgs mass matrix is as given above even at the very high SUSY scales, M S > ∼ O(100 TeV) that, as it will be seen later, are needed to consider tan β values close to unity 2 .
A second basic assumption of the hMSSM is that in the 2 × 2 matrix above for the radiative corrections, only the ∆M 2 22 entry is relevant, ∆M 2 22 ∆M 2 11 , ∆M 2 12 . In this case, one can simply trade ∆M 2 22 for the by now known mass value M h = 125 GeV using This assumption is valid in most cases as the by far dominant radiative correction from the stop-top sector that is quartic in the top quark mass, enters only in this entry [7,8]: which depends on, besides M S , the stop mixing parameter given by X t = A t −µ/ tan β with A t the stop trilinear coupling and µ the higgsino mass parameter. λ t = √ 2m t /(v sin β) is the top Yukawa coupling with v the standard vacuum expectation value v ≈ 246 GeV, and m t the running MS top quark mass to account for the leading two-loop corrections in a renormalisation group (RG) improved approach. 2 The validity of this approximation is currently studied by the LHC Higgs cross section working group [29]. Preliminary results in an effective two Higgs doublet model that is RG improved to resum the large logarithms involving MS, have been given in Ref. [30,31] and a more refined analysis is under way [29,32].
The maximal value of the h mass, M max h is given in the approximation above by and is obtained for the following choice of parameters [20]: a decoupling regime with a heavy pseudoscalar A boson, large enough tan β values that allow to maximize the treelevel term M 2 Z cos 2 2β → M 2 Z , heavy stop squarks with a sufficiently large M S value to enhance the logarithmic correction log(M 2 S /m 2 t ) and, finally, a stop mixing parameter such that X t = √ 6M S , the so-called maximal mixing scenario that maximizes the stop loops and hence M h . If the SUSY parameters are optimized as above, the maximal M h value can reach the level of M max h ≈ 130 GeV for M S of the order of the TeV scale, a range that is in general assumed in order to avoid a too large fine-tuning in the model. However, if tan β is small, the tree-level contribution M 2 Z cos 2 2β to the h mass squared becomes small as | cos 2β| → 0, thus requiring a substantial correction ∆M 2 22 to obtain a sufficiently large M h . To achieve this, eq. (2.3) shows that one has to substantially increase M S .
In Ref. [10], the approximation ∆M 2 11 , ∆M 2 12 ∆M 2 22 has been checked in various scenarios and found to be rather good if M S much larger than the other soft-SUSY breaking parameters that enter the subleading radiative corrections, such as the higgsino mass µ and the sbottom trilinear coupling A b or more generally the sbottom mixing parameter This assumption should be particularly justified at low and moderate tan β values where first, the bottom-Yukawa coupling is not strongly enhanced. In the approach of Ref. [5] to parameterize the correction matrix of eq. (2.1) including the dominant corrections from the stop and sbottom sectors (and which has been used in Ref. [10] to check this second hMSSM assumption), the entries ∆M 2 11 and ∆M 2 12 of the mass matrix when λ b is set to zero are simply given at lowest order by [4,8] They are proportional to µ/M S and hence, are small if |µ| < ∼ M S . Note that from the expressions above one can see that the two entries ∆M 2 11 and ∆M 2 12 are small not only for M S |µ|, but also when stop mixing is small, M S X t . For moderate tan β (and also at large tan β if the sbottom corrections can still be neglected), one has A t ≈ X t and the offdiagonal entry is further suppressed for maximal X t = √ 6M S . Thus, the approximation of retaining only the entry ∆M 2 22 for the radiative corrections should be good at least at low tan β where a very high SUSY scale is required to obtain a heavy enough h state, suggesting that one naturally has M S |µ| and eventually also M S X t . In this hMSSM approach the mass of the neutral CP even H particle and the mixing angle α that diagonalises the h, H states, will be given by the extremely simple expressions in terms of the inputs M A , tan β and the mass of the lightest h state M h = 125 GeV.
The mass of the charged Higgs boson is simply given by the tree-level relation as the SUSY radiative corrections in this particular case are known to be very small in general. According to Ref. [33] where a detailed analysis of the radiative corrections has been recently performed, the leading one-loop correction to M 2 H + reads when expanding in powers of the SUSY scale as it is justified at low tan β and is therefore very small for M S |µ|. In fact, even for M S ≈ |µ| one obtains ∆M 2 H ± ≈ −10 3 (250) GeV 2 for tan β ≈ 1(tan β 1) and, hence, a relative correction |∆M H ± /M H ± | that is only about 5% (1%) for M H ± ≈ 100 GeV and negligibly small for higher H ± masses. Hence, retaining only the tree-level relation eq. (2.7) as done in the hMSSM should be a very good approximation in this case.
A third assumption of the hMSSM is that all couplings of the Higgs particles to fermions and gauge bosons are given in terms of tan β and the mixing angle α only and, hence, the entire phenomenology of the Higgs particles is determined when the two inputs tan β and M A are fixed. This means that possible corrections not incorporated in the mixing angle α, such as direct vertex corrections, are assumed to have a small impact 3 . In particular, the couplings of the neutral Higgs bosons, collectively denoted by Φ, to up and down-type fermions and to massive gauge bosons (including the coupling of two Higgs and one gauge bosons) when normalized to the SM-Higgs couplings, are simply given by: The trilinear self-couplings among the Higgs bosons are also given in terms of β and α. This is clearly the case at tree-level but, to a good approximation, it remains true when radiative corrections are incorporated. Indeed, besides the corrections that affect the angle α as discussed above, the trilinear couplings receive direct corrections whose dominant component turns out to be simply the one that appears in the correction matrix ∆M 2 and hence, the correction ∆M 2 22 of eq. (2.2) [34]. Thus, the trilinear MSSM Higgs couplings are also fixed in terms of M A , tan β and M h to a good approximation. In units of λ 0 = −iM 2 Z /v, the Hhh and hhh self-couplings, which are the only ones that will matter for LHC phenomenology, will be then given by We note that at least for the hhh self-coupling, one should incorporate the radiative corrections in the same approximation that has been used for the Higgs masses. This would be the only way to achieve a consistent decoupling limit and to make that the λ hhh selfcoupling indeed reaches the SM value in this limit, λ hhh = 3M 2 h /M 2 Z for α = β − π 2 . For the sake of consistency, one should include the radiative corrections to the other self-couplings in the same approximation as for λ hhh . This then fully justifies the choice that we adopt in the hMSSM and the expression of eq. (2.10) for the Higgs self-couplings.
From the discussion above, one can conclude that the hMSSM approach has two very interesting aspects: its economy as only two input parameters are needed to describe the entire MSSM Higgs sector and its simplicity, as the Higgs masses and couplings are given by the very simple relations eqs. (2.6)-(2.10). This would allow to considerably simplify phenomenological analyses of the MSSM Higgs sector which, because of the large number of SUSY parameters to be taken into account, rely up to now either on large scans of the parameter space or resort to benchmark scenarios in which most of these parameters are fixed. Nevertheless, the most interesting aspect of the hMSSM is that it easily allows to describe scenarios with large values of the SUSY scale, M S 1 TeV, but weak-scale masses for the extended Higgs sector.
Because of the large log(M S /m h ) that occur, the high SUSY scale scenarios are notoriously difficult to analyze and, before resuming the large logarithms, the MSSM Higgs spectrum could not be calculated in a reliable way. Until very recently, this was the case of the numerical tools that deal with the MSSM, such as the renormalisation group program Suspect [35] or the program FeynHiggs [36] that is more specialized on the Higgs sector, which were not reliable at too high M S outside the decoupling regime. A new version of FeynHiggs in which some partial resummation of the large terms is performed has become available and allows to address low tan β values as will be discussed later. The hMSSM approach is currently being implemented in an updated version of the program Suspect.
An immediate advantage of the hMSSM is that it re-opens the possibility of studying the MSSM low tan β region [12], which was for a long time overlooked. Indeed, as only SUSY scales of the order of M S ≈ 1 TeV were assumed in the analyses performed in the past, one always had a too light h boson with a mass below the limit M h > ∼ 114 GeV derived from the negative searches of a SM-like h boson at the LEP2 collider [18,19]. For a scale M S = 1 TeV, the possibilities tan β < ∼ 3 and tan β < ∼ 10 were excluded for, respectively the maximal-mixing scenario X t = √ 6M S and the no-mixing scenario X t = 0. The situation became worse with the observation of the h state at the LHC and the determination that its mass is M h ≈ 125 GeV, i.e. well beyond the LEP limit. In fact, for M S ≈ 1 TeV, this relatively large M h value cannot be reached in a large part of the [tan β, M A ] parameter space that is being explored at the LHC in the search for the additional Higgs bosons.
Nonetheless, fixing the SUSY scale at M S ≈ 1 TeV is a very strong theoretical assumption and is currently challenged not only by the measured M h [38] but also by direct sparticles searches at the LHC [39], especially in constrained MSSM scenarios. In the search for the MSSM Higgs bosons at the LHC, one would like to avoid any such assumption and interpret the experimental results, for instance imposing the relevant experimental constraints in the absence of any evidence, in a way that is as model-independent as possible. The hMSSM approach, as no assumption on the SUSY scale nor on any other SUSY parameter (except eventually that they should be smaller than M S ) is made, is more suitable in this respect. In fact, one is considering simply in this case a two-Higgs doublet model of type II [37] where the MSSM relations eqs. (2.6-2.7) are enforced; the superparticles are assumed to be too heavy to have an impact on the Higgs sector (as it will be shown to be generally the case in the next section). The only price to pay is that when the very low tan β region is addressed, one is implicitly considering a very large SUSY-breaking scale, making the MSSM a very unnatural and fine-tuned scenario.
To illustrate this feature, we display in Fig. 1 contours in the [tan β, M S ] plane in which one obtains the value M h = 125 GeV for the h mass, as well as M h = 120 and 130 GeV. The latter examples are when one assumes that a possible mass shift of ∆M h = 5 GeV is missing from unaccounted for subleading corrections (e.g. the contributions of the charginos and neutralinos that we have ignored here) or unknown higher order terms (a theoretical uncertainty of ≈ 3 GeV in the determination of M h is usually assumed [22]). The limit M A ≈ M S and maximal stop mixing X t = √ 6M S are assumed. The figure has been in fact obtained from an analysis of the split-SUSY scenario where the large logarithms have been indeed resummed [40]. As can be seen, at high tan β, M S values in the vicinity of the TeV scale can be accommodate while in the low tan β region, extremely large values of the SUSY scale M S are necessary to obtain M h = 125 GeV. This is particularly the case for tan β close to unity where a value M S ≈ 400 TeV is required for tan β = 1. The situation becomes even worse for the more natural small mixing situation X t M S and outside the decoupling regime when the tree-level h mass is reduced by Higgs mixing. In both cases, huge M S values will be needed for tan β ≈ 1 to reach M h = 125 GeV. For tan β 2, the situation is less dramatic as in the configuration of Fig. 1, only M S ≈ 20 TeV is needed to reach M h = 125 GeV (or the target value M h ≈ 120 GeV if uncertainties are included). We thus expect that our hMSSM approach is valid down to value tan β ≈ 2, but we will extend its validity for tan β close to unity. Let us now illustrate the values that one obtains for the two outputs of the hMSSM, the CP-even H mass M H and the mixing angle α; the charged Higgs mass is simply given by eq. (2.7). These are shown in Fig. 2 as a function of M A for several representative tan β values, from unity to tan β = 30. One can see that at sufficiently high tan β values, tan β > ∼ 10, M H becomes very close to M A and the angle α close to β − 1 2 π, as soon as the pseudoscalar mass becomes larger than M A > ∼ 200 GeV. This is a reflection of the well known fact that the decoupling limit, in which the A and H states are degenerate in mass and have the same couplings to fermions and vanishing couplings to gauge bosons, is attained very quickly at high tan β. Hence, the hMSSM approach should be a good approximation as it describes correctly this decoupling regime. In turn, at low tan β, the mass difference M H −M A can be large and the angle α significantly different from β − 1 2 π even for M A ≈ 400 GeV meaning that the decoupling limit is reached slowly in this case. (For M A ≈ M h we are close to the regime in which the hMSSM is not valid and one gets M H → ∞ and α − π/2; this feature will be discussed shortly). This statement is made more explicit in Fig. 3 where contours for the heavier Higgs mass difference M H − M A and the square of the reduced H coupling to massive gauge bosons, g 2 HV V = cos 2 (β − α), which is a very good measure of the distance to the decoupling limit. At M A ≈ 500 GeV, the difference M H − M A is less than 1 GeV for tan β > ∼ 10 while it is about 10 GeV for tan β ≈ 2. However, even in this case, the mass difference represents about 2% of the A/H masses and, hence in view of the experimental resolution, one can still consider that the two states A and H are degenerate in mass.
There is one problem with the hMSSM at low tan β, though. If tan β ≈ 1, the denominator of eq. (2.2) that expresses the correction ∆M 2 22 in terms of M h becomes close to h and, at low M A , it approaches zero rendering eq. (2.2) ill defined. For tan β 1, the range M A < ∼ 160 GeV is inaccessible, shrinking to M A < ∼ M h for tan β > ∼ 3. As a lower bound M A > ∼ M Z has been set in the model-independent searches of the MSSM Higgs bosons at LEP [19], this area in which the hMSSM is not defined is rather small. We will show later that in fact, this area is entirely excluded 4 by H ± and A searches at the LHC as in both cases, the constraints can be interpreted only in terms of tan β and M A .
Finally, as already mentioned, the program FeynHiggs [36] for the MSSM Higgs sector has now an option that allows to consider the low tan β region in a reliable way (albeit with M h values far below M h ≈ 125 GeV at tan β < ∼ 2). We have used this new version to make the following comparison 5 : for a (tan β, M A ) set and given MSSM inputs (those of the M max h scenario e.g.), we calculate α, M H and M h using FeynHiggs and with the value obtained for M h , we recalculate the hMSSM values of α and M H . The relative difference between FeynHiggs and the hMSSM is shown in Fig. 4 in the [tan β, M A ] plane and is very impressive as, even at very low M A and tan β values, the outputs for M H and α differ by less than 1%. This proves once more that the second assumption of the hMSSM, i.e. that one can consider only the ∆M 22 radiative correction, is fully justified.  4 This low MA is also excluded by the measurement of the observed Higgs boson production and decay rates at the LHC. Indeed, for these MA values, we are very far from the decoupling limit in which the couplings of the h boson are close to their SM values, as the LHC data seem to strongly indicate [6]. However, we will refrain from using this argument and exclude this possibility, as we will prefer to perform the direct Higgs searches in a model independent manner, without relying on any indirect constraint. 5 We thank Pietro Slavich who, originally, made the suggestion to perform this comparison [29].

MSSM Higgs production and decays at the LHC
We come now to the discussion of the decays and the production at the LHC of the heavier A, H and H ± particles in the hMSSM. We will be mostly interested in the low tan β region, but we will first summarize the main features at high and moderate tan β.

Neutral Higgs decays
At high tan β values, say tan β > ∼ 10, the decay pattern of the heavier neutral H/A bosons is extremely simple [5,41] as a result of the strong enhancement of the couplings to downtype quarks and charged leptons that are proportional to tan β, not only for the A state but also for the H boson. Indeed, as in the decoupling limit M A M Z on has α → β − 1 2 π, the Hbb and Hτ τ couplings normalised to the SM Higgs coupling take the limit The neutral Φ = A/H states will decay almost exclusively into τ + τ − and bb pairs, with branching ratios of BR(Φ → τ τ ) ≈ 10% and BR(Φ → bb) ≈ 90%. This is a simple consequence of the fact that the partial widths are proportional to respectively (m τ tan β) 2 and 3(m b tan β) 2 when the color factor is included; m τ = 1.78 GeV and the MS bottom quark mass defined at the scale of the Higgs mass is m b ≈ 3 GeV, implying thus, At high tan β, all other decay channels of the H/A states are strongly suppressed. This is particularly the case of the decays into top quark pairs, despite of the large value m t m b , as the Higgs coupling to up-type quarks are inversely proportional to tan β, rendering very small the Φ = H, A partial widths, given in the Born approximation by where β t = (1 − 4m 2 t /M 2 Φ ) 1/2 and p = 3 (1) for the CP-even (CP-odd) Higgs boson. This is also the case of Higgs decays involving gauge and Higgs particles in the final state. In particular, one should have in principle also the decay modes H → V V with V = W, Z and H → hh in the case of the CP-even and A → Zh in the case of the CP-odd Higgs bosons. However, the partial decay widths of the H particle into massive gauge bosons Γ(H → V V ) are proportional to the square of the reduced coupling which becomes zero in the decoupling limit as is the case for the pseudoscalar A boson, that has no tree-level couplings to V V states as a result of CP-invariance. For the latter state, the possibility A → hZ for M A ≥ M h +M Z ≈ 220 GeV, i.e. near the decoupling limit, will have a suppressed rate as the coupling g AhZ = g HV V tends to zero at large tan β. Indeed, an expansion in terms of 1/M 2 A gives [12] g HV V = g AhZ (3.6) and, at high tan β, both sin 4β and sin 2β are proportional to cot β so that the limit g HV V → 0 is reached faster in this case. The same is true for the decay H → hh when M H ≥ 2M h as the trilinear Higgs coupling of eq. (2.10) for M H > ∼ 2M h reaches the limit g Hhh and is thus strongly suppressed at high tan β that implies sin 2β = 2 tan β/(1+tan 2 β) → 0. The situation is drastically different at low values of tan β when the heavy Higgs states are kinematically allowed to decay into top quark pairs, M H ≈ M A > ∼ 2m t [5]. Indeed, H/A → tt become by far dominant g Φtt ∝ m t / tan β is so strong that it leaves no chance to the other possible channels. This is clearly the case for the H/A → bb, τ + τ − rates which become negligibly small as the couplings g Φdd are not enhanced anymore and m t m b , m τ . This is also the case of the decays A → hZ and H → V V at large M Φ since the couplings approach zero in this case. For the H → hh decay, there is still a component of the g Hhh coupling of eq. (3.7), the one ∝ sin 2β ≈ 1 for tan β ≈ 1, that is non-zero in the decoupling limit. However, besides the fact that the Htt coupling is larger than the Hhh coupling, the partial decay width for the process H → hh decreases as . At low tan β and high M H values, one of the components of g HV V given in eq. (3.6) (the one ∝ sin 4β) vanishes while the other component tends to Because of the enhancement of the decay rate by M 3 H , one would have then a partial width Γ(H → V V ) that is suppressed by a power 1/M H only and hence, does not become completely negligible compared to H → tt even at very high M H . For instance, at M H ≈ 500 GeV, the branching ratios for the decays H → W W and H → ZZ are still at the 2% and 1% level respectively. This is appreciable and, at least in the case of the ZZ decay, it is of the same order of magnitude as the branching fraction of the observed 125 GeV h boson, with the advantage that the ZZ pair has a much larger invariant mass with a significantly smaller background (which compensates for the smaller Higgs production cross section as will be seen later).
If the H/A states have masses below the 2m t kinematical threshold, the two-body H/A →tt decays are not allowed anymore. Off-shell three-body decays A/H →tt * →tbW are possible, but the rates are suppressed by an additional electroweak factor and the virtuality of one of the top quarks [42]. The gauge and Higgs decays of the H/A states would then become significant at low tan β values. In the mass range M h +M Z < ∼ M A < ∼ 2m t , the decay A → hZ will be dominant: the reduced coupling g AhZ = g HV V ∝ M 2 h /M 2 A is only moderately suppressed and the full AhZ coupling is still substantial compared to the tiny Abb coupling. Likewise, for 2M h < ∼ M H < ∼ 2m t , the decay mode H → hh is dominant as the coupling g Hhh at low tan β will stay appreciable. The two-body decays into massive gauge bosons H → W W and ZZ are also significant below the 2m t threshold.
The bosonic decays will also be non-negligible at intermediate values of tan β, tan β ≈ m t /m b ≈ 5-10, when the A/H couplings to top quarks are suppressed while those to bottom quarks are not yet strongly enhanced. However, below the 2m t threshold, when the Higgs couplings to the bosonic states are not too suppressed and the only competition will be due to the Φ → bb decays that is only slightly enhanced, the rates for the H → W W, ZZ, hh and A → Zh channels will be smaller than at low tan β values.
The branching fractions for the various Higgs decays discussed above are displayed in Fig. 5 in the [tan β, M A ] plane assuming the hMSSM with M h = 125 GeV. The Fortran program HDECAY [41] where the hMSSM relations were implemented has been used. The color code is such that the red area is when the considered decay rates are large, while the blue area is when they are small. The white areas are when the decay rates are very small, below the minimal value of the scale in the color axis. As can be seen, the H/A → τ τ decays are important at high tan β values. The branching ratios for the decays H/A → bb follow that of τ τ final states when a factor 9 is included and are, hence, largely dominant. The decays H/A → tt are by far leading at low tan β for M A,H > ∼ 350 GeV (one notices that at least for A, they are also significant slightly below the tt threshold). The bosonic decays H → W W, ZZ, hh and also A → hZ have reasonable rates (with a color from light green to yellow) only at very low tan β values, tan β < ∼ 3, and below the 2m t threshold.
A final word should be devoted to the total Higgs decay widths, which are displayed for the A and H particles in Fig. 6, again in the [tan β, M A ] hMSSM plane. In the low and high tan β regimes, one can consider only the dominant fermionic decays of the Φ states and, to a good approximation, the total decay widths will be given by (up to phase space factors that become unity for M Φ 2m t and that we will ignore) For tan β ≈ 1 and tan β ≈ 60, one obtains a total decay width that is approximately Γ tot Φ ≈ 5%M Φ and, compared to the Higgs mass, it is not very large. Hence, to a good approximation the A/H states can be considered as narrow resonances in most cases.
Note that for the H state, as the branching ratios and total decay width are shown as a function of M A , some peculiar features can be observed. These are explained by the fact that there is a large splitting between M H and M A at low tan β and M A values which lead to, for instance, the opening of the H → tt mode already at M A < ∼ 200 GeV and, hence, suppressed H → W W, ZZ decays but a large total decay width for tan β ≈ 1.

Neutral Higgs production
Let us turn now to the production of the neutral MSSM Φ = H/A bosons at the LHC. Also in this case, the cross sections crucially depend on the considered tan β regime and in most cases, the two processes that play a leading role are the gluon-fusion mechanism gg → Φ which is initiated by a heavy quark loop [43] and the associated Higgs production with b-quarks, gg/qq → bb+Φ, which at high energies and if no-additional b-quark is considered in the final state, is equivalent to the fusion process bb → Φ [44]. All other processes, in particular vector boson fusion and associated production with a massive gauge boson for the CP-even H state, qq → Hqq and qq → HV , and associated production with top-quark pairs for both the H and A states, pp → Φtt, have much smaller rates as the couplings g HV V and g Φtt are suppressed and/or the available phase space is not favorable.
At leading order in perturbation theory, the partonic cross sections for the bb → Φ and gg → Φ processes can be written in terms of the partonic c.m. energyŝ and M Φ as [3] σ(bb → Φ) = π 12 In the case of the gg → Φ process, the quarks Q running in the loop should be taken to be the heavy bottom and top quarks with Higgs couplings given in eq. (2.9) and masses incorporated into the reduced variables τ Q = M 2 Φ /4m 2 Q . The form factors for spin-1 2 fermion loops in the case of a CP-even H and a CP-odd A bosons are given by where the function f (τ ) above and below the τ = 1 kinematical threshold is defined as While the amplitudes are real for M Φ ≤ 2m Q , they develop an imaginary part above the kinematical threshold. At very low Higgs masses, compared to the internal quark mass, the amplitudes for a scalar and a pseudoscalar states reach constant but different values Instead, in the opposite limit, M Φ 2m Q , chiral symmetry holds and the amplitudes for the CP-even and a CP-odd Higgs bosons are identical (as in the bb → Φ case), The maximal values of the amplitudes occur slightly above the kinematical threshold where one has for the real parts Re(A H 1/2 ) ∼ 2 and Re(A A 1/2 ) ∼ 5. At high tan β, the strong enhancement of the Higgs couplings to b-quarks and the suppression of the couplings to top quarks and gauge massive bosons makes that only these two processes are relevant, with the gluon-gluon fusion mechanisms dominantly generated by the bottom quark loop. The cross sections σ(gg → Φ) and σ(bb → Φ) are of the same order of magnitude and can be so large that they make the process pp → gg+bb → Φ → τ + τ − , with the branching fraction for the decay Φ → τ + τ − being of order 10% as seen previously, the most powerful LHC search channel for the heavier MSSM Higgs bosons. The pp → bb+H/A mode with A/H → bb, which has an order of magnitude larger rates in principle (if no high-p T b-quark from production is required), has also been considered but the sensitivity is smaller as this fully hadronic process is subject to a much larger QCD background.
At high tan β, the production rates are approximately the same for the H and A states in both the bb and gg fusion cases as discussed earlier. While σ(bb → Φ) is known up to NNLO in QCD perturbative theory [45], σ(gg → Φ) is instead known only up to NLO in the limit M Φ > ∼ 2m Q that we will be mainly interested in [46]. For the top-quark loop, we will nevertheless also include the NNLO QCD corrections [47] that are in principle only valid for M Φ < ∼ 2m Q as advocated in Ref. [48]. The precise values of the cross section times branching fractions σ(pp → Φ) × BR(Φ → τ + τ − ) for a given [tan β, M A ] MSSM point have been updated in Refs. [48,49] and the associated theoretical uncertainties from missing higher order perturbative contributions, the parametrisation of the parton distribution functions and uncertainties on the inputs α s and b-quark mass, have been estimated to be of the order of 25%. Any effect below this level, such as the SUSY effects that we will be discussed later in this section, should be considered as small.
Again, at low tan β, the situation is very different. The cross sections for the bb → Φ processes are now very small as the Φbb coupling is not enhanced anymore. For M A > ∼ 200 GeV, this is also the case of the associated production with tt pairs as a result of a the small phase-space and, in the case of the H state, of the vector boson fusion qq → Hqq and associated production with a gauge boson qq → HV as a result of the suppressed HV V coupling (the A state cannot be produced in these two processes as there is no AV V coupling). The only process which would have a reasonable production cross section would be the gluon-fusion process gg → Φ with, this time, the leading contribution being generated by loops of top quarks that have couplings to the Higgs bosons that are only slightly suppressed compared to the SM Higgs case. The production cross sections σ(gg → Φ + X) and σ(bb → Φ + X) with Φ = A (left) and Φ = H (right) are displayed at the LHC in the [tan β, M A ] hMSSM parameter space for √ s = 8 TeV (top) and 14 TeV (bottom); the MSTW parton distribution functions [50] have been used. The rates for bb → Φ have been obtained using the program SuShi [51] and an adapted version of HIGLU [52] has been used for gg → Φ. As can be seen, the cross sections are rather large in particular at high tan β and, for gg → Φ, also at low tan β when the Higgs couplings to b-or t-quarks are strong and at relatively low M A when the phase space is not too penalizing. Even for M A = 500 GeV (1 TeV), the production rates are significant at √ s = 8 TeV (14 TeV), if tan β is sufficiently high or low.

The case of the charged Higgs boson
A final word should be devoted to the case of the charged Higgs boson, whose coupling to fermions is proportional to The coupling is large at low tan β when the component m u / tan β is not suppressed and at very large tan β when the component m d tan β is enhanced, so that many aspects discussed for the pseudoscalar Higgs boson hold also in this case [5]. At low mass, M H ± < ∼ 160 GeV, which corresponds to M A < ∼ 140 GeV, the charged Higgs boson can be produced in the decay of top quarks that are copiously produced at the LHC, gg + qq → tt with one top quark decaying into the dominant t → bW mode and the other into t → bH + . For M H ± ≈ 140 GeV, the latter channel has a branching ratio ranging from order ≈ 10% for tan β ≈ 1 or tan β ≈ 60 to order ≈ 1% for tan β ≈ 7-8 when the Higgs couplings are the smallest. In this low mass range above, the H ± boson will decay almost exclusively into τ ν final states but some some competition with the hadronic decay channel H + → cs will occur at very low tan β. The H ± branching fractions are shown in Fig. 8 as a function of M H ± for two representative tan β values, tan β = 2 and tan β = 30.
BR(H ± )   At higher masses, the H ± state will be mainly produced in the three-body production process pp → tbH ± which, at high energies, is equivalent to the two-body channel gb → H ± t if no additional final state b quarks are detected [53]. Again, significant rates occur only at very low or very large values of tan β when the H ± tb coupling of eq. (3.17) is large (some small additional contributions from the tree-level qq → H + H − and loop induced gg → H + H − pair and associated qq → H ± +A/h/H production modes are also possible [5]). The cross sections have been derived in Ref. [54] where the two possibilities for the process, pp → tbH − and gb → tH − , are properly matched and some numerical grids have been provided for the MSSM. The output of these grids for the production rates at √ s = 8 and 14 TeV is shown in Fig. 9 in the [tan β, M H ± ] plane. At high tan β, as shown in Fig. 8, the H ± decay branching fractions are BR(H + → τ ν) ≈ 10% and BR(H + → tb) ≈ 90% exactly for the same reasons discussed previously for the H/A particles. All the other decay channels can be safely ignored so that the main search channel would be pp → H ± t(b) production with H + → τ ν. As will be seen later, the process is however less powerful in probing the MSSM parameter space than the pp → H/A → τ τ channel discussed earlier.  At low tan β, the dominant decay channel will be by far the H + → tb mode with a branching ratio close to unity for a sufficiently heavy H ± state when phase-space effects are irrelevant (one should note though, that slightly below the m t + m b kinematical threshold, the three-body decay with an off-shell top quark, H + → bt * → bbW , is also important in the case tan β ≈ 2 as shown in Fig. 8). The main search channel in this case would be the pp → H ± t(b) → ttbb mode. This process, which is sensitive to the same area of [tan β, M A ] parameter space as the H/A → tt channels discussed before i.e. the low tan β and high M A regions, has been considered in the past and found to be of limited use as it is subject to a large QCD background [55]. However, a recent CMS analysis [56] gave interesting and more optimistic results that we will discuss in the next section.
Finally, at H ± masses above M H ± > ∼ 160 GeV, an interesting decay channel would occur, namely H ± → W h. Nevertheless, and in contrast to the similar A → hZ decay mode discussed previously, this channel has to compete from the beginning with the dominant H ± → tb decay. Only at moderate tan β and low M H ± that this decay has a sizeable branching ratios, of the order of a few percent, allowing for H ± searches in the interesting channel pp → tbH ± → tbW h which, experimentally, has not been considered so far.

Impact of the SUSY spectrum and dark matter
An important question would be if the MSSM Higgs production times decay rates are not affected by the presence of supersymmetric particles. These could have two impacts: first, they could contribute virtually to the processes and modify the production cross sections and decay branching ratios. This issue is directly related to the third assumption of the hMSSM, namely that the couplings of the Higgs bosons are simply given by eqs. (2.9)-(2.10) and no direct correction is involved. Second, SUSY particles could appear in the decays of the Higgs particles and alter the branching ratios for the standard channels that are searched for. This possibility would also invalidate the simple hMSSM approach as some SUSY-breaking parameters would be then required to describe Higgs phenomenology. Both issues have been discussed e.g. in Ref. [12] and below, we simply summarise the main points with details concerning the low tan β region and the decays H/A → tt.
For what concerns the production processes, besides the standard top and bottomquark loops, there are also squark (and mainly stop) loops [57] that contribute to the production of the CP-even H boson in the gluon-fusion channel, gg → H; the CP-odd A states does not couple to identical sfermions and there is no-squark contribution to gg → A at lowest order. However, as the Higgs-squark couplings are not proportional to squark masses, the contributions are damped by powers of the squark mass squared ∝ 1/m 2 Q and should be small for sufficiently heavy squarks. This is particularly the case at high tan β values where the standard bottom-quark contributions are so strongly enhanced that the impact of squarks becomes negligible. At low tan β values, as one needs a large SUSY scale M S 1 TeV in order to accommodate an h boson with a mass M h ≈ 125 GeV, the impact of the too heavy squarks should also be negligible in the gluon-fusion process. Hence, in most cases, these SUSY loop contributions can be ignored in the production modes.
SUSY particles can have a large impact also through the Higgs boson couplings. Indeed, besides the radiative corrections that affect the Higgs mass matrix eq. (2.1), there are additional one-loop vertex corrections that modify the Higgs-fermion couplings and which are not described by the hMSSM. These corrections are in general only important in the case of b-quarks and only at high-tan β and large µ values, since they grow as µ tan β. The dominant components are due to the contributions to the Higgs-bb vertices from the strongly interacting sbottoms and gluinos and the weakly interacting higgsinos with top squarks. They can be approximated by [58] They affect mainly the heavier Higgs couplings that become in the limit M A M Z , For the lighter h state, the coupling g hbb is not affected in this limit and stays SM-like. Nevertheless, as already discussed in many places including Refs. [12,49], this correction has only a limited impact in the case of the full pp → Φ = H/A → τ τ process as the correction appears in both the production cross section σ(gg + bb → Φ) ∝ (1 + ∆ b ) −2 and in the τ τ decay branching fraction, BR(Φ → τ τ ) = Γ(Φ → τ τ )/[(1 + ∆ b ) −2 Γ(Φ → bb) + Γ(Φ → τ τ )], and it largely cancels out in the product of the two Hence, only when the ∆ b correction is huge and larger than unity (a feature that might put in danger the perturbative series) that its impact on the pp → τ τ cross section times decay rate is of the order of the theoretical uncertainty, about 25% as discussed earlier. At low tan β values, and eventually also at intermediate tan β values, the ∆ b correction is not enhanced and its effects should be rather small. Nevertheless, this is not the case of all process and, in particular, the search channel pp → H/A with H/A → bb in which the ∆ b impact is in fact doubled in the production times decay rates, and the pp → H ± bt mode with H + → τ ν at low M H ± and H + → tb at high M H ± . We will see, however, that these processes do not play a leading role in MSSM Higgs searches at the LHC at high tan β. In summary, and to first approximation, one can thus consider that the ∆ b correction has a limited impact on the Higgs searches in the hMSSM.
For the second option, namely that light SUSY particles could contribute to the decays of the Higgs bosons, the situation is also relatively simple; see Ref. [5] for review. At very high tan β, the partial widths of the H/A → bb, τ + τ − as well as H + → tb, τ + ν decay modes are so strongly enhanced, that they leave no room for the SUSY decay channels.
At low tan β, high values of the SUSY scale are required, resulting in large squark masses (at least in universal models in which the squark masses of the three generations are related) that make the Higgs decays into squarks kinematically closed for reasonable M A values. If the masses of the sleptons are disconnected from the SUSY scale and are made small enough for the decays of the heavier Higgs bosons into slepton pairs, H →˜ i˜ j and A →˜ 1˜ 2 (again, CP invariance forces the A boson not to couple to identical sfermions), to occur. Nevertheless, except for the Higgs-stau couplings at sufficiently high tan β values when the competition from the standard channels is though, the Higgs-sleptons couplings are in general small making these channels very rare and their impact limited. Thus, only decays into charginos and neutralinos could play a role and affect significantly the Higgs branching fractions in the standard channels. Let us briefly comment on these channels.
Three conditions must be fulfilled in order to have significant rates for Higgs decays into charginos and neutralinos, H/A → χ 0 i χ 0 j (with i, j = 1· · ·4), H/A → χ ± i χ ∓ j (with i, j = 1, 2) and H ± → χ ± i χ 0 j [5,59]. First, one needs that some of the χ states are light, M Φ > ∼ 2m χ , in order to allow for some decay channels to be kinematically open. Second, one needs to have significant Φχχ couplings; these couplings are maximal when the χ final states are mixtures of higgsinos and gauginos, a feature which requires comparable higgsino and gaugino mass parameters, µ ≈ M 2 . Last but not least, one needs that the standard Higgs decay modes are not enhanced and hence, not too low or too large values of tan β where, respectively, the Higgs-top and the Higgs-bottom couplings are enhanced.
The maximal Higgs decay rates into charginos and neutralinos are obtained at moderate tan β when all χχ channels are kinematically accessible. In this case, as a consequence of the unitarity of the diagonalizing chargino and neutralino mixing matrices, the sum of the partial widths do not involve any of the elements of these matrices in the asymptotic regime M Φ 2m χ where phase space effects can be neglected. The sum of the branching fractions of the three Higgs bosons Φ = H/A/H ± decaying into the various χχ final states is then simply given by (θ W is the electroweak mixing angle) [59] when only the leading tt, bb and τ τ modes for the neutral and the tb and τ ν modes for the charged Higgs bosons are included in the total widths. This is approximately the case as we are close to the decoupling limit when these SUSY channels are accessible and the other standard decay modes such as H → V V, hh and A → hZ are suppressed.
The branching ratios when all ino states are summed up are shown for the three MSSM Higgs states in Fig. 10 as a function of tan β for M A = 600 GeV. The hMSSM relations for the Higgs sector have been enforced and the other relevant SUSY parameters are fixed to µ = M 2 = 200 GeV, assuming that the bino and wino soft SUSY-breaking parameters are related by the unification condition M 2 ≈ 2M 1 . One can see that the branching ratios for the three Higgs particles are indeed similar and that they do not dominate at low nor at high tan β. For instance, they are less than 25% (which is the magnitude of the theoretical uncertainty on the production cross sections) for tan β < ∼ 2 and tan β > ∼ 30 as can be seen from the figure. In contrast, the χχ branching rations can be large for intermediate values of tan β when the Higgs couplings to top (bottom) quarks are suppressed (not strongly enhanced) and, for instance, they reach the level of ≈ 70% at tan β ≈ 5-10.  Nevertheless, this possibility with large Higgs decay rates into SUSY particles seems very unlikely. First, low and comparable values of the wino and higgsino mass parameters M 2 ≈ µ < ∼ 300 GeV that would lead to light charginos and neutralinos in the decay products of not too heavy H/A/H ± bosons, are constrained by the direct searches for these particles at LEP and the LHC [18,60,61]. In particular, the associated production of the lighter chargino and the next-to-lightest neutralino qq → χ ± 1 χ 0 2 would lead to large cross sections at √ s = 8 TeV and the decays χ ± 1 → W χ 0 1 and χ 0 2 → Zχ 0 1 , with leptonic gauge boson decays, would have significant branching fractions. The search for leptons plus missing energy at the first run of the LHC, in particular the clean trilepton events from the chain pp → χ ± 1 χ 0 2 → W Zχ 0 1 χ 0 1 → E mis T , imposes severe restrictions on the parameter space. For instance, for M 1 ≈ 100 GeV, the area µ ≈ M 2 < ∼ 200 GeV that corresponds to the choice adopted for Fig. 10 is by now excluded by the LHC data 6 [61].
To evade these experimental bounds, one needs either to increase the parameters µ 2 ≈ M 2 well above 200 GeV with a consequence that the phase space for the Higgs decays will be limited, or to make that the light χ states are either pure higgsinos (µ M 2 ) or pure gauginos (µ M 2 ), which then suppresses the Zχχ couplings e.g. and hence the trilepton signals, leading to Higgs couplings to the kinematically accessible charginos and neutralinos that are too small. Thus, in all these cases, the SUSY decays are suppressed and do not jeopardize the Higgs signals in the standard search channels.
These arguments become stronger in the case where the lightest neutralino χ 0 1 is forced to be the candidate for the dark matter in the universe, with a cosmological relic density as measured by the WMAP/Planck teams, 0.09 ≤ Ωh 2 ≤ 0.12 [62] with h being the reduced Hubble constant. Traditionally, four regions of the MSSM parameter space have been advocated to fulfill this condition for the LSP neutralino [63] and we list them below.
i) The "well tempered neutralino" region, with a mixed gaugino-higgsino LSP with significant couplings to gauge and Higgs bosons that allow for a good LSP annihilation rate into these finals states, χ 0 1 χ 0 1 → W W, ZZ, hZ. For low M 2 and µ, this is the region discussed above that is constrained by the multi-lepton plus missing energy searches.
ii) The bino like neutralino region, µ M 1 , where the main LSP annihilation channel is into third generation tau leptons, with the exchange of light sleptons in the t-channel, χ 0 1 χ 0 1 → τ + τ − , and theτ co-annihilation region in which the lightestτ 1 state is almost mass degenerate with the LSP, mτ 1 ≈ m χ 0 1 , and the correct relic density is provided by the processτ 1τ1 → SM particles. If the SUSY scale is high and a kind of universality is assumed for sfermions, theτ 1 state will be too heavy and both channels become inoperative.
iii) The regions where the LSP is almost a pure higgsino or gaugino and hence with small couplings to H/A/H ± . The correct relic density is provided by the co-annihilation of χ ± 1 and χ 0 2 that need to be very heavy and inaccessible in the decays of TeV Higgs bosons. iv) Finally, there is the Higgs-pole region [64] in which an efficient LSP annihilation into SM particles is provided by the exchange of an almost on-shell A boson in the schannel; one thus needs M A ≈ 2m χ 0 1 (the possibility of h-boson exchange [65] leading to m χ 0 1 ≈ 60 GeV is by now unlikely). In the past, the high tan β region was favored and the most discussed annihilation channel was χ 0 1 χ 0 1 → A → bb. At low tan β, a new possibility opens up, the channel χ 0 1 χ 0 1 → A → tt which can also lead to the correct relic density. In Fig. 11, we display the areas of the [M 2 , µ] parameter space in which the relic density of the lightest neutralino, calculated using the program micrOMEGAs [66], is as 6 Note that we have adopted the same choice of gaugino-higgsino parameters as in the benchmark scenario of Ref. [21] that has been used to interpret the experimental limits in the pp → τ τ searches made by the ATLAS and CMS collaborations [16,17]. Hence, this choice leads to a large branching fraction for Higgs decays into χχ states for MA > ∼ 300 GeV and weakens the experimental constraints that can be obtained from H/A searches in the pp → τ τ process, while it is apparently excluded by direct SUSY searches. determined by the WMAP/Planck collaborations, i.e. 0.09 ≤ Ωh 2 ≤ 0.12 [62]. We have chosen M A = 500 GeV and tan β = 2 for the hMSSM inputs, and assumed very heavy sfermions. In the left plot we have fixed the bino mass to M 1 = 100 GeV, while in the right plot, we used the unification condition M 2 = 2M 1 . Outside the areas excluded by LEP and LHC ino searches at M 1 = 100 GeV ≈ m χ 0 1 , only two areas lead to a correct relic density: small M 2 or µ values ( < ∼ 150 GeV) that allow for χ 0 1 χ 0 2 and χ 0 1 χ ± 1 co-annihilation or a mixed bino-higgsino LSP. Another area opens up if m χ 0 1 ≈ 1 2 M 2 ≈ 1 2 M A as can be seen in the right-figure: the A funnel in which the LSP efficiently annihilates through the channel χ 0 1 χ 0 1 → A → tt. In most of this area, the decay A → χ 0 1 χ 0 1 is kinematically closed (or phase-space suppressed) and, because χ 0 1 is the LSP, so are all Higgs decays into superparticles.
Hence, in all cases, the requirement that the lightest neutralino is the dark matter in the universe with the correct relic density makes that the decays of the Higgs bosons into charginos and neutralinos should not occur, or at least should not dominate. One concludes from all the discussions of this subsection that it is rather unlikely that the SUSY particles make a significant impact in the phenomenology of the MSSM Higgs bosons, either in their virtual contributions to the production and/or decay processes (in particular since the SUSY effects should be larger than the ≈ 25% theoretical uncertainty that affects the production rates) or in the direct appearance in the decays of the heavier Higgs states. One can thus assume that the superparticles are very heavy and/or too weakly coupled and that they decouple from the MSSM Higgs sector, except of course in the radiative corrections to the CP-even mass matrix eqs. (2.1). This is, in fact, another way of stating the third assumption of the hMSSM discussed in section 2 and one can thus consider this effective approach as a very good benchmark. Ignoring the SUSY effects is a rather reasonable attitude since, besides the tremendous simplifications that the hMSSM introduces in the description of the Higgs sector, it leads to a straightforward interpretation of the experimental constraints, that do not need to be "de-convoluted" from these complicated effects when they are included.

Interpretation of the fermionic Higgs decay modes in the hMSSM
As discussed earlier, the most efficient channels that allow to probe the MSSM parameter space at the LHC are the search for charged Higgs bosons coming from top quark decays, the process t → bH + followed by the decay H + → τ + ν and its charge conjugate H − process, and the search for high mass resonances decaying into tau-lepton pairs, the processes pp → H/A → τ + τ − . Both channels have been considered by the ATLAS and CMS collaborations and we briefly summarize below the resulting constraints on the hMSSM.
The CMS H ± search [15] was performed with the 19.7 fb −1 data collected at √ s = 8 TeV with the τ -leptons decaying fully hadronically. 95%CL upper bounds have been set on the product of branching ratios BR(t → bH + )×BR(H + → τ + ν) from 1.2% at M H ± = 80 GeV (about the exclusion limit obtained on M H ± from LEP2 searches [18,19]) to 0.16% at M H ± = 160 GeV (beyond which phase space effects start to be too penalizing). The search excludes the entire range M H ± < ∼ 140 GeV for all values of 1 ≤ tan β ≤ 60. For larger H ± masses, the areas where tan β ≈ 8 at M H ± = 140 GeV to tan β ≈ 5-15 at M H ± = 160 GeV, in which the m t / tan β component of the H ± tb coupling is suppressed while the m b tan β component is not yet enhanced, remain viable at the 95%CL. The ATLAS search for H ± states [14] was also performed with the full 19.5 fb −1 data recorded at √ s = 8 TeV; the same channel as above, i.e. t → bH + → bτ + ν with the taulepton decaying hadronically, has been used. Similar 95%CL upper limits than CMS have been obtained on the product BR(t → bH + )×BR(H + → τ + ν). Compared to the previous limits, a small additional area of the [tan β, M H ± ] plane, at tan β ≈ 8 for M H ± ≈ 100 GeV and to tan β ≈= 6-10 at M H ± ≈ 90 GeV, remains unexcluded by the ATLAS analysis, as a result of the presence of a large tt, W/Z+jet backgrounds in this mass bin. These limits can be turned into bounds in the [M A , tan β] parameter space assuming the usual relation M 2 This is what is illustrated by the dark blue area of Fig. 12 in which the constraints on the [M A , tan β] plan are shown: we take the limits on BR(t → bH + )×BR(H + → τ + ν), that we calculate using the program SDECAY [67], and interpret them in the hMSSM. We see that in our case, the entire area M A < ∼ 140 GeV is excluded at low tan β values, reducing to M A < ∼ 130 GeV at high tan β. In contrast to ATLAS and CMS, which use different means to calculate the product of branching ratios, we do not have the holes at the extreme values of M H ± .
In the previous references, both experiments performed also searches for heavier H ± states with M H ± > ∼ m t +m b ≈ 180 GeV, by considering the process pp → H ± t(b) with again H + → τ + ν. The areas tan β > ∼ 45 (60) at M H ± ≈ 200 (250) GeV are excluded by the CMS collaboration (the ATLAS group assumed BR(H + → τ + ν) = 100% in this area while it should be only 10%). These limits are much less powerful than those obtained from the τ τ search as will be seen shortly. However, there was in interesting CMS search in the channel H + → tb performed with 19.7 fb −1 data at √ s = 8 TeV [56]. Limits on σ(pp → tbH ± ) assuming BR(H + → tb) = 1 have been set and surprisingly, one is not far from being sensitive to the very low tan β area and with about a factor of two more data, one would have probed tan β ≈ 1 for M H ± ≈ 200 GeV. We note that these limits from H ± searches exclude a substantial part of the low [tan β, M A ] area in which the hMSSM is ill defined, i.e. the region at the left of the thick black solid line in Fig. 12. The exclusion is valid in this area as all the ingredients used to obtain the limits depend only on M H ± and tan β and do not involve the CP-even Higgs parameters M H or α that were undefined. The only assumption is that the relation M 2 H ± = M 2 A + M 2 W remains valid. Hence, one problematic issue within the hMSSM is in a sense partly solved by these "model-independent" exclusion limits (we do not address here the theoretical issue of the validity of the entire model at tan β values close to unity).
The most important search mode in the MSSM is certainly the pp → H/A → τ τ channel. The ATLAS collaboration has searched for this signal using the 19.5-20.3 fb −1 data collected at 8 TeV [16] while CMS has used the full 24.6 fb −1 data collected at 7+8 TeV [17]. Both collaborations consider the leptonic (τ e τ µ ), semi-leptonic (τ τ had ) and hadronic (τ had τ had ) τ decays and CMS also considers the case where an additional b-quark is present in the final state. Limits at the 95%CL on σ(pp → τ + τ − ) as a function of the invariant mass M τ τ of the tau-lepton pair have been given by the two experiments.
Our procedure to interpret these limits in the hMSSM is as follows. First, we combine the ATLAS and CMS 95% CL exclusion limits on σ(pp → τ + τ − ). We then compare them with the numbers that we obtain for the rates, namely the H/A production cross sections calculated with the programs HIGLU [52] and SuShi [51] and the branching ratios BR(H/A → τ τ ) calculated using HDECAY [41]. These are derived assuming that the parameters M H and α given by eqs. The result is shown Fig. 12 by the light blue areas of the [M A , tan β] plane and is truly impressive. The largest area is the one that excludes all values above tan β ≈ 8 for M A < ∼ 300 GeV, extending to tan β ≈ 20 (40) for M A < ∼ 700 (900) GeV.
The ATLAS and CMS observed limits, when interpreted in the M max h benchmark scenario, are also displayed in Fig. 12. We observe that below M A ≈ 200 GeV, our limit is less restrictive, the reason being that we do not make use of any refinements in order to treat the regions in which the three Higgs bosons have a comparable mass and, also, to deal with the observed signal of the lighter h boson. In contrast, our limit is stronger at masses above M A ≈ 300 GeV, the reason being that while in our case there is no SUSY decays of the Higgs bosons, the M max h scenario leads to H/A decays into charginos and neutralinos that are significant. Indeed, for the gaugino and higgsino mass parameters of the M max h benchmark (and the slightly modified ones), µ = M 2 = 2M 1 = 200 GeV, the LSP has a mass of m χ 0 1 ≈ 100 GeV while the heavier neutralinos and the charginos have masses of the order of ≈ 200 GeV, so that many SUSY decays occur starting from M A ≈ 300 GeV and all of them will be present for M A ≈ 500 GeV and above. These decays will have substantial branching fractions, in particular in the intermediate tan β ≈ 10 range where they become dominant, as can be seen from Fig. 10. Hence, the H/A → τ τ branching ratios are suppressed in this this case, resulting in a weaker exclusion limit (which is unfortunate since this parameter configuration is almost certainly excluded by the direct searches [61]).
A second step is to extrapolate the ATLAS and CMS limits to the low tan β region, which as discussed earlier, can be described within the hMSSM approach in contrast to the benchmark scenarios used by the collaborations. Two islands were discovered during the exploration. A first and substantial area is at very low tan β and M A , tan β < ∼ 2 and M A < ∼ 230 GeV. Here, because part of the area is ill defined, we consider only the production and the decay of the A state that depend only on tan β and M A as the decay channel A → hZ, which introduces a dependence on the angle α through the AhZ coupling, is not yet kinematically open. In this domain, as discussed earlier, both the gg → A cross section (dominated by the top-quark loop) and the branching ratio for the decay A → τ τ (which, together with the one for bb and cc, is the only significant one to occur) is substantial. Hence, despite of the fact that we have only one resonance, the cross section times branching fraction is large enough to generate an observable signal. The excluded area from this search removes the small residual part of the ill-defined hMSSM region, that was left after imposing the exclusion limit from the H ± searches discussed above.
More surprising at first sight, we discovered another smaller island at M A ≈ 350 GeV and tan β ≈ 2-4. It turns out that, around the tt threshold, there is a very strong increase of the gg → A amplitude as the form-factor Re(A A 1/2 ) ≈ 5 is maximal at the 2m t threshold 7 , τ = M 2 A /4m 2 t ≈ 1. At the same time, BR(A → τ τ ) ≈ is substantial being a few percent, as the other decays except for bb are slightly suppressed, A → hZ by the coupling ≈ cos(β −α) and A → tt by phase space effects (only the three-body decay channel is kinematically open and it is suppressed). Hence, there would have been a substantial surplus of events from the gg → A → τ τ process in this limited area that is excluded by the search.

Interpretation of the bosonic Higgs decay modes in the hMSSM
We now turn to the constraints that can be imposed on the [tan β, M A ] plane by considering the bosonic decay channels of the heavier H and A states. In contrast to the H/A → τ τ and H ± → τ ν searches, no interpretation of these modes has been done in the context of the MSSM by the ATLAS and CMS collaborations. In the following, we will therefore adapt the constraints that have been obtained either in the context of the SM but with a heavier Higgs state than the observed one, or in extensions of the SM other than the MSSM. We will focus on the experimental analyses that provide the most stringent constraints.
The . The high mass range was analyzed and the events corresponding to the observed state with a mass of 125 GeV were considered as a background. In the ZZ channel, an additional CMS analysis in the H → ZZ * → 2 2q channel has been made with the 19.6 fb −1 data collected at 8 TeV [68]. All these analyses exclude a significant area in the hMSSM parameter space at low and moderate tan β.
In the case of the H → W W → νν CMS search, when all production channels are included (there is a dominance of the gg → H mode of course) and the various final state topologies are summed up, a Higgs particle with a SM-like coupling to gauge bosons is excluded from M H ≈ 200 GeV to ≈ 600 GeV. The 95%CL upper limit as a function of M H and relative to the SM expectation can be easily turned into an exclusion area in the [M A , tan β] plane by considering the production and decay rates of the MSSM H state discussed in section 3. The result is shown in Fig. 13 where the area excluded by this search, interpreted in the context of the hMSSM, is depicted in dark green.
The exclusion area starts at relatively high tan β values, tan β > ∼ 10, and light A, M A ≈ 140 GeV (below this limit, we enter the domain in which the model is ill defined, a domain that extends to M A ≈ 160 GeV and tan β ≈ 1) where one has an H state with a mass M H > ∼ 160 GeV and a coupling g HV V = cos(β − α) that is not very small as shown in the right-hand side of Fig. 3, allowing for substantial H production times decay rates. For tan β ≈ 1, the excluded region extends to M A ≈ 250 GeV, when other decay channels such as H → hh and even H → tt open up and suppress the massive gauge boson decay modes.
As a result of its clean final state and despite of the low statistics, the H → ZZ → 4 search turns out to be more constraining at high mass and excludes a SM-like Higgs boson up to M H ≈ 800 GeV (with a search domain extending to 1 TeV). While for low M H values, H → ZZ is less powerful than the companion H → W W mode as a consequence of the reduced phase space, it clearly becomes the leading channel for M H > ∼ 250 GeV. In fact, because of the higher statistics, the most severe constraint is obtained in the combination of the H → ZZ → 4 , 2 2ν, 2 2q topologies that was performed in Ref. [68]. Here, the 95%CL exclusion of a Higgs state with SM-like couplings extends to a mass close to 1 TeV.
The area excluded at 95%CL by the non observation of these ZZ final states at the LHC outside the M h ≈ 125 GeV mass window is given by the light green area of Fig. 13. It extends from M A ≈ 160 GeV to M H ≈ 280 GeV and concerns all values tan β < ∼ 5. An additional small area around M A ≈ 300 GeV and tan β < ∼ 2, in which M H is close to the 2m t threshold and the gluon-fusion amplitude A H 1/2 is maximal thus enhancing the gg → H cross section, is also excluded.
One should note that in a dedicated MSSM search, not only this H → ZZ channel but also the H → W W mode will lead to more effective constraints as the SM and MSSM Higgs particles have total decay widths that are completely different at high masses [5]. Indeed, while the SM state would have been a very wide resonance, the MSSM H boson is a relatively narrow resonance as shown in Fig. 6, allowing to select smaller bins for the V V invariant masses that lead to a more effective suppression of the various backgrounds. The dashed area is the one that is ill-defined in the hMSSM.
The resonant H → hh channel, which is important in the mass range between 250 and slightly above 350 GeV has been considered by both the ATLAS and CMS collaborations with the ≈ 20 fb −1 of data collected at √ s = 8 TeV. The main focus was on the γγbb signature [25,26] but additional searches in the 4 b-quark final state have been recently reported [69,70]. However, neither collaborations has interpreted the 95%CL exclusion limits in these channels in the context of the MSSM, the main reason being again that the low tan β area in which these signals occur is not theoretically accessible in the usual benchmark scenarios used for the MSSM Higgs sector. The interpretation is however straightforward in the hMSSM as the trilinear selfcoupling λ Hhh that controls the H → hh decay rate is simply given, as shown in eq. (2.10), in terms of the angles α, β and the radiative correction matrix element ∆M 2 22 that is fixed in terms of tan β and M A if the constraint M h = 125 GeV is used. We have adapted the constraints from these analyses to the hMSSM case and the resulting excluded domain in the [tan β, M A ] plane is shown in purple in Fig. 13. It covers the very low tan β region, tan β < ∼ 2, for the mass range between M A ≈ 180 GeV (which implies M H > ∼ 250 GeV for these low tan β values) and M A ≈ 380 GeV, i.e. slightly above the 2m t threshold.
Similarly to the previous channel, the A → hZ mode has only been considered in the context of two Higgs doublet models [37] and not in the MSSM. A CMS analysis considered the final state bb + − with the ≈ 20 fb −1 collected in 2012 at 8 TeV [27]. A search of both the A → hZ and H → hh channels has been performed by CMS again in the multi-lepton and eventually photon finale states [71]. The impact of the 95%CL exclusion limits of these studies, when interpreted in the context of the hMSSM, is illustrated by the yellow area of Fig. 13. As in the H → hh case, the ranges tan β < ∼ 2 and M A ≈ 230-350 GeV should be in principle excluded with the present data.  In fact, the entire area in which the hMSSM is not mathematically defined, and which is delineated by the solid line in the figure, is excluded by these H ± and A searches that do not involve the undefined CP-even H boson mass M H and the mixing angle α.
These constraints, if no new signal is observed, can be vastly improved at the next phase of the LHC with a center of mass energy up to √ s = 14 TeV and with one or two orders of magnitude accumulated data. More optimistically, this implies that the 2σ sensitivity for a heavier MSSM Higgs boson will be drastically enhanced at the next LHC phase. Starting from the expected median 95%CL exclusion limits that have been given by the ATLAS and CMS collaborations in the various searches performed at 8 TeV with ≈ 20 fb −1 , we have made an extrapolation to this next LHC phase with √ s = 14 TeV and 300 fb −1 data. We have naively assumed that the sensitivity will simply scale with the square root of the number of expected events and did not include any additional systematical effect. This comes from the observation that the results of the experimental analyses are limits on the signal cross section at a given c.m. energy for a given resonance mass bin, R S √ s (M A ), for a channel that is subject to a given background rate R B √ s (M A ) at this mass bin, when the integrated luminosity is fixed at a value L √ s . Knowing the sensitivity limit R S 8 (M A ) at √ s = 8 TeV, one derives the associated limit at √ s = 14 TeV using Having the knowledge of only the signal cross sections σ S √ s (M A ) for the various points and not the corresponding background rates, we assume that the latter simply and very naively scale like the signal cross sections. This is the case of some channels of interest, such as gg → H/A → tt whose main background is gg → tt and as both are gg initiated processes, they roughly scale with the gg luminosity at higher energies. However, for many other channels such as H → W W, ZZ or A/H → τ τ , the irreducible background is mostly due to qq annihilation which increases more slowly with energy than the initiated gg signal processes. This makes our approach rather conservative.
With this assumption, one obtains for the sensitivity at √ s = 14 TeV, R S 14 (M A ), needed to set the exclusion limit, that we turn into a 95%CL sensitivity, for a given M A The output of this procedure is presented in the [tan β, M A ] hMSSM plane in Figs. 15 for the fermionic (left) and bosonic (right) Higgs search channels. In the former case, we have included in addition the channel pp → tbH + → tbtb which now shows some sensitivity a low tan β and not too high M H ± values. The combined expected 95%CL sensitivities are shown in Fig. 16 and, as can be seen, a vast improvement of the current sensitivity to the MSSM parameter space is foreseen in all channels. This is particularly the case of the A/H → τ τ channels which alone, closes the entire region below M A < ∼ 350 GeV for any tan β value, while the H → W W, ZZ modes which show sensitivity up to M A ≈ 600 GeV at very low tan β. In the Higgs mass range in which they are relevant, i.e. below the tt threshold, the channels H → hh and A → hZ start to probe rather high tan β values, tan β ≈ 10 and tan β ≈ 6, respectively. Nevertheless, there will remain an area of the hMSSM parameter space, at tan β < ∼ 4 and masses above M A ≈ 400 GeV to name it, which will not be accessible by the channels that have been considered so far in the search of the heavier H/A and H ± states. To probe this area, the high luminosity option of the LHC with L = 3 ab −1 data or a higher energy pp collider, such as the presently discussed Fcc-pp at √ s ≈ 100 TeV will be necessary.
However, as it was discussed in many instances in this paper, this virgin area is the ideal territory to perform searches in the gg → H/A → tt channel to which we turn our attention now.

Including the pp → H/A → tt channel
As it was discussed at length in the previous section, for low tan β and high M A values, the decay modes H/A → tt of the heavier MSSM scalar and pseudoscalar Higgs states will largely become the dominant ones while the gg → H/A cross sections are still substantial thanks to the large Higgs coupling to the top quarks that mediate the production process. Hence, the search for resonances decaying into tt final states will be mandatory in order to probe these areas of the [M A , tan β] parameter space at the LHC. However, a peak in the invariant mass distribution of the tt system, that one generally expects in the narrow-width approximation, is not the only signature of a Higgs resonance in this case. Indeed, the gg → H/A signal will interfere with the QCD tt background which, at LHC energies, is mainly generated by the gluon-fusion channel, gg → tt. The signal-background interference will depend on the CP nature of the Φ = H/A boson and on its mass and total decay width; it can be either constructive or destructive, leading to a rather complex signature with a peak-dip structure of the tt invariant mass distribution.
These aspects are known since already some time and have especially been discussed in the context of a heavy SM Higgs state [72] and, hence, for the CP-even Higgs case. The slightly more involved MSSM situation, as there are one CP-even and one CP-odd resonances that are close in mass, has been addressed only in a very few places; see for instance Refs. [73,74]. Dedicated analyses have been performed at the parton-level only and do not make use of recent developments like boosted heavy quark techniques [75] that could allow to enhance the observability of the Higgs signal. The ATLAS and CMS collaborations have performed searches for heavy states decaying into tt pairs [76,77] but did not specifically address the complicated Higgs situation as only electroweak spin-one resonances, like new neutral gauge bosons or electroweak Kaluza-Klein excitations, were considered. In these two cases, the main production channel is qq annihilation and there is no interference with the (colored) QCD qq → tt background and the resonance signal simply appears as a peak in the invariant mass distribution of the tt pair.
A full and realistic Monte-Carlo simulation of the gg → H/A → tt process including the effects of the interference and taking into account reconstruction and detector aspects is beyond the scope of this paper, and will be postponed to a future publication [78]. Here, we will simply make a very crude estimate of the sensitivity that can be achieved in this channel, relying on previous ATLAS [76] and CMS [77] analyses performed at √ s = 8 TeV c.m. energy in the spin-one resonance context mentioned above. We will naively consider the number of signal and background events, applying very simple kinematical cuts and ignoring the complicated interference effects, and delineate the area in the [tan β, M A ] hMSSM parameter space in which one has N signal / N bkg ≥ s. The significance s = 5 would correspond to a 5σ observation of the Higgs signal while s = 2 would be a first hint of the new effect; in the absence of any effect, s = 2 would correspond to the 95%CL exclusion limit of the phenomenon. To further simplify our analysis, we will assume that the two heavy A and H states are mass degenerate so that the signal rate is simply the sum of the A and H production cross section times the respective branching ratios in their decays into tt pairs (which, as we have already seen, is a good approximation).
The main ingredients of the analysis are as follows. The normalization of the Higgs signal has been obtained using the programs HIGLU for the production cross sections and HDECAY for the decay branching ratios. The total cross section of the SM background (which will serve as a normalization) has been obtained using the program Top++ [79]. For the input m t = 173.2 GeV one obtains for the background rate at the first stage of the LHC with √ s = 8 TeV when the renormalisation and factorization scales are fixed to µ R = µ F = m t . In this equation, the first error is the one due to the scale variation within a factor of two from the central scale, and the second one the PDF+α s uncertainty. This value for the cross section is obtained at NNLO in QCD including the resummation of next-to-next-to-leading logarithmic (NNLL) soft gluon terms and it turns out that it is only 3% larger than the value of the cross section when evaluated at NNLO [81]. Note that at √ s = 14 TeV, using the same approximation and ingredients, one would obtain for the cross section Using the program MadGraph5 [82], we have generated the signal and background cross sections for the process pp → tt. The differential cross section as a function of the invariant mass of the tt system, dσ/dm tt , is shown at √ s = 8 TeV in the upper part of Fig. 17 where mass bins of 10 GeV have been assumed. We overlay on the continuum QCD background distribution (in black solid line), the distributions for the A signal only (the colored lines) with tan β = 1 and three possible mass values, M A = 400, 600 and 800 GeV. In order to see the signals in the figure, we have multiplied the distributions by a factor of 5, 50 and 300, respectively. In order to enhance the significance s, one could apply very basics kinematical cuts that suppress the background while leaving the signal almost unaffected. In the left and right-hand sides of Fig. 17, we show two distributions (as we are interested in the shapes only, the distributions have not been re-weighted with the correct K-factors etc.. and the integrated areas thus correspond to the Monte-Carlo cross sections). The first one is 1/σ × dσ/d cos θ * where θ * the helicity angle between the off-shell Higgs boson boosted back into the top quark pair rest frame and the top quark pair direction (left). As can be seen, while the signal distribution is almost flat, the background is peaked in the forward and backward directions; a cut | cos θ * | ≤ 0.8 for instance would remove a large sample of background events. A second distribution per 10 GeV bin is in terms of the transverse momentum of the top quarks, 1/σ × dσ/dp T (right). They show a characteristic behavior for the signal events, with a pronounced peak and then a sharp drop. One grossly estimates that, for the mass value M A = 800 GeV for instance, a cut on the p t T distribution could allow to suppress the background by a factor of ≈ 6.
Assuming that when applying all kinematical cuts, one could suppress the tt QCD background by an order of magnitude compared to the Higgs signal, we delineate in Fig. 18 the regions of the [tan β, M A ] plane in which one would expect N signal / 6N bkg ≥ 2, 3, 4, 5.

Conclusions
In this paper, we have addressed the issue of covering the entire parameter space of the MSSM Higgs sector at the LHC by considering the search of the heavier H, A and H ± states that are predicted in the model, in addition to the already observed lightest h boson. These searches should not only be restricted to the channels that have been considered so far by the ATLAS and CMS collaborations, namely those with a surplus of τ ν events and those with high mass resonances decaying into τ lepton pairs, which would signal the presence of new contributions from the t → bH + → bτ ν and pp → H/A → τ τ processes, which are mainly relevant for the high tan β region of the MSSM Higgs sector. Search for heavier Higgs bosons should also be conducted in channels that are more appropriate for the probing of the low tan β region and which, until now, have been overlooked.
We have first discussed and refined the hMSSM approach introduced in Ref. [10] in which the dominant radiative corrections to the MSSM Higgs sector, that introduce a dependence on numerous soft SUSY-breaking parameters, are traded against the measured mass M h = 125 GeV of the Higgs boson which was observed at the LHC, thus allowing to describe again the entire Higgs sector of the model with only two input parameters. This simple, economical and "model independent" approach permits to reopen the low tan β region, at the expense of considering the possibility that the scale of SUSY-breaking is extremely high, M S 1 TeV, and that the model is severely fine-tuned. The hMSSM is expected to be viable down to values tan β ≈ 2 and, for higher tan β values, reproduces to a very good approximation the standard results of the MSSM Higgs sector. This is particularly true if the higgsino mass parameter is much smaller than the SUSY-breaking scale, µ M S , an assumption that is natural at low tan β values which imply a very high SUSY-breaking scale. Thus, searches for new signals in the MSSM Higgs sector can be performed in the entire [tan β, M A ] parameter space, in a reliable way for tan β > ∼ 2.
Nevertheless, in an effective approach, one can eventually extrapolate to values of tan β very close to unity, despite of the fact that the scale M S required to reach this value is so high that its renders the model not only too fine-tuned but also potentially inconsistent.
We have then analyzed the production and decay modes of the H, A and H ± particles at the LHC, with a special attention to the low tan β region in which the top quark plays a prominent role, as its couplings to the Higgs bosons are not strongly suppressed compared to the SM case. We have first shown that the searches that are presently conducted by ATLAS and CMS can also be relevant at low tan β. This is for instance the case of the pp → A → τ τ and pp → tbH + → tbtb processes at low to moderate M A values. We have then shown that search channels such as H → W W, ZZ, hh and A → hZ, when interpreted in the context of the hMSSM, can also probe the low tan β and not too high M A regions. for any value of tan β could be probed up to M A ≈ 400 GeV, when combining the searches in the usual fermionic channels and in the additional bosonic channels discussed here. An important message conveyed by the present paper is that, in order to fill or close the gap in the MSSM [tan β, M A ] plane left by the fermionic and bosonic searches mentioned above, one should definitely consider the pp → H/A → tt process. Indeed, at low tan β and for Higgs masses above the tt kinematical threshold, the decays H/A → tt become the dominant ones, suppressing the rates for the other decay channels to a very low if not negligible level. On the other hand, the gg → H/A production mode has a still significant cross section as the top quark that generates this loop process has substantial couplings to the H/A states at sufficiently low tan β values. This is not a very easy search channel in view of the formidable pp → tt QCD background. Nevertheless, it exhibits very special and interesting features such as an interference with the QCD background that leads to a rather involved peak-dip structure of the signal.
We have not performed a detailed and realistic study of this process but attempted to roughly quantify the observation of a signal at the LHC, relying on present ATLAS and CMS analyses in searches for heavy (non Higgs) resonances decaying into top quark pairs at 8 TeV center of mass energies, and discussed its possible implications. It appears that the channel gg → H/A → tt, would be capable of covering partly the area at low tan β and high M A , hence allowing for a full coverage of the [tan β, M A ] plane of the MSSM up to Higgs masses M A ≈ 600 GeV with 300 fb −1 data at √ s = 14 TeV. At the high luminosity option of the LHC with 3000 fb −1 data, one could reach a full coverage of the MSSM parameter space for pseudoscalar masses closer to M A ≈ 1 TeV. More refined analyses are required in order to firmly establish the viability of the various processes discussed here, in particular the H/A → tt channel. In view of the important role that it could play in the probing of the MSSM parameter space, the latter process is worth investigating in a more realistic way, including the interference between the Higgs signal and the QCD background. This is what we plan to do in a forthcoming publication [78].