The Higgs boson discovery of 2012 [1–4] at the CERN Large Hadron Collider (LHC) has led to the confirmation of the Standard Model (SM) as the proper theory of the Electro-Weak (EW) scale. However, there is much evidence that the SM is not appropriate at all scales, rather it should be viewed as an effective low-energy realization of a more complete and fundamental theory on setting beyond the EW regime. Among the many proposals for the latter, one can list theories with some new symmetry, e.g., Supersymmetry (SUSY), or an enlarged particle content (e.g., in the Higgs sector), or both. Following the aforementioned discovery, no new particle has however been seen at the LHC, implying that new physics at the EW scale should be weakly interacting or that strongly interacting particles, if present, should lead to signatures involving soft decay products or in channels with overwhelming (ir)reducible backgrounds. We shall adopt here the first assumption.

Many SM extensions possess in their spectra additional neutral and/or charged Higgs states. Amongst these, SUSY [5] is indeed considered the most appealing one as it addresses several shortcomings of the SM, including the problem of the large hierarchy between the EW and Planck scales as well as the dark matter puzzle. While the search for SUSY was unsuccessful during the first LHC run, the increase in the Center-of-Mass (CM) energy of the machine from 8 to 13 TeV plus the additional luminosity of the second run are improving greatly the sensitivity to the new superparticles which are predicted. While the jury is still out on these, we remind here the reader that SUSY also requires at least two Higgs doublets for a successful EW Symmetry Breaking (EWSB) pattern. For exactly two such fields, yielding the so-called Minimal Supersymmetric Standard Model (MSSM), also having the same gauge group structure of the SM, one obtains four physical Higgs particles, in addition to the discovered SM-like one with the observed mass of 125 GeV. In fact, the same Higgs mass spectrum also belongs to a generic 2-Higgs Doublet Model (2HDM), i.e., one not originating from SUSY. In neither case, though, there exists a precise prediction of the typical masses of the new Higgs states, though we know already that the MSSM allows for one to be lighter than the 125 GeV state in a very small region of parameter space [6], whereas the 2HDM generally does so over a significantly larger expanse of it [7, 8]. Either way, the presence of extra physical Higgs boson states alongside the SM-like one is thus one of the characteristics of Beyond the SM (BSM) physics, whether within SUSY or otherwise. Hence, looking for these additional states in various production and decay channels over a wide range of kinematic regimes is an important part of the physics programme of the multi-purpose LHC experiments, ATLAS, and CMS. Specifically, the discovery of a (singly) charged Higgs boson would point to a likely additional Higgs doublet (or a Higgs field with higher representation, such as triplet). Hence, we concentrate on this Higgs state here.

The two Higgs doublet fields pertaining to the MSSM are required to break the EW symmetry and to generate the isospin-up and -down type fermions as well as the W^± and Z boson masses [9–11]. The Higgs spectrum herein is given by the following states: two charged H^±'s, a CP-odd A and two CP-even Higgses h and H, with m_h < m_H (conventionally, wherein h is the SM-like Higgs state). The tree-level phenomenology of the Higgs sector of the MSSM is described entirely by two input parameters, one Higgs mass (that can be taken to be that of the CP-odd Higgs state, m_A) and the ratio tan β of the Vacuum Expectation Values (VEVs) of the two Higgs doublet fields. Note that one of the most powerful prediction of SUSY is the existence of a light Higgs boson that could be produced at colliders. In the MSSM, at the tree-level, the light CP-even h is predicted to be lighter than the Z boson. However, it is well-known that loop effects could raise the h mass upper bound to 135 GeV for a large soft breaking trilinear parameter, A_t, and/or a heavy scalar top [12–22]. After the Higgs boson discovery at the LHC, MSSM benchmark scenarios have been refined to match the experimental data and to reveal characteristic features of certain regions of the parameter space [23–25]. Of the many MSSM frameworks presented in literature, we consider in this work the so-called mhmod+ [24] and hMSSM [26–30] ones, which will be described in the coming section. As for the 2HDM, one ought to specify the Yukawa sector, in order to proceed to study phenomenologically its manifestations. While SUSY enforces this in the form of a so-called Type-II, this is only one of four Ultra-Violet (UV) complete realizations of a generic 2HDM, the others been termed Type-I, -X, and -Y. The difference between these four scenarios is the way the fermionic masses are generated. We define as Type-I the model where only one doublet couples to all fermions, Type-II is the scenario where one doublet couples to up-type quarks and the other to down-type quarks and leptons, the Type-X is the model where one doublet couples to all quarks and the other to all leptons while a Type-Y is built such that one doublet couples to up-type quarks and to leptons and the other to down-type quarks. In all such cases, the number of free parameters at tree-level is seven to start with, hence it becomes more cumbersome than in SUSY to map experimental results onto theoretical constraints. Yet, in virtue of the fact that a 2HDM is the simplest realization of a BSM scenario based solely on doublet Higgs fields, its study is vigorously being pursued experimentally.

So far, the non-observation of any Higgs signal events in direct searches above and beyond those of the SM-like Higgs state constrains the parameter space of the underlying physics model. Specifically, in the case of the H^± boson, wherein the relevant phenomenological parameters are mH± and tan β in whatever scenario, one can pursue the study of its production and decay modes in a model independent way, which results can a posteriori be translated to exclude the relevant parameter space in a given scenario (whether it be the MSSM, 2HDM, or something else). This recasting is conveniently done on the (m_A, tan β) and (mH±,tanβ) planes for the MSSM and 2HDM, respectively, so that we will map our findings in the same way.

At hadron colliders, there exists many production modes for charged Higgs bosons which are rather similar in the MSSM and 2HDM. For a light charged Higgs, i.e., with mass mH±+mb<mt, its production comes mainly from top decay. At the LHC, the production of top quark pairs proceeds via Quantum Chromo-Dynamics (QCD) interactions and, when kinematically allowed, one top could decay into a charged Higgs state and a bottom quark in a competition with the SM decay into a W^± boson and again a bottom quark. Therefore, the complete H^± production mechanism qq̄,gg→tt̄→tb̄H- provides the main source of light charged Higgs bosons at the LHC and offers a much more copious signature than any other form of direct production. After crossing the top-bottom threshold, i.e., when mH±+mb>mt, a charged Higgs (pseudo)scalar can be produced through the process gb → tH⁻ [31–34]. In fact, these two mechanisms can be simultaneously captured via the process gg→tb¯H- [35, 36], which again makes it clear that one should expect large H^± cross sections induced by QCD interactions also in the heavy H^± mass range¹.

In the MSSM, and also in a variety of 2HDM Types, light charged Higgs bosons would decay almost exclusively into a (hadronic or leptonic) τ lepton and its associated neutrino for tan β ≳ 1. When the top-bottom channel is kinematically open, then H+→tb̄ would compete with H^± → hW^±, HW^±, AW^± decays as well as various SUSY channels in the MSSM. In the latter, H+→tb̄→bb̄W+ is the dominant channel and the bosonic decays H^± → hW^±, HW^±, AW^± (also yielding bb̄W+ final states) are subleading. In the 2HDM, if none of these bosonic decays is open, then H+→tb̄ is the dominant mode. At the LHC Run-1, lighter charged Higgs bosons were probed in the decay channels τν [37, 38], cs [39, 40] and also cb [41]. No excess was observed and model independent limits are set on the following product of Branching Ratios (BRs): BR(t → H⁺b) × BR(H⁺ → τν). At Run-2, mainly the decay modes τν [42, 43] and tb [44] are explored in the mass range mH±=200–1,000 GeV, in the latter mode using multi-jet final states with one electron or muon from the top quark decay. No significant excess above the background-only hypothesis has been observed and upper limits are set on the pp → tbH^± production cross section times BR(H^± → tb). Several interpretations of these limits have eventually been given in benchmark scenarios of the MSSM, including those mentioned above. Note that current ATLAS and CMS bounds are significantly weakened in the 2HDM once the exotic decay channels into a lighter neutral Higgs, e.g., H^± → hW^± or H^± → AW^±, are open. This scenario could also happen in the MSSM if one of the SUSY decay channels of charged Higgs bosons are open (such as into chargino-neutralino pairs). In the 2HDM, the possibility of producing a light charged Higgs boson from top decay with a subsequent step H^± → hW^± or H^± → AW^± was studied in Arhrib et al. [45] and it was shown that it can lead to sizable cross sections at low tan β. We stress here that there exist several recent analyses dedicated to 2HDM phenomenology [34, 46, 47] that we consulted. However, unlike Akeroyd et al. [34], Arbey et al. [46], and Bernon et al. [47] only concentrates on neutral Higgs phenomenology and discuss the charged Higgs contribution only to flavor physics observables without singling out the relevant charged Higgs production and decay channels at the LHC, which is indeed one of the aims of this analysis.

In this paper, we analyze the allowed σ(pp→tb̄H++c.c.)×BR(H±→anything) rates by taking into account both theoretical and experiments constraints on the underlying BSM model, the latter including the latest ATLAS and CMS results for SM-Higgs (h) and other Higgs (H, A, H⁺) searches with the full set of 36.5 fb⁻¹ data collected in the second LHC phase. We will then interpret these results under the proposed scenarios to quantify the magnitude of the available parameter space to be covered by future LHC analyses. In doing so, we will extract several Benchmarks Points (BPs) that could lead to detectable signals, all of which are consistent with the best fit regions in both the MSSM and 2HDM.

The paper is organized as follows. In the second section we review the MSSM and introduce the benchmark scenarios that we will discuss. The 2HDM, with its parameterizations and Yukawa textures, is described in the third section. The fourth section is devoted to a discussion of the theoretical and experimental constraints used in our study. Results and discussions for the MSSM and 2HDM are presented in the fifth section and we finish with our conclusions.

In the MSSM, due to the holomorphy of the superpotential, one introduce two Higgs doublets Φ_{1, 2} in order to give masses to up-type quarks as well as down-type quarks and leptons, respectively. Both Higgs fields acquire VEVs, denoted by v_{1, 2}. After EWSB takes place, the spectrum of the model contains the aforementioned Higgs states: h, H, A and H^±. The MSSM Higgs sector is parameterized at tree-level by tan β = v₂/v₁, and e.g., the CP-odd mass m_A. One of the interesting features of the MSSM is the prediction, at the tree-level, of a light CP-even Higgs h with a mass m_h ≤ m_Z. However, such tree-level prediction is strongly modified by radiative corrections at one- and two-loop level [12–21]. It has been shown in Degrassi et al. [22], on the one hand, that the loop effects can make the light CP-even mass m_h reach a value of 135 GeV and, on the other hand, that the theoretical uncertainties due to unknown high order effects can be of the order of 3 GeV. In fact, these large loop effects are welcome in order to shift the light CP-even Higgs mass to the measured experimental value m_h ≈ 125 GeV. Note also that the loop effects will modify not only the tree-level Higgs mass relations but also the Higgs self-couplings and the Higgs coupling to SUSY particles. Therefore, beside the tree-level parameters tan β and m_A, the top quark mass and the associated squark masses and their soft SUSY breaking parameters enter through radiative corrections [12–21, 48, 49]. In fact, when trying to push the light CP-even mass from m_Z to 125 GeV through loop effects, one needs to introduce a large SUSY scale with large soft trilinear parameter A_t. Such a large SUSY scale puts automatically the SUSY spectrum at the TeV scale, which is consistent with negative searches for SUSY particles at the LHC.

To compute the masses and couplings of Higgs bosons in a given point of the MSSM parameter space we use the code FeynHiggs [50, 51] for the mhmod+ scenario and the program HDECAY for the hMSSM case [52, 53]. Both codes include the full one-loop and a large part of the dominant two-loop corrections to the neutral Higgs masses. Since the theoretical uncertainty on the Higgs mass calculation in the FeynHiggs code has been estimated to be of the order of 3 GeV, we consider as phenomenologically acceptable the points in the MSSM parameter space where FeynHiggs predicts the existence of a scalar state with mass between 122.5 and 128.5 GeV and with approximately SM-like couplings to gauge bosons and fermions. In addition to the tree-level scalar potential parameters, tan β and m_A, when taking into account high order corrections, the MSSM parameters most relevant to the prediction of the masses and production cross sections of the Higgs bosons are: the soft SUSY-breaking masses for the stop and sbottom squarks, which, for simplicity, we assume all equal to a common mass parameter M_SUSY, the soft SUSY-breaking gluino mass mg~, the soft SUSY-breaking Higgs-squark-squark couplings A_t and A_b, the superpotential Higgs(ino)-mass parameter μ and the left-right mixing terms in the stop and sbottom mass matrices (divided by m_t and m_b)

respectively. Since the (approximate) two-loop calculation of the Higgs masses implemented in FeynHiggs and the Next-to-Leading Order (NLO) calculation of QCD corrections to the production cross section implemented in SusHi [54, 55] employ the same renormalization (on-shell) scheme, the input values of the soft SUSY-breaking parameters can be passed seamlessly from the Higgs mass to the cross section calculations.

In the light of the latest LHC data on the discovered Higgs-like boson, and given the fact that the MSSM contains many independent parameters which makes it a fastidious task to perform a full scan, there have been many studies which lead to several benchmarks that could fit the observed Higgs boson as well as be tested at the future LHC with higher luminosity [24, 26–29]. As intimated already, in this study, we will concentrate on two of these benchmark scenarios: the mhmod+ and hMSSM ones, which we will describe hereafter.

In this section, we define the scalar potential and the Yukawa sector of the general 2HDM. The most general scalar potential which is SU(2)_L⊗U(1)_Y invariant is given by [62, 63]

The complex (pseudo)scalar doublets Φ_i (i = 1, 2) can be parameterized as

with v_{1, 2} ≥ 0 being the VEVs satisfying v=v12+v22, with v = 246.22 GeV. Hermiticity of the potential forces λ_{1, 2, 3, 4} to be real while λ₅ and m122 can be complex. In this work we choose to work in a CP-conserving potential where both VEVs are real and so are also λ₅ and m122.

After EWSB, three of the eight degrees of freedom in the Higgs sector of the 2HDM are eaten by the Goldstone bosons (G^± and G) to give masses to the longitudinal gauge bosons (W^± and Z). The remaining five degrees of freedom become the aforementioned physical Higgs bosons. After using the minimization conditions for the potential together with the W^± boson mass requirement, we end up with seven independent parameters which will be taken as

where, as usual, tan β ≡ v₂/v₁ and β is also the angle that diagonalizes the mass matrices of both the CP-odd and charged Higgs sector while the angle α does so in the CP-even Higgs sector.

The most commonly used versions of a CP-conserving 2HDM are the ones that satisfy a discrete Z₂ symmetry Φi→(-1)i+1Φi (i = 1, 2), that, when extended to the Yukawa sector, guarantees the absence of Flavor Changing Neutral Currents (FCNCs). Such a symmetry would also require m122=0, unless we tolerate a soft violation of this by the dimension two term m122 (as we do here).

The Yukawa Lagrangian can then be written as

where QLT=(uL,dL) and LLT=(lL,lL) are the left-handed quark doublet and lepton doublet, respectively, the Ykf's (k = 1, 2 and f = u, d, l) denote the 3 × 3 Yukawa matrices and Φ~k=iσ2Φk* (k = 1, 2). The mass matrices of the quarks and leptons are a linear combination of Y1f and Y2f, Y1,2d,l and Y1,2u. Since they cannot be diagonalized simultaneously in general, neutral Higgs Yukawa couplings with flavor violation appear at tree-level and contribute significantly to FCNC processes, such as ΔM_{K, B, D} as well as Bd,s→μ+μ- mediated by neutral Higgs exchanges. To avoid having those large FCNC processes, one known solution is to extend the Z₂ symmetry to the Yukawa sector. When doing so, we ended up with the already discussed four possibilities regarding the Higgs bosons couplings to fermions [63].

After EWSB, the Yukawa Lagrangian can be expressed in the mass eigenstate basis as follows [64, 65]:

We give in Table 1 the values of the Yukawa couplings κfϕ (ϕ = h, H, A), i.e., the Higgs boson interactions normalized to the SM vertices introduced in David et al. [66], in the four 2HDM Types. The couplings of h and H to gauge bosons W^±, Z are proportional to sin(β − α) and cos(β − α), respectively. Since these are gauge couplings, they are the same for all Yukawa types. As we are considering the scenario where the lightest neutral Higgs state is the 125 GeV scalar, the SM-like Higgs boson h is recovered when cos(β − α) ≈ 0. As one can see from Table 1, for all 2HDM Types, this is also the limit where the Yukawa couplings of the discovered Higgs boson become SM-like. The limit cos(β − α) ≈ 0 seems to be favored by LHC data, except for the possibility of a wrong sign limit [67, 68], where the couplings to down-type quarks can have a relative sign to the gauge bosons ones, thus oppositely to those of the SM. Our benchmarks will focus on the SM-like limit where indeed cos(β−α) ≈ 0.

We end this section by noticing that we have used the public program 2HDMC [69] to evaluate the 2HDM spectrum as well as the decay rates and BRs of all Higgs particles. We have used 2HDMC to also enforce the aforementioned theoretical constraints onto both BSM scenarios considered here.

In order to perform a systematic scan over the parameter space of the two MSSM configurations and the four 2HDM Types, we take into account the following theoretical³ and experimental constraints.

In this section, we present our findings for the MSSM and 2HDM in turn.

We have studied charged Higgs boson phenomenology in both the MSSM and 2HDM, the purpose being to define BPs amenable to phenomenological investigation already with the full Run-1 and -2 datasets and certainly accessible with the Run-3 one of the LHC. They have been singled out following the enforcement of the latest theoretical and experimental constraints, so as to be entirely up-to-date. Furthermore, they have been defined with the intent of increasing sensitivity of dedicated (model-dependent) H^± searches to some of the most probable parameter space configurations of either scenario. With this in mind, we have listed in two tables their input and output values, the former in terms of the fundamental parameters of the model concerned and the latter in terms of key observables (like, e.g., physical masses and couplings, production cross sections and decay BRs). We have also specified which numerical tools we have used to produce all such an information, including their settings.

For the MSSM we have concentrated on two popular scenarios, i.e., the hMSSM and mhmod+ ones. It was found that the hMSSM case still possesses a rather large available parameter space, here mapped over the (m_A, tan β) plane, while the mhmod+ one is instead much more constrained. In terms of the largest production and decay rates, in the hMSSM scenario one finds that the most copious channels, assuming pp → tH⁻ + c.c. production, are via the decay H+→tb̄ followed by H⁺ → τν whereas for the mhmod+ scenario the decay modes H+→tb̄ and H+→χ~1+χ~0 offer the largest rates. In both cases, only mH±>mt values are truly admissible by current data.

Within the 2HDM, we have looked at the four standard Yukawa setups, known as Type-I, -II, -X, and -Y. Because of B¯→Xsγ constraints, the profile of a charged Higgs in the 2HDM Type-II and -Y is a rather heavy one, with a mass required to be more than 580 GeV. While this puts an obvious limit to LHC sensitivity owing to a large phase space suppression in production, we have emphasized that H±→bb̄W± channels should be searched for, with intermediate contributions from the AW^± and tb modes (including their interference [138]), alongside H^± → τν. In the case of the 2HDM Type-I and -X, a much lighter charged Higgs state is still allowed by data, in fact, even with a mass below that of the top quark. While the configuration mH±<mt is best probed by using tt̄ production and decays into τν, the complementary mass region, i.e., mH±>mt (wherein pp → tH⁻ + c.c. is the production mode), may well be accessible via a combination of H+→tb̄ and H^± → AW^± (in Type-I) plus H^± → τν (in Type-X).

The datasets analysed in this study can be found in the HiggsBounds repository: https://higgsbounds.hepforge.org.

Since the original submission of this paper, several new experimental analyses have been carried out by ATLAS and CMS using the full Run-2 data sample of ≈ 139 fb⁻¹. Some of these, covering both measurements of the SM-like Higgs Boson and the search for new (pseudo)scalar Higgs states, both charged and neutral, have been captured by the latest versions of HiggsBounds and HiggsSignals, HiggsBounds-5.3.2beta and HiggsSignals-2.2.3beta, respectively. Likewise, further analyses by LHCb of flavor observables have been carried out since and most of these have been captured by the latest version of SuperIso. Hence, we have repeated our scans using all such tools and found negligible differences between our original results and the new ones. Further, we have investigated which ones of the full Run-2 data set analyses were not incorporated in the above codes and found that their ad-hoc application to our analysis did not change our results either.

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.