Accurate predictions for charged Higgs production: closing the $m_{H^{\pm}}\sim m_t$ window

We present predictions for the total cross section for the production of a charged Higgs boson in the intermediate-mass range ($m_{H^{\pm}}\sim m_t$) at the LHC, focusing on a type-II two-Higgs-doublet model. Results are obtained at next-to-leading order (NLO) accuracy in QCD perturbation theory, by studying the full process $pp\to H^\pm W^\mp b \bar b$ in the complex-(top)-mass scheme with massive bottom quarks. Compared to lowest-order predictions, NLO corrections have a sizeable impact: they increase the cross section by roughly 50\% and reduce uncertainties due to scale variations by more than a factor of two. Our computation reliably interpolates between the low- and high-mass regime. Our results provide the first NLO prediction for charged Higgs production in the intermediate-mass range and therefore allow to have NLO accurate predictions in the full $m_{H^{\pm}}$ range. The extension of our results to different realisations of the two-Higgs-doublet model or to the supersymmetric case is also discussed.

doublets [1]. Each of the four ways gives rise to rather different phenomenologies. In this work, we consider the so-called type-II 2HDM (although we will discuss how our results can be generalised to other types), in which one doublet couples to up-type quarks and the other to down-type quarks and charged leptons.
The dominant production mode for a charged Higgs boson depends on the value of its mass with respect to the top-quark mass, and can be classified into three categories. Light charged Higgs scenarios are defined by Higgs-boson masses smaller than the mass of the top quark, where the top-quark decay t → H + b is allowed and the charged Higgs is light enough so that top-quark off-shell effects can be neglected (typically experimental analyses consider masses up to m H ± 160 GeV). The cross section for the production of a light charged Higgs boson is simply given by the product of the top-pair production cross section and the branching ratio of a top quark into a charged Higgs boson, see Fig. 1 (a). The former is known up to next-to-next-to-leading order in perturbative QCD [2] and displays a 3% QCD scale uncertainty, while the NLO branching ratio for t → H + b [3][4][5][6][7][8][9][10][11][12][13] is affected by a 2% scale uncertainty due to missing higher-order QCD contributions. Thus the theoretical ac-   curacy for the production of a light charged Higgs boson is at the few % level. The model-independent bounds on the branching ratio of a light charged Higgs boson [14] are transformed into limits in the (m H ± , tan β) plane, with tan β being the ratio of the vacuum expectation values of the two Higgs doublets. Direct searches at the LHC, with a centre-of-mass energy of 7 TeV [15][16][17][18] and 8 TeV [19,20] set stringent constraints on the parameter space with a light charged Higgs boson.
Heavy charged Higgs boson scenarios, on the other hand, correspond to charged Higgs masses larger than the top-quark mass (typically m H ± 200 GeV). In this case, the dominant charged Higgs production channel is the associated production with a top quark, 1 see Fig. 1 (b). Theoretical predictions at NLO(+PS) have been computed both at the inclusive and fully-differential level in the five-flavour scheme (5FS) [21][22][23][24][25][26][27][28] and in the four-flavour scheme (4FS) [29,30,28] 200 GeV). In this region, finite top-width effects as well as the interplay between top-quark resonant and non-resonant diagrams cannot be neglected. Therefore, the full process pp → H ± W ∓ bb (with massive bottom quarks), see Fig. 2, including non-resonant, singleresonant and double-resonant contributions, has to be considered, to perform a reliable perturbative computation of the charged Higgs cross section. The intermediate-mass range has not been searched for at the LHC to date, mostly due to the lack of sufficiently accurate theoretical predictions, and the consequent shortage of specific strategies devised to increase the sensitivity to the signal. Despite the fact that some studies exist on the intermediate mass-range, they are either only LO-accurate, thus affected by large theoretical uncertainties [36][37][38], or based on an incoherent sum of the pp → tt and pp → t H − production mechanisms [24,27], and neglecting interferences between the two. With this work, where we compute the cross section for the pp → H ± W ∓ bb process at NLO accuracy, we provide for the first time precise and theoretically consistent predictions in the intermediate-mass range, which are an essential ingredient for H ± searches at Run II of the LHC. We leave it to further work in collaboration with our experimental 1 In the four-flavour scheme there is also an explicit bottom quark in the final state.
colleagues to devise appropriate cuts and selection strategies that would maximise the sensitivity to this particular mass range. Despite the fact that indirect exclusion bounds from flavour physics for a type-II Higgs doublet model are now very strong and exclude charged Higgs boson lighter than 380 GeV, 2 the intermediate-mass region is not excluded for type-I models nor for models that embed the 2HDM-II at tree-level. Indeed, the intermediate-mass range has recently gained extra attention in the model-building community. For example, supersymmetric scenarios where the heavy Higgs boson of the spectrum has a mass of 125 GeV and the light Higgs can possibly act as a mediator to the dark-matter sector lead to a charged Higgs-boson mass similar to the top-quark mass [43,44]. In fact, at tree-level, the Higgs-fermion Yukawa couplings of the MSSM and p-MSSM follow the 2HDM-II pattern. However, when radiative corrections are included, the Yukawa couplings are modified by supersymmetry-breaking effects, thus leading to a different phenomenology. It is important to notice that such modifications of the Yukawa coupling can be included in our calculation, as it is explicitly spelled out in the following.
Our computation employs a chain of automatic tools in the MadGraph5_aMC@NLO + NLOCT framework [45,46], developed to study the phenomenology of new physics models at NLO accuracy. In this framework, NLOCT automatically computes the R 2 rational terms and the ultraviolet counterterms used in the virtual amplitudes, and relies internally upon FeynRules [47] and FeynArts [48]. The one-loop matrix elements are computed using the MadLoop module [49], which employs CutTools [50] and Ninja [51][52][53] for loop reduction at the integrand level and IREGI [54] for tensor integral reduction. All methods are complemented by an in-house implementation of the OpenLoops [55] algorithm. For the factorisation of the IR poles in the real-emission phase-space integrals, the resonance-aware MadFKS [56,57] module is used. We work in the four-flavour scheme, where the bottom-quark mass regulates any soft or collinear divergence related to finalstate bottom-quark emissions, making it possible to compute the total cross section without having to impose artificial cuts on the final state particles. In a 5FS version of this computation (bb → H ± W ∓ ), non-, single-and double-resonant contributions are included at different accuracies. In particular the double-resonant contributions only enter at NNLO (and beyond). Even in that case, these contributions would be effectively included only at lowest Table 1 LO and NLO total cross sections (in pb) and K -factors for the pp → H + W − bb process, for tan β = 1, 8, 30 at the 13 TeV LHC. The first quoted uncertainties are from scale variations, the second from PDFs (both in per cent of the total cross section). The statistical uncertainty from the numerical phase-space integration is of the order of 1% or below. 160 14.  [29,28], without being spoiled by large logarithms. For consistency, we use the four-flavour set of the PDF4LHC15 parton distributions [58][59][60][61], and the correspond- The identification of the hard scales in a complex process, such as the one at hand, is not necessarily a trivial task. One has to bear in mind, however, that in the intermediate region it is desirable to have a matching to the scale in the pp → tt cross section for light charged Higgs masses, where the natural choice is of the order of the top-quark mass (or below [62]), and for larger masses to the scale in the heavy charged Higgs cross section, where the scale μ = (m t + m H ± + m b )/3 is typically applied in 4FS computations.
We therefore fix our renormalisation and factorisation scales (μ r and μ f ) to μ = 125 GeV, which matches the numerical value used for the heavy charged Higgs production at m H ± = 200 GeV, while it satisfies the requirement of being in between m t /2 and m t for the light charged-Higgs case.
The top-quark mass and Yukawa coupling are renormalized on-shell, while we use a hybrid scheme for the bottom-quark mass: kinematical bottom-quark masses are treated with an onshell renormalization, but the MS renormalisation scheme is employed for the bottom-quark Yukawa coupling. For the numerical values we follow the recommendations of the LHC Higgs Cross Section Working Group [63], which implies m OS Since the pp → H ± W ∓ bb process involves resonant top-quark contributions, the width of the top quark has to be included in the computation without spoiling gauge invariance. This is achieved by employing the complex-mass scheme [65,66], where the top-quark mass (and Yukawa coupling) are regarded as complex parameters.
For a given charged Higgs mass and tan β, we compute the corresponding top-quark width at the same perturbative order in α s as the cross section. The charged Higgs boson and the W boson are kept on-shell.
Compared to calculations of similar complexity (e.g. the pp → W + W − bb process in the 4FS [67,68]), the technical challenges of this process lie in the interplay between the non-, single-and double-resonant contributions, which can have a different hierarchy depending on m H ± . On top of this, the cross section receives contributions with different powers of the bottom-quark Yukawa coupling, and therefore its running cannot be accounted for through an overall factor. Unlike in previous computations [69,28] H , A) and their coupling to bottom quarks, see Fig. 2 (d). We refrain from including these contributions in our computation at NLO, but briefly comment on the size of their effects below. To be able to make quantitative statements we must make some assumptions regarding the 2HDM parameters. We use the so-called "alignment" region (cos(β − α) 0, with α the mixing angle of the two CP even scalars), where the 125 GeV Higgs boson discovered at the LHC corresponds to the light scalar h [70]. 3   the low-mass prediction and amount to 10%-15% of the pp → tt cross section depending on the specific value of tan β. In contrast, looking at the matching of the intermediate-mass predictions to the heavy charged Higgs cross section, we observe a 5%-10% gap for tan β = 8 and tan β = 30, while there is essentially no gap for tan β = 1. Such a gap originates from the non-resonant part of the pp → H ± W ∓ bb amplitude, which, because of the chiral structure of the H + tb and W tb vertices, is enhanced (suppressed) for large (small) values of tan β. At 145 and 200 GeV, the size of the scale uncertainty in the intermediate region and the side-bands is slightly different. These discontinuities are related to missing subleading terms in the predictions used in the low and high-mass regions, i.e. mostly single-resonant and non-resonant, respectively, although it is difficult to pin down exactly the origin of the discontinuities because of the non-trivial separation of these contributions beyond leading order. Finally, we note that the K -factor in the intermediate region interpolates very well the ones in the low and high-mass range.
We now discuss how to generalise our results at a single tan β value in order to obtain the charged Higgs boson cross section in the intermediate-mass range for any value of tan β or in a type-I 2HDM by means of reweighting. As discussed in Ref. [28], the cross section for charged Higgs production receives contributions proportional to y 2 b , y 2 t and y b y t . In a type-II 2HDM, while the y b y t contribution does not depend on tan β, the y 2 b and y 2 t ones scale as tan β 2 and 1/ tan β 2 , respectively. Conversely, in a type-I 2HDM, all contributions (and therefore the total cross section) scale as 1/ tan β 2 . We point out that a naive reweighting, such as the one proposed in Ref. [28] for a heavy charged Higgs boson, is bound to fail in our case, since it will miss effects due to the tan β dependence of the top width. We verified that, if the top-width dependence is included as an overall factor, we are able to reproduce our tan β = 1 and tan β = 30 NLO cross sections and uncertainties starting from the numbers at tan β = 8 with an accuracy of 1% or better, using the relation (the dependence on m H ± is understood) (1) This also shows that effects due to the width-dependent complex phase of y t are very small. Concerning how to extend our results in a type-I 2HDM, we first point out that for tan β = 1, the crosssection is identical to the type-II case. Then, the cross-section for any other value of tan β can be simply obtained as Exploiting Eqs. (1) and (2) [29].
In conclusion, we have presented predictions for the production of an intermediate-mass charged Higgs boson. While we have focused on the case of a type-II 2HDM, our results can be easily extended to other scenarios, such as a type-I 2HDM or supersymmetry. For the first time theoretically consistent predictions at NLO QCD accuracy have been made available in this mass range. To this end, we have studied the pp → H ± W ∓ bb process in the complexmass scheme, including finite top-width effects and contributions with resonant top quarks. Our results provide a reliable interpolation of low-and high-mass regions and make it possible to finally extend direct searches for charged Higgs bosons to the m H ± ∼ m t region, so far unexplored by LHC experiments. The central value of the NLO total cross section is well-approximated by a factor of about 1.5-1.6 times the LO cross section, with only a very mild dependence on the charged Higgs mass and tan β. The results presented in paper constitute an important step in filling a gap in the available theoretical predictions for charged Higgs boson production at next-to-leading order in QCD. Current results could be further improved by including model-dependent sub-leading contributions that may become dominant in case of large width of heavy neutral Higgses, and by considering differential distributions. We leave it to future work to study if this factorisation of the NLO corrections also holds at the same level for differential distributions, employing modern techniques developed to take into account internal resonances when matching NLO computations with parton showers [74,57,75].