Abstract
We study the prospects for charged Higgs boson searches in the decay channel. This loopinduced decay channel can be important if the charged Higgs is fermiophobic, particularly when its mass is below the threshold. We identify useful kinematic observables and evaluate the future Large Hadron Collider sensitivity to this channel using the custodialfiveplet charged Higgs in the GeorgiMachacek model as a fermiophobic benchmark. We show that the LHC with 300 fb of data at 14 TeV will be able to exclude charged Higgs masses below about 130 GeV for almost any value of the SU(2)triplet vacuum expectation value in the model, and masses up to 200 GeV and beyond when the triplet vacuum expectation value is very small. We describe the signal simulation tools created for this analysis, which have been made publicly available.
1 Introduction
The discovery of the Higgs boson at the CERN Large Hadron Collider (LHC) Aad:2012tfa ; Chatrchyan:2012xdj represents the first experimental evidence for a (possibly) fundamental scalar particle. This naturally raises the question of whether there are more fundamental scalars; in particular, whether the Higgs sector is the minimal one predicted in the Standard Model (SM) or whether there are additional Higgs bosons.
Most extensions of the SM Higgs sector contain electricallycharged Higgs bosons , which require very different experimental search strategies than do neutral Higgs bosons. The standard charged Higgs searches at the LHC exploit the charged Higgs couplings to SM fermion pairs, which are expected in models in which the charged Higgs comes from an additional SU(2) doublet of scalars. These searches comprise charged Higgs production in top quark decays with the charged Higgs decaying to Aad:2014kga ; Khachatryan:2015qxa , Aad:2013hla ; Khachatryan:2015uua , or Sirunyan:2018dvm , as well as associated production of a charged Higgs and a top quark with the charged Higgs decaying to Aaboud:2016dig ; Khachatryan:2015qxa or Aad:2015typ ; Khachatryan:2015qxa . Searches for a charged Higgs produced in the decay of a heavier neutral Higgs have also been proposed for the LHC deFlorian:2016spz .
Fermiophobic charged Higgs bosons appear in a number of models including the GeorgiMachacek (GM) model Georgi:1985nv ; Chanowitz:1985ug , the Stealth Doublet model Enberg:2013ara ; Enberg:2013jba , and certain parameter regions of the Aligned twoHiggsdoublet model (2HDM) Pich:2009sp . The fermiophobic charged Higgs in the GM model, denoted because it is a member of a fiveplet of the custodial symmetry, couples at tree level to with strength proportional to the vacuum expectation value (vev) of the SU(2) triplets in the model. Dedicated searches have been performed at the LHC for produced in vector boson fusion and decaying to Sirunyan:2017sbn ; Aaboud:2018ohp ; these have focused on charged Higgs masses above 200 GeV.
In this paper we study the prospects for charged Higgs boson searches in the decay channel . This decay first appears at one loop Arhrib:2006wd ; Enberg:2013jba ; Ilisie:2014hea ; Degrande:2017naf , and hence its branching ratio is typically very small if treelevel decays to fermion pairs or are available. However, for a fermiophobic charged Higgs with mass below the threshold, the branching ratio into can dominate Enberg:2013jba ; Ilisie:2014hea ; Degrande:2017naf , especially if the coupling to is suppressed due to a small triplet vev in the GM model or induced only at one loop as in the Stealth Doublet model and the Aligned 2HDM. We will therefore focus on charged Higgs masses below 200 GeV.^{1}^{1}1Previous LHC searches for new resonances in the final state have been performed only for masses above 200 GeV Aad:2014fha .
This paper is organized as follows. In Sec. 2 we examine the general form of the loopinduced vertex and derive the key kinematic distribution that we will use to discriminate the charged Higgs decay from backgrounds. We also discuss the possible contributions to the loopinduced effective couplings that control this distribution. In Sec. 3 we choose the fermiophobic in the GM model as a concrete benchmark. After a brief description of the model to set our notation, we summarize the relevant decay modes and discuss the most important charged Higgs production processes in the lowmass region. We focus on DrellYan production of in association with another member of the scalar custodial fiveplet because of its large cross section even in the small triplet vev limit and its independence from the choice of model parameters.
In Sec. 4 we perform a sensitivity study for the channel and evaluate the exclusion reach for 300 fb at the 14 TeV LHC. We describe our implementation of the loopinduced decays via effective couplings in a new Universal FeynRules Output (UFO) Degrande:2011ua model file to be used with version 1.4.0 of the model calculator GMCALC Hartling:2014xma (these have been made publicly available). We simulate the dominant backgrounds and give an optimized set of cuts. Our main result is a projection for the 95% confidence level upper limit on the signal fiducial cross section as a function of the charged Higgs mass, which we then interpret as an upper limit on BR() and an exclusion reach in the GM model parameter space. In particular, we find that the LHC with 300 fb of data at 14 TeV will be able to exclude masses below about 130 GeV for almost any value of the triplet vev, and masses up to 200 GeV and beyond when the triplet vev is very small. Finally in Sec. 5 we summarize our conclusions. Details of our choice of the parameter benchmark in the GM model and the form factors in the limit of small triplet vev are given in Appendices A and B, respectively.
2 decay
The decay amplitude for is forced by electromagnetic gauge invariance to take the form Ilisie:2014hea
(1) 
where and are the fourmomenta and and are the polarization vectors of the boson and the photon, respectively.
The form factors and for have been computed in 2HDMs in Refs. Arhrib:2006wd ; Enberg:2013jba ; Ilisie:2014hea (Ref. Arhrib:2006wd also considered the Minimal Supersymmetric Standard Model (MSSM)) and in the GM model in Ref. Degrande:2017naf . In a CPconserving theory, the scalar form factor receives contributions from loops of fermions, scalars, and gauge bosons, while the pseudoscalar form factor receives contributions only from loops of fermions; this implies that for a fermiophobic charged scalar, . Furthermore, while and are complex in general, their imaginary parts arise only if a contributing loop diagram can be cut yielding an onshell treelevel twobody decay. While we maintain full generality in this section, it will be useful to keep in mind the fact that the decay is most interesting phenomenologically when competing decays to onshell twobody final states and to fermion pairs are absent, i.e., when both form factors are real and .
The vertex in Eq. (1) leads to the decay partial width
(2) 
where is the mass of and is the mass of the boson.
2.1 Differential distributions
In practice, the boson will be reconstructed from its decay products, providing an additional experimental handle on the structure of the vertex via the polarization. Allowing the boson to decay leptonically to , the square of the matrix element takes the form
(3)  
where , , and are the fourmomenta of the finalstate lepton , neutrino, and photon, respectively. Here we have assumed that the boson is emitted on shell and the propagator dependence in is omitted, which for an onshell is just an overall multiplicative factor. We have also neglected the finalstate fermion masses.
In particular, the square of the matrix element can be expressed as a quadratic polynomial in the experimentallyobservable kinematic invariant , the kinematicallyaccessible range of which is . It is convenient to reparameterize the form factor and momentum dependence of the kinematic distribution in Eq. (3) in terms of the ratios
(4) 
where a fermiophobic charged Higgs corresponds to . The kinematic distribution in Eq. (3) can then be rewritten as
(5) 
This function is a parabola in with its minimum at
(6) 
We plot the ideal differential decay distribution in Fig. 1 for various real values of between and , as a function of the experimental observable . Note that dividing Eq. (5) by an overall factor of yields the exact same distribution with ; therefore the differential distribution for real values outside the range can be obtained trivially from Fig. 1 by using this substitution. For concreteness, we set GeV; choosing different values of the charged Higgs mass only rescales the range of the axis in Fig. 1.
2.2 Possible values of
We now consider the possible values that can take.
The pseudoscalar form factor can be generated only by loops of fermions. Therefore, for a purely fermiophobic charged Higgs, . Phenomenologically, this is the most interesting situation because then the decays to light fermion pairs are absent and the branching ratio of can be significant. This is the case for of the GM model, which we will discuss further in the next section.
When is not fermiophobic, and both receive contributions from loops involving top and bottom quarks. also generically receives contributions from loops involving scalars and/or gauge bosons. Ignoring the bosonic loops, we can study the behaviour of due only to the top and bottom quark loops. This is shown in Fig. 2, where we implement only the top/bottom quark loop contributions to and using the calculation of Ref. Degrande:2017naf for the fermiophilic charged Higgs in the GM model. The fermion couplings of follow the same pattern as in the TypeI 2HDM. We also generalize to the TypeII 2HDM using the results of Ref. Ilisie:2014hea for the Aligned 2HDM, with the couplings as given in Table 1 Pich:2009sp .
Aligned 2HDM  

TypeI 2HDM  
GM model  
TypeII 2HDM 
In the left panel of Fig. 2 we plot the real and imaginary parts of including the top/bottom quark loop only and taking the couplings of as in the TypeI 2HDM or the GM model. Dependence on or cancels out in the ratio , so depends only on the mass. The threshold at which opens up is clearly visible. Below this threshold, is real and lies between and . Above this threshold, treelevel decays to compete with the loopinduced decay to , making the latter phenomenologically much less interesting.
In the right panel of Fig. 2 we plot the real and imaginary parts of including the top/bottom quark loop only, this time taking the couplings of as in the TypeII 2HDM with . The threshold at which opens up is much less obvious, but still visible. In this case, Re() is close to over a wide range of masses. now depends on the value of : TypeII couplings with lead to values nearly (but not exactly) identical to the left panel of Fig. 2.
In a realistic model, also receives contributions from loops involving scalars and/or gauge bosons. These can have either sign – in particular, in the GM model with small , the sign of the scalar loop contribution is controlled by the sign of the trilinear scalar coupling parameter [see Eq. (10)]. Therefore, the scalar and/or gauge boson contributions to can interfere constructively or destructively with the fermion contribution, and can even change the sign of . This means that Re() can be larger or smaller in magnitude than shown in Fig. 2, and can even change sign.
The general conclusion that we can draw from experimental detection of a nonzero value of from the shape of the distribution is therefore rather limited: nonzero tells us only that the fermion loop contribution is nonnegligible. This implies that is not fermiophobic and can also be searched for via its fermionic decay products, and (for masses below the top quark mass) its production in top quark decays.
3 A benchmark scenario
For the remainder of this paper we adopt the GM model as a prototype in order to study in more detail the future LHC sensitivity to the decay channel of a fermiophobic charged Higgs.
3.1 The GeorgiMachacek model
The scalar sector of the GM model Georgi:1985nv ; Chanowitz:1985ug consists of the usual SM complex scalar doublet with hypercharge^{2}^{2}2We use the convention . , together with a real triplet with and a complex triplet with . In order to avoid stringent constraints from the electroweak parameter, custodial symmetry is introduced by imposing a global SU(2)SU(2) symmetry upon the scalar potential. The isospin doublet is written as a bidoublet under SU(2)SU(2) and the two isospin triplets are combined into a bitriplet in order to make the symmetry explicit,
(7) 
The vevs are given by
(8) 
where is the unit matrix and the and boson masses give the constraint,
(9) 
The most general gaugeinvariant scalar potential involving these fields that preserves custodial SU(2) is given, in the conventions of Ref. Hartling:2014zca , by:
(10)  
where and
(11) 
The matrix , which rotates into the Cartesian basis, is given by
(12) 
The physical fields can be organized by their transformation properties under the custodial SU(2) symmetry into a fiveplet, a triplet and two singlets:
Fiveplet:  
Triplet:  
Singlets:  (13) 
where
(14) 
Within the fiveplet and triplet, the masses are degenerate at tree level, and are given in terms of the parameters of the scalar potential by
(15) 
The two custodial singlets will mix by an angle to give the two mass eigenstates and ,
(16) 
where and . The mixing is controlled by the mass matrix,
(17) 
where
(18) 
3.2 Fermiophobic decays and parameter choices
The custodialfiveplet states have no doublet component, and hence are fermiophobic at tree level. The fiveplet states do, however, couple at tree level to massive vector boson pairs with a coupling proportional to . They also take part in gauge couplings of the form and , where or ; in what follows we will assume that , in which case there are no decays of into other scalar states. The remaining possible decay channels for the states are listed in Table 2, including the loopinduced decays involving one or more photons.
Particle  Decay channels  Comment 

:  Loopinduced  
Suppressed by , offshell  
:  Loopinduced  
Loopinduced, phase space disfavored  
Suppressed by , offshell  
Suppressed by , offshell  
:  Suppressed by , offshell 
The decay width for is naturally small because this process is loop suppressed. This decay therefore can become important only when the competing treelevel decay is sufficiently suppressed. This can happen in two ways: (i) when is small, suppressing the coupling; and/or (ii) when is below the threshold, where the decay is offshell and hence kinematically suppressed. These two parameter regions are illustrated in Fig. 3, where we show the dependence of BR() on and , taking GeV and fixing the other parameters according to (see Appendix A)^{3}^{3}3We will adopt the choice of parameters in Eq. (19) for the remainder of this paper, keeping , , and as free parameters whose values we will specify.
(19) 
This choice of parameters ensures that the full range of and shown in Fig. 3 satisfies the theoretical constraints from perturbative unitarity of twototwo scalar scattering amplitudes, boundedness from below of the potential, and the absence of deeper alternative minima Hartling:2014zca . The kinematic threshold below which the competing channel goes off shell is clearly visible. Guided by this, we will concentrate on the region with GeV and fairly small. We have chosen and to be large so that we can (conservatively) ignore their contributions to production, which we discuss in the next subsection.
The amplitude for the loopinduced decay receives contributions from loop diagrams involving charged scalars and , and bosons, and mixed diagrams involving both scalars and gauge bosons Degrande:2017naf . The amplitudes for the gauge and mixed loop diagrams are all proportional to , and hence are suppressed when is small. This leaves the diagrams involving scalars in the loop, which are not suppressed at small . Instead, at small , these diagrams are all proportional to the trilinear scalar coupling parameter , and depend also on the masses and of the scalars in the loop (details are given in Appendix B). With our choice , the loops involving become small, and the partial width for essentially becomes a function of only and at small . The partial width for the competing treelevel decay is proportional to . Thus, for a given mass and not too large, the branching fractions of are determined entirely by and .
3.3 production processes
Because the states are fermiophobic, we focus on gaugebosoninitiated production processes. The relevant interactions of with one or two gauge bosons have the following coupling strengths:
(20) 
Note that all the couplings of to two gauge bosons are proportional to , while the couplings of two scalars ( or ) to one gauge boson are either a gauge coupling or a gauge coupling times . Therefore for , the cross sections for single production (via vector boson fusion or associated production with a vector boson) will be suppressed by , while DrellYan processes that produce a pair of states (or ) will be unsuppressed, with cross sections controlled only by the relevant gauge coupling and the masses of the finalstate scalars.
Taking , we can ignore the contribution from associated production.^{4}^{4}4We also ignore the possible contribution from . The coupling (which also controls ) is suppressed in the small limit. The most important production channels for are then , , and . The Feynman diagrams are shown in Fig. 4. These cross sections depend only on , as illustrated in Fig. 5 for TeV and between 80 and 200 GeV. These are calculated at leading order in QCD with MadGraph52.4.3 Alwall:2014hca , using the NNPDF23 parton distribution set Ball:2013hta and the model implementation described in the next section. has the largest cross section, reaching above a picobarn for GeV. The cross section for is smaller by a factor of , due entirely to the different couplings in Eq. (20). The smallest is , reaching a little over 200 fb for GeV. While these DrellYan cross sections drop rapidly with increasing , they offer plenty of events at low mass if the signal is sufficiently clean.
4 Search prospects at the LHC
We now study the search prospects for the charged Higgs in the channel. We focus on the mass range GeV and project the exclusion reach for 300 fb at the 14 TeV LHC.
4.1 Model implementation
The whole GM model at leading and nexttoleading orders in QCD has previously been implemented in FeynRules Alloul:2013bka and a UFO Degrande:2011ua model file produced for simulation purposes. We extend the leading order FeynRules implementation to include effective vertices of the form given in Eq. (1) for all loopinduced decays of the scalars into gauge boson pairs that are not present at tree level GMUFO . The oneloop calculations of these effective vertices were already implemented in GMCALC 1.3.0 for the purpose of calculating decay branching ratios; we adapt GMCALC to write the effective coupling form factors in a param_card.dat file for use by MadGraph5 Alwall:2014hca . (This adaptation is included in the public release of GMCALC 1.4.0.) This implementation allows us to accurately simulate the kinematics of the loopinduced scalar decays.
4.2 Simulation and selection cuts
In order to determine the sensitivity of a charged scalar search in the channel, we perform a cutbased Monte Carlo analysis of the inclusive signal. In particular, we require at least one lepton ( or ) and at least one photon in the final state. Signal and background events are generated at leading order in QCD using MadGraph5 Alwall:2014hca , showered and hadronized using Pythia Sjostrand:2014zea ; Sjostrand:2006za , and then passed to Delphes deFavereau:2013fsa for the detector simulation.
The signal processes, as discussed in Sec. 3.3, are
(21) 
We generate the inclusive signal requiring at least one lepton and at least one photon (with kinematic requirements given below). While we will vary BR() in order to extract limits on this branching ratio, we have to make some assumptions about the decay branching ratios of the other states produced in association. In our simulation we assume that and . The first of these is a safe assumption because this is the only possible twobody decay of when . The second is a conservative assumption because the additional photons from introduce combinatoric background and reduce the signal efficiency. Finally, for the channel, we allow the second to decay into either or , taking . Again, this is a safe assumption so long as .
We simulate the following SM processes as backgrounds:
(22) 
has the largest cross section before cuts, but it can be easily suppressed by the cuts described below. The dominant background after cuts is , followed by and . When calculating the signal significance, we include an overall 10% systematic error on the background cross section.
We begin by requiring at least one lepton with transverse momentum GeV and pseudorapidity and at least one photon with GeV and . To reduce combinatoric backgrounds from mispairings of the lepton and photon in signal events, we take the following strategy. When more than one lepton passes the and requirements, we choose the highest lepton as most likely to have come from the decay of . This is mostly an issue for the signal process; because the must decay to two bosons, they are more likely to be offshell than the from , and hence their decay leptons are generally softer. When more than one photon passes the and requirements, we choose the photon with the smallest separation (where is the azimuthal separation in radians) from our chosen lepton. This is mostly an issue for with , as well as for when both charged Higgs bosons decay to . Because the DrellYan scalar pair production process is wave, the scalars tend to be somewhat boosted, making the selection based on sufficiently effective.^{5}^{5}5Choosing the photon with highest is not a good strategy, because the photons from tend to have higher than the photon from .
We then apply additional cuts on each of the following variables:

, the number of reconstructed jets with GeV, and , the number of the jets that are tagged as jets by Delphes; in all cases we require and . This helps to reduce the background;

, the missing transverse energy;

, the scalar sum of the of all visible objects;

, the vector sum of the of our chosen lepton and photon together with the missing transverse momentum. In events with only one neutrino, this is equal to the transverse momentum of ;

, the dot product of the fourmomenta of our chosen lepton and photon, which was identified as a useful variable in Sec. 2.1.
The distributions of the last two variables for each signal and background process are shown in Fig. 6 for GeV.
The cuts are optimized for the best signal significance for each value of .^{6}^{6}6Note that when is close to , the photon coming from becomes soft and the partonlevel upper limit of becomes close to zero, making reconstruction of the correct lepton and photon difficult and leading to numerical instabilities in the automatic optimization of the cuts. To avoid this, for GeV we fix the cuts at the values obtained for GeV. For example, for GeV, we take

GeV,

,

,

.
The expected cross section of each signal and background process before and after applying these cuts is listed in Table 3 for GeV assuming for the signal processes.
Process  

BR [fb] (before cuts)  57.29  38.19  19.07  856  23000  30  120  65  25 
BR [fb] (after cuts)  4.21  1.01  0.95  0.49  0.09  0.05  0.38  0.28  0.05 
Because each production process has a different efficiency to pass the cuts and because the contribution to the signal rate of the process depends nonlinearly on BR(), we first present the expected upper limit on the fiducial cross section as a function of in the left panel of Fig. 7. The fiducial cross section is defined as
(23) 
Here and stands for the efficiency of the cuts for the process . This efficiency is shown for each signal process in the right panel of Fig. 7. As the mass of the scalar approaches the threshold of the channel, the efficiency drops to near zero. This is due to the photon becoming too soft to pass the initial selection as well as the variable losing its discriminative ability when is close to . The upturn in the efficiency for in the right panel of Fig. 7 is due to a (counterintuitive) rise in the number of photons passing the minimum threshold in our simulation as the is pushed off shell. Because the form factor for the vertex that we use in our calculation is computed assuming onshell external particles, we will consider our results reliable only for GeV. As we will see in Sec. 4.4, lower values are mostly well covered by searches for .
The DrellYan cross section for production of pairs of scalars in the GM model depends only on the mass of . Thus the interpretation of the LHC exclusion in this model depends only on the branching fraction of . The projected upper limit on is shown in the left panel of Fig. 8, where the nonlinear dependence on the branching fraction of the total cross section in Eq. (23) has been taken into account. The projected exclusion ranges from of about 2% for GeV to about 12% for GeV.
In the right panel of Fig. 8 we show the projected 95% confidence level upper limit on the process alone. The axis shows the projected upper bound on . This can be used to estimate the sensitivity of the search in other models, as well as in scenarios in which the final state is produced resonantly through the decay of a heavier scalar particle. (We note however that the kinematic distribution from such a decay will be different than that from DrellYan production, resulting in different selection efficiency.)
4.3 Constraint on the GM model parameter space
The projected upper bound on BR() shown in the left panel of Fig. 8 can be reinterpreted as a constraint on the GM model parameter space. The dependence of BR() on the underlying parameters is remarkably simple when . We show this as a function of and in Fig. 9, for GeV (left) and 150 GeV (right) and the remaining model parameters chosen as in Eq. (19).^{7}^{7}7For the sake of illustration, to populate the full range of these plots we ignore the theoretical constraints on the GM model parameters Hartling:2014zca . The theoretical constraints will be satisfied in the low region that we focus on below.
For small enough and fixed , BR(