Low-mass fermiophobic charged Higgs phenomenology in two-Higgs-doublet models

After the recent discovery of a Higgs-like boson, the possibility of an enlarged scalar sector arises as a natural question. Experimental searches for charged scalars have been already performed with negative results. We analyze the phenomenology associated with a fermiophobic charged Higgs (it does not couple to fermions at tree level), in two-Higgs-doublet models. All present experimental bounds are evaded trivially in this case, and one needs to consider other decay and production channels. We study the associated production of a charged Higgs with either a $W$ or a neutral scalar boson, and the relevant decays for a light fermiophobic charged Higgs. The interesting features of this scenario should result encouraging for the LHC collaborations to perform searches for such a particle.


Introduction
The recent discovery of a boson with mass around 125 GeV by the ATLAS [1][2][3][4], CMS [5][6][7], DØ and CDF [8,9] collaborations is the first direct hint of the electroweak symmetrybreaking mechanism. The experimental data confirm that it is a Higgs-like scalar with couplings compatible with the Standard Model (SM) predictions. However, this new particle could belong to an enlarged scalar sector.
In order to give mass to fermions and gauge bosons while preserving gauge invariance, the SM assumes the presence of one SU(2) electroweak scalar doublet with a non-zero vacuum expectation value. However, no fundamental principle or symmetry forbids the presence of additional scalar doublets. The simplest extension of the SM is the two-Higgs-doublet model (2HDM) [10,11], which leads to a richer scalar sector and very interesting phenomenological implications . Generic multi-Higgs doublet models give rise to unwanted flavour-changing neutral current (FCNC) interactions, which are found to be very suppressed experimentally.
The FCNCs can be eliminated at tree level by requiring the alignment in flavour space of the Yukawa matrices [15]. The so-called aligned two-Higgs-doublet model (A2HDM) contains as particular cases the different versions of 2HDMs with discrete Z 2 symmetries while at the same time introduces new sources of CP violation beyond the CKM phase.
The main feature of the 2HDM is the presence of three neutral and one charged Higgs bosons. Finding extra neutral or charged scalar bosons would be a clear signal of an extended scalar sector. The ATLAS [45,46] and CMS collaborations [47] have performed direct searches for a charged Higgs particle. However, since no excess has been found over the SM background, this only allows us to further constrain the parameter space of the various types of 2HDMs; recent analyses within the A2HDM have been performed in [12][13][14]. In their searches, both collaborations assume that the charged Higgs is produced in a top-quark decay (t → H + b) and that it decays dominantly into fermions; i.e., H + → q uqd , l + ν l . However, all experimental bounds would be trivially evaded for a fermiophobic charged Higgs, i.e., a charged scalar which does not couple to fermions at tree level. In order to probe such scenario, other production channels and decay rates would have to be considered. Although such analyses have not been yet performed by the LHC collaborations, they become more compelling as the experimental bounds on a non-fermiophobic charged Higgs are getting stronger, at least in the low mass region. The fermiophobic scenario is a simplified model that, if it turns out to be the one preferred by Nature, would allow us to measure (or at least estimate) for the first time the parameters of the scalar potential. This is usually a rather difficult task in more generic 2HDM settings. It is also worth mentioning that a fermiophobic charged Higgs is present in the inert 2HDM [48,49], where one of the neutral scalars is a nice candidate for dark matter [32-35, 37, 50-58]. The discovery of a fermiophobic H ± particle could be interpreted in this case as an indirect signal of the presence of dark matter.
In this work, we shall focus our analysis on the search of a light fermiophobic charged Higgs H ± , with mass in the range M H ± ∈ [M W , M W + M Z ] so that only a few relevant decay modes are kinematically open. We will study the two most important production channels for a fermiophobic H ± : associated production with either a W ∓ boson or a neutral scalar. Due to their similarity with the SM Higgs production channels, one expects them to be experimentally accessible at LHC energies. Next-to-leading order (NLO) QCD corrections will be included for both cross sections, and the bounds on the various parameters of the model from the current LHC data [12] will also be taken into account. The main features of the A2HDM are briefly presented in section 2. Section 3 discusses the calculation of the various decay rates and production modes. Finally, in section 4 we perform a phenomenological analysis, assuming different scenarios for the scalar spectrum, and conclude in section 5 with a summary of our results. Some technical details are given in four appendices.

The Aligned Two-Higgs-Doublet Model
The 2HDM extends the SM with a second scalar doublet of hypercharge Y = 1 2 . The neutral components of the two scalar doublets acquire vacuum expectation values that are in general complex, 0|φ (0) a (x)|0 = 1 √ 2 v a e iθa (a = 1, 2), although only the relative phase θ ≡ θ 2 − θ 1 is observable. It is convenient to perform a global SU(2) transformation in the scalar space (φ 1 , φ 2 ), characterized by the angle β = arctan (v 2 /v 1 ), and work in the so-called Higgs basis (Φ 1 , Φ 2 ), where only one doublet acquires a vacuum expectation value: where G ± and G 0 denote the Goldstone fields. Thus, Φ 1 plays the role of the SM scalar doublet The physical scalar spectrum contains five degrees of freedom: the two charged fields H ± (x) and three neutral scalars ϕ 0 i (x) = {h(x), H(x), A(x)}, which are related with the S i fields through an orthogonal transformation ϕ 0 i (x) = R ij S j (x). The form of the R matrix is fixed by the scalar potential [14], which determines the neutral scalar mass matrix and the corresponding mass eigenstates. A detailed discussion is given in appendix A. In general, the CP-odd component S 3 mixes with the CP-even fields S 1,2 and the resulting mass eigenstates do not have a definite CP quantum number. If the scalar potential is CP symmetric this admixture disappears; in this particular case, A(x) = S 3 (x) and Performing a phase redefinition of the neutral CP-even fields, we can fix the sign of sinα. In this work we adopt the conventions M h ≤ M H and 0 ≤α ≤ π, so that sinα is positive.
The most generic Yukawa Lagrangian with the SM fermionic content gives rise to FCNCs because the fermionic couplings of the two scalar doublets cannot be simultaneously diagonalized in flavour space. The non-diagonal neutral couplings can be eliminated by requiring the alignment in flavour space of the Yukawa matrices [15]; i.e., the two Yukawa matrices coupling to a given type of right-handed fermions are assumed to be proportional to each other and can, therefore, be diagonalized simultaneously. The three proportionality parameters ς f (f = u, d, l) are arbitrary complex numbers and introduce new sources of CP violation. In terms of the fermion mass-eigenstate fields, the Yukawa interactions of the A2HDM read [15] where P R,L ≡ 1±γ 5 2 are the right-handed and left-handed chirality projectors, M f the diagonal fermion mass matrices and the couplings of the neutral scalar fields are given by: As in the SM, all scalar-fermion couplings are proportional to the corresponding fermion masses, and the only source of flavour-changing interactions is the Cabibbo-Kobayashi-Maskawa (CKM) quark mixing matrix V [59,60]. The usual models with natural flavour conservation, based on discrete Z 2 symmetries, are recovered for particular (real) values of the couplings ς f [15].
The full set of interactions among the gauge and scalar bosons is given in [14]. The coupling of a single neutral scalar with a pair of gauge bosons takes the form (V = W, Z) which implies g 2 hV V + g 2 Thus, the strength of the SM Higgs interaction is shared by the three 2HDM neutral bosons. In the CP-conserving limit, the CP-odd field decouples while the strength of the h and H interactions is governed by the corresponding cosα and sinα factors.
In the following analysis we are also going to need the coupling of a neutral scalar with a pair of charged Higgses. We have parametrized the corresponding interaction as: Explicit expressions for the reduced cubic couplings λ ϕ 0 i H + H − , in terms of the generic Higgs potential parameters, can be found in [14].
The symmetries of the A2HDM protect in a very efficient way the flavour-blind phases of the alignment parameters from undesirable phenomenological consequences. The experimental upper bounds on fermion electric dipole moments provide the strongest constraints on Im(ς f ), but O(1) contributions remain allowed at present [20]. For simplicity, in section 4, we will restrict our analysis to the CP-conserving limit and, therefore, will consider real alignment parameters. The LHC data require the gauge coupling of the 125 GeV boson to have a magnitude close to the SM one. Assuming that it corresponds to the lightest CP-even scalar h of the CP-conserving A2HDM, the measured Higgs signal strengths imply | cosα| > 0.90 (0.80) at 68% (90%) CL [12][13][14]. Direct searches for a heavier neutral scalar (H) provide upper bounds on | sinα| as a function of M H , which at present result in a weaker constraint on the mixing angle [12].
In the following we will explore the intriguing possibility that the charged scalar could be fermiophobic, i.e., that its tree-level couplings to fermions vanish (ς u,d,l = 0). All current experimental bounds are then trivially avoided, in particular the flavour constraints [16]. The Yukawa couplings of the h(125) boson scale in this case, with respect to the SM ones, with the same factor as the gauge couplings: y h f = R 11 = cosα. The global fit to the Higgs signal strengths results in the slightly improved bound | cosα| > 0.86 at 90% CL [12].
A more specific version of the fermiophobic scenario is provided by the inert 2HDM [48,49], which assumes a discrete Z 2 symmetry in the Higgs basis such that all SM fields and Φ 1 are even In this restricted case, there is no mixing between the CP-even neutral scalars h and H; i.e., cosα = 1. The spectrum of the inert 2HDM is described in appendix A.1.

Decay and Production modes
We are going to analyse the possibility of having a fermiophobic charged Higgs with a mass in the restricted interval M H ± ∈ [M W , M W + M Z ]. In this region, the only relevant decay rates are H + → W + γ and H + → W + ϕ 0 i . We are mainly interested in the one-loop suppressed decay H + → W + γ, the only two-body kinematically allowed decay mode, but we need to account also for the tree-level decay into a W + boson and a neutral scalar, which cannot be both on-shell simultaneously for the whole considered kinematical region. Thus, we shall consider three-body decays like H + → W + ff mediated by the neutral scalars ϕ 0 i and H + → ϕ 0 i f ufd mediated by a virtual W + , where f ufd stands for quark pairs q uqd , or lepton-neutrino pairs l + ν l .
The loop-induced decay H + → f ufd has a strong Yukawa suppression m 2 f /v 2 and, therefore, it is irrelevant for this discussion. When surpassing the M W + M Z threshold, the one-loop decay H + → W + Z would enter the game and we would also be close to the top-quark production threshold. The analysis of these two extra decay modes lays beyond the goal of this paper.

H
The first process that we are going to analyse is H + (k + q) → W + (k) γ(q). Owing to the conservation of the electromagnetic current, the decay amplitude must adopt the form: where S andS are scalar form factors. To obtain this expression, we have considered the most general Lorentz structure for the effective Γ µν vertex, and have imposed the electromagnetic current conservation condition q µ Γ µν = 0. All terms proportional to q µ and k ν have been also eliminated, as they cancel when contracted with the polarization vectors of the photon and the W boson. Note that, accidentally, the Ward-like identity k ν Γ µν = 0 also holds for (7).
In the unitary gauge, the decay proceeds at one loop through the three sets of diagrams shown in Fig. 1: fermionic loops (set 1), scalar loops (set 2) and loops with both gauge and scalar bosons (set 3). Each set is transverse by itself, i.e., of the form given in (7). We can then decompose the result into the three separate contributions: S = S (1) + S (2) + S (3) andS =S (1) (the only contribution to the structure µναβ k α q β comes from the fermionic loops). One can further simplify the calculation of S (j) by only considering the terms of the transverse set j that contribute to the structure k µ q ν . In order to calculate these contributions, one only needs to compute diagrams 1.a and 1.b for the first set, 2.a for the second set and 3.a for the third one. We obtain the following expressions for the form factors: with s W ≡ sin θ W . The calculation of S (3) has been also performed in the Feynman (ξ = 1) gauge, where additional diagrams with Goldstone bosons are present, verifying that these expressions Figure 2: H + → W + ff process mediated by the virtual neutral scalars ϕ 0 i (left) and H + → ϕ 0 i f ufd mediated by a virtual W + (right).
are gauge independent. Our results are in agreement with the recent calculation of the H + W − γ effective vertex in Ref. [62]. This calculation was also done many years ago by several groups [63][64][65][66] using a somewhat different notation.
The H + → W + γ decay width is easily found to be: This one-loop decay rate is in general much smaller than the tree-level decay rates of a charged Higgs into fermions. However, it becomes relevant if the charged Higgs is fermiophobic (ς f → 0).
In this case, the first set of diagrams (which has only been presented for completeness) does not contribute.

H
The H + decay rate to on-shell W + and ϕ 0 i bosons is given by with the usual definition of the lambda function λ(x, y, z) ≡ The corresponding three-body decay rate to W + ff , with off-shell neutral scalars (Fig. 2, left), takes the form: where N f C stands for the number of colours of the fermion f , 3 for quarks and 1 for leptons, s 23 is the square of the fermion-antifermion invariant mass and Obviously, the b-quark contribution will dominate because of the global factor m 2 f . Therefore, we will neglect the other fermionic final states.
For the decay H + → ϕ 0 i f ufd , with an of-shell W + (Fig. 2, right), we are going to consider all possible final states, quarks and leptons. We exclude the top quark, since this process is well below its production threshold. Neglecting the final fermion masses, the sum over all kinematically-allowed decay modes amounts to a global factor where the unitarity of the CKM matrix has been used. The total decay width can be expressed as an integral over the fermion-antifermion invariant-mass squared:

Charged-Higgs Production
In order to see if the fermiophobic scenario can be experimentally probed, one needs an estimation of the production cross sections for different channels. Here we will consider two possibilities, the associated production with a neutral scalar and the associated production with a W boson (Fig. 3). The q uqd → H + ϕ 0 i production process is by far the most interesting channel, as it requires the least number of new parameters. For initial-state massless quarks, the leading-order (LO) partonic cross section readŝ whereŝ is the partonic invariant-mass squared. The NLO QCD corrections are available and can be expressed in a very simple form, as shown in appendix C. The associated production with a W boson can proceed through either qq or gg fusion. The partonic LO cross section for the qq fusion process, is given bŷ with the reduced amplitudes We have kept the dependence on the initial quark masses, since otherwise the qq Yukawa coupling vanishes. This implies a strong suppression of this production mechanism by a factor The gluon fusion mechanism dominates by far the previous one. The corresponding LO cross section at the partonic level takes the form where T F = 1/2 is the SU (3) colour group factor and the reduced amplitudes G ij are given by with x q ≡ 4m 2 q /ŝ. The explicit expressions of the different loop functions are: with We have regulated the propagator poles with the term iM ϕ 0 i Γ ϕ 0 i , both in Eqs. (20) and (22), because in our analysis one of the neutral scalars will, most likely, reach the on-shell kinematical region. NLO QCD corrections to the gluon fusion channel are also available and will be taken into account; the details are given in appendix D.

Phenomenology
In the following phenomenological analysis, besides the fermiophobic charged-Higgs assumption (ς f → 0), we are also going to consider that the Higgs potential is CP-conserving. The consequence of this last hypothesis is that the CP-odd neutral Higgs A will also be fermiophobic, as we have mentioned before in section 2; moreover λ AH + H − = 0. This means that the decay H + → W + A * → W +f f does not occur and A does not contribute either to H + → W + γ. The charged-Higgs production amplitudes mediated by a virtual A also vanish. The CP-odd scalar can contribute to H ± production in a direct way through the q uqd → W * → H + A production channel or, in an indirect way, by modifying the total decay rate Γ ϕ 0 i , which regulates the pole in the CP-even scalar propagators (ϕ 0 i = h, H), through decays like ϕ 0 i → AA or ϕ 0 i → AZ. The decay H → Ah cannot occur at tree level because all cubic vertices of the scalar potential involving an odd number of A fields vanish in the CP-conserving limit. The total decay width Γ ϕ 0 i is the sum of all the decay rates explicitly presented in appendix B. In our particular case, the expressions for the Yukawa couplings simplify and become equal to the reduced scalar couplings to two gauge bosons. They are given by Even within the restricted range of charged-Higgs masses we are interested in, , the possible phenomenological signals depend on the choice of masses for the remaining scalars. In the following subsections, we will therefore consider different scenarios for the scalar spectrum. The first part of the analysis will be dedicated to the study of the various decay modes of the charged Higgs and the second part will focus on estimating the production cross sections.

Decay rates and branching ratios
One of the two CP-even scalars should correspond to the Higgs boson discovered at the LHC, but a broad range of masses is allowed for the other two neutral scalars. We will consider the following four scenarios, which cover the different possibilities:

First Scenario
In the first scenario the mass of the lightest CP-even scalar is set to M h = 125 GeV. Therefore, the strong constraint on the scalar mixing angle, from the global fit to the light Higgs boson signal strengths using the LHC data, must be used: | cosα| > 0.9 at 68% CL [12][13][14]. The masses of the remaining neutral scalars are considered to be greater than M W + M Z so that decays of a charged Higgs into an on-shell H or A are kinematically forbidden. In the limit cosα → 1, the only surviving decay amplitude (not proportional to sinα) is the contribution of H to the amplitude S (2) . Thus, in this limit the branching ratio of H + → W + γ is 100%; all the other decay channels vanish.
If we set cosα = 0.9, λ hH + H − = λ HH + H − = 1, vary the charged Higgs mass in the region The corresponding total decay widths are shown in the bottom-right panel (top-left) and total decay width (bottom-right) shown in Fig. 4. The width of the branching ratio bands reflects the variation of the input parameters in the mentioned ranges. The same consideration is valid for the following scenarios. The decay channel H + → W + γ dominates for M H ± M h . When the charged Higgs is kinematically allowed to decay into an on-shell h, then H + → hf ufd rapidly becomes the dominant channel as M H ± grows. The remaining H + → W + bb branching ratio stays at a few percent level or less for the whole allowed region.
The total decay width approximately grows from 10 −14 up to 10 −8 GeV, in the region dominated by the radiative H + → W + γ decay, and sizeably increases up to 10 −5 GeV, once the hf ufd production threshold is reached. The tree-level decay rates are significantly larger than the loop-induced one. Flipping the sign of cosα leads to an equivalent solution with a sign flip of the coupling λ hH + H − . This is also valid for the next scenarios.
If, instead, we consider all the previous settings but taking this time λ hH + H − = λ HH + H − = 0, then the only amplitude that contributes to the H + → W + γ decay channel is S (3) , which is suppressed by a factor sinα. As shown in Fig. 4 (Fig. 4, bottom-left).

Second Scenario
In It is worth mentioning that, just as in the previous scenario, the W bb branching ratio can be sizeably increased by decreasing the W γ decay width through a sign flip of the λ ϕ 0 i H + H − couplings, creating destructive interference among the various loop contributions. The same consideration is also valid for the next scenario.

Third Scenario
In this scenario the mass of the lightest CP-even scalar is also set to M h = 125 GeV, while    The H ± decay into an on-shell h boson has a global relative suppression factor of tan 2α with respect to the decay into an on-shell H and sin 2α with respect to the decay into an on-shell A.
Therefore, when hf ufd competes with Hf ufd , the later one dominates as cosα → 0.99 (Fig. 6, upper-left). When all three channels compete, the decay rate into the heaviest scalar boson grows the slowest and, therefore, brings a sub-dominant contribution to the branching ratios.

Fourth Scenario
In this last scenario we are going to set the mass of the heavy CP-even scalar to M H = 125 GeV; therefore, the LHC bounds translate into | sinα| > 0.9 at 68% CL. The mass of the light CP-even scalar will be set to M h = 90 GeV. As for the CP-odd one, we will consider three possible values: M A = 150, 140 and 110 GeV.
In order to safely avoid the stringent constraints on light scalar masses from LEP [67,68], we need to have very suppressed decay and production channels. In our particular case with ς f = 0, CP-conserving potential, and M A > M h (therefore the decays h → AA and h → AZ are forbidden), we have the simple relation Γ h = cos 2α Γ SM h . Here Γ h is the total decay rate of the light CP-even scalar boson with M h < M H = 125 GeV, and Γ SM h the corresponding decay rate in the SM for a Higgs boson with the same mass M h . The cos 2α suppression factor is common to all allowed h → ff decay modes, and cancels out in the branching ratios. The same suppression factor appears in the LEP production rate, so that the signal strengths, relative to the SM, are then given by with X = bb and τ + τ − . Thus, we have a global suppression factor cos 2α . The LEP constraints from the τ + τ − channel, which are the strongest ones in our case, can then be avoided by setting cos 2α ≈ 0.02 (sinα ≈ 0.99). The OPAL collaboration has also performed a decaymode-independent search for a light neutral scalar and found the upper limits cos 2α < 0.1 (1) for M h < 19 (81) GeV [67], which are weaker (in our case).
It is worth mentioning that in (26) we have ignored the charged-Higgs contribution to the h → γγ decay rate. If however, we choose to enhance it through the H ± loop contribution, it would only further suppress the fermionic branching ratios, weakening the bound on sinα.
With all this being said, we set sinα = 0.99. In Fig. 7   sin 2α ∼ 1. The suppression factor appears now in the Hf ufd decay mode with a partial decay width proportional to cos 2α . This is why, when M A ∼ M H , the decay into an on-shell A boson dominates over the decay into an on-shell H. Both A and H contributions are, however, very suppressed due to their heavy masses. It is also worth mentioning that a small variation of M A can produce a significant change (roughly, one order of magnitude) in Br(H + → Af ufd ), as can be seen in Fig. 7 (top-left and top-right).
For the last case we set M A to 110 GeV. The perturbativity bounds on neutral scalar couplings to a pair of charged Higgses, for the considered region of the charged Higgs mass, are roughly given by |λ ϕ 0 i H + H − | ≤ 5 (here ϕ 0 i = h, H) [14]. In order to see the impact of these two parameters on the H ± branching ratios, we will vary both independently in this region. The result, shown in Fig. 7 (bottom-left), is that W γ and hf ufd compete, even after crossing the h production threshold. Since M A is lighter than in the previous two cases, the H + → Af ufd branching ratio can also reach higher values. The total decay rate for this configuration is also shown in Fig. 7 (bottom-right).
As we have seen, in the four proposed scenarios, the configuration of the H ± branching ratios depends very sensitively on the chosen parameters. However, we can draw some important conclusions. There are only a few decay channels to be analysed and the largest decay widths are the tree-level ones, corresponding to the on-shell production of scalar bosons. Thus, the number of decay channels decreases as the number of neutral scalar bosons that are heavier than the charged Higgs (i.e., M ϕ 0 i > M H ± ) increases. The W γ decay mode can bring sizeable contributions below and close to the the on-shell production threshold of a scalar boson. Short after this threshold is reached, as M H ± grows, the H + → W + γ branching ratio rapidly decreases. As we have shown, the H + → W + bb decay can dominate over H + → W + γ in some cases, depending on the values of the λ ϕ 0 i H + H − couplings. If a fermiophobic charged Higgs is finally discovered in this mass range, the precise values of its mass and branching ratios would provide priceless information about all other parameters. The masses of the remaining scalars would also be highly constrained by the electroweak oblique parameters. These constraints were used in our second scenario, because they put an upper bound on M H ; we did not mention them in the other cases, since they do not bring additional constraints. The mean lifetime of a fermiophobic charged scalar is short, ranging from 10 −11 to 10 −23 s, making its direct detection very compelling at the LHC.

Production cross sections
In order to estimate the total hadronic cross sections for the various production channels, we need to convolute the partonic cross sections with the corresponding parton distribution functions (PDFs). Here we will use the MSTW set [69]. Moreover, we will compute the cross sections at the NLO; i.e., including the LO QCD corrections, for which simple analytical expressions can be obtained [70,71]. For the q uqd → H + ϕ 0 i associated production, the O(α s ) contributions simply correspond to the QCD corrections to the Drell-Yan process q uqd → W * , integrating over the virtuality of the W boson. As for the H + W − associated production, the needed QCD corrections can be easily extracted from the SM Higgs production channels qq → h and gg → h. At the LHC, gg → H + W − production dominates over qq → H + W − . For typical LHC hadronic center-of-mass energies, i.e., √ s ∼ 14 TeV, the latter only corresponds at LO to a few percent of the total pp → H + W − cross section, so we can safely neglect it.
The detailed expressions of the hadronic cross sections and the QCD corrections are given in appendices C and D. In order to estimate the theoretical uncertainty of the QCD enhancement factor K ≡ σ N LO /σ LO , we vary the factorization (µ F ) and renormalization (µ R ) scales for σ N LO , keeping both scales fixed at their central value µ F = µ R =ŝ for σ LO .
When one of the intermediate scalar bosons reaches its on-shell kinematical region, one needs to estimate also its total decay rate. The explicit expressions for the tree-level scalar decay rates are presented in appendix B.

H + ϕ 0 i associated production
Assuming the most general scalar potential, the LO partonic cross section, given in Eq. (18), is proportional to the combination of rotation matrix elements R 2 ≡ (R 2 i2 + R 2 i3 ). We take away the explicit dependence on the scalar-potential parameters, plotting in Fig. 8 (left) the ratio As expected, the cross section reaches higher values for lower scalar masses. The most interesting case is of course M ϕ 0 i =125 GeV, which could constitute a very good detection channel, since we already know that there is one scalar with that mass. If we consider ϕ 0 i to be the light CP-even scalar of the theory, the cross section is suppressed by a factor R 2 = sin 2α . The measurement of this production channel can be experimentally challenging due to the small value of the cross section.
QCD corrections provide a mild enhancement of the cross section. The resulting QCD K factor is shown in Fig. 8 (right), for M ϕ 0 i = 125 GeV and different choices of µ R and µ F . Its central value is around 1.2, similarly to other cross sections of the Drell-Yan type.

H + W − associated production
For this specific production channel we are going to consider two alternative possibilities: we can either identify the 125 GeV boson with the lightest CP-even scalar h, or with the heaviest one H. In the first case (M h = 125 GeV), the scalar H can be heavy enough to reach the on-shell region and, therefore, it is necessary to regulate the propagator pole with its total

Conclusions
The recent discovery of a Higgs-like boson has confirmed the existence of a scalar sector, which so far seems compatible with the SM predictions. As it is widely known, an enlarged scalar sector is not forbidden by the symmetries of the electroweak theory, and there exists a broad range of possibilities satisfying all experimental constraints. The direct discovery of another scalar particle would represent a major break-through in particle physics, opening a window into a new high-energy dynamics and providing priceless information on which type of extension, amongst many theoretical models of the scalar sector, is preferred by Nature.
Here we have focused on a particular 2HDM scenario, characterized by a fermiophobic The two most important production channels for a fermiophobic charged scalar have been investigated, including NLO QCD corrections: the associated production with either a neutral scalar or a charged W ; i.e., q uqd → H + ϕ 0 i and gg → H + W − . The predicted cross sections are small in most of the parameter space, making the experimental search challenging, but they become very sizeable (≥ 10 −3 pb) for large values of the mass of the heavy neutral scalar H.
In some extreme cases, cross sections between 0.1 and 1 pb are obtained. Thus, the detection of a fermiophobic H ± at the LHC seems plausible in the near future. The interesting features of this possible scenario should encourage specific experimental searches for such a particle in the LHC data.

A Scalar Potential
In the Higgs basis, the most general scalar potential takes the form The hermiticity of the potential requires all parameters to be real except µ 3 , λ 5 , λ 6 and λ 7 ; thus, there are 14 real parameters. The minimization conditions 0|Φ T The potential can then be decomposed into a quadratic term plus cubic and quartic interactions The mass terms take the form with and where λ R i ≡ Re(λ i ) and λ I i ≡ Im(λ i ). The symmetric mass matrix M is diagonalized by an orthogonal matrix R, which defines the neutral mass eigenstates: where we have introduced the shorthand matrix notation Since the trace remains invariant, the masses satisfy the relation The minimization conditions allow us to trade the parameters µ 1 and µ 3 by v and λ 6 . The freedom to rephase the field Φ 2 implies, moreover, that only the relative phases among λ 5 , λ 6 and λ 7 are physical; but only two of them are independent. Therefore, we can fully characterize the potential with 11 parameters: v, µ 2 , |λ 1,...,7 |, arg(λ 5 λ * 6 ) and arg(λ 5 λ * 7 ). Four parameters can be determined through the physical scalar masses [14]. The matrix equation relates the scalar masses and mixings. Summing the second row with (−i) times the third row, one obtains the identity (imaginary parts included): This proves in full generality that Taking instead the first row, one gets: which generalizes the usual relation determining tanα in the CP-conserving limit (R 13 = R 23 = 0). It also proves that the following identity holds in general Here, similarly to Eq. (6), we have parametrized the Goldstone terms of V 3 in the form These identities generalize the ones from [72], that are valid only in the CP-conserving limit of the scalar potential. They turn out to be very useful if one works in R ξ gauges with a fully general potential.
The first identity reproduces in complex form the last two real equations in (42). Separating the real and imaginary parts of the last two relations, one gets:

A.1 Inert 2HDM
Imposing a discrete Z 2 symmetry such that all SM fields remain invariant under a Z 2 transformation, while one makes the second scalar doublet inert: linear interactions of Φ 2 with the SM fields are odd under a Z 2 transformation, and thus forbidden [48,49]. In particular, Φ 2 is fermiophobic. This Explicit expressions for these couplings can be found in [14]. Here we didn't consider the off-shell ϕ 0 i → ϕ 0 * j ϕ 0 * j decay mode because in addition to its kinematical suppression it also depends on the unknown parameter λ ϕ 0 i ϕ 0 j ϕ 0 j and would not bring useful information. The last two processes that must be taken into account are ϕ 0 i → ϕ 0 j Z and ϕ 0 i → H + W − . We have Again, the scalar couplings to gauge bosons are taken from [14].
C QCD corrections to pp → H + ϕ 0 i For the H + ϕ 0 i associated production, we write the LO hadronic cross section as where we have introduced the shorthand notationσ LO ≡σ(q uqd → H + ϕ 0 i ), for the partonic cross section given in Eq. (18). As usual, the partonic invariant-massŝ must be expressed as a fraction of the hadronic center-of-mass energy s, that isŝ = τ s. The lower integration limit is given by τ 0 = (M H ± + M ϕ 0 i ) 2 /s. The PDFs q i (x, µ F ), for a given quark flavour 'i', depend on the momentum fraction x and the factorization scale µ F .