Abstract
We discuss the implications of assuming a four-zero Yukawa texture for the properties of the charged Higgs boson within the context of the general 2-Higgs Doublet Model of Type III.
We begin by presenting a detailed analysis of the charged Higgs boson couplings with heavy quarks and the resulting pattern for its decays. The production of charged Higgs bosons is also sensitive to the modifications of its couplings, so that we also evaluate the resulting effects on the top decay t → bH + as well as on 'direct' cb → H + + c. c. and 'indirect' qq, gg →tbH + + c. c. production. Significant scope exists at the Large Hadron Collider for several H ± production and decay channels combined to enable one to distinguish between such a model and alternative 2-Higgs doublet scenarios.
I. INTRODUCTION
Detecting a charged Higgs boson during the imminent Large Hadron Collider (LHC) experimental running would constitute a clear evidence of physics beyond the Standard Model (SM) [1] .
Charged Higgs bosons appear in many well motivated extensions of the SM, whose phenomenology has been widely studied over the years [2, 3, 4] . In particular, 2-Higgs Doublet Models (2HDMs), in both Supersymmetry (SUSY) and non-SUSY versions [5, 6] , can be considered as a prototype of a Higgs sector that includes a charged Higgs boson (H ± ). It is expected that the LHC will allow us to test the mechanism of Electro-Weak Symmetry Breaking (EWSB) and, in particular, to probe the properties of charged Higgs bosons, which represent a unique probe of a weakly-interacting theory, as is the case of the Minimal Supersymmetric Standard Model (MSSM) [5] and general 2HDMs of Type I, II, III and IV (2HDM-I, 2HDM-I, 2HDM-III and 2HDM-IV) [7] , or whether strongly-interacting scenarios are instead realized, like in the old Technicolor models or similar ones discussed more recently [8] . Ultimately, while many analyses in this direction can be carried out at the LHC, it will be a future International Linear Collider (ILC) [9] or Compact LInear Collider (CLIC) [10] which will have the definite word about exactly which mechanism of mass generation and which realization of it occurs in Nature.
The 2HDM-II has been quite attractive to date, in part because it coincides with the Higgs sector of the MSSM, wherein each Higgs doublet couples to the u-or d-type fermions separately 1 .
However, this is only valid at tree-level [12] . When radiative effects are included, it turns out that the MSSM Higgs sector corresponds to the most general version of the 2HDM, namely the 2HDM-III, whereby both Higgs fields couple to both quarks and leptons. Thus, we can consider the 2HDM-III as a generic description of physics at a higher scale (of order TeV or maybe even higher), whose low energy imprints are reflected in the Yukawa coupling structure. With this idea in mind, some of us have presented a detailed study of the 2HDM-III Yukawa Lagrangian [13] , under the assumption of a specific texture pattern [14] , which generalizes the original model of Ref. [15] .
Phenomenological implications of this model for the neutral Higgs sector, including Lepton Flavour
Violation (LFV) and/or Flavour Changing Neutral Currents (FCNCs) have been presented in a previous work [16] . Here, we are interested in extending such an approach to investigate charged Higgs boson phenomenology: namely, we want to study the implications of this Yukawa texture for the charged Higgs boson properties (masses and couplings) and discuss in detail the resulting pattern of charged Higgs boson decays and main production reactions at the LHC.
Decays of charged Higgs bosons have been studied in the literature, including the radiative modes W ± γ, W ± Z 0 [17] , mostly within the context of the 2HDM-II or its SUSY incarnation (i.e., the MSSM), but also by using an effective Lagrangian extension of the 2HDM [18] and, more recently, within an extension of the MSSM with one Complex Higgs Triplet (MSSM+1CHT) [19, 20] . Charged Higgs boson production at hadron colliders was studied long ago [21] and, more recently, systematic calculations of production processes at the LHC have been presented [22] .
Current bounds on the mass of a charged Higgs boson have been obtained at Tevatron, by studying the top decay t → b H + , which already eliminates large regions of the parameter space [23] , whereas LEP2 bounds imply that, approximately, m H + > 80 GeV [24, 25] , rather model independently. Concerning theoretical limits, tree-level unitarity bounds on the 2HDM Higgs masses have been studied in generic 2HDMs and in particular an upper limit for the charged Higgs mass of 800 GeV or so can be obtained, according to the results of Ref. [26] . This paper is organized as follows. In section II, we discuss the Higgs-Yukawa sector of the 2HDM-III, in particular, we derive the expressions for the charged Higgs boson couplings to heavy fermions. Then, in section III, we derive the expressions for the decays H + → f ifj and numerical results are presented for some 2HDM-III scenarios, defined for phenomenological purposes. A discussion of the main production mechanisms at the LHC is presented in section IV. These include the top decay t → b H + as well as s-channel production of charged Higgs bosons through cb(cb)-fusion [27] and the multi-body more qq, gg → tbH − + c.c. (charge conjugated). These mechanisms depend crucially on the parameters of the underlying model and large deviations should be expected in the 2HDM-III with respect to the 2HDM-II. Actual LHC event rates are given in section V.
Finally, we summarize our results and present the conclusions in section VI. Notice that in carrying out this plan, unlike other references [28, 29] , where the 2HDM-II and the 2HDM-III appear as different structures, we shall consider here that, under certain limits, the 2HDM-III reduces to the 2HDM-II and, therefore, that the properties of the charged Higgs bosons change continuously from one model to the other.
II. THE CHARGED HIGGS BOSON LAGRANGIAN AND THE FERMIONIC

COUPLINGS
We shall follow Refs. [13, 16] , where a specific four-zero texture has been implemented for the Yukawa matrices within the 2HDM-III. This allows one to express the couplings of the neutral and charged Higgs bosons in terms of the fermion masses, Cabibbo-Kobayashi-Maskawa (CKM) mixing angles and certain dimensionless parameters, which are to be bounded by current experimental constraints. Thus, in order to derive the interactions of the charged Higgs boson, the Yukawa Lagrangian is written as follows:
where Φ 1,2 = (φ + 1,2 , φ 0 1,2 ) T refer to the two Higgs doublets,Φ 1,2 = iσ 2 Φ * 1,2 , Q L denotes the lefthanded fermion doublet, u R and d R are the right-handed fermions singlets and, finally, Y u,d
1,2 denote the (3 × 3) Yukawa matrices. Similarly, one can write the corresponding Lagrangian for leptons.
After spontaneous EWSB and including the diagonalizing matrices for quarks and Higgs bosons 2 , the interactions of the charge Higgs boson H + with quark pairs acquire the following form:
where V CKM denotes the mixing matrices of the quark sector (and similarly for the leptons). The term proportional to δ ij corresponds to the contribution that would arise within the 2HDM-II, while the terms proportional toỸ d 2 andỸ u 1 denote the new contributions from the 2HDM-III. These contributions, depend on the rotated matrices:
and n = 2 when q = d ), where O q is the diagonalizing matrix, while P q includes the phases of the Yukawa matrix. In order to evaluateỸ q n we shall consider that all Yukawa matrices have the Hermitian four-zero texture form [14] , and the quark masses have the same form, which are given by:
This is called a four-zero texture because one assumes that the Yukawa matrices are Hermitian, therefore each u and d type Yukawa matrix contains two independent zeros. According to current analyzes this type of texture satisfies the experimental constraints and at the same time it permits to derive analytical expressions for the Higgs boson fermion couplings.
To diagonalize M q , we use the matrices O q and P q , in the following way [14] :
Then, one can derive a better approximation for the product
In order to perform our phenomenological study, we find it convenient to rewrite the Lagrangian given in Eq. (2) in terms of the coefficients [χ q n ] ij , as follows: 
where S ij and P ij are defined as:
As it was discussed in Ref. [13] , most low-energy processes imply weak bounds on the coefficients χ q ij , which turn out to be of O(1). However, some important constraints on tan β have started to appear, based on B-physics [30] . In order to discuss these results we find convenient to generalize the notation of Ref. [31] and define the couplingsū i d j H + and u idj H − in terms of the matrices X ij , Y ij and Z ij (for leptons). In our case these matrices are given by:
where X lj and Y il are related with S ij and P ij defined in the Eq. (8) as follows:
The 33 elements of these matrices reduce to the expressions for the parameters X,Y,Z (= Other constraints on the charged Higgs mass and tan β, based on ∆a µ , the ρ parameter, as well as B-decays into the tau lepton, can be obtained [34, 35] . For instance, as can be read from Ref. [36] , one has that the decay B → τ ν, implies a constraint such that for m H + = 200 (300) GeV, values of tan β less than about 30 (50) are still allowed, within MSSM or 2HDM-III: However, these constraints can only be taken as estimates, as it is likely that they would be modified for 2HDM-III. In summary, we find that low energy constraints still allow to haveχ 
III. DECAYS OF THE CHARGED HIGGS BOSON
Let us now discuss the decay modes of the charged Higgs boson within our model. Hereafter, we shall refer to four benchmark scenarios, namely.
We have performed the numerical analysis of charged Higgs boson decays by taking tan β = 0.1, 1, 15, 70 and varying the charged Higgs boson mass within the interval 100 GeV ≤ m H ± ≤ 1000 GeV, further fixing m h 0 = 120 GeV, m A 0 = 300 GeV and the mixing angle at α = π/2.
The condition
in the frame of the 2HDM-II implies
However, in the 2HDM-III we have that
, we have checked numerically that this leads to 0.08 < tan β < 200 when |χ u 33 | ≈ 1 and 0.3 < tan β < 130 as long as |χ u 33 | → 0 recovering the result for the case of the 2HDM-II [7, 38] . In this sense, if we consider the constraints imposed by the perturbativity bound, a portion of the low tan β appearing in some graphs would be excluded. However, we have decided to keep that range both to show the behaviour of the quantities of interest, and also because we have to keep in mind that such criteria (perturbativity) should be taken as an order of magnitude constraint.
The expressions for the charged Higgs boson decay widths H + → u idj are of the form:
where λ is the usual kinematic factor λ(a, b, c) = (a − b − c) 2 − 4bc. When we replaceχ ud → 0, the formulae of the decays width become those of the 2HDM-II: see, e.g., Ref. is also very different from the 2HDM-II and becomes an interesting phenomenological consequence of the 2HDM-III. We can also observe that, for m H + > 180 GeV, the decay mode tb is dominant (as in the 2HDM-II). Now, from Fig. 2(b) , where tan β = 1, we find that the dominant decay mode is into τ + ν τ for the range m H + < 175 GeV, again for 175 GeV < m H + < 180 GeV the mode ts is the leading one, but for 180 GeV< m H + < 600 GeV, the decay channel W + h 0 becomes relevant, whereas for the range 600 GeV < m H + the mode W + A 0 is dominant. It is convenient to mention that this sub-scenario is special for the mode tb, because its decay width is zero at the tree-level, since the CKM contribution is canceled exactly with the terms of the four-zero texture implemented for the Yukawa coupling of the 2HDM-III. Then, see 
GeV< m H + < 400 GeV, the decay channel tb becomes the leading one, whereas for the range 400
GeV < m H + , the mode W + h 0 is again dominant.
Scenario B. In Fig. 3 , we present the BRs of the channels H + → tb, cb, ts, τ Fig. 3(a) , we observe that for tan β = 0.1, when m H + < 175 GeV, the dominant decay of the charged Higgs boson is the mode cb, with BR(H + i → cb) ≈ 1. When 175 GeV < m H + < 180 GeV the mode ts is important and for m H + > 180 GeV the decay mode tb becomes the leading one. From Fig. 3(b) , we see that, for tan β = 1, the dominant decay mode is now into τ + ν τ for m H + < 175 GeV, while in the range 175 GeV < m H + < 180 GeV the mode ts is relevant. For 180 GeV< m H + < 500 GeV the decay channel tb becomes the leading one, whereas for the range 500 GeV < m H + the mode W + A 0 is dominant. From Fig. 3 Fig. 4(c) , the relevant decay channels are: τ + ν τ in the range m H + < 180 GeV, tb when 180 GeV < m H + < 300 GeV, W + h 0 for 300 GeV < m H + . In Fig. 4(d) , for tan β = 70, we observe that τ + ν τ dominates when m H + < 180 GeV, but when 180
GeV < m H + < 900 GeV the mode tb is the leading one, whereas for 900 GeV < m H + the mode we obtain that, when m H + < 230 GeV, the mode τ + ν τ becomes the most important one but, for 230 GeV < m H + < 800 GeV, the channel tb is the leading one, whereas, for 800 GeV < m H + , the mode W + h 0 is the dominant one. In order to cover further the Higgs sector in our analysis, it is appropriate to also mention how the previous results change with m h 0 , m A 0 and α. Regarding the former two, clearly, the lather the neutral Higgs boson mass the later the corresponding H ± decay channel will onset.
Regarding the latter, we adopted two further choices, α = β and 0, in all scenarios previously studied. In general the behavior of the decay modes of the charged Higgs boson is similar to the cases presented above, except for the decay channel W h 0 . For α = 0, this mode has BR < 10 −3 when tan β is large. However, for tan β < 1, it becomes the dominant one. In the case α = β, the decay channel W h 0 can be dominant with a BR that could be O(1).
As a general lesson from this section, and distinctive features of our 2HDM-III, we can see that both decay modes W + h 0 and cb become very relevant phenomenologically, effectively of O (1) for some of the scenarios considered. Therefore, we want to study next the general behaviour of shows up clearly in the pattern of charged Higgs boson decays, which can be very different from the 2HDM-II case and thus enrich the possibilities to search for H ± states at current (Tevatron) and future (LHC, ILC/CLIC) machines.
IV. CHARGED HIGGS BOSON PRODUCTION AT LHC
The production of charged Higgs bosons at hadron colliders has been evaluated in early [21] (also for the Superconducting Super Collider, SSC) and more recent [22] (for the LHC) literature, mainly for the 2HDM-II and its SUSY realization (i.e., the MSSM). In these two scenarios, when kinematically allowed, the top quark decay channel t → bH + is the dominant H ± production mechanism. Instead, above the threshold for such a decay, the dominant H ± production reaction is gluon-gluon fusion into a 3-body final state, i.e., gg → tbH ±4 . Both processes depend on the coupling H − tb and are therefore sensitive to the modifications that arise in the 2HDM-III for this vertex. However, detection of the final state will depend on the charged Higgs boson decay mode, which could include a complicated final state, that could in turn be difficult to reconstruct. For these reasons, it is very important to look for other production channels, which may be easier to reconstruct. In this regard, the s-channel production of charged Higgs bosons, through the mechanism of cb-fusion, could help to make more viable the detection of several charged Higgs boson decay channels [27] .
Here, we shall evaluate the predictions of the 2HDM-III for the t → bH + (and sH + ) decay rate plus the cb-as well as the gg-fusion mechanisms (hereafter, referred to as 'direct' and 'indirect'
H ± production, respectively).
A. The decays t → H
We shall discuss here the charged Higgs boson interactions with heavy quarks (t, b, c, s) and their implications for charged Higgs boson production through top quark decays. In order to study the top quark BRs, besides the SM decay mode t → bW + , we need to consider both decays t → bH + and t → sH + , because these modes could both be important for several parameter configurations within our model. The decay width of these modes takes the following form:
where λ is the usual kinematic factor λ(a, b, c) = (a − b − c) 2 − 4bc, j = 2 for the mode sH + and j = 3 for the mode bH + . Furthermore, we shall neglect the decay width for the light fermion generations. If one takesχ i,j → 0, the formulae for the decay width reduce to the 2HDM-II case:
see, e.g., [2] .
We have explored several theoretically allowed regions within our scenario, which are constrained by using the bounds on the BR(t → bH + ). In the so-called "tauonic Higgs model" [23] , the decay mode (H + → τ + ν τ ) dominates the charged Higgs boson decay width, and BR(t → bH + ) is constrained to be less than 0.4 at 95 % C.L. [23] . However, if no assumption is made on the charged Higgs boson decay, BR(t → bH + ) is constrained to be less than 0.91 at 95 % C.L. [23] .
However, the combined LEP data exclude a charged Higgs boson with mass less than 79.3 GeV at 95 % C.L., a limit valid for an arbitrary BR(H + → τ + ν τ ) [25] . Thus, in order to perform our analysis, we need to discuss all the charged Higgs boson decays following the steps of our previous paper [19] . In the present section, we take all charged Higgs boson decays relevant for masses below that of the top quark, thus including the modes τ + ν τ , ts, cb, W + h 0 , W + A 0 . As usual, we refer to our four benchmark scenarios.
Scenario A. Remember that this scenario was defined by takingχ u ij = 1 andχ d ij = 1, while for tan β we considered the values tan β = 0.1, 1, 15, 70. In Fig. 9 we present plots of BR(t → b H + ) vs. models and a simplified version of the 2HDM-III. Then the SUSY case was discussed in [39] and [40] .
Here we perform a detailed study of this mechanism within the 2HDM-III, paying special attention to the effects induced by the assumed Yukawa texture on the charged Higgs boson couplings.
Defining the H ±q q ′ coupling here as C L
2 , we can express the total cross section for H + direct production at hadron colliders as [27] 
where The lines correspond to: tan β = 0.1, tan β = 1, tan β = 15, tan β = 70.
and
We notice here that Eqs. (15) and (16) reduce to the case of the 2HDM-II if one takes
and
In Fig. 13 , we present plots for the total cross section rates of process As we have already said, the expected integrated luminosity at LHC is of the order 10 5 pb −1 , hence we also conclude from this figure that in the framework of the 2HDM-II we obtain production rates for the charged Higgs boson via cb-fusion that may be detectable at LHC energies.
C. Indirect production of charged Higgs bosons at the LHC
We have found that, in some of the 2HDM-III scenarios envisaged here, light charged Higgs bosons could exist that have not been excluded by current experimental bounds, chiefly from LEP2
and Tevatron. Their discovery potential should therefore be studied in view of the upcoming LHC and we shall then turn our attention now to presenting the corresponding hadro-production cross sections via an indirect channel, i.e., other than as secondary products in (anti)top quark decays and via cb-fusion, considered previously.
As dealt with so far, if the charged Higgs boson mass m H ± satisfies m H ± < m t − m b , where m t is the top quark mass and m b the bottom quark mass, H ± particles could be produced in the decay of on-shell (i.e., Γ t → 0) top (anti-)quarks t → bH + and the c.c. process, the latter being in turn produced in pairs via qq annihilation and gg fusion. We denote such a H ± production channel as qq, gg → tt → tbH − + c.c. large top decay width (Γ t ≥ 1.5 GeV) and due to the additional diagrams which do not proceed via direct tt production but yield the same final state tbH − + c.c. [43, 44, 45] , charged Higgs bosons could also be produced at and beyond the kinematic top decay threshold. The importance of these effects in the so-called 'threshold' or 'transition' region (m H ± ≈ m t ) was emphasized in various Les Houches proceedings [46, 47] as well as in Refs. [42, 48, 49, 50] , so that the calculations of Refs. [43, 44] (based on the appropriate qq, gg → tbH ± description) are now implemented in HERWIG [51, 52, 53, 54] and PYTHIA [55, 56] . A comparison between the two generators was carried out in Ref. [48] . For any realistic simulation of H ± production with m H ± > ∼ m t , as can well be the case here, the use of either of these two implementations is of paramount importance.
Here, we use HERWIG version 6.510 in default configuration, by onsetting the subprocess IPROC = 3839, wherein we have overwritten the default MSSM/2HDM couplings and masses with those pertaining to the 2HDM-III: see Eqs. (7)- (8) . The production cross sections are found in Altogether, by comparing the qq, gg → tbH − i + c.c. cross sections herein with, e.g., those of the MSSM in [6] or the 2HDM in [49, 57] , it is clear that the 2HDM-III rates can be very large and thus the discovery potential in ATLAS and CMS can be substantial, particularly for a very light H ± , which may pertain to our 2HDM-III but not the MSSM or 2HDM-II. However, it is only by combining the production rates of this section with the decay ones of the previous ones that actual event numbers at the LHC can be predicted.
V. EVENT RATES OF CHARGED HIGGS BOSONS AT THE LHC
To illustrate the type of charged Higgs signatures that have the potential to be detectable at the LHC in the 2HDM-III, we show in Tabs. I and II the event rates of charged Higgs boson through the channels qq, gg → tbH − i + c.c. and cb → H + + c.c., alongside the corresponding production cross sections (σ's) and relevant BRs, for a combination of masses, tan β and specific 2HDM-III parameters amongst those used in the previous sections (assuming m h 0 = 120 GeV, m A 0 = 300 GeV and the mixing angle at α = π/2 throughout). In particular, we focus on those cases where the charged Higgs boson mass is above the threshold for t → bH + , for two reasons. On the one hand, the scope of the LHC in accessing t → bH + decays has been established in a rather model independent way. On the other hand, we have dealt at length with the corresponding BRs in section III. (As default, we also assume an integrated luminosity of 10 5 pb −1 .)
To illustrate these results, let us comment on one case within each scenario. From Table I , we can see that for Scenario A, with (χ u ij = 1,χ d ij = 1) and tan β = 15, we have that the H ± is heavier than m t −m b , as we take a mass m H + = 400, thus precluding top decay contributions, so that in this case σ(pp → tbH + ) ≈ 2.2×10 −1 pb, while the dominant decays are H + → tb, τ + ν τ W + h 0 , W + A 0 which give a number of events of 7040, 46, 13860, 374, respectively. In this case the most promising signal is H + → W + h 0 . However, when tan β = 70 we have that all event rates increase substantially.
Here, the signal H + → W + h 0 is still the most important with an event rate of 15480.
Then, for Scenario B (χ u ij = 0.1,χ d ij = 1), we have that H ± is again above the threshold for t → H + b. So, for the declared values of the relevant parameters, we take a charged Higgs boson mass m H + = 600 for tan β = 1 and tan β = 70, respectively. In such a case the decay H + → W + h 0 can reach significant numbers for the LHC. We obtain a number of events of 3960 and 2703, respectively. The other decay that has a large BR is H + → W + A 0 , and in these cases the number of events ranges over 1200-3500.
Next, we discuss the Scenario C (χ u ij = 1,χ d ij = 0.1) for tan β = 15. Here, we obtain that the signals H + → tb and W + h 0 are the most relevant ones, with a number of events about 34560 and 26240, respectively.
Finally, for scenario D (χ u ij = 0.11,χ d ij = 0.1) the dominant decays are H + → tb, τ + ν τ and W + h 0 , which give a spectacular number of events: 269800, 68400 and 34200, respectively. Here, we have set tan β = 70.
All these rates correspond to the case of indirect production. The contribution due to direct production is in fact subleading, especially at large m H ± values. Nonetheless, in some benchmark cases, they could represent a sizable addition to the signal event rates. This is especially the case for Scenario A with tan β = 15 or 70 and Scenario C with tan β = 15. In general though, also considering the absence of an accompanying trigger alongside the H ± ,i.e. for instance a top quark produced in gb → H − t could help to identify the signal. Thus, we expect that the impact of cb-fusion at the LHC will be more marginal that that of gg-fusion for large Higgs masses, in fact, at times even smaller that the contribution from qq-annihilation. 
VI. CONCLUSIONS
We have discussed the implications of assuming a four-zero Yukawa texture for the properties of the charged Higgs boson, within the context of a 2HDM-III. In particular, we have presented a detailed discussion of the charged Higgs boson couplings to heavy fermions and the resulting pattern for its decays. The latter clearly reflect the different coupling structure of the 2HDM-III, e.g., with respect to the 2HDM-II, so that one has at disposal more possibilities to search for H ± states at current and future colliders, ideally enabling one to distinguish between different Higgs models of EWSB. We have then concentrated our analysis to the case of the LHC and showed that the production rates of charged Higgs bosons at the LHC is sensitive to the modifications of the Higgs boson couplings. We have done so by evaluating 2HDM-III effects on the top decay t → bH + as well as in the s-channel production of H ± through cb-fusion and the multibody final state induced by gg-fusion and qq-annihilation. Finally, we have determined the number of events for the most promising LHC signatures of a H ± belonging to a 2HDM-III, for both cb → H + + c.c.
and qq →tbH + + c.c. scatterings (the latter affording larger rates than the former). Armed with these results, we are now in a position to carry out a detailed study of signal and background rates, in order to determine the precise detectability level of each signature. However, this is beyond the scope of present work and will be the subject of a future publication.

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