Département de Mathématiques, Faculté des Sciences et Techniques B.P 416

We study the complete one loop contribution to $H^\pm\to W^\pm V$, $V= Z, \gamma$, in the Minimal Supersymmetric Standard Model (MSSM). We evaluate the MSSM contributions taking into account $B\to X_s\gamma$ constraint as well as experimental constraints on the MSSM parameters. In the MSSM, we found that in the intermediate range of $\tan\beta \la 10$ and for large $A_t$ and large $\mu$, where lightest stop becomes very light and hence squarks contribution is not decoupling, the branching ratio of $H^\pm \to W^{\pm} Z$ can be of the order $10^{-3}$ while the branching ratio of $H^\pm \to W^{\pm} \gamma$ is of the order $10^{-5}$. We also study the effects of the CP violating phases of Soft SUSY parameters and found that they can modify the branching ratio by about one order of magnitude.


Introduction
The Standard Model (SM) of electroweak interactions is very successful in explaining all experimental data available till now. The cornerstone of the SM, the electroweak symmetry breaking mechanism, still has to be established and the Higgs boson has to be discovered. The main goals of future colliders such as LHC and ILC is to study the scalar sector of the SM. Moreover, the problematic scalar sector of the SM can be enlarged and some simple extensions such as the Minimal Supersymmetric Standard Model (MSSM) and the two Higgs doublet model (2HDM) [1,2] are intensively studied. Both in the 2HDM and MSSM the electroweak symmetry breaking is generated by 2 Higgs doublets fields Φ 1 and Φ 2 . After electroweak symmetry breaking we are left with 5 physical Higgs particles (2 charged Higgs H ± , 2 CP-even H 0 , h 0 and one CP-odd A 0 ). The charged Higgs H ± , because of its electrical charge, is noticeably different from the other SM or 2HDM/MSSM Higgs particles, its discovery would be a clear evidence of physics beyond the SM. In this study, our concerns is charged Higgs decays H ± → W ± V , V = Z, γ , we will first review the production mechanisms of charged Higgs.
The charged Higgs can be copiously produced both at hadrons and e + e − colliders. In hadronic machines, the charged Higgs bosons can be produced in many channels: (i) The production of tt pairs may offer a source of charged Higgs production. If kinematically allowed m H ± m t , the top quark can decay to H +b , competing with the SM decay t → W + b. This mechanism can provide a larger production rate of charged Higgs and offers a much cleaner signature than that of direct production.
(iii) Single charged Higgs production in association with W ± gauge boson via gg → W ± H ∓ or bb → W ± H ∓ [4] and also single charged Higgs production in association with A 0 boson via qq, gg → A 0 H ∓ [5].
At e + e − colliders, the simplest way to get a charged Higgs is through H ± pair production. Such studies have been already undertaken at tree-level [7] and one-loop orders [8] and shown that e + e − machines will offer a clean environment and in that sense a higher mass reach.
Experimentally, the null-searches from L3 Collaborations at LEP-II derive the lower limit of about m H ± 80 GeV [9], a limit which applies to all models (2HDM or MSSM) in which BR(H ± → τ ν τ ) + BR(H ± → cs) = 1. DELPHI has also carried out search for H ± → A 0 W ±1 topologies in the context of 2HDM type I and derive the lower limit of about m H ± 76 GeV [10]. Recently and for relatively small tan β 1 and for a specific SUSY spectrum, CDF Run II can excluded a charged Higgs mass in the range 80 < m H ± < 160 GeV [14]. While for intermediate range of tan β CDF has no limit. If the charged Higgs decay exclusively toτ ν, the BR(t → H + b) is constrained to be less than 0.4 at 95% C.L. On the other hand if no assumption is made on charged Higgs decay, the BR(t → H + b) is constrained to be less than 0.91 at 95% C.L.
At the LHC, the detection of light charged Higgs boson with m H ± m t is straightforward from top production followed by the decay t → bH + . 2 Such light charged Higgs (m H ± m t ) can be detected also for any tan β in the τ ν decay which is indeed the dominant decay mode [15]. However, for heavy charged Higgs masses m H ± m t which decay predominantly to tb, the search is rather difficult due to large irreducible and reducible backgrounds associated with H + → tb decay. However, it has been demonstrated in [16] that the H + → tb signature can lead to a visible signal at LHC provided that the charged Higgs mass below 600 GeV and tan β is either below 1.5 or above 40. Ref. [17], proposed H ± → τ ν as an alternative decay mode to detect a heavy charged Higgs, even if such decay is suppressed for heavy charged Higgs it has the advantage being more clean than H + → tb.
An other alternative discovery channel for heavy charged Higgs is its decay to charged gauge boson and lightest CPeven Higgs: H ± → W ± h 0 , followed by the dominant decay of h 0 to bb [18]. Since the branching ratio of H ± → W ± h 0 is suppressed for High tan β, this channel could lead to charged Higgs discovery only for low tan β where the branching ratio of H ± → W ± h 0 is sizeable. In MSSM, at tree level, the coupling H ± → W ± γ is absent because of electromagnetic gauge invariance U(1) em . While the absence of H ± → W ± Z is due to the isospin symmetry of the kinetic Lagrangian of the Higgs fields [19]. Therefore, decays modes like H ± → W ± γ , H ± → W ± Z are mediated at one loop level and then are expected to be loop suppressed [20][21][22][23][24]. We emphasize here that it is possible to construct models with an even larger scalar sector than 2 Higgs doublets, one of the most popular being the Higgs triplet model (HTM) [25]. A noteworthy difference between 2HDM and HTM is that the HTM contains a tree level ZW ± H ∓ coupling.
Motivated by the fact that there is no detailed study about H ± → W ± V , V = Z, γ , in the framework of MSSM in the literature which take into account left-right squarks mixing, b → sγ and other electroweak and experimental constraints. 1 Note that in the 2HDM it may be possible that the decay channel H ± → W ± A 0 is open and even dominate over τ ν mode for m H ± m t [11][12][13]. 2 Note that at Tevatron run II, the charged Higgs is also searched in top decay [14].
We would like to reconsider and update the existing works [20][21][22][23][24] on the charged Higgs boson decays into a pair of gauge boson: H ± → W ± γ, W ± Z both in 2HDM and MSSM with and without CP violating phases. Although these decays are rare processes, loop or/and threshold effects can give a substantial effect. Moreover, once worked out, any experimental deviation from the results within such a model should bring some fruitful information on the new physics and allow to distinguish between models. We would like to mention also that, those channels have a very clear signature and might emerge easily at future colliders. For instance, if H ± → W ± Z is enhanced enough, this decay may lead to three leptons final state if both W and Z decay leptonically and that would be the corresponding golden mode for charged Higgs boson. Charged Higgs decays: H ± → W ± γ, W ± Z, have received much more attention in the literature. H ± → W ± Z has been studied first in the MSSM in [20]. Ref. [21] has considered both H ± → W ± γ and H ± → W ± Z in the MSSM and show that the rate of H ± → W ± γ is very small while the rate of H ± → W ± Z can be enhanced by heavy fermions particles in the loops. The fourth generation contribution was given as an example. Although the squarks contribution has been considered in Ref. [21]. Left-right squarks mixing which could give substantial enhancement has been neglected. In contrast to Ref. [21] which argue that the squarks contributions decouple, we will show that there is non-decoupling effects originating from squarks contributions at large A t and large μ limit. H ± → W ± γ was also studied in [22] within the MSSM, but the pure SUSY contribution from charginos, neuralinos and squarks has been neglected. Later on, Ref. [23] studied the possibility of enhancing H ± → W ± Z by the non-decoupling effect of the heavy Higgs bosons in the context of 2HDM, substantial enhancement was found [23]. Recently, H ± → W ± γ was also studied in 2HDM type II [24]. All the above studies has been carried out either in unitary gauge [20,21] or in the nonlinear R ξ -gauge [24]. The analysis of [22] and [23] have been performed in 't Hooft-Feynman gauge without any renormalization scheme. It has been checked in [22,23] that the sum of all Feynman diagrams: Vertex, tadpoles and vector bosonscalar mixing turns out to be ultraviolet finite.
In the present study, we will still use 't Hooft-Feynman gauge to do the computation. However, the amplitudes of H ± → W ± γ and H ± → W ± Z are absent at the tree level, complications like tadpoles contributions and vector bosonscalar mixing require a careful treatment of renormalization. We adopt hereafter the on-shell renormalization scheme developed in [26].
The Letter is organized as follows. In Section 2, we describe our calculations and the one-loop renormalization scheme we will use for H ± → W ± Z and H ± → W ± γ . In Section 3, we present our numerical results and discussions, and Section 4 contains our conclusions.

Charged Higgs decay: H ± → W ± V
As we have seen in the previous section, in MSSM, at tree level, the coupling H ± → W ± γ and H ± → W ± Z do not exist.
They are generated at one loop level and then are expected to be loop suppressed [20][21][22][23][24]. Hereafter, we will give the general structure of such one loop couplings and discuss the renormalization scheme introduced to deal with tadpoles and vector boson scalar boson mixing.

One loop amplitude H
The amplitude M for a scalar decaying to two gauges bosons V 1 and V 2 can be written as where V i are the polarization vectors of the V i . According to Lorenz invariance, the general structure of the are form factors, and μνρσ is the totally antisymmetric tensor. The form factor F 1 has dimension 2 while the other are dimensionless.
For [21]. This means that only F 2 and F 3 will contribute to the decay H ± → W ± γ . In case of H ± → W ± Z, there is no such constraint on form factors.
In terms of an effective Lagrangian analysis, from gauge invariance requirement we can write: μ V μ is dimension three and the last two operators H ± F μν V F W μν and μνρσ H ± F μν V F Wρσ are dimension five. One conclude that g 2,3 (respectively, g 1 ) must be of the form g(R)/M (respectively, Mg(R)) with M a heavy scale in MSSM, g(R) a dimensionless function and R is a ratio of some internal masses of the model under studies. Therefore, it is expected that in case of H ± → W ± Z decay, F 1 will grow quadratically with internal top quark mass while F 2,3 will have only logarithmic dependence [21]. 3 A contrario, for H ± → W ± γ decay, the electromagnetic gauge invariance relates F 1 and F 2 and then the amplitude of H ± → W ± γ will not grows quadratically with internal masses. One expect that the decay H ± → W ± γ is less enhanced compared to H ± → W ± Z.

On-shell renormalization
We have evaluated the one-loop induced process H ± → W ± V in the 't Hooft-Feynman gauge using dimensional regularization. Since we are interested in the charged Higgs decay to SM particles like W ± Z and W ± γ , at one-loop level, dimensional regularization will give the same result as dimensional reduction. In fact, we have checked numerically that the dimensional regularization and dimensional reduction gives the same result.
Note that the mixing H ± -W ± (Figs. 1.12, 1.13, 1.14) vanishes for an on-shell transverse W gauge boson. There is no contribution from the W ± -G ∓ mixing because γ G ± H ∓ and ZG ± H ∓ vertices are absent at the tree level. All the Feynman diagrams have been generated and computed using FeynArts and FormCalc [27] packages. We also used the Fortran FFpackage [28] in the numerical analysis.
Although the amplitude for our process is absent at the tree level, complications like tadpole contributions and vector boson-scalar mixing require a careful treatment of renormalization. We adopt, hereafter, the on-shell renormalization scheme of [29], for the Higgs sector, which is an extension of the on-shell scheme in [30]. In this scheme, field renormalization is performed in the manifest-symmetric version of the Lagrangian. A field renormalization constant Z Φ 1,2 is assigned to each Higgs doublet Φ 1,2 . Following the same approach adopted in [26], the Higgs fields and vacuum expectation values v i are renormalized as follows: With these substitutions in the scalar covariant derivative Lagrangian of the Higgs fields (in the convention of [1]), followed by expanding the renormalization constants Z i = 1 + δZ i to the one-loop order, we obtain all the counter-terms relevant for our process: where k denotes the momentum of the incoming W ± and Denoting the one particle irreducible (1PI) two point function for W ± H ± (respectively, G ± H ± ) mixing by ±ik μ Σ W ± H ± (k 2 ) (respectively, iΣ G ± H ± (k 2 )) where k is the momentum of the incoming W ± (respectively, G ± ), and H ± is outgoing. The renormalized mixing will be denoted byΣ .
In the on-shell scheme, we will use the following renormalization conditions: • The renormalized tadpoles, i.e. the sum of tadpole diagrams T h,H and tadpole counter-terms δ h,H vanish: These conditions guarantee that v 1,2 appearing in the renormalized Lagrangian L R are located at the minimum of the one-loop potential. • The real part of the renormalized non-diagonal self-energŷ Σ H ± W ± (k 2 ) vanishes for an on-shell charged Higgs boson: H ± = 0. This renormalization condition determines the term Δ to be The last renormalization condition is sufficient to discard the real part of the H ± -G ± mixing contribution as well. Indeed, using the Slavnov-Taylor identity [31] (11) which is valid also for the renormalized quantities together with Eq. (9), it follows that (12) eΣ H ± G ± m 2 H ± = 0. In particular, the Feynman diagrams depicted in Fig. 1.9 will not contribute with the above renormalization conditions, being purely real valued.
To make the amplitude of Fig. 1 ultra violet finite we need to add the following counter-terms: Counter-terms for γ W ± H ∓ and ZW ± H ∓ vertices Fig. 2.2a, a counter-term for the W ± -H ∓ mixing Figs. 2.2b, 2.2d, and a counter-term for the G ± -H ∓ mixing Fig. 2.2c.

Numerics and discussions
In our numerical evaluations, we use the following experimental input quantities [32]: α −1 = 129, m Z , m W , m t , m b = 91.1875, 80.45, 174.3, 4.7 GeV. In the MSSM, we specify the free parameters that will be used as follow: (i) The MSSM Higgs sector is parameterized by the CP-odd mass m A 0 and tan β, taking into account one-loop radiative corrections from [33], and we assume tan β 3. (ii) The chargino-neutralino sector can be parameterized by the gaugino-mass terms M 1 , M 2 , and the higgsino-mass term μ. For simplification GUT relation M 1 ≈ M 2 /2 is assumed. (iii) Sfermions are characterized by a common soft-breaking sfermion mass M SUSY ≡M L = M R , μ the parameter and the soft trilinear couplings for third generation scalar fermions A t,b,τ . For simplicity, we will take When varying the MSSM parameters, we take into account also the following constraints: (i) The extra contributions to the δρ parameter from the Higgs scalars should not exceed the current limits from precision measurements [32]: |δρ| 0.003.  (ii) b → sγ constraint. The present world average for inclusive b → sγ rate is [32] We keep the B → X s γ branching ratio in the 3σ range of (2.1-4.5) × 10 −4 . The SM part of B → X s γ is calculated up to NLO using the expression given in [34]. While for the MSSM part, the Wilson coefficient C 7 and C 8 are included at LO in the framework of MSSM with CKM as the only source of flavor violation and are taken from [35]. (iii) We will assume that all SUSY particles Sfermions and charginos are heavier than about 100 GeV; for the light CP even Higgs we assume m h 0 98 GeV and tan β 3 [36].
As the experimental bound on m h 0 is concerned, care has to be taken. Since we are using only one-loop approximation for the Higgs spectrum, and as it is known, higher order corrections [37] may reduce the light CP-even Higgs mass in some cases. It may be possible that some parameter space points, shown in this analysis, which survive to the experimental limit m h 0 98 GeV with one loop calculation may disappear once the higher order correction to the Higgs spectrum are included.
The total width of the charged Higgs is computed at tree level from [2] without any QCD improvement for its fermionic decays H ± →f f . The SUSY channels like H + →f if j and H + →χ 0 iχ + j are included when kinematically allowed. In Fig. 3, we show branching ratio of H ± → W ± Z (left) and H ± → W ± γ (right) as a function of charged Higgs mass for tan β = 16 and 25. In those plots, we have shown both the pure 2HDM 4 and the full MSSM contribution. As it can be seen from those plots, both for H ± → W ± Z and H ± → W ± γ the 2HDM contribution is rather small. Once we include the  SUSY particles, we can see that the branching fraction get enhanced and can reach 10 −3 in case of H ± → W ± Z and 10 −5 in case of H ± → W ± γ . The source of this enhancement is mainly due to the presence of scalar fermion contribution in the loop which are amplified by threshold effects from the opening of the decay H ± →t ib * j . It turns out that the contribution of charginos neutralinos loops does not enhance the branching fraction significantly as compared to scalar fermions loops. The plots also show that, the branching fraction is more important for intermediate tan β = 16 and is slightly reduced for larger tan β = 25. This tan β dependence is shown in Fig. 4 both for H ± → W ± Z and H ± → W ± γ for three representative values of A t . It is obvious that the smallest is tan β the largest is the branching fraction. Increasing tan β from 5 to about 40 can reduce the branching fraction by about one or two order of magnitude. As one can see from those figures, the plots stops for tan β ≈ 24 for A t = 500 GeV, this is due to b → sγ constraint. For A t = 1400 GeV, only tan β ∈ [16,26] is allowed, the reason is that for tan β 16 the light stopt 1 becomes lighter than the experimental limit of 98 GeV and for tan β 26, the exper-imental limit on the light CP even Higgs h 0 is violated. As indicated above, we assume a conservative limit of m h 0 98 GeV rather than m h 0 114 GeV which should be used in the case of decoupling limit where ZZh 0 coupling mimic the SM one.
We also show a scatter plot Fig. 5 for H ± → W ± Z (left) and H ± → W ± γ (right) in (m H ± , tan β) plane for A t = −μ = 1 TeV, M SUSY = A t and M 2 = 175 GeV. As it can be seen from Fig. 5 there is only a small area for tan β 10 where the branching ratio of H ± → W ± Z can be in the range 10 −5 -10 −3 .
We now illustrate in Fig. 6 the branching fraction of H ± → W ± Z (left) and H ± → W ± γ (right) as a function of A t = A b = A τ = −μ for M SUSY = 500 GeV and M 2 = 200 GeV. Since b → sγ favor A t and μ to have opposite sign, we fix μ = −A t and in this sense also μ is varied when A t is varied. Both for H ± → W ± Z and H ± → W ± γ , the charginoneutralino contribution which is rather small decrease with μ = −A t , the largest is A t the smallest is chargino-neutralino contribution. As one can see from those figures, the plots stops for A t = 1.1 TeV and tan β = 3 because for larger A t δρ constraint is violated. For tan β = 1 and 20, the plots stops for the same reason.  In case of H ± → W ± Z, for A t 1 TeV it is the pure 2HDM contribution which dominate and that is why it is almost independent of A t while for large A t the branching ratio increase with A t . It is clear that the largest is A t the largest is the branching ratio which can be of the order of 10 −3 for H ± → W ± Z with tan β = 10. As we know from h 0 → γ γ and h 0 → γ Z in MSSM [38], the squarks contributions decouple except in the light stop mass and large A t limit [38]. In H ± → W ± V case, the same situation happen. As we can see from Fig. 6  We have also studied the effect of the MSSM CP violating phases on charged Higgs decays. Similar study has been done for the single charged Higgs production at hadron collider [39]. It is well known that the presence of large SUSY CP violating phases can give contributions to electric dipole moments of the electron and neutron (EDM) which exceed the experimental upper bounds. In a variety of SUSY models such phases turn out to be severely constrained by such constraints, i.e. Arg(μ) < (10 −2 ) for a SUSY mass scale of the order of few hundred GeV [40]. For H ± → W ± Z and H ± → W ± γ decays which are sensitive to MSSM CP violating phases through squarks and charginos-neutralinos contributions, it turns out that the effect of MSSM CP violating phases is important and can enhance the rate by about one order of magnitude. For illustration we show in Fig. 7 the effect of A t,b,τ CP violating phases for M SUSY = 500 GeV, A t,b,τ = −μ = 1 TeV and M 2 = 150 GeV. For simplicity, we assume that μ is real. As it is clear, the CP phase of A t,b,τ can enhance the rates of both H ± → W ± Z, γ by more than an order of magnitude. The observed cuts in the plot are due to b → sγ constraint. The CP violating phases can lead to CP-violating rate asymmetry of H ± decays, those issues are going to be addressed in an incoming paper [41].

Conclusion
In the framework of MSSM we have studied charged Higgs decays into a pair of gauge bosons namely: H ± → W ± Z and H ± → W ± γ . In the MSSM we have also studied the effects of MSSM CP violating phases. In contrast to previous studies, we have performed the calculation in the 't Hooft-Feynman gauge and used a renormalization prescription to deal with tadpoles, W ± -H ± and G ± -H ± mixing. The study has been carried out taking into account the experimental constraint on the ρ parameter, b → sγ constraint. Numerical results for the branching ratios have been presented. In the MSSM, we have shown that the branching ratio of H ± → W ± Z can reach 10 −3 in some cases while H ± → W ± γ never exceed 10 −5 . The effect of MSSM CP violating phases is also found to be important.
Those branching ratio of the order 10 −3 might provide an opportunity to search for a charged Higgs boson at the LHC through H ± → W ± Z.
At the end, we would like to mention that some effects shown in this study may be ruled out by the experimental bound on the light CP-even mass if we take into account 2-loop radiative corrections on the Higgs spectrum which have the tendency to reduce the light CP-even mass in some cases. On the other hand, the inclusion of high effects for H ± → W ± V like bottom and top quarks mass running as well as tan β resummation could affect the rate of H ± → W ± V .