Enhancement of the $H^\pm W^\mp Z$ vertex in the three scalar doublet model

We compute one-loop induced trilinear vertices with physical charged Higgs bosons $H^\pm$ and ordinary gauge bosons, i.e., $H^\pm W^\mp Z$ and $H^\pm W^\mp \gamma$, in the model with two active plus one inert scalar doublet fields under a $Z_2(\text{unbroken})\times \tilde{Z}_2(\text{softly-broken})$ symmetry. The $Z_2$ and $\tilde{Z}_2$ symmetries are introduced to guarantee the stability of a dark matter candidate and to forbid the flavour changing neutral current at the tree level, respectively. The dominant form factor $F_Z$ of the $H^\pm W^\mp Z$ vertex can be enhanced by non-decoupling effects of extra scalar boson loop contributions. We find that, in such a model, $|F_Z|^2$ can be one order of magnitude larger than that predicted in two Higgs doublet models under the constraints from vacuum stability, perturbative unitarity and the electroweak precision observables. In addition, the branching fraction of the $H^\pm \to W^\pm Z$ $(H^\pm \to W^\pm \gamma)$ mode can be of order 10~(1)\% level when the mass of $H^\pm$ is below the top quark mass. Such a light $H^\pm$ is allowed by the so-called Type-I and Type-X Yukawa interactions which appear under the classification of the $\tilde{Z}$ charge assignment of the quarks and leptons. We also calculate the cross sections for the processes $H^\pm \to W^\pm Z$ and $H^\pm \to W^\pm \gamma$ onset by the top quark decay $t\to H^\pm b$ and electroweak $H^\pm$ production at the LHC.


I. INTRODUCTION
From the above reasons, it is clear that the strength of the H ± W ∓ Z vertex measures the effects of the violation of the custodial symmetry in the model embedding it. Therefore, by measuring this vertex, we can indirectly observe such a new physics effect.
Feasibility studies to measure the H ± W ∓ Z vertex have been performed in Ref. [13] for the LHC and in Ref. [14] for future linear colliders.
In this paper, we calculate the magnitude of the H ± W ∓ V (V = Z, γ) vertices at the oneloop level in the 3-Higgs Doublet Model (3HDM), in which the Higgs sector is composed of two active (with a non-zero Vacuum Expectation Value (VEV)) and one inert (without a non-zero VEV) doublet scalar fields. In this model, the scalar bosons from the inert doublet field give an additional contribution to the H ± W ∓ V vertex with respect to the top/bottom quarks and scalar bosons from the active doublet loop contributions. As a phenomenological application, we also discuss how such new contributions change the decay branching fractions of the H ± → W ± Z and H ± → W ± γ modes and, consequently, the production cross sections involving these decay processes at the LHC. This paper is organized as follows. In Sec. II, we define the Lagrangian of the 3HDM, i.e., the scalar potential and the Yukawa interactions. In Sec. III, we introduce the form factors of the H ± W ∓ V vertices and discuss relationships between these form factors and effective operators. We then explain how to calculate these form factors at the one-loop level. In Sec. IV, we summarise various constraints on the parameters of our model. From the theoretical point of view, we consider vacuum stability and perturbative unitarity. As experimental constraints, we take into account the bounds from the Electro-Weak (EW) S, T and U parameters, the flavour experiments and direct searches for H ± states from LEP-II and the LHC Run-I. In Sec. V, we show numerical results for the form factors of the H ± W ± V vertices, branching fractions of H ± and their signal cross sections at the LHC. Our conclusion is given in Sec. VI. In Appendix, we present the full analytic expressions for the form factors of the H ± W ∓ V vertices.

II. THE MODEL
We give a brief review of the 3HDM 3 of which the Higgs sector is composed of two active and one inert isospin doublet scalar fields [15,16]. We represent the active doublets as Φ 1 and Φ 2 whereas the inert doublet as η. Such an inert nature can be realised by assuming an unbroken Z 2 (Z 2 ,Z 2 ) charge Mixing factor symmetry in the scalar potential, in which only η has an odd parity while all the other fields are assigned to be even. One of the important consequences of imposing such a Z 2 symmetry is that the lightest neutral scalar component in η can be a DM candidate, because it cannot decay into SM particles.
In addition to the Z 2 symmetry, we impose another Z 2 symmetry, denoted byZ 2 to distinguish it from the above one, which is required to forbid the Flavour Changing Neutral Current (FCNCs) at the tree level. This prescription is the same as that in the 2HDM [19]. For theZ 2 symmetry, we consider the softly-broken case, since avoidance of tree level FCNCs can already be achieved in this case. Under theZ 2 symmetry, four independent types of Yukawa interactions (Type-I, -II, -X and -Y) [20][21][22] are allowed depending on the assignment of theZ 2 charge to the SM fermions.
In Tab. I, we show the charge assignments required by the Z 2 andZ 2 symmetries for the three scalar doublets Φ 1 , Φ 2 and η and all the SM fermions, where L L (e R ) is the left (right)-handed lepton doublet (singlet) and Q L (u R , d R ) is the left (right)-handed quark doublet (up-type and down-type quark singlets). and ξ f in each type of Yukawa interactions are listed in Tab. I.
It is important to mention here that there is the so-called SM-like limit or alignment limit defined by sin(β − α) → 1 [23,24]. In this limit, all the h coupling constants to the SM particles become the same values as those of the SM values. In fact, the ratios of hff and hV V couplings in our model to those in the SM are respectively given as ξ f h given in Eq. (15) and sin(β − α).
where V µν V is written in terms of the following three dimensionless form factors: with p µ W and p µ V being the incoming momenta for W ± and V , respectively. For the case of V = γ, the Ward identity guarantees the following relation; From this relation, the form factor F γ is written as where we use p 2 W = m 2 W and (p W + p γ ) 2 = m 2 H ± . In our model, the H ± W ∓ V vertices do not appear at the tree level, just like in the 2HDM. This is clearly seen by introducing the so-called Higgs basis of the active scalar doublets defined as where with h ′ 1 = H cos(β −α)+h sin(β −α) and h ′ 2 = −H sin(β −α)+h cos(β −α). The kinetic Lagrangian for Φ 1 and Φ 2 is then rewritten as where D µ is the covariant derivative. Since the gauge-gauge-scalar type vertex is proportional to the Higgs VEV v, these vertices come from the |D µ Φ| 2 term as only Φ has a non-zero VEV.
However, the physical charged Higgs bosons H ± are contained in the |D µ Ψ| 2 term. Therefore, the H ± W ∓ Z vertex is absent at the tree level 4 . The above statement can be generalised to a model with N active doublet scalar fields. In that case, we can also define a base transformation similar to the one of Eq. (21). Regarding the H ± W ∓ γ vertex, it does not appear at tree level in any models based on the SU (2) L × U (1) Y → U (1) em gauge theory, because of the U (1) em invariance and the consequent Ward identity.
The form factors defined in Eq. (18) are introduced from the following effective Lagrangian [10,11]: where F µν W and F µν V are the field strength tensors for W ± and V , respectively. It can be seen that the coefficient f Z has mass dimension one whereas g H ± W V and h Z have mass dimension minus one. Hence, the coefficient f Z can be proportional to a squared mass (M 2 i ) of a particle running in the loop according to a dimensional analysis: where F is a dimensionless function. Typically, it is expressed by the logarithmic function of M 2 i . On the other hand, g Z and h Z can be expressed as where G is another dimensionless function of M 2 i . Therefore, only the coefficient f H ± W Z can be enhanced significantly due to the M 2 i dependence, so that the form factor F Z gives the dominant contribution to the H ± W ∓ Z vertex. In fact, it has been pointed out in Ref. [10] that the top/bottom loop contribution to the form factor F Z is proportional to m 2 t only, as m t ≫ m b . The origin of the quadratic dependence can be understood in terms of the Yukawa coupling H + tb, which is proportional to m t /v as in Eq. (14), and of another m t coming from the chirality flipped effect.
Similarly, the quadratic mass dependence appears in the extra Higgs boson loop contribution as discussed in Ref. [11]. This too can be understood, as the trilinear H ± SS ′ (S and S ′ being extra scalar bosons) couplings can be rewritten by squared masses of extra scalar bosons.
Another important reason for the appearance of a M 2 i dependence in F Z is in relation to a violation of the custodial SU (2) V symmetry. As it has been discussed in Ref. [11], the dimension three term in Eq. (24) comes from the following operator 5 where Φ = (Φ c , Φ) and Ψ = (Ψ c , Ψ) with Φ c = iσ 2 Φ * and Ψ c = iσ 2 Ψ * are the 2 × 2 representation form of the Higgs doublets. They are translated under the SU (2) L × SU (2) R symmetry by Φ → where U L and U R are respectively the SU (2) L and SU (2) R unitary transformation matrices. We can see that the operator given in Eq. (27) is not invariant under the SU (2) R transformation, so that this operator breaks the SU (2) R invariance. Since the custodial SU (2) V symmetry corresponds to the remaining symmetry after the EW symmetry breaking, i.e., SU (2) L × SU (2) R → SU (2) V and a violation of the SU (2) R symmetry means a violation of the Therefore, the quadratic mass dependence in F Z can be understood as a result of the custodial symmetry breaking. In fact, it has been known that the mass difference between the top and bottom quarks gives the violation of the custodial symmetry in the Yukawa sector. In addition, that between A and H ± also gives the violation of the custodial symmetry in the Higgs potential [26].
Since the top quark mass is already known by experiments, the top quark loop contribution to the H ± W ∓ Z vertex is determined by its mass 6 . In contrast, parameters in the scalar sector have not yet determined by experiments except for the Higgs boson mass of about 125 GeV, so that we can expect a sizable enhancement of the H ± W ∓ Z vertex from scalar boson loop effects in suitable regions of the 3HDM parameter space.
In the following, we discuss how we calculate the form factors of the H ± W ∓ V vertices. We can separately consider the one-loop contributions to the vertices from the 1PI diagrams and the counter terms as where X 1PI V and δX V are respectively the 1PI and the counter term contributions to the form factor X V (X = F, G and H). Their analytic expressions are given in App. A.
The counter term contributions are obtained as follows. First, we define the renormalized two point function for the W ± -H ± mixing aŝ where p µ is the incoming four momentum of H ± . The renormalised form factorΓ W H is given bŷ where δ GH is the counter term for the G ± -H ± mixing, and Γ 1PI W H is the 1PI diagram contribution to the W ± -H ± mixing. The analytic expression of Γ 1PI W H is given in App. A. The counter term is obtained by the shift of the charged NG boson field G ± : By imposing the on-shell renormalisation condition [27,28] we can determine the counter term We then obtain the counter term contribution to the H ± W ∓ V vertex as where s W = sin θ W and c W = cos θ W with θ W being the weak mixing angle. From Eqs. (33) and (34), δF V is given by We then obtain the finite results for the form factors of the H ± W ∓ Z and H ± W ∓ γ vertices. In the case of sin(β − α) = 1, m H = m A ≫ m H ± and m η H = m η A ≫ m η ± , we obtain where the first, second and third terms correspond to the contributions from t-b, active and inert scalar boson loops, respectively. From the above expression, we can clearly see the quadratic mass dependences m 2 t , m 2 A and m 2 η A . However, as it will be discussed in the next section, the case considered in the above, i.e., m H = m A ≫ m H ± and m η H = m η A ≫ m η ± also gives the similar quadratic dependence in the EW T parameter. Therefore, too large mass difference between H ± and A (with m H = m A ) and that between η ± and η A (with m η H = m η A ) are not allowed. Instead of taking the above case, we can consider the case with sin(β where the contribution to the T parameter from extra scalar boson loops is cancelled. We then obtain where N c = 3 is the color factor, and the function F is given by This function has the following asymptotic behavior: In this case, although the quadratic dependence m 2 A and m 2 η A disappears, there still remains their logarithmic dependence.

A. Vacuum stability
The stability condition for the Higgs potential is given by requiring that the Higgs potential is bounded from below in any direction of the scalar boson space. The necessary and sufficient condition to guarantee such a positivity of the potential has been derived in Ref. [16] as

B. Unitarity
Some combinations of scalar quartic couplings are constrained from perturbative unitarity. In the 3HDM, the s wave amplitude matrix for all the 2-to-2 body scalar boson elastic scatterings have been calculated in Ref. [29] in the high energy limit. We obtain the following independent eigenvalues or sub-matrices for the s wave amplitude matrix as We then require the following condition: where x i are the eigenvalues of X 1 , X 2 and X 3 .

C. S, T and U parameters
The EW oblique parameters S, T and U given in the case with sin(β assuming m A ≃ m H ± , and The general expression is given in Ref. [29]. From the global fit of the EW precision data, ∆S and ∆T are extracted by fixing ∆U = 0 as with the correlation coefficient of +0.91 [31]. In Fig. 1, we show the constraint from the S and T parameters on the m A -m η A plane. We take sin(β − α) = 1, m H = m A and m η ± = m η H = m A /2, which is also taken in the numerical results shown in Sec. V. In the left and right panel, m H ± is fixed to be 150 GeV and 200 GeV, respectively.
We can see that, for m η A ≃ m η ± , a magnitude of the mass splitting between A and H ± to be larger than about 75 GeV is excluded by the T parameter due to the quadratic dependence of the mass splitting shown in Eq. (55). In this case, the contribution to ∆S is almost zero as it is seen in Eq. (59). Conversely, in the case of m η A ≫ m η ± , the positive logarithmic contribution to ∆S appears as shown in Eq. (57) and a too large mass splitting between η A and η ± is excluded by ∆S.
However, the constraint from ∆S is getting milder when there is a positive contribution to ∆T , because of the positive correlation between ∆S and ∆T . Therefore, in order to have a large mass splitting between η A and η ± , which is required to obtain a significant contribution to the H ± W ∓ Z vertex, we need a mass splitting between A and H ± .

D. Flavour constraints
We can apply the same constraints from the B physics measurements as those in the 2HDM to our 3HDM, because of the same structure of the active sector. From the b → sγ process, the mass bound of m H ± 322 GeV is given at 95% confidence level (CL) in models with the Type-II and Type-Y Yukawa interactions with tan β 2 via the next-to-next-to-leading order calculation performed in Refs. [32,33]. This bound is getting stronger when a smaller value of tan β is considered. In models with Type-I and Type-X Yukawa interactions, the constraint from b → sγ is only important in the small tan β case. For instance, the lower limit on m H ± is given to be about 100, 200 and 800 GeV at 95% CL in the cases of tan β = 2.5, 2 and 1, respectively [33].
The B 0 -B 0 mixing also gives a bound on m H ± , especially for small tan β's. In the case of tan β = 1, m H ± 500 GeV is excluded at 95% CL in models with all the types of Yukawa interactions [34], which is stronger than the constraint from b → sγ for the Type-II and Type-Y cases. This bound becomes rapidly weaker when we consider tan β 1, e.g., for tan β = 1.5 (2), the limit is m H ± 300 (100) GeV at 95% CL.

E. Direct search at LEP II
At the LEP II experiment, charged Higgs bosons have been searched via the e + e − → Z * /γ * → H + H − process [35]. From the non-observation of a significant excess, the lower mass limit has been taken to be about 80 GeV at 95% CL under the assumption of BR(H ± → τ ± ν) + BR(H ± → cs) = 1. The slightly stronger bound m H ± 90 GeV can be obtained assuming BR(H ± → τ ± ν) = 1. (Right) Excluded parameter regions on the tan β-m H ± plane in the Type-I and Type-X 2HDMs/3HDMs.
Regions inside from each curve are excluded at 95% CL by the measurement of top decay t → H ± b → τ ± bν.
The solid and dashed curves are the results using the upper limit on BR(t → H + b)×BR(H + → τ + ν) to be 0.23% and 1.3%, respectively.

F. Direct search at LHC Run-I
At the LHC, H ± searches have been performed for the two cases: the low mass region m H ± < m t +m b and the high mass region m H ± > m t +m b . For the low mass case, the t → H + b decay is used as the H ± production mode and the full process pp → tt → bbH ± W ∓ with the H ± → τ ± ν decay has thus been analysed. Using the data obtained at √ s = 8 TeV after 19.5 fb −1 of the integrated luminosity, the upper limit on the product of branching ratios BR(t → H ± b)×BR(H ± → τ ± ν) has been obtained to be between 0.23% and 1.3% at 95% CL for m H ± in the range of 80 GeV to 160 GeV [36].
In the left panel of Fig. 2, the above product of branching ratios is shown as a function of tan β in the Type-I and Type-X 2HDMs. Because the light H ± scenario, i.e., m H ± < m t , in the Type-II and Type-Y 2HDMs has already been excluded by b → sγ data as explained in Sec. IV-D, we here only show the Type-I and Type-X cases. In the Type-X 2HDM, the product of the branching fractions is slightly larger than that in the Type-I 2HDM. This can be understood in such a way that in the Type-X 2HDM the branching fraction of H ± → τ ± ν is enhanced as tan β is increased, while it does not depend on tan β in the Type-I 2HDM. For example, BR(H + → τ + ν) can be almost 100% when tan β 3 in the Type-I 2HDM, but it is about 40% in the Type-I 2HDM. In contrast, the branching ratio of t → H + b is given by the same value in both Type-I and Type-X 2HDMs. Therefore, a bit stronger bound on tan β for a fixed value of m H ± is obtained in the Type-X 2HDM. For example, if we use the stronger bound for BR(t → H ± b)×BR(H ± → τ ± ν), i.e., 0.23%, tan β 6 (4) and 15 (10) are excluded for m H ± = 100 and 150 GeV in the Type-X (Type-I) 2HDM.
For the high H ± mass region, i.e., m H ± > m t , the production process gb → tH ± (i.e., H ±strahlung) can be used instead of the top quark decay 7 . The 95% CL upper limit on the cross section times branching ratio σ(pp → tH ± + X) × BR(H ± → τ ± ν) has been given to be between 0.76 pb and 4.5 fb in the range of m H ± =180 GeV to 1 TeV [36]. This limit gives an upper limit on tan β for a fixed value of m H ± in the 2HDMs. For example, tan β 50 (60) at m H ± = 200 (230) GeV can be excluded at 95% CL in the MSSM [36], where a similar bound is expected to be obtained in the Type-II 2HDM because of the same structure of the Yukawa interaction 8 . In the Type-I and Type-X 2HDMs, the production cross section of pp → tH ± + X is significantly suppressed by a factor cot 2 β, so that we cannot expect to obtain an important bound in the high mass region.

V. NUMERICAL RESULTS
In this section, we perform numerical evaluations for the H ± W ∓ V vertices and related observables. In particular, we focus on the light H ± case, i.e, m H ± = O(100) GeV, because of its phenomenological interest. As we discussed in Sec. IV, such a scenario is allowed in the Type-I and Type-X Yukawa interactions from flavour constraints, so that we consider these types only in 7 Notice that we have emulated both the top quark production and the decay as well as H ± -strahlung through the single gg → tbH ± mode, in the spirit of [37]. 8 In the Type-Y 2HDM, although the same production cross section of pp → tH ± + X is obtained as in the Type-II case, the branching fraction of H ± → τ ± ν is significantly suppressed due to the enhancement of the decay rate of the H ± → bc mode [38]. Therefore, the bound in the Type-Y 2HDM can be much weaker than that in the Type-II case.  In our model, there are 16 independent parameters in the potential given in Eq. (1), namely, µ 2 1-3 , µ 2 η , λ 1-5 , λ η , ρ 1-3 and σ 1-3 . They are divided into 8 parameters in the active sector (µ 2 1-3 and λ 1-5 ) and the remaining 8 parameters (µ 2 η , λ η , ρ 1-3 and σ 1-3 ). After the tadpole conditions are imposed, the former 8 parameters can be expressed by v, tan β, sin(β − α) m h , m H , m A , m H ± and M 2 . Two of the 8 parameters, v and m h , should be used to reproduce the gauge boson masses and the observed Higgs boson mass, i.e., v ≃ 246 GeV and m h ≃ 125 GeV. Furthermore, the Higgs boson search data at the LHC suggests that the observed Higgs boson is SM-like [1][2][3][4], so that taking sin(β − α) ≈ 1 gives a good benchmark scenario as we explained in Sec. II. We thus take sin(β − α) = 1 in the following calculation.
Regarding the latter 8 parameters, we proceed as follows. First, we take λ η = 0, as this gives a four-point interaction among the inert scalar bosons that does not affect the following analysis.
Second, we take ρ 1 and σ 1 so as to satisfy the vacuum stability condition given in Eqs. (41) and (42) for given values of ρ 2,3 and σ 2,3 : Finally, the remaining 5 parameters can be expressed in terms of three masses of the inert scalar bosons (m η ± , m η A and m η H ) and the ρ 2 and ρ 3 parameters. In this parametrisation, the σ 2 and σ 3 parameters are given as the outputs: Therefore, to recap, we are left with 5 new parameters in the active sector (tan β, m H ± , m A , m H and M 2 ) and 5 new ones in the inert sector too (m η ± , m η A , m η H , ρ 2 and ρ 3 ) and we will scan over these. Regarding the SM inputs, we use the following values [39,40]: where V ij are the Cabibbo-Kobayashi-Maskawa matrix elements, and the quark masses m b and m c are given at the m Z scale as quoted from Ref. [40].
The form factors depend on the three momenta p µ W , p µ V and q µ = p µ W + p µ V for W , V (= Z, γ) and H ± , respectively. In the numerical calculation, when m H ± ≥ m W + m Z , we take p 2 W = m 2 W , p 2 Z = m 2 Z and q 2 = m 2 H ± while when m H ± < m W + m Z , we take p 2 W = (m H ± − m Z ) 2 , p 2 Z = m 2 Z and q 2 = m 2 H ± (thereby allowing for below threshold H ± decays too). For the H ± W ∓ γ vertex, we take p 2 W = m 2 W , p 2 γ = 0 and q 2 = m 2 H ± .

A. Form factors
We start by showing the numerical results of the form factors of the H ± W ∓ Z and H ± W ∓ γ vertices. In order to see how the inert scalar boson loops can change the prediction, we first show the result in the 2HDM under the constraints from unitarity, vacuum stability and the EW parameters as discussed in Sec. IV. Then, we move on to the 3HDM.
In Fig. 3, the values of |X Z | 2 (X = F, G and H) and GeV is excluded by the constraint from the S parameter at 95% CL. We also note that only the fermion loop contributes to H Z and H γ .
We can see that the value of |F Z | 2 is the biggest of all the form factors as we expected in Sec. III, because of the m 2 t dependence. Typically, |F Z | 2 is more than one order of magnitude larger than |G Z | 2 and |H Z | 2 . In addition, all the squared form factors decrease as tan β is getting larger, because the top Yukawa coupling is proportional to cot β. The maximal allowed value of GeV. For the H ± W ∓ γ vertex, the maximal allowed values of |G γ | 2 and |H γ | 2 are order of 10 −6 at tan β ≃ 2.
Regarding the 3HDM, as we see from Eq. (37), F Z is logarithmically enhanced by m η A in the case of m η ± = m η H . However, a too large mass difference between η A and η ± is excluded by the S parameter as shown in Fig. 1 in the case of m H ± = m A = m H or ∆T = 0. We thus take a mass difference between H ± and A/H with m H = m A to avoid the constraint by the effect of non-zero ∆T . From the above reason, we consider the following parameter conditions in the forthcoming calculations: We note that, in this setup, η H corresponds to the DM candidate. The measured relic abundance of DM 9 can be satisfied by the resonant process of η H η H → A/H → ff .
In Fig. 4, the values of |X Z | 2 (X = F, G and H) and Remarkably, at tan β = 2, we obtain |F Z | 2 ≃ 10 −3 , which is one order of magnitude larger than |F Z | 2 in the 2HDM.
In Fig. 5, we show the m η A dependence of the squared form factors in the case of tan β = 2.5.
We take m H ± = 150 (200) GeV in the left (right) panel. The description of the objects in the figure is the same as in Fig. 4. Clearly, we can see that only |F Z | 2 is enhanced as m η A is getting larger. The maximal allowed value of |F Z | 2 is about 10 −3 at m η A ≃ 500 GeV.
B. Branching fractions of H ± Next, we discuss the decay branching ratios of H ± . As we see in Figs. 4 and 5 that the form factor F Z is much larger than G Z and H Z , we only keep the term proportional to |F Z | 2 for the When m H ± > m W + m Z , the on-shell decay of H ± → W ± Z opens and its decay rate is calculated as where If m H ± is smaller than m W + m Z , the off-shell decay modes H ± → W ± Z * and H ± → W ± * Z are allowed. The decay rate with three body final states is given by where We note that the argument y * is for the ratio of squared masses of a virtual gauge boson to that of H ± , e.g., for the H ± → W ± * Z case, we should use F 3 (m 2 Z /m 2 H ± , m 2 W /m 2 H ± ). The decay rate for H ± → W ± γ is given by In Fig. 6, we show the branching fractions of H ± as a function of m η A in the 3HDM with the Type-I Yukawa interaction. We take m H ± = 150 (left), 170 (center) and 200 GeV (right). The  value of tan β is fixed to be 2.5 in all the panels. In these plots, we scan the values of ρ 2 and ρ 3 in the range of −10 to +10 and extract the set of (ρ 2 , ρ 3 ) combinations giving the maximal value of the decay rate Γ(H ± → W Z). Further, for the case of m H ± < m W + m Z , we show the branching fraction of H ± → W ± Z as the sum of the branching fractions of H ± → W ± Z * and H ± → W ± * Z.
In all the plots, the behavior of m η A in the H ± → W ± Z decay is similar to that of |F Z | 2 shown in In Fig. 7, we also show the branching fraction of H ± in the Type-X Yukawa interaction with   tan β = 2.5. Although we observe a similar behavior of BR(H ± → W ± Z) and BR(H ± → W ± γ) as seen in Fig. 6, their maximal values are smaller than those in the case of the Type-I Yukawa interaction. This is because in the Type-X Yukawa interaction, the decay rate of the H ± → τ ± ν mode is enhanced by tan 2 β. Here, the maximal value of BR(H ± → W ± Z) is about 0.2%, 2% and 0.3% in the cases of m H ± = 150, 170 and 200 GeV, respectively.

C. Cross sections at the LHC
Finally, we discuss the signature of the H ± → W ± Z and H ± → W ± γ decays at the LHC. If the H ± mass is below the top quark mass, the top decay t → H ± b is the dominant production mode of H ± while above it H ± -strahlung becomes dominant. In reality, the latter is never significant as a means of enabling H ± → W ± Z and H ± → W ± γ detection, so we only concentrate on the former. We then expect the signature pp → bbH ± W ∓ → bbW ± W ∓ V . The signal cross section of this process σ top S is estimated by where σ tt is the top quark pair production cross section at the LHC. In Ref. [41], σ tt = 923.0 pb has been obtained with m t = 171 GeV and √ s = 14 TeV at the next-to-next-to leading order using CTEQ6.6 parton distribution function [42]. As alternative production modes of H ± states, especially helpful when the charged Higgs mass is larger than the top quark mass, one should also count the EW productions, e.g., pp → H ± A, pp → H ± H and pp → H + H − whose cross sections are determined by the masses of extra Higgs bosons. The cross sections for H ± A and H ± H productions are the same as long as we take m A = m H and sin(β − α) = 1. By using these production modes, we can consider pp → H ± A/H ± H → W ± V + X 0 and pp → H + H − → W ± V + X ∓ , where X 0 and X ± are respectively the decay product of A/H and H ± . The signal cross section via the EW production modes are estimated by For the EW processes, the reduction of the production cross section (σ H ± A , σ H ± H and σ H + H − ) is milder than that of the top decay process (σ tt ×Br(t → H ± b)). Therefore, the signal cross section of the EW processes become larger than the top decay process at m H ± = 170 GeV. Finally, we note that the signal cross sections in the Type-X case is more than one order of magnitude smaller than those in the Type-I case. VI We have thus computed the ensuing cross sections in all cases and shown that the LHC Run-II has the potential to access H ± → W ± Z and/or H ± → W ± γ decays, certainly for light H ± 's (at standard luminosity) and possibly for heavy H ± 's (at very high luminosity). To establish one or the other such signals at the CERN machine may represent circumstantial evidence of a 3HDM sector, as opposed to a 2HDM. In order to express loop functions, we use the Passarino-Veltman functions [43]. Here, we give the integral formulae of some of the functions which we use in the following discussion: where In Eq. (A1), ∆ is given by where ǫ appears in the D(= 4 − 2ǫ) dimensional integral, µ is an arbitrary dimensionful parameter and γ E is the Euler constant. In the four dimension limit ǫ → 0, ∆ is divergent. We note that this divergent part ∆ appears in the following expressions, but it is exactly cancelled in the renormalized variables such as X Z and X γ (X = F, G and H). We use the shorthand notations like B i (p 2 ; A, B) = B i (p 2 ; m A , m B ) and C i, ij (A, B, C) = C i, ij (p 2 1 , p 2 2 , q 2 ; m A , m B , m C ). The fermion loop contribution to X 1PI Z is given by The boson loop contribution is given by  − λ H + η − η H (C 12 + C 23 )(η H , η ± , η ± ) − λ H + η − η A (C 12 + C 23 )(η A , η ± , η ± ) , and H 1PI Z,B = H 1PI γ,B = 0, where We note that the above expressions are obtained by extracting the coefficient of the scalar trilinear vertex, i.e., L = +λ φ 1 φ 2 φ 3 φ 1 φ 2 φ 3 + · · · .
The fermion and boson loop contributions to the W ± -H ± mixing, i.e., Γ 1PI W H (p 2 ) F and