Charged Higgs contribution to $\bar{B}_s \rightarrow \phi \pi^0 $ and $\bar{B}_s \rightarrow \phi \rho^0 $

We study the decay modes $\bar{B}_s\to \phi \pi^0$ and $\bar{B}_s\to \phi \rho^0$ within the frameworks of two-Higgs doublet models type-II and typ-III. We adopt in our study Soft Collinear Effective Theory as a framework for the calculation of the amplitudes. We derive the contributions of the charged Higgs mediation to the weak effective Hamiltonian governing the decay processes in both models. Moreover we analyze the effect of the charged Higgs mediation on the Wilson coefficients of the electrowek penguins and on the branching ratios of $\bar{B}_s\to \phi \pi^0$ and $\bar{B}_s\to \phi \rho^0$ decays. We show that wthin two-Higgs doublet models type-II and type-III the Wilson coefficients corresponding to the electroweak penguins can be enhanced due to the contributions from the charged Higgs mediation leading into enhancement in the branching ratios of $ \bar{B}_s\to \phi \pi^0$ and $\bar{B}_s\to \phi \rho^0 $ decays. We find that, within two-Higgs doublet models type-II, the enhancement in the branching ratio of $\bar{B}_s\to \phi \pi^0$ can not exceed $18\%$ with respect to the SM predictions. For the branching ratio of $\bar{B}_s\to \phi \rho^0$, we find that the charged Higgs contribution in this case is small where the branching ratio of $\bar{B}_s\to \phi \rho^0$ can be enhanced or reduced by about $4\% $ with respect to the SM predictions. For the case of the two-Higgs doublet models type-III we show that the branching ratio of $\bar{B}_s\to \phi \pi^0$ can be enhanced by about a factor $2$ of its value within two-Higgs doublet models type-II. However no sizable enhancement with respect to the SM predictions can be obtained for both $\bar{B}_s\to \phi \pi^0$ and $\bar{B}_s\to \phi \rho^0$ decays.


INTRODUCTION
Within Standard Model (SM) flavour-changing neutral current (FCNC) decays are generated at the one loop level. As a result they are highly suppressed and can serve as a sensitive probe of possible New Physics(NP) beyond SM. Of particular interest are the purely isospinviolating decaysB s → φρ 0 andB s → φπ 0 that are dominated by electroweak penguins [1].
They have been studied within SM in different frameworks such as QCD factorization as in Refs. [2,3], in PQCD as in Ref. [4] and using Soft Collinear Effective Theory (SCET) as in Refs. [5,6]. In Ref. [3] the study has been extended to include NP models namely, a modified Z 0 penguin, a model with an additional U(1) ′ gauge symmetry and the MSSM using QCDF. Their results showed that the additional Z ′ boson of the U(1) ′ gauge symmetry with couplings to leptons switched off can enhance the electroweak penguin amplitude sizably leading to an enhancement in their branching ratios by up to an order of magnitude.
This finding makes these decay modes are very interesting for LHCb and future B factories searches [3]. Motivated by this possibility we extend the study to the two Higgs doublet models (2HDMs).
In 2HDMs, the Higgs sector of the SM can be extended to include extra SU(2) L scalar doublet. Accordingly, the simplest picture of the SM Higgs coupling to the quarks and leptons can be modified by the presence of the extra Higgs doublet. This results in several classes of 2HDMs such as 2HDMs type-I, type-II, type-III, type-X and type-Y [7][8][9][10][11][12]. For 2HDMS type-I and type-II an investigation of the effect of the charged Higgs contributions to the electrweak penguins has been done in Ref. [13] where the interest was to explore their significance to B → Kπ decay modes. Their conclusion is that the significant contributions to the electrweak penguins are favored for small charged Higgs mass and cot β = 1. However taking into account B → X s γ constraints rule out this possibility.
In the present work we derive the new contributions to the electrweak penguins that are proportional to m b tan 2 β/m t which were neglected in Ref. [13]. These new contributions become dominant when tan β becomes large as we will show in the following. Moreover the charged Higgs mediation at tree-level can lead to a set of new operators that can not be generated in the SM. We derive their contributions to the effective Hamiltonian governs the process under consideration and calculate their corresponding Wilson coefficients. Having all these new contributions we will give the predictions for the branching ratios ofB s → φπ 0 andB s → φρ 0 within 2HDMs type-II which has not been calculated in Ref. [13]. In addition we extend our study to include 2HDMs type-III which has generic Yukawa structure that can allow for sizable effects in FCNC processes as shown in Ref. [8] and can also enhance CP violation in charm sector [14].
In this work we adopt SCET as a framework for the calculation of the amplitudes [15][16][17][18].
SCET provides a systematic and rigorous way to deals with the processes in which energetic quarks and gluons have different momenta modes such as hard, soft and collinear modes.
The power counting in SCET reduces the complexity of the calculations. In addition, the factorization formula given by SCET is perturbative to all powers in α s expansion. This paper is organized as follows. In Sec. II, we briefly review the decay amplitude for B → M 1 M 2 within SCET framework. Accordingly, we give a brief review of the SM contribution to the branching ratios ofB s → φπ 0 andB s → φρ 0 decays within SCET framework. Then we derive the Wilson coefficients in the case of Two Higgs-doublets models type II and type III and analysis their contributions to the branching ratios ofB s → φπ 0 andB s → φρ 0 in section III. Finally, we give our conclusion in Sec. V. light mesons can be written as given in ref. [5] corresponding to the two solutions obtained from the χ 2 fit. For the light cone distribution amplitudes we use the same input values given in ref. [22]. Following our work in Ref. [6], the amplitudes ofB s → φπ 0 andB s → φρ 0 decays corresponding to solution 1 of the SCET parameters are given as while for solution 2 of the SCET parameters we have [6] A(B 0 s → φπ 0 ) × 10 6 ≃ (−5.1C 10 − 0.3C 10 + 9.3C 7 − 9.3C 7 + 1.
here C i andC i are the Wilson coefficients that can be expressed as if in the √ λ expansion [19]. Here f refer to d and s quarks and i = 1, 2, ... These are the only relevant operators as higher order operators will be suppressed due to the smallness of the scaling parameter λ that is defined if operators and thus the amplitude given in Eq.(1) is for P P , P V and for two longitudinally polarized vector mesons, B → V V , [19].
Here P and V stands for pseduscalar and vector mesons respectively.
In Refs. [23,24] it was pointed out thatB → V ⊥ V ⊥ decays can be enhanced by the presence of an enhanced O(m b ) electromagnetic operator. This operator can lead to a contribution that are m b /Λ enhanced compared to the amplitudes for B → V V , but which are, on the other hand, also α em suppressed due to the exchanged photon [19]. Thus, numerically, the contribution from the electromagnetic operator can be expected to be smaller than the O(m 0 b ) terms in Eq.(1) [19]. Hence at leading order the only contributions to B → V ⊥ V ⊥ can arise from nonperturbative charming penguins A cc [20], which does not contribute tō B s → φρ 0 decay, while the other terms are either 1/m b or α em 0 m b /Λ suppressed [19]. The predictions for the branching ratios ofB 0 s → φπ 0 andB s → φρ 0 within SM are presented in Table I. As can be seen from Table I, the SCET predictions for the branching ratios are smaller than PQCD and QCDF predictions. This can be explained as the predicted form factors in SCET are smaller than those used in PQCD and QCDF [5].
As can be seen from Table I, the branching ratios ofB 0 s → φρ 0 are larger than the branching ratios ofB 0 s → φπ 0 . BothB 0 s → φρ 0 andB 0 s → φπ 0 decays are generated via thē B s → φ transition. Thus they have the same non perturbative form factors ζ Bφ and ζ Bφ J . However, using a non-polynomial model for the light cone distribution amplitude φ ρ (u) in the case ofB 0 s → φρ 0 decay can lead to a slightly different result from using the polynomial model for the light cone distribution amplitude φ π (u) in the case ofB 0 s → φπ 0 decay as pointed out in ref. [22]. Another reason for this difference is that the Wilson coefficients C 7 and C 8 enter the hard kernels, T 1ζ (u) and T 1J (u, z) ofB 0 s → φρ 0 with opposite signs to the case inB 0 s → φπ 0 [6].  In the two Higgs doublet models type-III both Higgs can couple to up and down type quarks and upon taking some limits we restore back two Higgs doublet model type-II as we will show in the following. Thus the Yukawa sector of this model will allow for FCNC at tree level not only by the charged Higgs mediation but also with the exchanging of neutral Higgs particles. One can avoid the unwanted FCNC at tree level by imposing strong constraints on the new couplings from several observables in some processes as we show in the following. However some new couplings can still escape these constraints and thus can lead to interesting results as explaining the B → D * τ ν anomaly which can not be explained in 2HDMs type-II [7]. In addition these new coupling can be in general complex and thus can lead to new sources of weak CP violating phases which can enhancedirect CP asymmetries comparing to the SM.
The Yukawa Lagrangian of the 2HDMs type-III can be written as [7,25] : where ǫ ab is the totally antisymmetric tensor, and ǫ q ij parameterizes the non-holomorphic corrections which couple up (down) quarks to the down (up) type Higgs doublet. After electroweak symmetry breaking the two Higgs doublets H u and H d result in the physical Higgs mass eigenstates A 0 (CP-odd Higgs), H 0 (heavy CP-even Higgs), h 0 (light CP-even Higgs) and H ± . In our study we follow Refs. [7,25] and assume a MSSM-like Higgs potential and thus the charged Higgs mass is given by where the W boson mass, m W , is related to the the vacuum expectation values of the neutral component of the Higgs doublets, v u and v d , via and the mass m A 0 is treated as a free parameter. It should be noted that in the limit v << m A 0 all heavy Higgs masses (m H 0 , m A 0 and m H ± ) are approximately equal [8].
The effective Lagrangian L ef f Y gives rise to the following charged Higss-quarks interaction Lagrangian: with [7] Γ H ± LR eff Here V is the CKM matrix and tan β = v u /v d . Using the Feynman-rule given in Eq. (8) Where the the loop functions I 1,2,3 (x) are given by and [13] with x = m 2 t /m 2 H ± . In Eq.(10), we neglected the small contributions to the Wilson coefficients from the terms that are proportional to ǫ u 13 and ǫ u 23 due to the strong constraints on these parameters from b → dγ and b → sγ respectively arising at the one loop-level [8].
The charged Higgs mediation can give rise to new set of Wilson coefficients corresponding to flipping the chirality in the effective Hamiltonian from left to right: As before, in the above equation, we neglected the small contributions to the Wilson coefficients from the terms that are proportional to ǫ d ⋆ 32 and ǫ d ⋆ 12 due to the strong constraints on these parameters from tree-level contributions to FCNC process [8].
The charged Higgs mediation at tree level can lead to the following weak effective Hamil- where C H i are the Wilson coefficients obtained by perturbative QCD running from M H ± scale to the scale µ relevant for hadronic decay and Q H i are the relevant local operators at low energy scale µ ≃ m b . The operators can be written as And the corresponding Wilson coefficients C H i are given as

IV. NUMERICAL RESULTS AND ANALYSIS
In order to estimate the enhancements in the full Wilson coefficients C 7 and C 9 due to the charged Higgs contribution we define the ratios: where C i are the SM Wilson coefficients. These ratios will give us an indication about the magnitudes of the charged Higgs Wilson coefficients compared to the SM ones and thus can give a hint of the expected enhancement or reduction in the branching ratios of our decay channels. We also define the ratios where M = π, ρ, i = 1, 2 refers to solutions 1, 2 for the SCET parameter space for which the corresponding amplitudes are given in Eqs. (2,3) and BR SM +H ± (B s → φM) and BR SM (B s → φM) are the branching ratios obtained when we consider the total contributions including charged Higgs and the SM contributions alone respectively. These ratios will give us the size of the enhancement or reduction to the branching ratios of our decay modes compared to the contribution from the SM.

A. Two Higgs doublet model type-II
We start by considering two Higgs doublets models type II. In this case the Wilson coefficients can be obtained from Eqs.(10,13) by setting ǫ u 33 = ǫ d 22 = ǫ d 33 = 0. The requirement for the top and bottom Yukawa interaction to be perturbative results in a constraint on tan β namely, 0.4 < ∼ tan β < ∼ 91 [26]. LEP has performed a direct search for a charged Higgs in 2HDM type-II and they have set a lower limit on the mass of the charged Higgs boson of 80 GeV at 95% C.L., with the process e + e − → H + H − upon the assumption BR(H + → τ + ν) + BR(H + → cs) + BR(H + → AW + ) = 1 [27]. If BR(H + → τ + ν) = 1 the bound on the mass of the charged Higgs is 94 GeV [27]. Recent results on B → τ ν obtained by BELLE [28] and BABAR [29] have strongly improved the indirect constraints on the charged Higgs mass in type II 2HDM [30]: Other experimental bounds can be applied on the (tan β, m H ± ) plane such as the bounds [8] and the bounds from ATLAS [32] and CMS [33] collaborations coming from pp → tt → bbW ∓ H ± (→ τ ν).
We note from Eq.(10), after setting ǫ u 33 = 0, that the dependency of the Wilson coefficients C Higgs mass m H ± = 380 GeV . This mass is the lower limit of the charged Higgs mass allowed by B → X s γ constraints [31]. In the left diagram the blue (red) curve corresponds to R H ± 7 (R H ± 7 ) while in the right diagram it corresponds to R H ± 9 (R H ± 9 ). As expected from Eq.(10) the Wilson coefficients C (H ± ) 7,9 vary inversely with tan 2 β which can is clear in Fig.(1). where the dependency in this case will be directly on Turning now to the Wilson coefficients C H 11 − C H 14 given in Eq. (16). By setting ǫ u,d ij = 0 we find that C H 11 , C H 13 and C H 14 will be suppressed by the smallness of the product of quark masses m 2 u , m u m s and m u m b respectively. For C H 12 we find that it is proportional to m s m b tan β which can be enhanced for large values of tan β in a similar manner to C D 11 resulted from the charged Higgs mediation in the MSSM with large tan β considered in Ref. [34]. Since all these Wilson coefficients have to be multiplied by the CKM factor λ s u they should be compared to the tree level Wilson coefficient of the SM. Clearly safely drop and only C H 12 can be comparable with the SM tree level Wilson coefficients only when tan β is large. However due to the constraints from B + → τ + ν τ , one find that C H 12 is roughly one order of magnitude smaller than C SM 2 as can be read from Eq. (24) in Ref. [34].

Thus larger values of C
Thus we can also safely drop C H 12 in our analysis.
In Fig.(2) we plot R π b 1 (R π b 2 ) , blue(red) curve, as a function of tan β for m H ± = 380 GeV and m H ± = 1000 GeV . For the lower bound on tan β = 0.4 and for m H ± = 380 GeV we find that R π b 1 ≃ 18%, R π b 2 ≃ 14% which means charged Higgs contributions to the branching ratio ofB s → φπ can reach a maximum value 18% of the SM prediction. For m H ± = 1000 GeV and tan β = 0.4 the charged Higgs contributions to the branching ratio ofB s → φπ can reach 3% and 0.64% of the SM prediction corresponding to solutions 1 and 2 of the SCET parameter space respectively as shown in the plot. We note from Fig.(2 for all values of tan β. This can be explained by noticing that their denominators are BR SM 1 (B s → φπ) and BR SM 2 (B s → φπ) and form Table I we have BR SM 2 (B s → φπ) > BR SM 1 (B s → φπ). Another remark is that R π b 2 varies with tan β and can have positive, zero and negative values. The reason is as follows: for tan β < 5 we see from Fig.(1) . Note also C (H ± ) 7 has similar sign to C SM 7 and thus it leads to instructive effect and enhance the amplitude.
For values of 5 < tan β < 20 we find that the term in the amplitude proportional to C (H ± ) 7 starts to be non zero and have opposite sign to the total Wilson coefficient C 7 leading to a destructive effect and almost Higgs contributions become negligible and thus we get R π b 2 = 0. For tan β ≥ 20 we find that C term in the amplitude corresponding to solution 1 is smaller than its corresponding one in solution 2. This explains why we do not have zero and negative values for R π b 1 as we have for R π b 2 as shown in Fig.(2). So far we have applied only the constraints from the requirement that the top and bottom Yukawa interaction to be perturbative to just give an estimation of the maximum enhancement can be obtained in 2HDMs type-II. We have selected two values of the charged Higgs mass and found that for the two values of the charged Higgs mass m H ± = 380 GeV and m H ± = 1000 GeV the maximum enhancement can be 18% of the SM prediction and correspond to solution 1 of the SCET parameter space. Thus for charged Higss masses smaller than 380 GeV and vales of tan β < ∼ 0.4 the enhancement in the branching ratio ofB s → φπ can exceed 18%. This result motivates us to determine the regions in the (tan β, m H ± ) plane which the enhancement in the branching ratio ofB s → φπ can be 18% or more of the SM prediction. In Fig.(3) we plot this region in the (tan β, m H ± ) plane.
In Ref. [8], see Figure 1, an updated study of the possible constraints imposed on the (tan β, m H ± ) plane of the two Higgs doublet model type-II from the experimental measurements in B → sγ, B → Dτ ν, B → τ ν, K → µν/π → µν, B s → µ + µ − and B → D * τ ν showed that no region in the (tan β, m H ± ) plane is compatible with all these processes.
Explaining B → D * τ ν requires large values of tan β and very small Higgs mass and thus together with B → sγ constraints excludes the green region in Fig.(3). Thus we conclude that the enhancement in the branching ratio is always less than 18% for the allowed regions in the (tan β, m H ± ) plane assuming no constraints from the anomaly in B → D * τ ν observed by BABAR. However if this anomaly is confirmed in the near future by other experiments, such as Belle-II experiment, then taking into account B → D * τ ν and B → sγ will rule out the whole parameter space of the charged Higgs in the two Higgs doublet model type-II.
As can be seen from Table I the errors of the SM predictions to the branching ratios are approximately 40% and thus it is clear that the enhancement in the branching ratio by 18% with respect to the SM predictions due to the charged Higgs mediation will be invisible within the theoretical uncertainties in the SM predictions.
Turning now to the branching ratios ofB s → φρ, we find that they can be enhanced or reduced by the charged Higgs contribution. However the enhancement or the reduction are always less than 4% of the SM prediction for the allowed regions in the (tan β, m H ± ) plane.

B. Two Higgs doublet model type-III
We turn now to the case of two Higgs doublet models type III. In this case the Wilson coefficients are those given in Eqs. (10,13) and the parameter space contains extra parameters which are the couplings ǫ q ij where q = u, d appears in the Yukawa Lagrangian. We start our analysis by discussing the constraints on the parameters ǫ u 33 , ǫ d 22 and ǫ d 33 relevant to our decay modes. Possible constraints on these parameters can be obtained by applying the naturalness criterion of 't Hooft to the quark masses. According to this criterion the smallness of a quantity is only natural if a symmetry is gained in the limit in which this quantity is zero [7]. Hence applying this criterion to the quark masses in the 2HDM of type III we find that for i ≥ j [8] |v u(d) ǫ As can be seen from the above equation that ǫ d 22 will be severely constrained by the small mass of the strange quark. In addition the constraints are expected to become more stronger with increasing the value of tan β due to the inverse dependency on v u = v sin β which increase with increasing tan β. However we find that v u changes slightly with varying The constraints imposed on ǫ u 33 by applying the naturalness criterion of 't Hooft to the top quark mass is expected to be even weaker than those obtained for ǫ d 33 due to the so large top quark mass compared to the bottom quark mass. Moreover we expect that the constrains becomes more loose with increasing the value of tan β due to the inverse dependency on v d = v cos β which decrease significantly with increasing tan β. Thus we can not rely on the naturalness criterion of 't Hooft to constrain ǫ u 33 . In Ref. [8] an extensive study of the flavor physics in the context of two Higgs doublet model type-III has been performed to constrain the model both from tree-level processes and from loop observables. It is shown that possible constraints on ǫ u 33 can be obtained from B s −B s mixing and B → X s γ. Moreover the constraints on ǫ u 33 from B → X s γ are the most important ones. For instance, applying B → X s γ constraints, for m H ± = 500 GeV and tan β = 50 the coupling ǫ u 33 should satisfy | ǫ u 33 |≤ 0.55 and the constrains become more strong for smaller values of m H ± and large values of tan β. Thus in our analysis we take into account the constraints imposed on ǫ d 33 and ǫ u 33 discussed in Ref. [8]. In 2HDMs type-III the constraints on the charged Higgs mass from B → X s γ become weaker comparing with their corresponding constraints in 2HDMs type-II. This because the off-diagonal parameter ǫ u 23 can lead to a destructive interference with the SM (depending on its phase) and thus reduces 2HDMs type-III contribution to the amplitude [8].  (16) and keeping the ǫ u,d ij parameters we still find that they are still suppressed either by the smallness of the quark masses or the constraints applied on the ǫ u,d ij parameters [8] and thus we drop their contributions in our analysis.
Turning now to the branching ratios ofB s → φπ 0 andB s → φρ 0 we note from Eqs. (2,3) that an enhancement in C 7 will enhance the branching ratio ofB s → φπ 0 and reduce at the same timeB s → φρ 0 due to the opposite sign of the terms proportional to C 7 . Since the enhancement is large for the case of tan β = 30 we find that R π b i can be enhanced by about 4% of the SM prediction for solution 1 while for solution 2 it is still very small about 1%. Comparing the branching ratio ofB s → φπ 0 corresponding to solution 1 in 2HDMs type-III with its value in 2HDMs type-II we find that R π b 1 is enhanced by about a factor 2. For smaller values of tan β where the constraints on ǫ u 33 becomes more weaker we find that the predictions for the branching ratios are close to their values for tan β = 30 as ǫ u 33 is multiplied by tan β and thus enhancement in ǫ u 33 will not be significant when it is multiplied by small value of tan β. Thus the branching ratios in 2HDMs type-III are approximately equal their values in 2HDMs type-II. For the case ofB s → φρ 0 we find that the reductions by the presence of ǫ u,d 33 terms are almost neglige. Thus we conclude that although the presence of ǫ u,d 33 terms enhance the branching ratio ofB s → φπ 0 by about a factor 2 of their values in 2HDMs type-II still the enhancement is not sizable compared to the SM predictions and will be also invisible within the theoretical uncertainties in the SM predictions for the branching ratios as for the case of 2HDMs type-II.

V. CONCLUSION
In this work we have studied the decay modesB s → φπ 0 andB s → φρ 0 within the frameworks of two-Higgs doublet models type-II and typ-III. We adopt in our study SCET Within two-Higgs doublet models type-II and type-III we find that the Wilson coefficients C 7 and C 9 can be enhanced due to the contributions from the charged Higgs mediation. As a consequence the branching ratios ofB s → φπ 0 andB s → φρ 0 decays are enhanced in turn.
Moreover we have shown that the charged Higgs mediation can lead also to new set of Wilson coefficients obtained from the weak effective Hamiltonian by changing the chirality from left to right. The presence of these new Wilson coefficients can also lead to enhancement of the branching ratios ofB s → φπ 0 andB s → φρ 0 decays.
We have shown that, within two-Higgs doublet models type-II, the enhancement in the branching ratio ofB s → φπ 0 can not exceed 18% with respect to the SM predictions for a charged Higgs mass 380 GeV . For the branching ratio ofB s → φρ 0 , we find that the charged Higgs contribution in this case is small where the branching ratio ofB s → φρ 0 can be enhanced or reduced by about 4% with respect to the SM predictions.
Turning to two-Higgs doublet models type-III we have shown for a value of the charged Higgs mass 300 GeV and tan β = 30 although the enhancement in BR (B s → φπ 0 ) can be about a factor 2 of its value within 2HDMs type-II however it is only 4% enhancement with respect to the SM predictions. For smaller values of tan β the predictions for the branching ratios are close to their predictions in 2HDMs type-II. We show also that, since the errors of the SM predictions to the branching ratios are approximately 40% forB s → φπ 0 , the enhancement in the branching ratios due to the charged Higgs mediation will be invisible within the theoretical uncertainties in the SM predictions. Clearly, charged Higgs contributions can not lead to a significant enhancement of the branching ratios ofB s → φπ 0 andB s → φρ 0 decays by one order of magnitude over their SM predictions making them possible for detection at LHC.