Photon Polarization in the b->s gamma processes in the Left-Right Symmetric Model

The circular-polarization of the photon in the radiative B decays is dominantly left-handed in the Standard Model (SM), but the right-handed polarization may be enhanced in some new physics models with right-handed currents, such as the Left-Right Symmetric Model (LRSM). In this article, we investigate how large this wrong polarization could be within the allowed parameter space of the LRSM. We show that in LRSM, the right-handed polarization of the photon in the $b\to s\gamma$ processes could be largely enhanced by the $W_L-W_R$ mixing contributions because of the helicity flip on the internal top quark line of the penguin diagrams and the enhancement by the CKM factor $V_{ts}^R/V_{ts}^L$. We discuss the sensitivity of the proposed methods to determine the $b\to s\gamma$ photon polarization to the LRSM as well as their complementary role compared to the direct search of right-handed charged gauge bosons at LHC.


Introduction
The circular-polarization of the photon in the radiative B decays has an unique sensitivity to the physics beyond the Standard Model (SM). There have been various methods proposed to measure this polarization but its precise measurement has never been achieved so far. Therefore, it is an important challenge for the on-going experiments such as LHCb or the future projects such as Belle II experiment. For this reason, there are many theoretical works in this subject recently [1][2][3][4][5][6][7][8][9][10].
The b → sγ processes come from the magnetic-operators, s L σ µν b R or s R σ µν b L . The photons induced by these operators have left-and right-handed polarization, respectively. In the SM, these operators are induced by the loop diagrams with the W bosons. Due to the fact that the W bosons couple only left-handedly to the quarks, the first operator receives a The left-handed fermions are SU (2) L doublets and SU (2) R singlets as in the SM while the right-handed fermions are SU (2) R doublets and SU (2) L singlets. Thus, the charge assignments (T L3 , T R3 ,Ỹ ) of left-and right-handed quarks and leptons yields: 2) The gauge symmetry in the LRSM is spontaneously broken in two steps 3) The first step SU (2) R × U (1)Ỹ → U (1) Y is parity and B − L violating while the second step is equivalent to the standard electroweak symmetry breaking. Let us first see the scalar multiplet Φ, which triggers the second step symmetry breaking. Consulting the Yukawa interaction of the form, Q L ΦQ R , Φ should be a 2 × 2 unitary matrix. Moreover, to be invariant under SU (2) transformation, this term requires Φ to be bi-doublet scalar fields with charge assignment: The scalar field Φ can not trigger the first step symmetry breaking because (i), Φ couples to both of SU (2) L and SU (2) R and does not distinguish these two groups. Therefore Φ could not break parity. (ii) Φ does not couple to U (1)Ỹ which would be unbroken and would leave a massless gauge boson which is not observed. Thus, we must introduce other scalar multiplets to break the parity symmetry, namely the SU (2) R , and also U (1)Ỹ . In particular, the scalar multiplet with charge B − L = 2 is attractive since it can generate right-handed Majorana neutrino masses [40][41][42]. As a result, we introduce the scalar triplet: Note that in this model another scalar field ∆ L ∼ (1, 0, 2) is also introduced in order to ensure the parity conservation at a high energy before the symmetry is broken. Now the symmetry breaking (2.3) can be undertaken by the non-zero vacuum expectation values (VEV). It is known that two of four complex phases can be eliminated and remaining two phases are assigned conventionally to the VEV of Φ, ∆ L and ∆ R [24]: where, v L,R , κ and κ are real numbers. The symmetry breaking of SU (2) R × U (1)Ỹ → U (1) Y is achieved at a high scale, i.e. multi-TeV, by non-zero ∆ R . If v L = v R , the leftright symmetry is spontaneously broken. The standard electroweak symmetry breaking is triggered by non-zero κ, κ .
There are some hierarchies among the vacuum expectation values κ, κ and v L,R . First of all, κ, κ , v L v R is needed to suppress the right-handed currents at low energy scales. On the other hand, we would expect another hierarchy v L v R in order to generate the neutrino masses through the see-saw mechanism, namely v L a few MeV [40][41][42] for v R ∼ multi-TeV. And also the constraints from electroweak ρ-parameter requires v L 10GeV [43]. In this work, we shall use the limit v L → 0, which is usually used in literatures. Note that the Higgs potential allows such a limit since v L ∝ κ 2 /v R [42]. Therefore the phase θ L has no physical consequence, while ω could trigger a spontaneous CP violation. According to the hierarchy described above, we introduce an expansion parameter as: where v = 174 GeV is the scale of the standard electroweak symmetry breaking and v R = O(TeV) as discussed above. The ratio of κ and κ is defined by the usual parameter β, i.e. κ = v sin β, κ = v cos β, tan β = κ κ . (2.7) Then tan β is a free parameter in this model. tan β = 1 is required by the difference of the masses of the fermions. On the other hand, the mass hierarchy m t m b implies large tan β. However, the large value of tan β as O( mt m b ) in some literatures, is disfavored by the electroweak precision observables [44]. 1 < tan β < 10 is used in [44]. In this work, we consider the range of tan β > 1.
In the low energy processes, the discrete left-right symmetry breaks down, then the gauge coupling constants g L and g R are in general unequal, g L = g R . As g L is the coupling constant in the SM, the ratio of r ≡ g R /g L is not allowed to be arbitrarily large, otherwise the interactions between right-handed gauge bosons and fermions would become nonperturbative. Having the latest direct search result, r m W 2 > 2.5 TeV [15,16], and assuming v R to be muti-TeV or higher, we use r less than 2 in this work.
The charged gauge bosons are mixture of the mass eigenstates, where the masses of W ± 1 and W ± 2 up to the order of O( 2 ) are 9) and the mixing angle The full Lagrangian for the neutral currents and the charged currents associated with gauge bosons and Goldstone bosons in the LRSM are given in Appendix A.
In the LRSM, there are charged Higgs bosons H ± and heavy flavor-changing neutral Higgs bosons H 0 and A 0 . In [24], it is shown that the masses of these bosons are nearly the same. To the leading order, their masses are equal to each other [44], For simplicity, we use M H representing the masses of charged and heavy neutral Higgs bosons. The Lagrangians for the interactions between H 0 , A 0 , H ± and fermions are given in Appendix A. As we will see later-on, the tree-level flavor changing neutral current due to H 0 and A 0 will affect the ∆F = 2 processes very much unless M H is sufficiently large [44].
In this work, we consider the cases of M H = 20TeV and 50TeV. For such heavy mass, the contributions on b → sγ and ∆F = 2 processes from charged Higgs in the loop diagrams become negligibly small. Concerning the quark mixing matrices (CKM matrices), we have one left-handed coupling V L CKM and one right-handed V R CKM for the left-and right-handed quarks respectively. We take V L CKM the same as the Standard Model one, defined by usual three rotation angles and one phase. In this way, the right-handed CKM matrix V R CKM is in general written by nine parameters remained after imposing the unitarity condition. There are many models on the right-handed CKM matrix. A general study of the quark sector is done in the minimal Left-Right Symmetric Model, where there is a symmetry between the left-and right-handed sector [17]. The authors in [17] shows that the only possibilities in the minimal LRSM are either P or C (LR+parity or LR+charge conjugation), with the consequences that V R CKM is equal or very close to V L CKM . These two cases correspond to the scenarios in the early works of LRSM with the ideas of spontaneous CP violation. In the so-called manifest LRSM [18,19], where the VEVs in the Higgs sector are all real, the right-handed matrix is exactly the same as the left-handed one, V R CKM = V L CKM , and hence there are only explicit CP violations from the phases in CKM matrices. While in the pseudo-manifest LRSM [20][21][22][23], where the Yukawa couplings are taken to be real, the right-handed CKM matrix is related to the left-handed one by diagonal phase matrices K u,d , V R CKM = K u V L CKM K d † , leading to spontaneous CP violation from the complex Higgs VEV. Both scenarios confront strong constraints from the the flavor and Higgs physics [24][25][26][27]. More general right-handed CKM matrix have been studies in [17,24,[44][45][46][47][48]. Motivated by the K 0 − K 0 mass difference, Langacker and Sankar proposed two simple formulae of right-handed CKM matrix [48], where c α ≡ cos α, s α ≡ sin α with α an arbitrary angle. As we will see in the following sections, for the purpose of this work, i.e. enhancing the right-handed polarization of the photon in b → sγ, the scenario (A) is more interesting as it keeps a possibility to have a large |V R ts |. In this work, we also investigate the CP violating observable, thus, we keep the CP violating phases with the following parametrization, where s 23 ≡ sin θ 23 , c 23 ≡ cos θ 23 , θ 23 represents the mixing angle between the second and the third generations of quarks. This formula is also a simple version of [44] in the limit of θ 12 = θ 13 = 0. So far, we have briefly introduced the LRSM. We summarize the free parameters used in this work: (i) the mass of heavy charged gauge boson W ± 2 , and the masses of heavy charged and neutral Higgs bosons, (ii) the right-handed CKM matrix elements, V R CKM , in our formula, they are (iii) the ratio of right-and left-handed gauge coupling constants, (iv) the ratio of VEVs in the scalar bidoublet φ, tan β, (2.17) and (v) the spontaneous CP violating phase ω. In this work, one will find that the observables we used are not very sensitive to ω, therefore we will fix it to be ω = 0 in the discussions for simplicity.
3 The C 7γ and C 7γ in the LRSM The effective Hamiltonian of b → sγ is [49] where the operators Q i are obtained from Q i by replacing P L ↔ P R . The leading contribution to the b → sγ decay is from the operators of Q 7γ = e 16π 2 m bs σ µν P R bF µν , Q 7γ = e 16π 2 m bs σ µν P L bF µν . (3.2) We can find that the polarization of the photon radiated from Q 7γ is left-handed, but righthanded from Q 7γ . In the SM, C 7γ /C 7γ ≈ m s /m b 1, and hence the photon polarization is dominantly left-handed. To test the wrong polarization in the LRSM, we will study C 7γ /C 7γ in this work.
Note that the photon polarization is not directly related to the ratio of C 7γ /C 7γ , but can also be affected by other operators. It was discussed in [50] that the wrong polarization in the SM could be as large as ten percent contributed by the operator of Q 2 =cγ µ P L bsγ µ P L c. Therefore, the new physics effects on the polarization could be determined in experiments only if the wrong polarization was larger than ten percent. Besides, the LRSM contributions to Q 2 and Q 2 are negligible due to the large mass of heavier charged gauge bosons. Hence, within the leading effects, we investigate the ratio of C 7γ /C 7γ to study the wrong polarization in the following discussions.
In the LRSM, the penguin diagrams with charged gauge bosons W ± 1 , W ± 2 and charged Higgs H ± in the loops, can contribute to the b → sγ processes. The full formulae of C 7γ and C 7γ are shown in Appendix B. Using the approximation of expansion to the order of O( 4 ) for C 7γ , the Wilson coefficients are , and the loop functions are shown in Appendix B. The first term in Eq.(3.3) is the SM contribution. Since the contribution of C 7γ to branching ratio is of order O( 4 ), we have to consider the LRSM contribution up to the order of O( 4 ) for C 7γ in which the contributions from LRSM and SM are interfered with each other. We neglect the contributions from the penguin diagrams with W 2 in the loops since they are at a higher order than 4 . Note that the dependence on the W 2 mass of Eqs. (3.3, 3.5) is from the mixing between W ± L and W ± R in W ± 1 in Eq.(2.10). In this work, the contribution from charged Higgs is also negligible unless tan β ∼ 1, because we choose its mass to be higher than several TeV. The loop functions A 1,2 H (y) are two or three orders . Besides, the charm contribution cannot be neglected in the LRSM as explained in the Appendix B.
The dominant contribution from the LRSM is C 7γ (µ W ) W 1 in Eq.(3.5), from the penguin diagram with W ± 1 and top quark in the loop, in which the W ± L couples to b quark and W ± R couples to s quark, as shown in Fig.1. It is enhanced by the factor of m t /m b since the helicity flip occurs in the internal top quark line which is resulted by the mass eigenstate W ± 1 as a mixture of W ± L and W ± R . Compared to the SM contribution, another effect of ts is the key element to have a large C 7γ /C 7γ . If we allow a large value of |V R ts |, this factor of enhancement could be as large as 1/λ 2 , where λ ≈ 0.2 is one of the parameters in the Wolfenstein formula of left-handed CKM matrix. In [51], the authors shows that the mechanism of enhancement of C 7γ /C 7γ in the LRSM is the same as the chromomagnetic one.

Meson mixings in LRSM
The effective Hamiltonian for ∆F = 2 transitions can be written in a general form [52,53] where the CKM matrix elements are involved in the Wilson coefficients C i . Here we factored out M W 1 for convenience. There are 8 operators responsible for oscillations [52] [53], taking Kaon mixing as example, We can readily write the corresponding operators for the B 0 d or B 0 s mixings by replacing s → b and d → d or s, respectively. In the SM, only the one operator, Q VLL 1 , contributes to this process. The amplitude for M − M mixing is then The hadronic matrix elements of these operators are shown in [53], in case of K 0 − K 0 mixing, is the chiral enhancement factor and B i are bag parameters which is given in Appendix D. The hadronic matrix elements of the operators V RR and SRR are the same as their corresponding V LL and SLL ones because the chiral conservation of QCD. It should be noted that the K 0 − K 0 system provides the most severe constraints on new physics models, due to the large enhancement factor 1/m s in R(µ) K comparing to the B d,s systems. At the high energy scale in the LRSM, there are three kinds of contributions to the ∆F = 2 processes: box diagrams with two charged gauge bosons W ± 1,2 , box diagrams with charged Higgs bosons H ± , and tree diagram with flavor-changing neutral Higgs bosons. We found that the contributions from box diagrams with charged Higgs bosons are always smaller than those with gauge bosons, because of M H > M W 2 . Therefore they are neglected in this work. Now, let us see the box diagram with W ± 1,2 . The Wilson coefficients for the box diagrams with two W 1 (i.e. SM contribution), one W ± 1 and one W ± 2 , and two W ± 2 are written respectively as In the LL and RR sectors, the GIM mechanism has been used and then the contribution from λ u is eliminated, while in the LR sector, the GIM mechanism is invalid because of the combination of left-and right-handed CKM matrix elements. The loop functions can be found in the Appendix C. We find that ∆ gauge C LR 2 dominates the LRSM contribution, while ∆ gauge C VRR 1 is negligible due to the high mass of W ± 2 . Our results agree with those in [17,24,44]. Considering the right-handed CKM matrix in (2.13) with some elements as zero, the dominant contributions to the box diagrams to Kaon, B d and B s mixing yeild: One more diagram with one charm quark and one top quark is also neglected due to the suppression of corresponding left-handed CKM matrix elements.
The tree-level contributions from the flavor-changing neutral Higgs (NH) are expected to dominate the ∆F = 2 processes. The corresponding Wilson coefficient is where the up-type quark masses are evaluated at the scale of µ H due to the renormalization of Yukawa couplings. Because of V R us = V R cd = V R td = 0 in our right-handed CKM matrix elements in (2.13), similarly to the case of box diagrams, only two terms contribute to Kaon mixing, and B 0 d mixing. Due to the tiny mass of up quark, the flavor-changing neutral Higgs contribution in the LRSM to B 0 d − B 0 d system is much smaller than the SM one, thus it can be neglected. Thereby we will not consider the B 0 d − B 0 d mixing in the following discussion in this work. For the Kaon mixing, the LRSM effects do not change the mass difference very much. However, since the indirect CP violation K is very small in the SM, the LRSM contribution to K has to be examined very carefully. The flavor-changing neutral Higgs (NH) contributions to the where the up-type quark masses are evaluated at the scale of µ H . With the same argument to the case of box diagrams above, we neglect two terms in the B 0 s − B 0 s mixing. As the quark masses don't change too much at µ H = (20 ∼ 50)TeV, we fix them as (4.15) 5 Numerical Analysis C 7γ and C 7γ are mainly constrained by the branching ratio of B → X s γ whose current world average value is [54] Br The SM prediction within next-to-next-to-leading order of QCD is [55] Br Adding the LRSM contributions, the C SM 7γ term is extended to: where the components of NP terms are already expressed in the previous section. Note that the left-handed and the right-handed polarization terms, i.e., C 7γ and C 7γ , do not interfere with each other. In the branching ratio of B → X s γ in the SM, the contribution from tree level operator, i.e. C SM 2 is very large. Therefore, the overlap of C to the high mass of the W ± 2 . For the LR and RL operators, they could contribute with the chirality flip by the charm quark mass which is much smaller than top quark mass. Thus, the tree level contributions from the LRSM can be safely neglected.
While we use the full equation in our numerical study, we show a simplified formula using typical input parameters for illustration. The branching ratio of b → sγ can be written in terms of new physics (NP) Wilson coefficients at the scale of µ b = 4.6GeV In this formula, we update the NLO analysis from [56] by the new results of quark masses and CKM matrix elements in [57], and by the NNLO SM result for this branching ratio from [55,58]. For the Wilson coefficients, we use the leading-order (LO) formulae for the W ± 1 and H ± contributions at the matching scale of µ W and µ H , and then evolve them down to the µ b scale separately at the leading logarithm (LL), as is done in [44]. The procedures of evolution are the same for C 7γ and C 7γ because QCD is conserved to the left-right symmetry. The mixing between O 7 and other operators are considered in the evolution. Besides, C 7γ and C 7γ are actually the effective C eff 7γ and C eff 7γ , but we use C ( ) 7γ in this article for simplicity without ambiguity.
Numerically, C 7γ (µ b ) and C 7γ (µ b ) can be expressed as a function of the parameters in the LRSM: The first term in C 7γ is from the SM contribution, while the second term in C 7γ is from the LRSM contribution which is suppressed by M 2 W 1 /M 2 W 2 , and can be neglected. So C 7γ can be taken as the SM one in an approximation, C 7γ C SM 7γ . Then We can find that |C 7γ /C 7γ | could be as large as 0.5 if sin θ 23 ∼ 1, M W 2 ∼ 2 TeV, g R /g L ∼ 1 and tan β ∼ 10.

Meson mixings
To constrain the LRSM, we use the mass difference ∆M s and the CP violating phase φ s in B 0 s − B The LRSM contribution to the tree-level two-body hadronic decays is suppressed by the high mass of W 2 and hence negligible. Therefore we take the weak phase difference φ ccs s between the B 0 s − B 0 s mixing and the b → ccs decay amplitudes as the phase in the mixing. The CP violation in the neutral Kaon meson decays is defined as [59][60] , and β CKM is the CP violating phase β in the left-handed CKM matrix, to be distinguished by the β parameter in the Higgs sector in the LRSM. We can find that the real part of contribution from the LRSM are much smaller than the SM one, and hence the LRSM would not affect ∆M K . However, the imaginary part of contributions from the LRSM are in the same order as the SM one, so that K can be significantly affected by the LRSM.
For the B 0 s − B where r ≡ g R /g L , η ≡ M 2 W 1 /M 2 W 2 , and η H ≡ M 2 W 1 /M 2 H . We can find that the contributions from W ± 2 and neutral Higgs bosons are at the same order of, or even larger than the SM contribution, within the region of values of parameter we discussed. Then the constraints from B 0 s − B 0 s mixing are severe. The CP violating phases φ 1 and φ 2 become important to provide solutions without fine-tuning.
In summary, concerning θ 23 , if θ 23 0 (or π), i.e. sin θ 23 0, we found that all the constraints from the meson mixings can be neglected as long as M W 2 , M H are not quite small, because M LRSM 12 (B s ) ∝ sin θ 23 , and the terms with cos θ 23 in M LRSM

12
(K) are without CP violating phase and hence lose the constraint from K . However in order to have a large |C 7γ /C 7γ |, we need sin θ 23 to be large. If so, the contributions from the second terms in Eq. (5.17,5.20) become dominant. Interestingly, these terms depend on the φ 1 and φ 2 . Thus, we can find some solutions in terms of φ 1,2 , which satisfy M SM 12 + M Box 12 + M NH 12 to be within the experimental bounds. Indeed such solutions exist as long as M H > 20 TeV and/or M W 2 > 2 TeV.

Impacts of the future photon polarization measurement on the LRSM
In this section, we attempt to estimate how large the |C 7γ /C 7γ | could be in the LRSM fully taking into account the various flavour constraints discussed in the previous sections. As we have already seen, |C 7γ /C 7γ | has a simple dependence on the parameters: it becomes larger hadronic parameters [61]. 0.82 ± 0.08 0.87 ± 0.08 1.15 ± 0.12 0.496 ± 0.047

CKM parameters [57]
|V L td | |V L ts | |V L tb | β CKM (9.44 ± 0.83) × 10 −3 (−46.6 ± 3.6) × 10 −3 0.89 ± 0.07 0.373 ± 0.013 Table 1. Input parameters for a lower W ± 2 mass and/or for a larger sin θ 23 . However, such a choice of parameter could make the LRSM contributions to the meson oscillation observables inconsistently large. Thus, we need a careful test of the whole parameter space of (θ 23 , M W 2 , M H , φ 1 , φ 2 ). Let us set our target to find the maximum possible |C 7γ /C 7γ | value for given values of M W 2 (here, we chose M W 2 = 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0 TeV), and M H (we fix it as 20 TeV or 50 TeV). Then, our first goal is to find a combination of φ 1 and φ 2 which satisfies all the experimental constraints while keeping sin θ 23 to be large.
It should be mentioned that in order to find that the allowed parameter space from the meson mixing and B → X s γ branching ratio, we consider not only the experimental errors but also the theoretical errors. The input parameters and their uncertainties taken into account here is listed in Table 1 1 . Some comments on the left-handed CKM matrix are in order. We assume that the W R contributions to the tree-level processes are negligible, which is reasonable for the large M W 2 considered here. Since V L ts is extracted by PDG [57] from B 0 s − B 0 s oscillation which is affected by the LRSM, we extract V L ts by using the unitarity relation of the left-handed CKM matrix, so that V L ts = −1/V L * tb (V L us V L * ub + V L cs V L * cb ). The central value and uncertainty of |V L tb | is taken as the experimental result in [57], which is also from tree-level top quark decays. |V L td | and the phase β CKM are both determined by the B 0 d −B 0 d oscillations, |V L td | from ∆M d , and β CKM from CP asymmetry. This process is also box diagrams mediated, but as discussed in the Section 4, due to the right-handed CKM matrix elements V R cd = V R td = 0, the effect of the LRSM in B 0 d system can be neglected. So that we could use the values of β CKM in [57]. Besides, |V L td | in [57] is obtained assuming |V L tb | = 1, but we use the experimental result of |V L tb | in this work, so we normalize |V L td | by |V L tb |. Now, we look for combinations of φ 1 and φ 2 which satisfy all the experimental constraints taking fully into account the uncertainties in these input parameters. The procedure to find these combinations is described in the appendix E. We fix the values of M W 2 and M H and scan the (φ 1 , φ 2 ) space, which is chosen in [−π, π] with π/50 interval (total of 10 4 are calculated by Lattice QCD, but their values are sometimes inconsistent by different lattice QCD groups, such as in [62] and [63], so that there are large uncertainties of these parameters. We take the center values of these parameters from [64] for Kaon system and [62] for B 0 s system, and take their uncertainty as almost 10%. The other B parameters are either determined very well such as B V LL 1 (K), or do not affect the observables in our work, so we do not consider their uncertainties. The bag parameters used in this work are shown in Appendix D.
points for each mass combination). Then, we find the minimum of the χ 2 defined in the appendix E. The number of points which did not exceed any of the experimental constraints or the input parameter ranges are (for M W 2 = (1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5) TeV, respectively): 01, 0.8, 0.8, 0.9, 0.9, 1.0, 1.1, 1.1, 1.2 As we have mentioned in the previous sections, M W 2 1.5 TeV is allowed only for a finetuned parameter combinations while for M W 2 > 2.0 TeV, we find sizable number of allowed points.
Using these allowed parameter points, we arrive to compute |C 7γ /C 7γ |. All other parameters are optimized to the values which give the minimum of χ 2 described in the appendix E. The LRSM corresponds approximately to the scenario II of [65] where C N P ∈ C (c.f. Eq. (5.5)). Then we could investigate the effect of the LRSM in a plan of Re[C 7γ /C 7γ ] and Im[C 7γ /C 7γ ], as shown in Figure 2. The left and right figures correspond to the cases that the heavy Higgs mass is 20 and 50 TeV respectively. The white circle represents the constraint from the measured branching ratio of B → X s γ with three standard deviation. The points with different colors represent the cases that the mass of W 2 is taken to be 1.5, 2.0, 2.5, 3.0, 3.5, 4.0, 4.5, 5.0 and 5.5 TeV respectively, with the circles from the outside to the inside in the figure. The points represents the solutions that are mainly constrained by K , ∆M s , φ s and Br(B → X s γ). |C 7γ /C 7γ | would be larger as the mass of W 2 decreases. One can see that |C 7γ /C 7γ | can reach to ∼ (0.5, 0.35, 0.25, 0.15, 0.1) for W 2 being (2.0, 2.5, 3.0, 3.5, 4.0) TeV.
Finally, we discuss briefly the discovery potential of the LRSM by the future photon polarization measurements. There have been various proposals for the experimental method to determine the photon polarization of the b → sγ. Here, following the discussion in [65], we study the three types of methods; i) The time-dependent CP -asymmetry of B → K S π 0 γ (S K S π 0 γ ), ii) The angular distribution of the B → K 1 γ → Kππγ decay (λ γ ), and iii) The ). Detailed descriptions of these observables are given in Appendix F. i) In the LRSM, S K S π 0 γ can be written as where β CKM is the CP violating phase from the left-handed CKM matrix. This formula is obtained by assuming C N P 7γ 0 and also absence of new physics contributions to the B 0 d − B 0 d mixing, which have been derived in the previous sections. In Figure.3(a), we overlap the curve representing the center value, 1σ, 2σ, 3σ, of the experimental result of S K S π 0 γ on top of the possible LRSM effects obtained in Figure  2. We can find that S K S π 0 γ is approximately a linear combination of Re C 7γ /C 7γ and Im C 7γ /C 7γ . Figure 2. Plots of real part and imaginary part of C 7γ /C 7γ in the LRSM. The left and right figures correspond to the cases that the heavy Higgs mass is 20 and 50 TeV respectively. The white circle represents the constraint from the measured branching ratio of B → X s γ with three standard deviation, in the scenario assuming C N P 7γ = 0, i.e. C 7γ = C SM 7γ , C N P 7γ ∈ C. The points from the outside to the inside with different colors represent the cases that the mass of W ± 2 is taken to be 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5 and 5.5 TeV respectively, with the circles from the outside to the inside in the figure. The points represent the solutions that are mainly constrained by K , ∆M s , φ s and Br(B → X s γ). |C 7γ /C 7γ | would be larger as the mass of W 2 decreases. We can find that |C 7γ /C 7γ | can be as large as 0.5 for M W2 = 2 TeV, and 0.35 for M W2 = 2.5 TeV.
ii) In the LRSM, the polarization parameter λ γ can be written as The future sensitivity of λ γ is shown in Figure.3(b). The λ γ constraint is circle and insensitive to the complex phases. The SM prediction stands at the central point. As stated in [65], in the region inside the circle of λ γ −0.8, i.e. C 7γ /C 7γ 0.3 we cannot distinguish the LRSM contribution from the SM one. Therefore, if the discovery of photon polarization using this method in experiments is distinguished from the SM, we can conclude that the mass of W ± 2 is smaller than 2.5 TeV in case of g R /g L ∼ 1 and tan β ∼ 10.
iii) In the LRSM, the transverse asymmetries can be written as These two observables can combine together to determine the magnitude and phase of C 7γ /C 7γ , shown in Figure.3(c) and 3(d).
(a) (b) (c) (d) Figure 3. Prospect of the future constraints on C 7γ /C 7γ in the LRSM. The four figures (a,b,c,d) T and A (im) T respectively, allowed by a ±3σ error of Br(B → X s γ) exp . The points in the figures are the same as those in Fig.2(b) with M H = 50TeV. The figure (a) is our result for S K S π 0 γ in a range including ±3σ. The uncertainty of measurement for S K S π 0 γ can be reduced down to 2% in the future Belle II experiment, in which the contours will be 10 times narrow than those in figure (a). In figures (b,c,d), the contours are the uncertainty for λ γ , A

Complementary to the direct search at LHC
The direct search of W R or W , which also provides an important constraint on the LRSM, is carried out at the high energy colliders, such as Tevatron and LHC. Relevant results have been obtained by the CDF and the D0 collaborations [66,67] and more recently by the ATLAS and the CMS collaborations [68][69][70][71] in terms of the W R or W bosons or heavy charged right-handed bosons searches (not necessarily LRSM). So far, most of the works are done in the Sequential Standard Model (SSM) in which the coupling of W W Z is set to be zero, and the W has the same couplings to fermions as the SM W boson[? ]. In the LRSM, the coupling of W 2 (or say W ) to W 1 Z 1 (or say W Z) is zero, because the left-and right-handed quantum numbers of Z 1 is zero, then it can not couple to W L W R and hence not to W 1 W 2 . In addition, if g R = g L , the coupling of W 2 to fermions is identical to the one of W 1 to fermions. Therefore,the LRSM in the case of g R /g L = 1 corresponds to the SSM. The current limit for M W in the case of the SSM is obtained as M W > 2.5 TeV at the 95% confidence level by ATLAS and CMS, using its decay into a charged lepton and a neutrino at √ s = 7 TeV [15,16]. Now, let us include this limit to our analysis in the previous sections. In Figure 4, we plot the dependence of |C 7 /C 7 | on the mass M W 2 and on the gauge coupling ratio g R /g L , i.e. Eq. (5.7). In this figure, we fix tan β = 10 and sin θ 23 = 1 (which is allowed by the ∆F = 2 observables). Looking at the horizontal line, g R /g L = 1, we find that the LHC limit M W > 2.5 TeV allows |C 7 /C 7 | to be 0.3, as we have seen in the previous section. As the LHC experiments will push this mass limit to a higher value, the possible size for |C 7 /C 7 | will decrease rapidly. Let's say, for M W > 5 TeV, |C 7 /C 7 | 0.1. On the other hand, in order to investigate the full range of the LRSM parameter space, we also need the mass limit for different values of g R /g L . For example, if we assume a larger value of g R /g L , a limit on |C 7 /C 7 | can be relaxed. In this sense, these two experiments are complementary. We should also note that so far, we have assumed tan β = 10 though |C 7 /C 7 | strongly depends on tan β (see Figure 5). From Figure 5, the odd behavior at tan β ∼ 1 is from the charged Higgs contribution to |C 7 /C 7 |, which is divergent at tan β = 1. Thus, measurements of |C 7 /C 7 | and the mass limit from the direct search should be used exhaustively to constrain the whole parameter space.

Conclusions
In this work, we investigate possible right-handed photon polarization in the b → sγ processes with the allowed range of parameters in the Left-Right Symmetric Model (LRSM). The circular-polarization of the photon in the radiative B decays are dominantly left-handed in the Standard Model (SM). But we find that in the LRSM, the right-handed polarization can be enhanced by the W L − W R mixing contributions because of the helicity flip on the internal top quark in the penguin diagrams and the enhancement by the CKM factor V R ts /V L ts . Note that for the right-handed CKM matrix, we extended the previously proposed parameterization in the literature by incorporating the additional CP violating phase.
We first constrained the parameter space of the LRSM by the observables of b → sγ and ∆F = 2 processes, such as the inclusive branching ratio of B → X s γ, the mass differences  In order to find the allowed parameter space, we consider not only the experimental errors, but also the theoretical uncertainties carefully. We find that the observables in the neutral meson mixing systems are significantly affected by the tree-level contributions from the heavy neutral Higgs bosons H 0 1 , A 0 1 , as well as the box diagrams with the charged gauge bosons W ± R . Our numerical analysis shows that the ratio of right-handed and left-handed polarization of the photon in the b → sγ processes, i.e. |C 7γ /C 7γ |, could be much larger than the SM prediction, with its value as large as (0.5, 0.35, 0.25) for the mass of W ± 2 as (2.0, 2.5, 3.0) TeV in the case of g R /g L = 1 and tan β = 10.
We also investigate the impact of the future photon polarization measurements on the LRSM, such as the time-dependent CP asymmetry in the decay of B → K S π 0 γ (S K S πγ ), the angular distribution in the B → K 1 γ → Kππγ decay (λ γ ), and the angular analysis ). We find that the LRSM can lead to large enough values of S K S πγ , λ γ , A T and A (im) T which are easily accessible by the LHCb and the future BELLE II experiments. We finally showed that the measurement of the b → sγ photon polarization plays a complementary role to the direct search of the right-handed charged gauge boson at the LHC.

A Interactions in the LRSM
The charged-current associated with W ± 1,2 bosons is where ζ is the mixing angle between W ± L and W ± R , P L,R ≡ 1∓γ 5 2 , g L,R are the gauge coupling constants in the SU (2) L and SU (2) R groups, V L,R ij are the left-and right-handed CKM matrices, and w is the spontaneous CP violating phase. The charged-current associated with charged Goldstone bosons is The charged-current with charged Higgs boson is where v, , and β are defined in Eqs. (2.6) and (2.7).
The Lagrangian of couplings between charged gauge bosons, charged Goldstone bosons and photons are The Yukawa coupling between heavy neutral Higgs and quarks are [24] L NH = ( The effective Hamiltonian of b → ccs in the LRSM at the tree level is B Wilson coefficients C 7γ and C 7γ In the LRSM, the Wilson coefficients for the b → sγ processes are A 1 H (y) = −5y 2 + 3y 6(y − 1) 3 + 3y 2 − 2y 3(y − 1) 2 ln y, (B.5) Note that there is a constant term in A LR (x), that is A LR (x → 0) = −10/3, because there is no GIM mechanism for the corresponding terms in Eqs.(B.1,B.2) with one lefthanded CKM matrix element and one right-handed CKM matrix element. Whilst the constant term in A SM (x) is canceled by the GIM mechanism. Therefore we have to consider the contribution from charm quarks in the loops, as shown in Eqs.(B.1,B.2). The up quark contribution is neglected due to its tiny mass. Numerically, in Eq.(B.1), m t V L ts A LR (x t 1 ) = 10.5, while m c V L cs A LR (x c 1 ) = −4.2, it is clear that the charm contribution has to be considered. We could neglect the charm contribution in Eq.(B.2) because of a suppression by m c V L cb .

C Box loop functions for LRSM
The box loop functions for the LRSM in Eqs.(4.6) are the following: ,(C.5) .(C.6)

D Bag parameters
The bag parameters are always calculated in the regularization-independent (RI) scheme by Lattice QCD. But the Wilson coefficients are always obtained in the naive dimension regularization (NDR) scheme in the perturbative theory. Therefore the bag paraemters obtained by the lattice QCD should be transformed into the NDR scheme. In this work, we use the bag parameters in the NDR scheme for Kaon mixing at µ = 2GeV given in [62] B with an approximation that the bag parameters are the same for B d and B s systems within the uncertainties.
E Procedure to find the allowed parameter space To simplify the task of finding the allowed parameter space, we introduce the following χ 2 including | K |, ∆M s , φ s , Br(B → X s γ) as well as the variables listed in Table 1: where the var i are |V td |, |V ts |, |V tb |, β CKM , B LR 2 (K), B V LL 1 (B), and B LR 2 (B). In addition, as our target is to maximize |C 7γ /C 7γ |, which can be achieved only when θ 23 ±π/2, we also include θ 23 to be a variable and vary in the region of θ 23 = − π 2 ± 0.3 (we found that the result does not change for ±π/2). For the given point on the (φ 1 , φ 2 ) space, which is chosen in [−π, π] with π/50 interval, we minimize this χ 2 . We chose the point which give the χ 2 to be less than a certain value as an allowed point. This value is chosen conservatively so that any components of χ 2 in Eq. (E.1) do not exceed its three sigma range.

F Observables to determine the photon polarization
We describe the three observables proposed in order to determine the photon polarization.
i) The time dependent CP -asymmetry, S K S π 0 γ: The time-dependent CP -asymmetry in the radiative decays of the neutral B mesons is an indirect method to measure the photon polarization. It arises from the interference between B → f CP γ and B → B → f CP γ amplitudes where f CP is a CP eigenstate. The amplitudes with the photon with left-and right-handed helicity can not interfere with each other since the photon polarization is measurable in principle. Then the CP -asymmetry is defined as [28,72] A where ξ(= ±1) is the CP -eigenvalue of f CP , and φ M is the phase in the B − B mixing, which in the SM is φ d = 2β CKM 0.75 and φ s 0 for the B 0 d and B 0 s mixing respectively. In the SM S f CP γ are always quite small because of the smallness of C SM 7γ . Specifically, the SM prediction for the B 0 → K ( * 0) (→ K S π 0 )γ decay is [73] S SM K S π 0 γ = −(2.3 ± 1.6)%, (F.3) while its current world average value is [54] S exp K S π 0 γ = −0.16 ± 0.23.

(F.4)
In the future Belle II experiment, the error of S K S π 0 γ will be significantly reduced down to 2%.
ii) The polarization parameter λ γ using the B → K 1 γ → Kππγ process: Compared to the CP -asymmetry, a direct method to measure the photon polarization is to study the three-body decays of K 1 (B → K 1 γ → Kππγ) [32,33]. The polarization can be studied by the angular distribution in the B → K 1 γ → Kππγ decay, extracting the polarization parameter λ γ (F. 5) In the SM, λ γ −1(+1) for B(B) decays. In the future, Belle II / LHCb experiment can measure this observable at the precision of σ(λ γ ) 0.2.
iii) The angular analysis of B → K * (→ Kπ) + − : Among various angular coefficients, the most interesting observables are the transverse asymmetries defined as [74][75] where q 2 is the + − invariant mass, I i (q 2 ) are differential decay amplitudes which can be found in [1][65]. In the limit of q 2 = 0, only the O ( ) 7γ operators contribute to this process, so that the above asymmetries are only related to C ( ) 7γ [65], In the future, Belle II / LHCb experiments can measure this observable at the precision of σ(A