Flavor physics in the multi-Higgs doublet models induced by the left-right symmetry

In this paper, we discuss the multi-Higgs doublet models, that could be effectively induced by the extended Standard Model (SM). In particular, we focus on the phenomenology in the supersymmetric model with left-right (LR) symmetry, where the down-type and the up-type Yukawa couplings are unified and the Yukawa coupling matrices are expected to be hermitian. In this model, several Higgs doublets are introduced to realize the realistic fermion mass matrices, and the heavy Higgs doublets have flavor changing couplings with quarks and leptons. The LR symmetry is assumed to break down at high energy to realize the Type-I seesaw mechanism. The supersymmetry breaking scale is expected to be around 100 TeV to achieve the 125 GeV Higgs. In such a setup, the flavor-dependent interaction of the Higgs fields becomes sizable, so that we especially discuss the flavor physics induced by the heavy Higgs fields in our work. Our prediction depends on the structure of neutrinos, e.g., the neutrino mass ordering. We demonstrate how the flavor structure of the SM affects the flavor violating couplings. In our analysis, we mainly focus on the four-fermi interaction induced by the scalar exchanging, and we propose a simple parameterization for the coefficients. Then, we find the correlations among the flavor observables and, for instance, see that our prediction for the $\mu \to 3 e$ process could be covered by the future experiment, in one case where the neutrino mass hierarchy is normal.


Introduction
There are a lot of candidates for new physics. Many possible extensions of the Standard Model (SM) have been considered to explain the origins of the parameters in the SM. For instance, the Grand Unified Theory (GUT) reveals the origin of the SM gauge symmetry; the left-right symmetry can resolve the strong CP problem [1]. Such new physics is often assumed to reside at the very high scale, so that we need to find out the fragments of the hypotheses at the low scale to verify them experimentally.
In Refs. [2,3], the authors propose that the extra gauge boson can be a good probe to test the SO(10) GUT in the high-scale supersymmetry (SUSY) scenario. In the GUTs, the unification of the SM gauge symmetries is elegantly achieved by considering the SU (5), SO(10) and E 6 gauge symmetries. On the other hand, the unified Yukawa couplings predicted by the GUT symmetry can not be compatible with the experimental results. A lot of mechanisms have been proposed to resolve this issue, and one simple solution is to add extra quark/lepton fields to the minimal setup. * In Refs. [2,3], we find that the flavor violating interaction of the extra gauge boson predicted by the SO(10) GUT reflects the mechanism. The detailed analysis of the flavor physics has been discussed in Ref. [3].
We can consider another way to realize the realistic Yukawa coupling. Simply, we introduce extra Higgs fields to the minimal setup, and write down several Yukawa couplings between the extra fields and the matter fields including quarks and leptons [7][8][9][10][11][12][13][14][15]. After the GUT symmetry breaking, many Higgs SU (2) L doublets are generated, and the light modes of the Higgs doublets contribute to the electroweak (EW) symmetry breaking. Then, the realistic mass matrices consist of the vacuum expectation values (VEVs) and the several Yukawa couplings. In this scenario, there is no reason that only one Higgs doublet exists around the EW scale. If anything, we can expect that there are additional Higgs doublets at the low scale. For instance, assuming that the minimal supersymmetric SM (MSSM) is effectively induced after the GUT symmetry breaking, the MSSM would lead the extended SM with one extra Higgs doublet, namely the type-II two Higgs doublet model (2HDM) below the SUSY breaking scale. † If we consider the left-right symmetric model (LR model), the LR symmetry may break down at high energy to generate the heavy neutrino Majorana mass. As discussed in Sec. 2, the several Higgs doublets remain the small mass scales even after the LR breaking, because they decouple with the LR breaking sector. Then, the induced effective models are namely multi-Higgs doublet models, where extra Higgs doublets couple to quarks and leptons. ‡ The couplings do not respect the condition for the minimal flavor violation, so that large tree-level flavor changing neutral currents (FCNCs) involving the Higgs doublets are predicted [22].
The extended SM with additional Higgs doublets have been discussed in the bottom-up approach, as well. There is a rich phenomenology even in the extended SM with an extra Higgs doublet (2HDM), so that a lot of aspects of 2HDM have been widely investigated.
In the bottom-up approach, we can simply classify 2HDMs according to the type of the Yukawa couplings between the two Higgs doublet fields and the SM fermions. For instance, in the type-II 2HDM, one Higgs doublet couples to the up-type and the other couples to the down-type quarks. This setup is known as the one that can forbid the FCNC at the tree level. On the other hand, we can consider a generic 2HDM, namely type-III 2HDM, where two Higgs doublet fields couple to both up-type and down-type quarks. In this case, tree-level FCNCs involving scalars are generally predicted and we need to assume that the FCNCs are enough suppressed to evade the strong bounds from flavor physics.
In this paper, we discuss multi-Higgs doublet models with the GUT constraint, where the realistic mass matrices for quarks and leptons are given by the linear combination of the VEVs of Higgs doublets and the several unified Yukawa couplings. If only two Higgs doublets reside around the EW scale, the Type-III 2HDM would be induced. It is interesting that the FCNCs of the scalars are written down by the mixing angles of the scalars, the CKM matrices and the fermion masses because of the GUT constraint. Then, we can derive the explicit predictions against flavor violating processes. In particular, we concentrate on the flavor physics in the model effectively induced by the supersymmetric LR model, to find out explicit predictions for the low-energy observables. In the supersymmetric LR model, the Higgs doublets can be expected to obtain the masses via the soft SUSY breaking terms, since they decouple to the LR breaking sector. The LR breaking scale may be large to generate large Majorana right-handed neutrino masses for the Type-I seesaw mechanism. Then, the effective model at the low energy is interpreted as a multi-Higgs doublet model with the tree-level FCNCs. As we see in Sec. 2, the bigger the hierarchy between the LR breaking scale and the EW breaking scale is, the larger FCNCs are predicted in the lepton sector. Although the many possibilities of the scalar mass spectrum and the scalar mixing would hinder the search for the explicit predictions of the FCNCs, we analyze the Wilson coefficients of the four-fermi interactions induced by the scalar exchanging and discuss the flavor physics only focusing on the the magnitudes of the new physics scales and the flavor structures of the coefficients.
In Sec. 2, we discuss how the realistic Yukawa couplings can be derived in the GUT models, especially in the LR model, and then study the induced Yukawa couplings of the extra Higgs doublets in the effective models. In Sec. 3, we study phenomenology, especially flavor physics, in the multi-Higgs doublet models with the GUT constraint of the LR model. Sec. 4 is devoted to summary. In the Appendix A, the alignment of the lepton Yukawa couplings required by the neutrino observations are shown. In the Appendix B, the supersymmetric LR model, that can realize the Type-I seesaw scenario, is introduced. The complementary discussions about the four-fermi interactions and the corrections from the renormalization group (RG) are summarized, in Sec. C, Sec. D and Sec. E.

Multi-Higgs doublet models effectively induced by the LR model
First of all, let us briefly explain how to realize the realistic Yukawa couplings in the extended SM, where the Yukawa couplings of the matter fields are unified. We begin with the brief introduction of the Yukawa interaction in the Standard Model. The Yukawa couplings for quark and lepton masses are given by where V is the CKM matrix and V R is the unitary matrix to rotate the right-handed quarks. Now, we extend the Higgs sector, assuming that the GUT is the underlying theory. In the GUT, the Yukawa couplings as well as the gauge couplings are unified at high energy; for instance, are predicted by the minimal setup of the SO(10) GUT. Note that the two VEVs of Higgs doublets can realize the mass hierarchy in one generation but cannot explain all observables. Even in the supersymmetric LR model, Y u ij = Y d ij is predicted as discussed below. This relation conflicts with the experimental results, so that we need some improvements to realize the realistic Yukawa couplings effectively. One simple way is to introduce extra Higgs fields and extra Yukawa couplings to the extended SM. Let us explain the idea in a (supersymmetric) LR model below.

The non-supersymmetric LR model
In the LR model, the right-handed up-type and down-type fermions are unified into SU (2) R doublet fields. Then the Yukawa couplings are described as introducing one bi-doublet field, Φ.Q j R andL j R denote (û j R ,d j R ) T and (n j R ,ê j R ) T . Φ can be decomposed as Φ = ( H u , H d ), where H u and H d are the SU (2) L doublets in this notation. We find that this simple structure predicts the unified Yukawa couplings as mentioned above. In the non-supersymmetric case, we can also write down the following couplings without conflict with the gauge symmetry: where Φ is defined as τ 2 Φ * τ 2 . Then, the effective Yukawa couplings for quark masses are given by the linear combinations of two types of Yukawa couplings: Note that the Yukawa couplings for leptons are also effectively generated and are discussed below.
If either Y 1 ij or Y 2 ij is vanishing, it is impossible to realize the CKM matrix and the mass differences between the up-type quarks and the down-type quarks. In other words, the both VEVs of H u and H d should be sizable. In addition, the VEV of H u should not be the same as the VEV of H d . This means that the SU (2) R breaking effect is required by the realistic Yukawa coupling. Once we assume such a vacuum alignment, we can principally derive the realistic Yukawa couplings. Now, we decompose H u and H d as where only h develops the non-vanishing VEV: v is defined as a real value: v 246 GeV. Then, we can derive a simple relation between Y u,d ij and Y 1,2 ij from the requirement for the realistic Yukawa couplings: θ is the CP phase that can be generally defined. Note that the other phase can be eliminated by rephasing the Higgs fields. In the LR symmetric model, the symmetry that exchangesQ i L andQ i R (L i andR i l ) is often required in addition to SU (2) R . Then, Y 1,2 and Y l 1,2 are hermitian and the strong CP problem can be resolved in this setup [1]. Besides, V R in Eq. (2) can be expected to be identical to the CKM matrix, V . Then, the predictability of the Yukawa couplings involving heavy scalar could become higher. This phase, however, breaks the LR symmetry, and causes the strong CP problem. In other words, θ originates V R = V in Y u . In this paper, we do not touch the detail of the strong CP problem, but we assume that all Yukawa couplings are (approximately) hermitian in our numerical study corresponding to the LR symmetry.
In addition to the Yukawa couplings involving h in Eq. (1), there are Yukawa couplings of H: In this notation, the Yukawa couplings of H are described as As discussed in Sec. 3, we cannot evade the tree-level FCNCs induced by these Yukawa couplings. If the 2 × 2 matrix in Eq. (11) is in the diagonal form, the Yukawa couplings for the heavy scalars do not have the off-diagonal elements in the mass eigenstate. We see that the limit cannot be realized in this non-supersymmetric LR model because of the tan β dependence in Eq. (11).
In the lepton sector, we can write down the Yukawa couplings for both of h and H, by analogy with the quark sector. After the SU (2) R breaking, the couplings in Eq. (4) and Eq. (5) lead The difference between the lepton sector and the quark sector is the existence of the Majorana mass term for the right-handed neutrino. When we define the Yukawa couplings for the neutrinos as the Yukawa couplings in the lepton sector are related to The Yukawa couplings involving the heavy scalar are given by If there is no Majorana mass term for the right-handed neutrino, Y ν ij is described as using the PMNS matrix, V P M N S . Note that U ν R is identical to V P M N S , when θ is vanishing and the LR symmetry is assumed. In this case, the active neutrino is the Dirac fermion m e 0.511 MeV [43] sin  Table 1: The input parameters for leptons in our analysis. The notation for the lepton mixing is following Ref. [43]. NO (IO) is short for the normal (inverted) ordering neutrino mass hierarchy. The central values are used in our study. and the tiny neutrino mass, m ν k , suppresses Y ν ij . Then, the couplings involving charged leptons are suppressed. The couplings involving neutrinos, on the other hand, could be relatively enhanced, since the Yukawa couplings of the heavy scalars and the neutrinos are governed by Y e . In our work, we concentrate on the Majorana neutrino case.
In the case that the Majorana mass term for the right-handed neutrino is effectively generated after the SU (2) R breaking, the mass term becomes another source to realize the PMNS matrix. Let us briefly discuss how to generate the Majorana mass term in the non-supersymmetric LR model. The SU (2) R symmetry breaking is, for instance, achieved by introducing one SU (2) R adjoint field ∆ R . The field can couple to the lepton fields and the bi-doublet fields: The first term induces the Majorana mass terms for the right-handed neutrinos and the second terms generate the mass terms for the Higgs doublets, according to the nonvanishing VEV of ∆ R . If we assume that the VEV of ∆ R is enough large to realize the tiny neutrino masses via the seesaw mechanism, the Higgs doublets would get very large masses from the second term. The Higgs doublets, however, need to develop the nonzero VEVs around the EW scale, so that we simply assume that the effective masses from the SU (2) R breaking effects are small but not too small compared to the EW scale. This hierarchy between the Majorana mass and the EW scale could be realized by the supersymmetric LR model. One illustrative setup is shown in Sec. 2.2 and Appendix B.
After the SU (2) R breaking, the Majorana mass terms, (M ν ) ij , would be effectively generated as Assuming that the magnitude of (M ν ) ij is very large compared to the EW scale, the tiny neutrino masses of the active neutrinos are given bŷ In our base, the Yukawa coupling for the charged lepton, Y e ij , is in the diagonal form, so thatm ν ij is described aŝ where Y ν ij is defined as Another important point is the relation between U ν and U ν R . If the LR symmetry is assumed to be conserved at high energy, Y ν ij would be the hermitian matrix if the radiative corrections can be safely ignored. § In such a case, we can simply estimate the sizes of Y ν and M ν . In Appendix A, Y ν and M ν are shown assuming the mass hierarchy in M ν . We assume that Y ν is a hermitian matrix in the base that M ν is in the diagonal form. We denote (M ν ) ii as M νi in the following. Then, we consider the three cases: In the each case, we can find that some elements of the Dirac neutrino Yukawa couplings are irrelevant to the observables concerned with the active neutrinos; e.g., In Fig. 7, the predictions for Y ν are summarized in the each case. The input parameters used to plot are summarized in Table 1. As we see, large off-diagonal elements of Y ν are predicted, depending on the mass hierarchy of M ν . Note that the Majorana phase and the lightest neutrino mass are vanishing in Fig. 7.
Y ν would be large if the Majorana mass is very heavy. The Majorana mass term is originated from the SU (2) R breaking, so that the high SU (2) R breaking scale, that is assumed in our study, leads sizable Yukawa couplings involving heavy scalars, according to Eq. (11). This prediction provides our model with smoking-gun signals. § There are also other contributions to the LR breaking effects: e.g., the one from the SU (2) L triplet which is introduced to respect the LR symmetry.

The supersymmetric LR model
We consider the supersymmetric extension of the LR model. In the supersymmetric LR model, the potential is described by the holomorphic function, namely superpotential. The superpotential for the visible sector is given by Here, two bi-doublet chiral superfields, Φ a (a = 1, 2), are introduced, in order to realize the realistic Yukawa coupling. This means that we obtain four Higgs doublet fields after the SU (2) R symmetry breaking. The third term effectively generates the Majorana mass term for the right-handed neutrino, and the last term corresponds to the µ-term of the Higgs superfields. In our analysis, the Yukawa couplings, Y a ij and Y l a ij , are defined in the base where µ ab is in the diagonal form: µ ab = µ a δ ab .
Let us consider the scenario that ∆ R develops the very large VEV for the very heavy right-handed neutrino. This can be easily realized by introducing a singlet field, S: We can find the supersymmetric vacuum that breaks down SU (2) R × U (1) B−L to U (1) Y . This type of model has been proposed in Ref. [46]. The other setup has been discussed in Ref. [47]. ¶ The matter contents and the charge assignment are summarized in Table 2.
Note that Φ a can not couple to ∆ R at the renormalizable level because of the U (1) B−L symmetry, so that the Higgs doublets from Φ a do not gain the large masses from the VEV of ∆ R . We could expect that the SU (2) R breaking effect is mediated by the mediators for ¶ See also [48,49].
the SUSY breaking effects. In our study, we simply assume that the SU (2) R breaking effect appears in the soft SUSY breaking terms and discuss the mass terms for the Higgs doublets which are associated with the SU (2) R breaking, below.
In this supersymmetric LR model, there are two up-type Higgs doublets (H a u ) and two down-type Higgs doublets (H a d ) originated from Φ a (a = 1, 2). The masses of the four Higgs doublets are given by not only the supersymmetric masses but also the soft SUSY breaking terms. Let us define the mass squared as , respectively. In this notation, (M 2 H ) IJ is given by Here, H a u and H a d are the supersymmetric mass eigenstates: µ ab = µ a δ ab . The other parameters in Eq. (25) denote the soft SUSY breaking parameters. In order to realize the EW symmetry breaking, sizable B ab is required. In addition, m 2 H a u and m 2 H a d should satisfy some conditions to cause the EW symmetry breaking and to avoid the unbounded-frombelow vacua. In our study, we do not discuss the origin of the SUSY breaking terms and simply assume that the conditions are satisfied, taking the bottom-up approach.
In this assumption, the VEVs ofĤ I are aligned as where U Ih is the four-dimensional vector that satisfies I U Ih U * Ih = 1. Finding the directions orthogonal to U Ih , we define another base for the Higgs doublets: where In this base, only h develops a non-vanishing VEV as shown in Eq. (8).
h would correspond to the mass eigenstate given by ( The other states, H A , could be also interpreted as the mass eigenstates of (M 2 H ) IJ , so that the mass squared for the Higgs fields is described aŝ Note that we may wonder how SUSY is broken and how the SU (2) R breaking effect is mediated. See, for instance, Ref. [50].
where the unitary matrix,Û H , is defined asÛ H = (U Ih U I1 U I2 U I3 ). Note that the exact masses of the heavy scalars would be deviated from M 2 H A , because of the contributions of 4-point couplings, e.g. |h| 2 |H A | 2 , to the masses squared. These contributions are, however, suppressed, compared to (M 2 H ) IJ , if M H A is much larger than the EW scale. Then, we discuss the phenomenology, assuming H A are the mass eigenstates with M H A . The mass differences among the scalars in each H A are negligible, in this assumption. Now, we write down the Yukawa couplings involving h and H A . The Yukawa couplings of h correspond to the SM Yukawa couplings, e.g., Y u ij and Y d ij . The relation between Y a and the realistic Yukawa couplings can be obtained by analogy with the non-SUSY case in Sec. 2.1: Note that the strong CP problem would arise if U Ih is complex. When the Yukawa couplings of H A with quarks are defined as Note that ∆ h is vanishing in the SU (2) R symmetric limit. Similarly, the Yukawa couplings of H A for the leptons, are related to Y ν ij and Y e ij : Compared to the non-SUSY case in Sec. 2.1, there are many parameters in the Yukawa couplings of the heavy scalars: U IA and U Ih . In addition, there are three mass parameters, M H A . The mass parameters could be expected to be around the SUSY breaking scale, since they are originated from the SUSY breaking terms. The mass spectrum, however, depends on the mediation mechanism. As we discuss in Sec. 2.3, if we focus on the four-fermi couplings, we find that those parameter dependences on the Yukawa couplings in Eq. (34) lead simple forms to the Wilson coefficients, that contribute to the flavor physics. We derive the coefficients in Sec. 3 and discuss the flavor physics, using the simplified parametrization of the Wilson coefficients.

The induced four-fermi couplings
Before the phenomenology, we derive the effective couplings induced by the heavy scalars with the Yukawa couplings in Eq. (32) and Eq. (34). Note that we discuss only the SUSY case below. Integrating out the heavy scalars, we obtain the four-fermi couplings. In our study, we assume that the components of H A in the supersymmetric LR models are degenerate. Then, the couplings by the heavy neutral scalar exchanging are given as follows: where f and F denote u, d, e or ν, respectively. Using the relation between (M 2 H ) IJ andÛ H , the coefficients in front of the four-fermi operators can be simplified. Let us demonstrate it in the down-type quark couplings, below.
Defining the dimensional parameters, Λ ab , we write down the down-type quark couplings in Eq. (35) as where We change the base of the down-type quark into the mass base denoted by . Note that Λ ab (a, b = u, d) are related to (M −2 H ) IJ and U Ih as follows: When we discuss the flavor violating processes, such as the ∆F = 2 processes, we find that Λ −2 uu is only relevant in our model according to Eq. (36). As shown in Eq. (37), Λ −2 uu is described as where (U 1 ⊥ ) I denotes the vector orthogonal to U Ih : The four-fermi coupling in the charged lepton sector has the structure similar to the one in the down-type quark sector. Replacing Y u and Y d with Y ν and Y e respectively, (C e 4 ) ij kl , that is the coefficient of (e i R e j L ) (e k L e l R ), is given by Note that Y ν is the source of the flavor violation in the charged lepton sector. This means that there is a possibility that the observable, the PMNS matrix, in the neutrino physics connects with the charged LFV processes. The detail is shown in Sec. 3.2.
In the non-supersymmetric case, we can see the more explicit dependence of the extra mass scale. The four-fermi couplings, (C d 4 ) ij kl and (C e 4 ) ij kl , are simply obtained, replacing The coefficients of the other four-fermi interactions, that induce the LFV decays of mesons, are given by where (C de 4 ) kl ij and (C ue 4 ) kl ij are described as and Here, Λ ue ab are related to M 2 H as follows: The other operators are shown in Appendix B.

The evaluation of the RG flow
Before the concrete study on flavor physics, let us discuss the LR breaking contributions from the renormalization-group (RG) to the Yukawa couplings. The LR breaking is induced by the SU (2) L × U (1) Y gauge interactions and the leptonic Yukawa interaction.
In particular, the LR breaking would be enhanced, if Y ν ij is large. In the supersymmetric case, the RG corrections to the leptonic Yukawa couplings (Y l a ij ) are only given by the wave faction renormalization factors: The each Z factor in the right-handed side corresponds to the wave faction renormalization factor of the each field denoted in the subscript. Even in the one-loop correction, the LR breaking effect appears since right-handed neutrinos are integrated out and the other fields such as SU (2) L triplet, ∆ L , may reside at low energy. Z L and Z R are not identical because of the effect, so that Y l a ij (µ) is not hermitian matrix below the LR breaking scale. Once the hermitian condition is violated, the strong CP phase of QCD may arise through the RG correction from the wave faction renormalization factors of Higgs doublets, Z H d ab and Z Hu ab . When we discuss the phenomenology in our model, we focus on the parameter region that the LR breaking effect in Y l a ij (µ) is approximately parameterized as using the hermitian matrix,Ŷ l a , and extra parameters Z l ij . Note that M ν is also renormalized to obtain the hermitian matrix,Ŷ l a , in this notation. Z l ij is generally a 3 × 3 matrix and the one-loop RG equation for Z l ij is (approximately) given by In our study on phenomenology, Z l ij is assumed to be in a diagonal form: Z l ij = Z l i δ ij . This situation can be realized, assuming that one element of Y l a ij is dominant in the each RG equation at the one-loop level. When only one element of Y l a ij is close to unit and the RG runs from 10 12 TeV to 10 2 TeV, the RG correction is about 20 %. The correction highly depends on the setup at the high scale. We simply treat Z i as real free parameters satisfying 0.8 ≤ Z i ≤ 1.2. * * One element ofŶ l a ij can be O(1) but the others are at most The RG corrections to the quark Yukawa couplings can be evaluated in the same manner: The contribution of the Yukawa couplings in Z Q L , Z u R and Z d R respects the LR symmetry at the one-loop level, ignoring the contribution of the SU (2) L × U (1) Y gauge interaction and Y l a interaction. The LR breaking effects induced by the gauge interaction are flavor universal, so that the flavor structure is not modified. Thus, we could expect that the prediction of the LR model, as in Eq. (9), is still valid at the low scale. In this approximation, the Yukawa couplings shown in Eq. (32) are modified by the RG corrections as Z EW is the EW contribution described as are defined in Appendix D. Thus, the new physics scales, discussed in Sec. 2.3, are redefined, taking the RG corrections into account. In our analysis on the phenomenology, the constraints from flavor physics on the scales and the Yukawa couplings will be discussed. The scales are the ones including the RG corrections, as shown in Eqs. (54) and (57).
The EW correction will give the difference between the scales for quark and for lepton. In our study, such a scale difference is ignored and the improved analysis, taking into account the RG corrections more precisely, will be given in the future. The strong CP problem will be also taken into consideration in the future work.

Flavor physics
In this section, we discuss the phenomenology, especially flavor physics, in our models. We simplify the RG corrections focusing on the cases (i), (ii) and (iii) as in Sec. 2.1 and Appendix A, and numerically study our predictions derived from the LR symmetry at high energy.
There are many parameters, e.g. the scalar masses and the mixing. In our study, we discuss the phenomenology using the dimensional parameters defined in Sec. 2.3. We do not touch the detailed setup concerned with the masses and the mixing. Then, the parameters relevant to our study about flavor physics are as follows: Λ ud = Λ * du is predicted, as shown in Eq. (39). The other parameters can be, in principle, independent each other, so that we discuss the constraints and the impacts of the each parameter on flavor physics. Note that we also assume that the SUSY breaking scale is O(100) TeV, to avoid the strong constraint on the SUSY particles from the LHC experiments and to obtain the 125 GeV Higgs mass [51][52][53][54][55][56][57][58][59]. This means that the extra scalar masses are also expected to be much higher than the EW scale, and then the flavor-violating processes induced by the one-loop diagrams, such as b → sγ, are safely negligible. Note that the branching ratio of b → sγ limits the new physics scale if the scalars are below 1 TeV: the lower bound is about 580 GeV in the Type-II 2HDM [60].
The first three parameters in Eq. (58) contribute to the ∆F = 2 processes, the LFV and the leptonic decays of the mesons. In particular, Λ uu is strongly constrained by the K-K mixing. In the LFV and the leptonic meson decay, Λ ud may significantly contribute to the observables.
The other three parameters, Λ dd , suppress the couplings between up-type quarks and leptons. Then, they contribute to the µ − e conversion process significantly. We comment on the other observables in flavor physics and the collider experiments.
We note that the RG correction from the LR breaking scale to the SUSY breaking scale around 100 TeV is approximately evaluated, as explained in Sec. 2.4. The correction, in fact, depends on the detailed setup, such as the mass spectrum. In our study, we simply multiply a numerical factor as the correction including not only the RG but also the threshold corrections. The RG correction from the SUSY breaking scale to the observed scale is evaluated at the one-loop level. The SUSY breaking scale is expected to be about 100 TeV to obtain the SM Higgs mass. Note that all gaugino masses are assumed to be 1 TeV, to introduce the dark matter candidate.
The Yukawa couplings for the quark and lepton masses are run from M Z to 100 TeV, and the Wilson coefficients of the four-quark interactions are evaluated at the scale, Λ uu . Then, the RG corrections from Λ uu to the low energy are taken into account at the one-loop level. In the four-lepton interactions, the RG corrections are ignored. In the four-fermi couplings concerned with the leptonic meson decays, the RG effect could be interpreted as the same as the one for the quark mass. We calculate the Wilson coefficients at 100 TeV, using the Yukawa couplings derived from the realistic quark mass matrices. † † Then, the RG corrections are included in our analysis. Below, we explain our results in the each process. † † A procedure for evaluating the Yukawa couplings for the quark and lepton masses at the 100 TeV is summarized in Appendix E.

∆F = 2 processes
First, we summarize our predictions for the ∆F = 2 processes. The ∆F = 2 processes are consistent with the SM predictions, although the predictions suffer from large uncertainties. In our model, the neutral Higgs exchanging modifies the SM prediction at the tree level: where the Wilson coefficient (C d 4 ) ij is given by where Y u is given in Eq. (2). Assuming that the LR symmetry is assigned, In the non-supersymmetric case, Λ 2 uu is given by Λ 2 uu | non−SU SY = m 2 H cos 2 2β, as shown in Eq. (43). ‡ ‡ We investigate the bound from the K-K mixing. In the K physics, K and ∆M K generally give stringent bounds. They are approximately evaluated as where κ and ϕ are κ = 0.94 ± 0.02 and ϕ = 0.2417 × π [64,65]. (∆M K ) exp is the experimental value given in Table 3 and M K 12 includes both the SM contribution and our prediction: The first term is the SM prediction described by (M K 12 ) SM , where x i and λ i denote (m u i ) 2 /M 2 W and V * is V id , respectively. η 1,2,3 correspond to the NLO and NNLO QCD corrections. The input parameters for the quark mixing and masses are summarized in Table 3. The input parameters for the ∆F = 2 processes are shown in Table 4. We use the central values to estimate the SM predictions. Note that the functions which appear in K-K mixing are defined as S(x, y) = −3xy 4(y − 1)(x − 1) − xy(4 − 8y + y 2 ) log y 4(y − 1) 2 (x − y)  Figure 1: Λ uu vs | K | in the LR models. The red line corresponds to our prediction. The light blue band is the SM prediction with 1σ errors of η 1,2,3 . The pink band corresponds to the experimental result [43] .
In Fig. 1, we draw our predictions for | K | in the supersymmetric LR models. When we assume that the LR symmetry is assigned to our models, the only parameter is Λ 2 uu . Fig. 1 shows the constraint on the scale, assuming V R = V . The red line corresponds to the prediction, and the blue band is the SM prediction with 1σ errors of η 1,2,3 . The pink band corresponds to the experimental result on | K |, given in Table 4. If we require that our prediction is within the 1σ-region of the SM prediction, the lower bound from K is Λ uu 200 TeV.
Similarly, we can discuss the B q -B q mixing, where q is d or s. The contributions to these observables are also governed by Λ 2 uu . In the same manner, we can evaluate the B d -B d and B s -B s mixing. The mass differences of the B mesons in our model are given by (M Bq 12 ) SM is given by the top-loop contribution: The time-dependent CP violations, S ψK and S ψφ , are evaluated as follows including the new physics contributions:  where ϕ Bq is the phase of M  Table 4, and the central values are used in our analyses. Note that S ψK and S ψφ are experimentally measured well: S ψK = 0.691 ± 0.017 and S ψφ = 0.015 ± 0.035 [43].
As far as Λ uu is larger than 200 TeV, the deviations of the observables concerned with the B q -B q mixing are enough suppressed to evade the conflicts with the experimental results. For instance, the deviations of the mass differences from the SM prediction are at most 0.2 % and the deviations of S ψK and S ψφ are much smaller. We conclude that the strongest bound comes from K , that is shown in Eq. (66).

Lepton flavor violation
Next, we discuss the charged LFV processes in our model. Λ uu plays an important role in the LFV processes, as well. Those processes are induced by the LFV four-lepton couplings: These operators predict the LFV µ and τ decays; e.g., µ → 3e and τ → eµµ. The Wilson coefficient (C e 4 ) ij kl , which contributes to the LFV decays, depends on the Yukawa couplings as Note that the charged leptons are mass eigenstates in this description. In the limit that the light leptons are massless, the branching ratio of the LFV process is estimated as This description can be applied to j = k case, such as µ → 3e and τ → 3e. In the  e i → e + k e k e j (j = k) processes, the branching ratios are given by The current experimental bounds are summarized in Table 5. Y ν ij in the coefficients originates the LFV decays. As discussed in Sec. 2.1 and Appendix A, Y ν ij has large off-diagonal elements to reproduce the neutrino mixing. In order to estimate our predictions explicitly, we focus on the three cases assuming that the lightest neutrino mass and the Majorana phases are vanishing. In particular, Y ν 11 and Y ν 22 , that are relevant to the LFV processes involving light leptons, can be large in the cases (i) and (ii).
To begin with, we estimate our predictions for µ → 3e in the cases (i) and (ii). As shown in Fig. 7, Y ν 12 can be also relatively large in those cases, depending on the size of the Majorana neutrino mass. In order to avoid too large RG corrections for the Yukawa couplings, we expect all Yukawa couplings to be less than unit. In Fig. 2 2 ) = (0.01, 1, 0) on the right panel, respectively. In those plots, Λ uu and Λ ud satisfies Λ uu = 200 TeV and Λ ud = 100 (1) TeV on the thick (dashed) lines. The light blue and red bands depict the 20 % corrections from the RG and the threshold. We note that the green region is excluded by the SINDRUM experiment [72]   prospect proposed by the Mu3e experiment [73]. As we discuss below, Λ ud is constrained strongly by B s → µµ, so that we can conclude that our model is not excluded as far as Λ uu is larger than 200 TeV. Interestingly, the NO case with |M ν2,3 | |M ν1 | predicts the sizable branching ratio of µ → 3e, as far as the heavy Majorana neutrinos reside above O(10 13 ) GeV. Our predictions can be covered by the future experiment [73]. If the right-handed neutrinos are lighter, Y ν 33 is smaller and the off-diagonal element Y ν 12 is also suppressed. In the case with vanishing M −1 ν2 (Case (ii)), Y ν 22 can be large and the predictions are different from the case (i), reflecting the difference between the neutrino mass spectrums. If the active neutrino is in the IO, the future experiment may cover our region, depending on the heavy Majorana mass scale.
Similarly, the LFV τ decay, such as τ → eµ + µ, is also predicted in our model. Although the prediction tends to be small compared to the future prospect of the experiments, it would be worth estimating the size of our prediction. Fig. 3 shows our prediction of τ → eµ + µ in the case (i) (left) and (ii) (right). The parameters are the same as in Fig.  2, on the each line. As we see, our prediction is at most O(10 −14 ), that is much below the current experimental bound and the future prospect in Table 5. The other LFV τ decays are also suppressed as in this process.

Leptonic meson decays
In this section, we discuss the leptonic meson decays, based on the results on the ∆F = 2 processes and the LFV processes. In our model, the leptonic meson decays are given by the following operators, The Wilson coefficients (C de 4 ) kl ij with i = j are obtained from the neutral scalars exchanging:  [72] and the dashed green line is the future prospect [73].
C ij SM is the SM prediction. Using the operators and the coefficients, the leptonic B q (q = s, d) decays are described as follows: where R Bq is defined as In the B q decays, the SM prediction is described as where x n is defined as x n = m 2 n /M 2 W and η Y corresponds to the NLO correction: η Y = 1.0113 [74]. Note that the lepton flavor violating decay is vanishing in the SM.
In our model, sizable Y ν and relatively small Λ ud may largely deviate the SM predictions in the leptonic B decays. The SM predictions are consistent with the experimental results [76,77].  Table 3 and Table 4. The relevant parameter in Y ν is only Y ν 22 , so that we can discuss the predictions using We stress on that this bound does not depend on the size of Y ν 22 . We note that the LFV decay of B q meson is also predicted but the prediction is below the current experimental bound [43], as far as Λ uu is larger than 200 TeV.
By analogy with the B q decay, we can obtain the leptonic K decays, e.g., K → ll . In Ref. [79], the short-distance contribution of Br(K L → µµ) has been studied: Br(K L → µµ) ≤ 2.5 × 10 −9 . Our prediction of the branching ratio is about 4.6 × 10 −10 , when Λ ud = 2 TeV and Λ uu = 200 TeV. Note that our SM prediction is 3.7 × 10 −10 . Thus, our model is safe for this leptonic K decay, as far as the constraints from the leptonic B q decays and K are satisfied. The LFV decay of K L is also experimentally constrained as Br(K L → eµ) ≤ 4.7 × 10 −12 [43]. In our model, the predicted branching ratio is about 1.0 × 10 −13 when Λ uu = 200 TeV and Y ν 12 = 1. Thus, we conclude that the leptonic B decay gives the stronger bound on our model.
We can also discuss the constraint from D → ll , but the current experimental bound is not too strong to draw crucial bounds on Λ bounds on the scales in the up sector may come from the Drell-Yan process at the LHC [80]. In the case (i) with vanishing M −1 ν1 , we obtain the lower limit as Note that we can also find the lower bound on Λ du from this process at the LHC [80]: Here, we estimate the cross sections, using CALCHEP [81], and adopt the conservative bound: the lower limit on the contact interaction scale normalized by √ 4π is 40 TeV [80].

µ − e conversion process in nuclei
Among the operators in Eq. (44), we find the lepton flavor violating coupling that induces the µ − e conversion in nuclei: As discussed in Sec. 2.3, the LFV processes induced by (C ue 4 ) kl ij are governed by Λ ue uu , Λ ue du and Λ ue dd , while (C de 4 ) kl ij depends on Λ uu and Λ du . We have found that Λ uu and Λ du are strongly constrained by the ∆F = 2 processes and the leptonic B s decay. Then, we expect that the µ − e conversion process dominantly depends on (C ue 4 ) kl ij . The branching ratio of the µ − e conversion can be calculated in our model, based on the results in Ref. [82]. We also study the typical values of Br(µ Au → e Au) and Future prospects Br(µ Al → e Al) in our model. Fig. 5 shows our predictions of the µ − e conversion processes. The thick and dashed blue lines corresponds to our predictions in the NO case with Λ du = 100 TeV and 5 TeV, respectively. Y ν 12 is evaluated assuming the mass hierarchy in the case (i) with M −1 ν1 = 0. On these lines, Y ν 33 satisfies Y ν 33 = 0.1. Y ν 11 is a free parameter in the case (i), but it is not relevant to this LFV process. Λ The green region is excluded by the SINDRUM experiment: Br(µ Au → e Au) < 3 × 10 −13 [83]. As we see the left panel in Fig. 5, the dashed line with Λ du = 5 TeV is already covered by the experiment. This could be more severe than the bound on Λ ud from B s → µµ, depending on Y ν ij . Following Eq. (39), We note that |Λ ud | is expected to be the same order as |Λ du |.
The future prospect for Br(µ Al → e Al) is O(10 −16 ) at the COMET-II experiment [84] and denoted by the dashed line on the right panel in Fig. 5, so that the parameter region with O(0.1) Yukawa couplings is expected to be covered by the future experiment. The dotted lines shows the Λ

Hadronic τ decay
We give a comment on the hadronic τ decay. The four-fermi interactions involving τ lepton and light quarks induce the hadronic τ decay; e.g., τ → µη and τ → µπ. Following Refs. [85][86][87], we obtain the predictions of our model. The branching ratio of the hadronic decay of τ is given by where M qq is the light meson: M ss = η and M sd = K. R M dd is defined as in Eq. (77), replacing m Bq with the meson mass. The light quark masses in R M qq should be m d + m s for K and m u +m d +4m s for η, instead of m b +m q in R Bq . Br(τ → lπ) is also given in the same manner: R π = m 2 π /(m u + m d ). Note that |(C ue 4 ) lτ uu | 2 should be added to |(C de 4 ) lτ dd | 2 in this case. In those processes, the large dimensional parameters, such as Λ uu and Λ ud , sufficiently suppress the branching ratios.

Leptonic meson decays in association with the active neutrinos
So far, we concentrate on the flavor violation induced by the neutral scalar exchanging. In our model, the leptonic meson decays associated with the active neutrinos are also deviated by the charged Higgs exchanging at the tree level. The charged currents, generated by the charged Higgs interactions, are written down in the following form; ( C de 4 ) kl ij and ( C ue 4 ) kl ij depend on (C de 4 ) kl ij and (C ue 4 ) kl ij as In addition, there may be couplings involving right-handed neutrinos, when the Dirac neutrino scenario is considered. The descriptions of the operators involving right-handed neutrinos are shown in Appendix C. Finally, let us discuss the deviations of the leptonic meson decays in association with active neutrinos in the final state. In our scenario, the right-handed neutrinos are very heavy, and then the leptonic charged B q decay can be described as where Br SM (B q ) is the SM prediction of the branching ratio and C bq ±SM = 4G F V * qb / √ 2 is defined. The interference between the SM prediction and the charged Higgs contribution may be large. The branching ratio of B q → e l ν can be approximately evaluated as where Y e = V † P M N S y e V P M N S is defined. The branching ratios of K → e l ν can be also obtained replacing R Bq y b d with R K y s d . In these processes, we can obtain the bound on Λ dd .
Especially, K → eν gives the stringent bound on our model. The prediction for the process is depicted in Fig. 6. The vertical line is the ratio between our prediction and our SM prediction of Br(K → eν). The process, K → eν, is experimentally measured well: Br(K → eν) = (1.582 ± 0.007) × 10 −5 [43]. Then, the deviation from the new physics should be less than about 0.4 %. The dashed black line depicts the upper bound from the experimental result. The thick blue line corresponds to our predictions in the NO case with Λ dd = Λ (ue) dd = Λ (ue) du / |Y ν 11 | = 100 TeV. As we see, the lower bound on |Λ du |/ |Y ν 11 | is about 6 TeV, that is the same as the one derived from the µ−e conversion. Λ dd and Λ ue dd are not so relevant to this process, because of the suppression from y e e . Thus, we obtain the lower bound on Λ du and Λ (ue) du :

Semileptonic meson decays
We can find the deviations from the SM predictions in the semi-leptonic decays, induced by the scalar exchanging. The coefficient, (C de 4 ) kl sb , in Eq. (74) contributes to the b − s transition in association with two leptons in the final state. Recently, the LHCb collaboration has reported the excesses in the observables of B → K ( * ) ll (l = e, µ) processes. One interesting result is about the lepton flavor universal violation of B → K ( * ) ll. The semi-leptonic decay processes, however, encounter the constraints from the leptonic decays discussed in Sec. 3.3. In particular, the contribution in the leptonic decay is enhanced by the lepton mass, while the semi-leptonic is not. Thus, we can not expect the large deviation from the SM prediction in our model, as discussed in Ref. [88].
The semi-leptonic K decay also constrains our model. Following Ref. [89], we can estimate the contribution of (C de 4 ) kl sb to K → πµµ. Λ uu and Λ du are, however, strongly constrained by the K and the leptonic B decay, so that the new physics contribution to the branching ratio is at most O(10 −15 ), even if Y ν 22 is O(1). Next, we investigate B → D e l ν processes, where the excesses are reported in the observables concerned with the lepton universality. One interesting possibility to explain the excesses is charged Higgs particle with large flavor changing current with quarks and leptons. In fact, ( C de 4 ) kl bc and ( C ue 4 ) kl bc may be able to improve the discrepancy between the theoretical predictions and the experimental results, since Λ dd , Λ (ue) uu and Λ (ue) du in the coefficients can evade the strong bounds from the flavor physics. On the other hand, large new physics contribution is required to explain the discrepancy, compared to the SM prediction. In our model, ( C de 4 ) kl bc and ( C ue 4 ) kl bc are suppressed by V * cb so that it seems that it is difficult to enhance the lepton universality of this decay. In fact, we can estimate the deviation of the lepton universality in B → Dτ ν and it is less than a few percent even if Λ dd , Λ

Summary and Discussion
The LR model is one of the attractive extended SMs. The extension can resolve the strong CP problem, and the phenomenology has been widely studied so far. The new physics contributions are sufficiently large, if the LR breaking scale is around TeV scale. Thus, the new particles predicted by the LR symmetry have been surveyed by the LHC experiments. Based on the current experimental results, the LR breaking scale seems to be much higher than the EW scale. Then, we may conclude that the new physics scale is extremely high compared to the energy scale that the LHC can reach.
In this paper, we assume that the LR breaking scale is much higher than the EW scale. The LR breaking induces the Majorana mass terms for the right-handed neutrinos, and then the tiny neutrino masses could be generated by the seesaw mechanism. In such a scenario, the neutrino Yukawa couplings are expected to be large, so that the sizable Yukawa interactions may be crucial to test our scenario. Interestingly, the LR symmetry predicts the explicit correlations between the neutrino Yukawa couplings and the Yukawa couplings for the charged leptons. Then, we have proposed that the sizable lepton flavor violation is predicted by the neutrino Yukawa couplings, even if the right-handed neutrinos are integrated out at the very high energy scale. The flavor violation is induced by the interaction via extra Higgs doublets, which are predicted by the LR symmetry. The Higgs doublets are expected to reside around the intermediate scale, i.e. SUSY breaking scale, so that the induced flavor violation is expected to be enough large to test the extended SM.
The motivation of this paper is to demonstrate how large the contributions of the extra Higgs doublets to the flavor physics can be. We simply assume that the RG corrections and the threshold corrections are at most O(10) %. Then, we derive the explicit relations between the observed fermion mass matrices and the predicted FCNCs. There are actually several free dimensional parameters, but we can derive the explicit predictions for the physical observables of the flavor physics. K gives the strongest bound to our model, and the lower limit of the scale where the extra Higgs doublets live is fixed around O(100) TeV. The energy scale can be compatible to the SUSY breaking scale in the high-scale SUSY scenario. In the LFV process, the µ → 3e is crucial to our model and our model can be tested if the right-handed neutrinos are heavier than O(10 13 ) GeV. The interesting point of this model is that the contributions of the extra Higgs doublets to flavor violation processes become negligible if the right-handed neutrino masses are light. This means that we can obtain the upper bound on the right-handed neutrino mass scale that corresponds to the LR breaking scale.
We also investigated the other processes, such as the LFV τ decay and the meson decays in association with leptons in the final state. One of the stringent constraints comes from B q → ll and the µ−e conversion process in nuclei. The future experiment can cover our prediction, depending on the dimensional parameters in our model. The search for the contact interaction at the LHC is also important, depending on the parameters. The obtained results are summarized in Table 6.
As mentioned above, our analysis has not yet explicitly included the LR breaking effects that are induced by the RG correction and the threshold correction. We focus on the parameter region where the LR breaking effect is at most O(10) %. If the LR breaking effects are quantitatively taken into account, we could survey the parameter region that predicts the larger flavor violation couplings. In addition, the flavor violating processes radiatively induced have not been studied in this work, assuming the mass scale of the scalars are enough high. In order to take into account both the LR breaking effects and the radiative corrections to flavor physics, we need to consider the mass spectrum and the mixing among the scalars as well. The detailed study will be given near future.
Before closing our discussion, let us comment on the other setup. We can also consider the case that the extra Higgs doublets are generated by the SO(10) GUT model. In the SO(10) GUT, all matter fields unified into a 16-representational field in each generation. Then, the realization of the realistic Yukawa couplings is, for instance, achieved by introducing several Higgs fields, 10, 126 and 120 [12][13][14][15]: In this setup, the several Higgs doublets are predicted after the GUT symmetry breaking. we could expect that the Higgs doublets originated from 10, 126 and 120 have flavor violating Yukawa couplings, so that we could discuss flavor physics in the effective multi-Higgs doublet model in the same manner. This possibility would be deserved to be discussed more clearly.  Table 6: We summarize bounds from processes considered in this paper.
In addition, we introduce the following term to break SU (2) R × U (1) B−L : W SB = m(S) T r ∆ R ∆ R + w(S).
Note that S is gauge singlet. There are also SU (2) L triplet fields, ∆ L and ∆ L , to respect the LR symmetry. Although the physics involving the SU (2) L triplets is not discussed in our paper, the couplings are given by m L is the soft LR breaking term, so that ∆ L and ∆ L are integrated out at some scale. Then, the F-terms and D-terms relevant to the SU (2) R triplets and S are as follows: Here, we simply assume that the MSSM fields do not develop VEVs. Now, we require that the vacuum does not break EW symmetry, so that the vacuum alignment should be Thus, The condition, |v| = |v| and ξ = 0, leads the SUSY conserving vacuum that breaks SU (2) R × U (1) B−L to U (1) Y . Note that we may wonder how break the SUSY and how mediate the SU (2) R breaking effect. See, for instance, Ref. [50].

C The summary of the induced four-fermi couplings
In this section, we summarize the four-fermi couplings, that are not mentioned in Sec. 2.3. Assuming that the components of H A (H) in the (non-)supersymmetric LR models are degenerate, the heavy neutral scalar exchanging gives the four-fermi couplings, We can also find the four-fermi couplings involving the light neutrinos. They become important in the Dirac neutrino case. In the Dirac neutrino case, the couplings relevant to the meson decays are (C uν 4 ) ij kl and (C dν 4 ) ij kl are described as

D RG flow of the LR breaking effects
In the supersymmetric model, the running Yukawa couplings for quarks are described as In the same manner, the running Yukawa couplings for leptons are given by In our study, we do not touch the detail of the setup and discuss the phenomenology. Then, we approximately parameterize the LR breaking contributions, decomposing the wave function renormalization factors as follows: Then, the running Yukawa couplings are described as Y l b kj and Y a ij are interpreted as the couplings in the LR symmetric limit andŶ l b kj is the hermitian matrix. Z EW φ (φ = Q L , u R , d R , L, e R ) denotes the contribution from SU (2) L × U (1) Y gauge interactions. Then, the RG equations at the one-loop level are where c φ Y and c φ 2 are the constants given by U (1) Y and SU (2) L gauge symmetries. When we assume thatŶ a ij is the only interaction that leads the LR breaking contribution, the LR breaking factors satisfy the following one-loop RG equations,

E RG flow to 100 TeV
For predicting the phenomenology, we calculate the Wilson coefficients of the four-fermi interactions at the integrated Higgs mass scale, here we assume it to be 100 TeV. These Wilson coefficients consist of the Yukawa couplings and dimensional parameters Λ ab . Thus, we evaluate the Yukawa couplings at 100 TeV. We evaluate the Yukawa couplings at 100 TeV under following steps. First, we evaluate running masses at the M Z scale by using the central values of experimental measurements summarized in Table 1 and 3. To evaluate the quark running masses we use Mathematica package RunDec [90] and to translate the lepton pole masses to the running masses we follow Ref. [91]. We calculate the Yukawa couplings at 1 TeV, using the SM RG running at the two-loop level [92]. In our scenario all gaugino has around 1 TeV masses, so that we translate the MS scheme into the DR scheme at 1 TeV by following Ref. [93]. Then, the RG correction from 1 TeV to 100 TeV includes the gaugino contributions [94].