1 Introduction


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

Syuhei Iguro, Yu Muramatsu, Yuji Omura and Yoshihiro Shigekami

Department of Physics, Nagoya University, Nagoya 464-8602, Japan


Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan, Hubei 430079, Peoples Republic of China


Kobayashi-Maskawa Institute for the Origin of Particles and the Universe,

Nagoya University, Nagoya 464-8602, Japan


Theory Center, IPNS, KEK, Tsukuba, Ibaraki 305-0801, Japan


In the models with extended gauge symmetry, extra fields that couple to quarks and leptons are often required to realize the realistic Yukawa couplings. For instance, some fields, in which Higgs doublets are embedded, are introduced to the Grand Unified Theory (GUT) and the realistic Yukawa couplings consist of some Yukawa couplings between the SM fermions and the lightest mode among the Higgs doubles. The GUT symmetry, in many cases, breaks down at the very high scale, and the Higgs doublets survive up to the electroweak (EW) scale or some intermediate scales, as far as they do not gain masses from the GUT symmetry breaking. In this paper, we discuss the multi-Higgs doublet models, that could be effectively induced by the extended Standard Model (SM) where the SM fermions are unified. In particular, we focus on the predictions in the supersymmetric left-right (LR) model, where the down-type and the up-type Yukawa couplings are unified and the Yukawa couplings are expected to be hermitian. Besides, the heavy Higgs doublets have flavor changing couplings with quarks and leptons corresponding to the realization of the realistic fermion mass matrices. The LR symmetry is assumed to break down at high energy, to realize the Type-I seesaw mechanism, and the EW symmetry breaking is radiatively realized. In this case, the flavor-dependent interaction of the Higgs fields is one promising prediction, 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 propose that our scenario could be proved by the process, in one case where the neutrino mass hierarchy is normal.

1 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 GUT in the high-scale supersymmetry (SUSY) scenario. In the GUTs, the unification of the SM gauge symmetries is elegantly achieved by considering the , and 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.***See, for instance, Refs. [4, 5, 6]. In Refs. [2, 3], we find that the flavor violating interaction of the extra gauge boson predicted by the 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 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. It would be a non-trivial issue to make the mass hierarchy between the extra Higgs doublet and the SUSY particles. One scenario, that is consistent with the recent experimental results, is introduced in Ref. [16]. 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.For instance, the Higgs fields induced by the LR model are summarized in Refs. [18, 17]. The similar analysis has been done in the supersymmetric LR model [19, 20]. 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 [21].

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 the bottom-up approach, there are many free parameters in the type-III 2HDM, so that we are sure that we can discuss interesting physics assuming some specific Yukawa-coupling alignments. For instance, the flavor-violating Higgs decays [22, 23, 24, 25, 26], the magnetic dipole moment of muon [22, 23], the physics [28, 27, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39] and the top physics [40, 41] have been studied in this framework, motivated by the experimental results. This approach, however, raises a question what is the underlying theory of the type-III 2HDM with the specific Yukawa couplings, even if the model is confirmed at some experiments.

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.

2 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 , , , and are the SM quarks and leptons in the interaction base. denotes the Higgs doublet, and is defined as . The neutral component of gains the mass about 125 GeV after the EW symmetry breaking. , and are the Yukawa couplings described as


where is the CKM matrix and 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 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, 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.

2.1 The non-supersymmetric LR model

In the LR model, the right-handed up-type and down-type fermions are unified into doublet fields. Then the Yukawa couplings are described as


introducing one bi-doublet field, . and denote and . can be decomposed as , where and are the 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 is defined as . 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 or 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 and should be sizable. In addition, the VEV of should not be the same as the VEV of . This means that the 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 and as


where only develops the non-vanishing VEV:


is defined as a real value: GeV. Then, we can derive a simple relation between and from the requirement for the realistic Yukawa couplings:


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 exchanges and ( and ) is often required in addition to . Then, and are hermitian and the strong CP problem can be resolved in this setup [1]. Besides, in Eq. (2) can be expected to be identical to the CKM matrix, . 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, originates in . 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 in Eq. (1), there are Yukawa couplings of :


In this notation, the Yukawa couplings of are described as


As discussed in Sec. 3, we cannot evade the tree-level FCNCs induced by these Yukawa couplings. If the 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 dependence in Eq. (11).

In the lepton sector, we can write down the Yukawa couplings for both of and , by analogy with the quark sector. After the 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 and as


The Yukawa couplings involving the heavy scalar are given by


If there is no Majorana mass term for the right-handed neutrino, is described as


using the PMNS matrix, . Note that is identical to , when is vanishing and the LR symmetry is assumed. In this case, the active neutrino is the Dirac fermion and the tiny neutrino mass, , suppresses . 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 . 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 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 symmetry breaking is, for instance, achieved by introducing one adjoint field . 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 non-vanishing VEV of . If we assume that the VEV of 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 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.

MeV [42] 0.321 [43]
MeV [42] (NO) 0.430 [43]
1776.86 MeV [42] (IO) 0.596 [43]
MeV [42] (NO) 0.02155 [43]
MeV [42] (IO) 0.02140 [43]
eV [43] (NO) 1.40 [43]
(NO) eV [43] (IO) 1.44 [43]
(IO) eV [43]
Table 1: The input parameters for leptons in our analysis. The notation for the lepton mixing is following Ref. [42]. NO (IO) is short for the normal (inverted) ordering neutrino mass hierarchy. The central values are used in our study.

After the breaking, the Majorana mass terms, , would be effectively generated as


Assuming that the magnitude of is very large compared to the EW scale, the tiny neutrino masses of the active neutrinos are given by


In our base, the Yukawa coupling for the charged lepton, , is in the diagonal form, so that is described as


where is defined as


Thus, is not identical to , unless is in the diagonal form.

Another important point is the relation between and . If the LR symmetry is assumed to be conserved at high energy, would be the hermitian matrix if the radiative corrections can be safely ignored.§§§There are also other contributions to the LR breaking effects: e.g., the one from the triplet which is introduced to respect the LR symmetry. In such a case, we can simply estimate the sizes of and . In Appendix A, and are shown assuming the mass hierarchy in . We assume that is a hermitian matrix in the base that is in the diagonal form. We denote as in the following. Then, we consider the three cases:

  1. , ,

  2. , ,

  3. , .

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.,

  1. does not depend on in the case (i),

  2. does not depend on in the case (ii),

  3. does not depend on in the case (iii).

In Fig. 7, the predictions for 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 are predicted, depending on the mass hierarchy of . Note that the Majorana phase and the lightest neutrino mass are vanishing in Fig. 7.

would be large if the Majorana mass is very heavy. The Majorana mass term is originated from the breaking, so that the high 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.

2.2 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, (), are introduced, in order to realize the realistic Yukawa coupling. This means that we obtain four Higgs doublet fields after the symmetry breaking. The third term effectively generates the Majorana mass term for the right-handed neutrino, and the last term corresponds to the -term of the Higgs superfields. In our analysis, the Yukawa couplings, and , are defined in the base where is in the diagonal form: .

Let us consider the scenario that develops the very large VEV for the very heavy right-handed neutrino. This can be easily realized by introducing a singlet field, :


We can find the supersymmetric vacuum that breaks down to . This type of model has been proposed in Ref. [45]. The other setup has been discussed in Ref. [46].See also [47, 48]. The matter contents and the charge assignment are summarized in Table 2.

Table 2: Matter contents and the charge assignment of the model.

Note that can not couple to at the renormalizable level because of the symmetry, so that the Higgs doublets from do not gain the large masses from the VEV of . We could expect that the breaking effect is mediated by the mediators for the SUSY breaking effects. Note that we may wonder how SUSY is broken and how the breaking effect is mediated. See, for instance, Ref. [49]. In our study, we simply assume that the breaking effect appears in the soft SUSY breaking terms and discuss the mass terms for the Higgs doublets which are associated with the breaking, below.

In this supersymmetric LR model, there are two up-type Higgs doublets and two down-type Higgs doublets originated from . 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


where is a hermitian matrix and () denotes (, , , )=(, , , ), respectively. In this notation, is given by


Here, and are the supersymmetric mass eigenstates: . The other parameters in Eq. (25) denote the soft SUSY breaking parameters. In order to realize the EW symmetry breaking, sizable is required. In addition, and should satisfy some conditions to cause the EW symmetry breaking and to avoid the unbounded-from-below 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 of are aligned as


where is the four-dimensional vector that satisfies . Finding the directions orthogonal to , we define another base for the Higgs doublets:


where satisfies


In this base, only develops a non-vanishing VEV as shown in Eq. (8).

would correspond to the mass eigenstate given by : . The other states, , could be also interpreted as the mass eigenstates of , so that the mass squared for the Higgs fields is described as


where the unitary matrix, , is defined as . Note that the exact masses of the heavy scalars would be deviated from , because of the contributions of 4-point couplings, e.g. , to the masses squared. These contributions are, however, suppressed, compared to , if is much larger than the EW scale. Then, we discuss the phenomenology, assuming are the mass eigenstates with . The mass differences among the scalars in each are negligible, in this assumption.

Now, we write down the Yukawa couplings involving and . The Yukawa couplings of correspond to the SM Yukawa couplings, e.g., and . The relation between 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 is complex. When the Yukawa couplings of with quarks are defined as


and are related to and as


where is defined. Note that is vanishing in the symmetric limit. Similarly, the Yukawa couplings of for the leptons,


are related to and :


Compared to the non-SUSY case in Sec. 2.1, there are many parameters in the Yukawa couplings of the heavy scalars: and . In addition, there are three mass parameters, . 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.

2.3 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 in the supersymmetric LR models are degenerate. Then, the couplings by the heavy neutral scalar exchanging are given as follows:


where and denote , , or , respectively. Using the relation between and , 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, , we write down the down-type quark couplings in Eq. (35) as


where is in Eq. (2) and satisfies . We change the base of the down-type quark into the mass base denoted by , so that corresponds to the coefficient of . Note that are related to and as follows:


When we discuss the flavor violating processes, such as the processes, we find that is only relevant in our model according to Eq. (36). As shown in Eq. (37), is described as


where denotes the vector orthogonal to : , , , ), , , ) satisfying . would denote one mass eigenstate of whose mass squared is , so that would be described by one linear combination of the other three mass eigenstates of with the masses, . Then, is positive and could not be vanishing, as far as all are not extremely large. As discussed in Sec. 3.1, large deviations from the SM predictions are actually derived in the processes.

The four-fermi coupling in the charged lepton sector has the structure similar to the one in the down-type quark sector. Replacing and with and respectively, , that is the coefficient of , is given by


Note that 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, and , are simply obtained, replacing with


The coefficients of the other four-fermi interactions, that induce the LFV decays of mesons, are given by