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We study the origin of fermion mass hierarchy and flavor mixing in the standard model, paying attention to flavor symmetries and fermion kinetic terms. There is a possibility that the hierarchical flavor structure of quarks and charged leptons originates from non-canonical types of fermion kinetic terms in the presence of flavor-symmetric Yukawa interactions. A flavor symmetry can be hidden in the form of non-unitary bases in the standard model. The structure of the Kähler potential could become a touchstone of new physics.

The origin of the fermion mass hierarchy and flavor mixing has been a big mystery, which comes from the fact that there is no powerful principle to determine Yukawa couplings in the standard model (SM). Yukawa couplings are expressed as general square matrices taking complex values, and they are diagonalized by bi-unitary transformations. Their eigenvalues become quark and charged lepton masses after multiplying the vacuum expectation value (VEV) of the neutral component in the Higgs doublet. The mixing of flavors occurs from the difference between mass eigenstates and weak interaction ones [

There have been many intriguing attempts to explain the values of physical parameters concerning fermion masses and flavor mixing matrices. Most of them are based on the top-down approach [

At present, any evidence from new physics except for neutrinos has not yet been discovered, and new physics might be beyond all imagination. Hence, it would be interesting to see flavor physics through a different lens, with the expectation that it offers some hints of a fundamental theory. We adopt several reasonable assumptions in a theory beyond the SM. (a)

Supposing that a flavor symmetry exists, we have several questions, such as “what type of symmetry exists?”, “what is the breaking mechanism?”, and “how is it hidden in the SM?” Here, we focus on the last one. There is a possibility that a flavor symmetry is hidden in the form of non-unitary bases, i.e., matter fields in the SM are transformed by non-unitary matrices. In

Our approach is summarized as follows. We suppose field variables respecting a flavor symmetry (that the corresponding transformation is realized by unitary matrices) and rewrite the Lagrangian density in the SM using such variables. We investigate the structure of terms violating the flavor symmetry, and attempt to conjecture physics beyond the SM. Although physics remains by a choice of field variables and representations, there can be a difference in the understandability of physical phenomena. For instance, in relativistic quantum mechanics, the Dirac representation of

In this paper, we study the origin of fermion mass hierarchy and flavor mixing in the SM, using the above-mentioned approach. We examine whether the hierarchical flavor structure of quarks and charged leptons can originate from specific forms of their kinetic terms in the presence of flavor-symmetric Yukawa interactions or not. We also propose a variant procedure based on the top-down approach.

The outline of this paper is as follows. In the next section, we review quark Yukawa interactions and a no-go theorem on flavor symmetries in the SM. We explore the origin of the hierarchical structure of quarks and charged leptons, paying attention to flavor symmetries and fermion kinetic terms in Sect. 3. In the last section, we give conclusions and discussions.

We review quark Yukawa interactions and the absence of exact flavor symmetries in the SM.

Let us start with the Lagrangian densities of the quark sector,

The Yukawa couplings are expressed by

Information on physics beyond the SM is hidden in

From Eqs. (

We find that there is a large hierarchy among the size of Yukawa couplings, and it has thrown up the big mystery of its origin. From Eq. (

We explain that there is no unbroken flavor-dependent symmetry respecting the

In the same way,

By multiplying both sides of each relation in Eq. (

From Eq. (

Then, we find that

Based on feasible assumptions in a theory beyond the SM that the field variables are not necessarily the same as those in the SM, there is a flavor symmetry broken down by the VEVs of flavons, and flavons couple to matter fields in matter kinetic terms dominantly, we rewrite the Lagrangian density in the SM using unitary bases of a flavor symmetry, investigate the structure of terms violating the flavor symmetry, and attempt to conjecture physics beyond the SM. Here, unitary bases mean sets of fields that are transformed by unitary matrices. For more details, see

We assume that a theory beyond the SM has a flavor symmetry^{1}

We denote unitary bases of a flavor group

The unitary bases of

Using new variables, the quark kinetic terms in the SM are rewritten as

Here,

We briefly give an alternative proof on the absence of exact flavor symmetries in the SM. Under the assumption that

From Eqs. (

Physical parameters, in general, receive radiative corrections, and the above values should be evaluated by considering renormalization effects and should match their counterparts at

To speculate on a theory of quarks beyond the SM, let us describe it by

When

As we have few hints on flavor symmetry, we study two examples, i.e., a case with a U(3) symmetry and that with an

In the case that a U(3) family symmetry is hidden in the SM, the Yukawa interactions are written by

Now, we conjecture the structure of the Kähler metric, based on Eqs. (

It is hard to derive semi-democratic forms (

Based on an

Let us re-examine a case with the

In the following, we examine whether the magnitudes of components in

By inserting the first relation of Eq. (

Using the formula ^{2}

If

In the same way, by inserting the second relation of Eq. (

Hence, we need

We study the lepton sector in the SM. In the absence of Majorana masses of right-handed neutrino singlets, the same argument as the quarks holds in the replacement of fields and couplings. Here, we consider the case with large Majorana masses and a flavor symmetry in a theory beyond the SM.

The lepton sector is described by the Lagrangian densities:

From Eq. (

We find that there is a hierarchy among charged lepton Yukawa couplings.

Using field variables

Here,

When a theory of leptons beyond the SM can be described by

In the case that the U(3) family symmetry exists and

We have developed a strategy taking the SM as a starting point. There are limitations on such a bottom-up approach. It is desirable to combine use of the bottom-up and top-down ones. Here, we propose a new procedure based on the top-down one, using knowledge and information obtained in the previous subsections.

First, we construct a theory with a flavor symmetry, extract fermion parts from it, write down a Lagrangian density as

Second, we diagonalize

Third, we change

Fourth, we diagonalize

Last, we examine whether the following relations hold or not:

Note that we need to diagonalize six Hermitian matrices in total by unitary transformations in our procedure. As explained in

As was described previously, we should consider renormalization effects when we match theoretical predictions to experimental data. We also need some modifications in the presence of mixing with extra particles, in the case with a large flavor symmetry and/or many matter fields.

We discuss whether realistic mass hierarchies and flavor mixing are realized or nor, based on a grand unification and a family unification.

First, we consider a model based on

In the case that

Next, we consider a model based on

Last, we consider the family unification based on a simple gauge group

We have studied the origin of fermion mass hierarchy and flavor mixing in the SM, using the bottom-up approach. The approach is based on the assumptions that the field variables in the SM are not necessarily the same as those in a theory beyond the SM, and there is a flavor symmetry and flavons couple to matter fields in the matter kinetic terms dominantly. We have supposed field variables respecting a flavor symmetry (unitary bases of a flavor symmetry) and rewritten the Lagrangian density in the SM using such variables. We have investigated the structure of terms violating the flavor symmetry, and conjectured physics beyond the SM. We have suggested that the hierarchical structure in the Yukawa interactions of quarks and charged leptons can originate from non-canonical matter kinetic terms in the presence of flavor-symmetric Yukawa interactions and a flavor symmetry can be hidden in the form of non-unitary bases in the SM. We have proposed a variant top-down procedure, using an insight and formulas obtained by our bottom-up approach.

In our approach, the problem of fermion masses and flavor mixing is deeply related to not only the determination of Yukawa coupling matrices but also the determination of matter kinetic terms and the VEVs of Kähler metric

We explain preceding works on the flavor physics based on matter kinetic terms, other than Refs. [

As fermion kinetic functions or the Kähler metric

Our approach would be useful as a complementary one to explore physics beyond the SM and it would be worth studying flavor physics model-dependently and/or independently by paying close attention to matter kinetic terms, because the structure of the Kähler potential could play a vital role as a key test of new physics.

This work was supported in part by scientific grants from the Ministry of Education, Culture, Sports, Science and Technology under Grant No. 17K05413.

Open Access funding: SCOAP

We give an illustration of a realization of U

The U

The previous U

We consider a SUSY model with the flavor symmetry

If

To obtain a semi-democratic form, we need

(a) Flavor-symmetric vacuum with

By inserting

(b) Broken vacuum of flavor symmetry with

From Eq. (

Then, by inserting these VEVs into Eq. (

For reference purposes, we explain an ordinary top-down procedure, starting from

First, we diagonalize the Kähler metrics by unitary transformations as

Second, we obtain the following Yukawa couplings from

Third, we diagonalize

Last, we examine whether the following relations hold or not:

^{1}The flavor structure of quarks and leptons has been studied intensively, based on various flavor symmetries [

^{2}In the case that