Yukawa textures from non-invertible symmetries

We start with the $M=3$ case. In this case, there are two classes to distinguish states: $[g^{0}]$ and $[g^{1}]$, each which corresponds to three generations of left- and right-handed fermions.

When the representation of the Higgs field corresponds to $[g^{0}]$, the Yukawa matrix is described by

Throughout this paper, we denote by $a,b,c,...,g$ complex numbers unless specified otherwise. The ordering of the fermions is $[g^{0}]$, $[g^{1}]$ for both rows and columns. As a result, this case leads to a diagonal matrix. The result holds for a conventional $\mathbb{Z}_{3}$ symmetry as well as a $\mathbb{Z}_{2}$ symmetry. In the following, we will discuss the case where the Higgs field has the different representation, $[g^{1}]$, but the order of the generations is still the same for both row and column, i.e., $[g^{0}]$, $[g^{1}]$. In this case, we find

Obviously, we can not derive this pattern by assuming $U(1)$ symmetry or a conventional $\mathbb{Z}_{M}$ for any $M$. Suppose that two left-handed (right-handed) fermions have $U(1)$ $Q_{1}$ and $Q_{2}$ ($q_{1}$ and $q_{2}$). Non-vanishing (1,2), (2,1), and (2,2) entries require

where this sum may correspond to the $U(1)$ charge of the Higgs fields. This equation leads to $Q_{1}=Q_{2}$ and $q_{1}=q_{2}$, but that is not consistent with the forbidden (1,1) entry. Thus, there is no solution of $U(1)$ charges leading to the above pattern. Also, this pattern is not consistent with a conventional $\mathbb{Z}_{M}$ symmetry for any $M$.

For $M=3$, we have just two class: $[g^{0}]$ and $[g^{1}]$. Two generations among three generations of fermions are degenerate in the class $[g^{k}]$. When $M=3$, Yukawa matrix of three generations can be obtained by

including possible permutations of rows and columns, or

including possible permutations of rows and columns. The former has 4 textures zeros, but this pattern can be derived by a conventional Abelian symmetry. The latter has one texture zero, and this pattern cannot be derived by a conventional symmetry.

Let us examine the $M=4$ case. In this case, there are three classes to distinguish the states: $[g^{0}]$, $[g^{1}]$, and $[g^{2}]$, which correspond to three generations of left- and right-handed fermions.

When the representation of the Higgs field corresponds to $[g^{0}]$, the Yukawa matrix is described by

The ordering of the generations is $[g^{0}]$, $[g^{1}]$, and $[g^{2}]$ for both rows and columns. The selection rule of the non-invertible symmetry leads to a diagonal matrix. The result holds for a conventional $\mathbb{Z}_{4}$. In the following, we will discuss the case where the Higgs field has a different representation such as $[g^{1}]$ and $[g^{2}]$, but the order of the generations is still the same for both rows and columns, i.e., $[g^{0}]$, $[g^{1}]$, and $[g^{2}]$. Yukawa matrices can be written by

when the Higgs field corresponds to $[g^{1}]$ and $[g^{2}]$, respectively.

The pattern $Y_{[g^{2}]}$ is equivalent to $Y_{[g^{0}]}$ by exchanging rows and columns. For $Y_{[g^{1}]}$, the restriction is more severe than $\mathbb{Z}_{2}$ symmetry, and some elements are zero. For example, the (1,3) and (3,1) components of $Y_{[g^{1}]}$ are allowed in $\mathbb{Z}_{2}$ symmetry, but not allowed in this non-invertible symmetry.

In this case, there are three different representations: $[g^{0}]$, $[g^{1}]$, and $[g^{2}]$, which correspond to three generations of left- and right-handed fermions. The notation is the same as above.

When the representation of the Higgs field corresponds to $[g^{0}]$, the Yukawa matrix is given by

The ordering of the generations is $[g^{0}]$, $[g^{1}]$, $[g^{2}]$ for both rows and columns. As a result, this case leads to a diagonal matrix. The result holds for a conventional $\mathbb{Z}_{5}$ or $\mathbb{Z}_{3}$. Although we will discuss the case where the Higgs field has other representations, the ordering of the generations is still $[g^{0}]$, $[g^{1}]$, $[g^{2}]$ for both rows and columns.

Next, we deal with the case where the representation of the Higgs field is given by $[g^{1}]$, which leads to the following configuration of the Yukawa matrix:

Remarkably, this reproduces the NNI form Branco:1988iq ; Branco:1994jx ; Branco:1999nb . The right-bottom $(2\times 2)$ submatrix is the same as $Y_{[g^{1}]}$ for $M=3$, which cannot be simply realized by a conventional symmetry. Similarly, one can not derive the NNI type simply by a conventional symmetry. Then, it is necessary to extend models by introducing new particles such as multi-Higgs. (See, for example, Ref. Kikuchi:2022svo .) Hence, the non-invertible symmetry allows us to derive mass matrices that cannot be derived by the conventional symmetry.

We move to the case where the representation of the Higgs field is $[g^{2}]$. The Yukawa matrix is described by

This pattern is also difficult to derive from the conventional symmetry argument. Note that there is a degree of freedom to change the order of the left-handed and right-handed fermions.

In this case, there are four types of representations, i.e., $[g^{0}]$, $[g^{1}]$, $[g^{2}]$, $[g^{3}]$. For the representation of Higgs field fixed as $[g^{k}]$, four left-handed and right-handed states can have the following patterns of couplings:

The ordering of both rows and columns is $[g^{0}]$, $[g^{1}]$, $[g^{2}]$, $[g^{3}]$. The patterns, $Y_{[g^{0}]}$ and $Y_{[g^{3}]}$ are equivalent by permutations of rows and columns. Also, $Y_{[g^{1}]}$ and $Y_{[g^{2}]}$ are equivalent by permutations.

In three-generation models, we pick up three generations from four rows and columns in the above matrices. From $Y_{[g^{0}]}$ and $Y_{[g^{3}]}$, we can obtain $(3\times 3)$ diagonal matrix including its permutations and the matrices with $\det Y=0$. From $Y_{[g^{1}]}$ and $Y_{[g^{2}]}$, we can obtain the following $(3\times 3)$ Yukawa matrices:

and their possible permutations of rows and columns as well as $(3\times 3)$ Yukawa matrices with $\det Y=0$. The former corresponds to $Y_{[g^{1}]}$ for $M=4$.

Here, the Yukawa matrices for $M=7$ are shown. In this case, there are four types of representations, i.e., $[g^{0}]$, $[g^{1}]$, $[g^{2}]$, $[g^{3}]$. We assign three of these combinations to the left-handed and right-handed fermions with 3 generations. We present four kinds of the Yukawa matrix in the ordering of $[g^{0}]$, $[g^{1}]$,$[g^{2}]$, $[g^{3}]$ as follows:

Notation is the same as the above, and we impose that the representations of Higgs fields are $[g^{0}]$, $[g^{1}]$, $[g^{2}]$ and $[g^{3}]$, respectively. $Y_{[g^{1}]}$ and $Y_{[g^{3}]}$ are equivalent by permutations of rows and columns.

In the three-generation models, we pick up three generations from four rows and columns in the above matrices. From $Y_{[g^{0}]}$, we can obtain $(3\times 3)$ diagonal matrix including its permutations of rows and columns and the matrices with $\det Y=0$. From $Y_{[g^{1}]}$ and $Y_{[g^{3}]}$, we can obtain the following $(3\times 3)$ Yukawa matrices:

and their possible permutations of rows and columns as well as $(3\times 3)$ Yukawa matrices with $\det Y=0$. From $Y_{[g^{2}]}$, we can obtain the following $(3\times 3)$ Yukawa matrices:

and their possible permutations of rows and columns as well as $(3\times 3)$ Yukawa matrices with $\det Y=0$.

Similarly, we can study the cases with larger $M$.

In the NNI basis, the quark Yukawa matrices are given as

where the coefficient of each element is complex in general ^∥∥∥Among ten phases, eight phases are removed by the redefinition of the quark fields. . This texture of Yukawa matrices is found in $Y_{[g^{1}]}$ of Eq. (3.9). That is the case of $M=5$ with Higgs representation $[g^{1}]$. Since this basis is also obtained by choosing a suitable weak basis transformation from general $3\times 3$ Yukawa matrices of down-type and up-type quarks Branco:1988iq , the Yukawa matrices of Eq. (4.1) is completely consistent with observed CKM matrices and quark masses. Thus, the non-invertible symmetry is compatible with the NNI basis. On the other hand, the complicated set-up is required to obtain the NNI basis in the conventional discrete flavor symmetry Kikuchi:2022svo .

On the other hand, a systematic study of texture zeros has been presented for the down-type quark mass matrix in the basis of diagonal up-type quark mass matrix in Ref. Tanimoto:2016rqy from the standpoint of “Occam’s Razor approach” Harigaya:2012bw , in which a minimum number of parameters is allowed. The down-type quark mass matrix was arranged to have the minimum number of parameters by setting three of its elements to zero, while at the same time requiring that it describes successfully the CKM mixing and CP violation without assuming it to be symmetric or hermitian.

We easily obtain a set of quark mass matrices:

where $M=5$ with the representation of the Higgs $H_{D}$, three generations of left-handed quarks, three generations of right-handed quarks for the down-type quarks being respectively assigned as $[g^{2}]$, $\{[g^{0}],[g^{1}],[g^{2}]\}$, $\{[g^{1}],[g^{1}],[g^{2}]\}$, while the representation of the Higgs $H_{U}$, and three generations of right-handed quarks for the up-type quarks are respectively assigned as $[g^{0}]$, $\{[g^{0}],[g^{1}],[g^{2}]\}$. The texture of Eq. (4.2) is equivalent to $M_{d}^{(7)}$ in the Appendix A of Ref. Tanimoto:2016rqy . However, this texture is excluded by the observed CKM mixing element $V^{\rm CLM}_{ub}$ in the strict sense although it predicts $V^{\rm CKM}_{ub}={\cal O}(\lambda^{3})$ consistent with the order of the observation where $\lambda$ is the Cabibbo angle.

The other one is:

where $Y_{D}$ corresponds to $M_{d}^{(2)}$ in Ref. Tanimoto:2016rqy . The texture of $Y_{D}$ is obtained in the case with $M=5$ by setting the representation of the Higgs $H_{D}$, three generations of left-handed quarks, three generations of right-handed quarks $[g^{2}]$, $\{[g^{1}],[g^{2}],[g^{2}]\}$, $\{[g^{2}],[g^{1}],[g^{0}]\}$, respectively. On the other hand, $Y_{U}$ is obtained by taking the representation of the Higgs $H_{U}$ being $[g^{0}]$ with the same assignments of up-type quarks as the down-type quarks then, $Y_{U}$ is not diagonal because of the degeneracy of the representation of the second and third generations. Since $|d_{U}|\gg|c_{U}|$ due to quark mass hierarchy, this set works well in the experimental data of the CKM matrix.

We discuss the texture zeros in the lepton sector. The neutrino mass matrix was investigated in the framework of the seesaw mechanism based on the Occam’s Razor approach Kaneta:2016gbq . Imposing four zeros in the Dirac neutrino Yukawa matrix gives the minimum number of parameters needed for the observed neutrino masses and lepton mixing angles, while the charged lepton Yukawa matrix and the right-handed Majorana neutrino mass matrix are diagonal ones. The low-energy neutrino mass matrix has only seven physical parameters. Among them, the CP phases are two Majorana phases which appear in the Majorana mass matrix. The matrices are given as:

where all parameters are real.

Those textures are obtained in the non-invertible symmetry. The Dirac neutrino texture is $Y_{[g^{1}]}$ in Eq. (3.9), where $M=5$ with Higgs $H_{U}$ representation $[g^{1}]$. On the other hand, the diagonal charged leptons are given by putting $M=5$ with Higgs $H_{D}$ representation $[g^{0}]$. The right-handed Majorana mass matrix is guaranteed to be diagonal.

The texture is completely consistent with the observed neutrino mass squared differences and three mixing angles with the normal mass hierarchy. The CP-violating Dirac phase is also consistent with recent data of NuFIT5.3 (2024) Esteban:2020cvm .