# The Minimal Flavor Structure from Decomposition of the Fermion Mass Matrix

Ying Zhang 111E-mail<EMAIL_ADDRESS>Corresponding author.
Institute of Theoretical Physics, School of Physics, Xi’an Jiaotong
University, Xi’an, 710049, China

###### Abstract

The minimal flavor structures for both quarks and leptons are proposed to
address fermion mass hierarchy and flavor mixings by bi-unitary decomposition
of the fermion mass matrix. The real matrix ${\bf M}_{0}^{f}$ is completely
responsive to family mass hierarchy, which is expressed by a close-to-flat
matrix structure. The left-handed unitary phase ${\bf F}_{L}^{f}$ provides the
origin of CP violation in quark and lepton mixings, which can be explained as
a quantum effect between Yukawa interaction states and weak gauge states. The
minimal flavor structure is realized by just 10 parameters without any
redundancy, corresponding to 6 fermion masses, 3 mixing angles and 1 CP
violation in the quark/lepton sector. This approach provides a general flavor
structure independent of the specific quark or lepton flavor data. We verify
the validation of the flavor structure by reproducing quark/lepton masses and
mixings. Some possible scenarios that yield the flavor structure are also
discussed.

family mass hierarchy; flavor mixing; Yukawa couplings

###### pacs:

12.15.Ff, 12.15.Hh, 14.60.Pq

##  I Introduction

Although the standard model (SM) has been proven as a successful description
of strong and electroweak interactions, there are some remaining unknown
issues underlying fermion flavor structure Feruglio:2015jfa ; Raidal:2008jk ;
ZZX2020PR . In the quark sector of the SM, all flavor information is encoded
into the complex Yukawa couplings ${\bf y}_{ij}^{f}$ (for $f=u,d$). After
electroweak symmetry breaking, quarks obtain masses ${\bf
M}_{ij}^{f}=v_{0}{\bf y}_{ij}^{f}/\sqrt{2}$ with Higgs vacuum expectation
value (VEV) ${v_{0}}/{\sqrt{2}}$. Complex Yukawa couplings ${\bf y}_{ij}^{u}$
and ${\bf y}_{ij}^{d}$ include $2\times(9+9)$ free parameters. However, they
correspond to only 10 phenomenological observables: 6 quark masses, 3 CKM
mixing angles and 1 CP violation (CPV). The redundancy implies that there must
be an unknown flavor structure hidden in the SM Yukawa couplings 2019Zupan .
Another insight comes from the similar flavor structure in the quark and
lepton sector. In the minimal extension lepton sector with 3 Dirac neutrinos,
the story of quark flavor is restated: charged lepton and neutrino masses are
provided from complex Yukawa terms; the total number of observables is also
10, namely, 6 lepton masses, 3 PMNS mixing angles and 1 Dirac CPV; and a
nonvanishing CPV also exists in neutrino mixings. This information inspires us
to seek the common flavor structures behind quarks and leptons that give rise
to fermion mass hierarchy and flavor mixings in a unified manner.

There are two popular approaches to investigate flavor structure. The first
approach is started by constructing a new physics model based upon some
symmetry or dynamical principle, which is referred to as top-down. Many models
and theories have addressed the flavor issue through this approach, including
examples based on discrete symmetry, GUT, extra dimension theory and string
theory 2016PetcovNPB ; Girardi2016NPB ; ZZXing2014PLB ; Dias2018PRD ;
King2016RPP ; Archer2012JHEP ; Nima2000PRD ; Bhattacharyya2012PRL ;
Martinez2018PRD . Extra particles including those of these models, such as
flavons, sterile neutrinos and extra Higgs, play roles of flavor symmetry
breaking and/or origination of the CPV. With the extension of the SM, the
particles, especially around the TeV scale, need be arranged subtly to
maintain the balance between new physics predictions and the SM observations.
Some challenging and complex experiments have been designed and even conducted
in the search for possible effects beyond the SM. Currently, however, there is
still no decisive signal. Another approach is the bottom-up approach, which is
inspired by the relation between mass and/or mixing angle values. These
experimental data can be interpreted as powers of one or more small flavor
breaking parameters, such as bi-maximal mixing (BM) and tri-bi-maximal mixing
(TBM) in lepton mixings 2012AltarelliFP ; 2012ZHZhaoPRD .

In addition to addressing hierarchal masses and flavor mixings with
nonvanishing CPV, an expected flavor structure should provide a common
mechanism that functions well in both the quark sector and lepton sector.
Furthermore, the mechanism does not depend on the special numerical results.
The most desired flavor structure should include only 10 parameters in the
Lagrangian corresponding to just 10 observables in the quark/lepton sector,
which is the so-called the minimal flavor structure. This struture can provide
a phenomenological platform to build flavor models and understand flavor
nature. In the paper, we adopt an approach that is different from top-down or
bottom-up strategies. Flavor structure is directly analyzed from bi-unitary
decomposition of fermion mass matrix. The method is rooted in the success of
the SM. Currently, almost all 10$+$10 independent parameters in the quark and
lepton sectors have been measured, except the Dirac CPV in neutrino mixing.
The known data provide a good checkpoint for the validation of possible flavor
structure behind fermion mixing and/or mass distribution.

Starting from the SM fermion mass matrix, it can be decomposed into the
production of a real matrix ${\bf M}_{0}^{f}$ and left- and right-handed
unitary matrix ${\bf F}_{L,R}^{f}$. We show that these matrixes fill different
roles in flavor physics. ${\bf M}_{0}^{f}$ is completely responsible for
fermion mass. ${\bf F}_{L}^{f}$ provides a complex phase in flavor mixing
transformation, which causes a CPV in CKM/PMNS. The minimal flavor structure
can be realized by parameterizing ${\bf F}_{L,R}^{\dagger}$ and ${\bf
M}_{0}^{f}$ to address mass hierarchies and flavor mixings. Inspired by
hierarchal eigenvalues that arise from a flat matrix with nondiagonal
corrections, we propose that the family mass hierarchy problem can be resolved
by a close-to-flat matrix structure in flavor space. The similar structure has
also been proposed as a quasi-democratic pattern from discrete symmetry with
complicated parameterization Fritzsch2017CPC ; Sogami1998PTP ; Fukuura1999PRD
. In the close-to-flat matrix, only three real parameters are needed. As a
byproduct, family universal Yukawa coupling can arise naturally
Shinohara1998PTP ; DSDu1993MPLA ; Branco1995PLB ; Teshima1997PTP ; it
determines the sum of a family mass. We also study the role of the unitary
matrix ${\bf F}_{L,R}^{f}$ in flavor mixing. Because of left-handed weak
$SU(2)_{L}$ symmetry, the CKM only requires left-handed ${\bf F}_{L}^{f}$ and
is blind to right-handed ${\bf F}_{R}^{f}$. The minimal parameterization of
${\bf F}_{L}^{f}$ requires two independent phases. We present that ${\bf
F}^{f}_{L}$ can be explained as a quantum phase between Yukawa interaction
states and weak interaction states. This provides a new origin of the CPV and
enables us to understand the role of CPV in many interesting issues of
particle physics and cosmology Branco2012RMP . We also generalize the minimal
flavor structure to the lepton sector with the normal hierarchy Dirac
neutrinos. All formulas and conclusions can be generalized from quarks to
leptons directly.

The remainder of the paper is organized as follows: in the next section, the
SM fermion mass matrixes are decomposed into bi-unitary form. The roles and
properties of ${\bf M}_{0}^{f}$ and ${\bf F}_{L,R}^{f}$ are analyzed. In Sec.
III, the minimal parameterization of ${\bf M}_{0}^{f}$ is proposed and the
physical roles of parameters are analyzed. All quark and lepton masses are
reproduced to fit experimental data. In Sec. IV, we focus on CPV in flavor
mixing. Only two phases are involved in ${\bf F}_{L,R}^{f}$. The physical
means of these phases are discussed as a quantum superposition effect. CKM and
PMNS are also reproduced successfully, which proves the validity of the
mechanism. In Sec. V, we discuss a possible new physics model that can
generate the minimal flavor structure. Finally, a short conclusion and
discussion are provided.

##  II Bi-unitary decompositon of the fermion mass matrix

In the SM, the quark Yukawa term is

$\displaystyle\mathcal{L}_{Y}=y^{d}_{ij}\bar{\Psi}_{L}^{i}\Phi{\psi^{d,j}_{R}}+y^{u}_{ij}\bar{\Psi}_{L}^{i}\tilde{\Phi}{\psi^{u,j}_{R}}+H.c.$
(1)

with generation index $i,j$, left-handed quark doublet $\Psi_{L}^{i}$, right-
handed up-type quark $\psi^{u,i}_{R}$, down-type quark ${\psi^{d,i}_{R}}$ and
Higgs doublet $\Phi$. After electroweak symmetry breaking, the Higgs obtains
VEV $\langle{\Phi}\rangle=v_{0}/\sqrt{2}$ and the quark mass matrix becomes

${\bf M}_{ij}^{f}=\frac{v_{0}}{\sqrt{2}}{\bf y}_{ij}^{f},~{}~{}f=u,d.$

Making bi-unitary transformations

$\displaystyle\psi_{L}^{f}=({\bf
U}^{f}_{L})^{\dagger}\psi^{f,m}_{L},~{}~{}~{}~{}\psi_{R}^{f}=({\bf
U}^{f}_{R})^{\dagger}\psi^{f,m}_{R},$ (2)

to mass eigenstate $\psi^{f,m}_{L,R}$, ${\bf M}^{f}$ is diagonalized as

$\displaystyle{\bf M}^{f}\rightarrow{\bf U}_{L}^{f}{{\bf M}^{f}}({\bf
U}_{R}^{f})^{\dagger}={\rm diag}(m^{f}_{1},m^{f}_{2},m^{f}_{3})$ (3)

In mass basis, the weak charged current interaction becomes

$\displaystyle\frac{g}{\sqrt{2}}\bar{\psi}^{u,m}_{L}\gamma^{\mu}{\bf
U}_{CKM}\psi^{d,m}_{L}W_{\mu}^{+}+H.c.$ (4)

The CKM matrix is determined by bi-unitary transformations as ${\bf
U}^{u}_{L}{{\bf U}^{d}_{L}}^{\dagger}$. A conventional parameterization of CKM
is ClauPRL1984

$\displaystyle{\bf U}_{CKM}=\left(\begin{array}[]{ccc}1&0&0\\\
0&c_{23}&s_{23}\\\
0&-s_{23}&c_{23}\end{array}\right)\left(\begin{array}[]{ccc}c_{13}&0&s_{13}e^{-i\delta_{CP}}\\\
0&1&0\\\
-s_{13}e^{i\delta_{CP}}&0&c_{13}\end{array}\right)\left(\begin{array}[]{ccc}c_{12}&s_{12}&0\\\
-s_{12}&c_{12}&0\\\ 0&0&1\end{array}\right)$

with $c_{ij}\equiv\cos\theta_{ij},s_{ij}\equiv\sin\theta_{ij}$.
(Alternatively, Wolfenstein parameterization can be defined in
WolfensteinPRL1983 .)

The above formula can also be generalized to the lepton sector with three
right-handed Dirac neutrinos. The lepton mixing PMNS matrix has form ${\bf
U}_{PMNS}={\bf U}_{L}^{\nu}{{\bf U}^{e}_{L}}^{\dagger}$. Because nonvanishing
CPV in both CKM and PMNS has been confirmed by experiments, the complex phase
requires complex Yukawa couplings in the SM.

Using eq. (3), a general decomposition of ${\bf M}^{f}$ can be expressed in a
bi-unitary form

$\displaystyle{\bf
M}^{f}=({\bf{F}}_{L}^{f})^{\dagger}{\bf{M}}_{0}^{f}{\bf{F}}_{R}^{f}$ (6)

with unitary matrix ${\bf F}_{L,R}^{f}$. Here ${\bf M}_{0}^{f}$ can be defined
from diagonalized mass matrix by a real orthogonal transformation ${\bf
U}_{0}^{f}$

$\displaystyle{\bf U}_{0}^{f}{\bf{M}}_{0}^{f}({\bf U}_{0}^{f})^{T}={\rm
diag}(m_{1}^{f},m_{2}^{f},m_{3}^{f}).$ (7)

The bi-unitary transformations in Eq.(2) can be constructed by

$\displaystyle{\bf U}_{L}^{f}={\bf U}_{0}^{f}{{\bf{F}}_{L}^{f}},~{}~{}~{}{\bf
U}_{R}^{f}={\bf U}_{0}^{f}{{\bf{F}}_{R}^{f}}.$ (8)

Thus, the real matrix ${\bf{M}}_{0}^{f}$ completely encodes fermion mass
information.

In the quark sector, flavor mixings are expressed into

$\displaystyle{\bf U}_{CKM}={\bf U}^{u}_{L}{{\bf U}^{d}_{L}}^{\dagger}={\bf
U}^{u}_{0}{\bf{F}}^{u}_{L}({\bf{F}}_{L}^{d})^{\dagger}{{\bf
U}^{d}_{0}}^{\dagger}.$ (9)

The production ${\bf{F}}^{u}_{L}({\bf{F}}_{L}^{d})^{\dagger}$ provides the
origin of complex phases between different flavors that is required by
nonvanishing CPV in CKM. In the lepton sector, the neutrino mixing matrix has
the similar form

$\displaystyle{\bf U}_{PMNS}={\bf U}^{e}_{L}{{\bf U}^{\nu}_{L}}^{\dagger}={\bf
U}_{0}^{e}{\bf{F}}^{e}_{L}({\bf{F}}_{L}^{\nu})^{\dagger}{{\bf
U}_{0}^{\nu}}^{\dagger}.$ (10)

The leptonic CPV is generated from complex
${\bf{F}}^{e}_{L}({\bf{F}}_{L}^{\nu})^{\dagger}$. After the mass matrix
decomposition, we have found that ${\bf M}_{0}^{f}$ and unitary matrix ${\bf
F}_{L,R}^{f}$ play different roles with respect to flavor structure:

  * (1)

real matrix ${\bf M}_{0}^{f}$ completely determines fermion mass eigenvalues;

  * (2)

left-handed ${\bf F}_{L}^{f}$ provides complex phases, which is a necessary
condition for CPV in CKM/PMNS.

##  III Family mass hierarchy

Quarks, charged leptons and even normal hierarchy (NH) neutrinos all exhibit
an important characteristic: family mass hierarchy, i.e.,
$h_{13}^{f},h^{f}_{12}\ll 1$ for $f=u,d,l,\nu$ with hierarchy definition
$h^{f}_{ij}=m^{f}_{i}/m^{f}_{j}$. Due to physical masses as eigenvalues of
${\bf M}_{0}^{f}$, the family mass hierarchy problem must be tracked from
${\bf M}_{0}^{f}$. In this section, we focus on the minimal parameterization
of real matrix ${\bf M}_{0}^{f}$ and its verification.

###  III.1 The minimal parameterization of ${\bf M}_{0}^{f}$

In mathematics, hierarchical eigenvalues can naturally arise from the close-
to-flat matrix. Considering a $2\times 2$ matrix with symmetric nondiagonal
element

$\displaystyle\left(\begin{array}[]{cc}1&1-\delta\\\
1-\delta&1\end{array}\right).$ (13)

$\delta$ stands for a small breaking effect. Its two eigenvalues $\chi_{i}$
are

$\displaystyle\chi_{1}$ $\displaystyle=$ $\displaystyle\delta$ (14)
$\displaystyle\chi_{2}$ $\displaystyle=$ $\displaystyle 2-\delta$ (15)

A very large order $h_{12}$ can be generated by perturbation $\delta$

$\displaystyle
h_{12}=\frac{\chi_{1}}{\chi_{2}}\simeq\frac{\delta}{2}+\mathcal{O}(\delta^{2}).$
(16)

Inspired by the property, we assume that the fermion mass structure is driven
by the same mechanism. Defining a $3\times 3$ matrix

$\displaystyle{\bf
I}_{\Delta}^{f}=\left(\begin{array}[]{ccc}1&1+\delta_{12}^{f}&1+\delta_{13}^{f}\\\
1+\delta_{12}^{f}&1&1+\delta_{23}^{f}\\\
1+\delta_{13}^{f}&1+\delta_{23}^{f}&1\end{array}\right)$ (20)

with flavor-dependent nondiagonal perturbations $\delta_{ij}^{f}$, the real
matrix ${\bf M}_{0}^{f}$ has a form

$\displaystyle{\bf M}_{0}^{f}=\frac{m_{\Sigma}^{f}}{3}{\bf I}_{\Delta}^{f}$
(21)

Here, $m_{\Sigma}^{f}\equiv\sum_{i}m_{i}^{f}$ is total family mass. Due to
flavor symmetry breaking being taken by perturbations $\delta_{ij}^{f}$,
$m_{\Sigma}^{f}$ hints that a family universal coupling can be introduced as
$y^{f}=\frac{\sqrt{2}}{v_{0}}\frac{m_{\Sigma}^{f}}{3}$. With $\delta_{ij}^{f}$
vanishing, the matrix ${\bf I}_{\Delta}^{f}$ becomes flat, which means that
all flavors are symmetric and that no flavor is prioritized. The case in which
${\bf I}_{\Delta}^{f}$ has eigenvalues $0,0,3$ corresponds with the case in
which only the 3rd generation is massive and the first two generations are
massless, which happens to coincide with results from Weinberg’s models of
lepton and quark masses 2020Weinberg . When flavor symmetry is broken by
nonvanishing $\delta_{ij}^{f}$, the two light flavors obtain small masses.
Here, only two free parameters are needed to control $m_{1}^{f}$ and
$m_{2}^{f}$. At 1-order approximate, the physical masses are

$\displaystyle m^{f}_{1,2}$ $\displaystyle=$
$\displaystyle\frac{y^{f}v_{0}}{\sqrt{2}}\left(\frac{1}{3}S^{f}\mp\frac{2}{3}\sqrt{Q^{f}}\right)+\mathcal{O}(\delta^{2})$
(22) $\displaystyle m^{f}_{3}$ $\displaystyle=$
$\displaystyle\frac{y^{f}v_{0}}{\sqrt{2}}\left(3-\frac{2}{3}S^{f}\right)+\mathcal{O}(\delta^{2})$
(23)

with parameters $S^{f},Q^{f}$ as

$\displaystyle S^{f}$ $\displaystyle\equiv$
$\displaystyle-\delta^{f}_{12}-\delta^{f}_{23}-\delta^{f}_{13}$ (24)
$\displaystyle Q^{f}$ $\displaystyle\equiv$
$\displaystyle(\delta^{f}_{12})^{2}+(\delta^{f}_{23})^{2}+(\delta^{f}_{13})^{2}-\delta^{f}_{12}\delta^{f}_{23}-\delta^{f}_{23}\delta^{f}_{13}-\delta^{f}_{13}\delta^{f}_{12}$
(25)

Using eqs. (22) and (23), $S^{f}$ and $Q^{f}$ can be expressed into
hierarchies

$\displaystyle S^{f}$ $\displaystyle=$
$\displaystyle\frac{9}{2}(h^{f}_{23}+h^{f}_{12}h^{f}_{23}-(h^{f}_{23})^{2})+\mathcal{O}(h^{3})$
(26) $\displaystyle Q^{f}$ $\displaystyle=$
$\displaystyle\frac{81}{16}(h^{f}_{23})^{2}+\mathcal{O}(h^{3})$ (27)

The two quantities $S^{f}$ and $Q^{f}$ split family masses, and one of
$\delta_{ij}^{f}$ is free. This can be understood from $SO(2)$ symmetry.
Defining
$R^{f}=(\delta_{12}^{f})^{2}+(\delta_{23}^{f})^{2}+(\delta_{13}^{f})^{2}$, it
is $SO(2)$ invariant along the direction of $(1,1,1)$ and is fixed by mass
hierarchies

$\displaystyle
R^{f}=\frac{1}{3}((S^{f})^{2}+2Q^{f})=\frac{81}{4}(h_{23}^{f})^{2}\Big{(}\frac{1}{2}-\frac{h_{12}^{f}}{3}\Big{)}+\mathcal{O}(h^{4})$
(28)

Fig. 1 explains the relationship between mass hierarchy and perturbations
$\delta_{ij}^{f}$. Eq. (24) represents a plane $S1$ in the space
$(\delta^{f}_{12},\delta^{f}_{23},\delta^{f}_{13})$. Eq. (25) represents a
surface $S2$ that is determined by hierarchy $h_{23}^{f}$. Their intersection
line corresponds to the parameter space allowed by fermion mass data. The
intersection forms a circle with invariant distant $R^{f}$ to the origin.

Figure 1: Parameter space and family mass hierarchies. Surfaces $S1$ (rainbow
colors) and $S2$ (gray) are shown at $h_{12}^{f}=0.05$ and $h_{23}^{f}=0.02$.
The intersection line (deep blue) is allowed by the fermion mass hierarchies.
The invariance $R^{f}$ and axis $(1,1,1)$ are labeled by black dashed and red
solid lines, respectively.

###  III.2 Reproduction of fermion mass hierarchies

Two of three $\delta_{ij}^{f}$ are bounded by two hierarchies, and one is
free. In the order of $\mathcal{O}(\delta_{ij}^{f})$, $S^{f},Q^{f}$ is fixed
by experiment data in eqs. (26) and (27). We choose $\delta_{13}^{f}$ as a
free parameter, and $\delta_{12}^{f},\delta_{23}^{f}$ can be solved from eq.
(24,25) with fixed mass hierarchy $h_{ij}^{f}$. There are two kinds of methods
to find the parameter space of $\delta_{ij}^{f}$:

  * 1.

Parameter space scanning: Initializing three $\delta_{ij}^{f}$ in the range of
$(-S^{f},0)$ in terms of eq. (24), the eigenvalues of mass matrix eq. (21) can
be calculated to match fermion mass data. This method is adopted for the quark
sector to investigate the $1\sigma$ range in the quark sector.

  * 2.

Solution from hierarchies: For fixed $S^{f},Q^{f}$, the perturbations
$\delta_{12}^{f}$ and $\delta_{23}^{f}$ can be solved from each input
$\delta_{13}^{f}$ as

$\displaystyle\delta_{12,23}^{f}$ $\displaystyle=$
$\displaystyle-\frac{\delta^{f}_{13}+S^{f}}{2}\pm\frac{1}{2\sqrt{3}}\sqrt{-9(\delta^{f}_{13})^{2}+4Q^{f}-6\delta^{f}_{13}S^{f}-(S^{f})^{2}}.$

When $\delta_{13}^{f}$ adopts the entire possible range, the complete
parameter space is obtained. We adopt this method in the lepton sector due to
the high precision of the charged lepton mass data.

#### Quark mass hierarchies

The current quark mass data are listed in Tab. (4). Setting
${y^{f}v_{0}}/{\sqrt{2}}=(m^{f}_{1}+m^{f}_{2}+m^{f}_{3})/3$, the quark masses
can be determined by calculating ${\bf M}_{0}^{f}$ eigenvalues by scanning the
complete possible range. If mass eigenvalues satisfy the experimental data,
the set of perturbations is recorded. The numerical results show that all
allowed data are distributed on a circle in Fig. 2. The distribution thickness
is controlled by the $1\sigma$ range of $h^{u}_{23}$.

Figure 2:  Parameter spaces
$(\delta_{12}^{f},\delta_{23}^{f},\delta_{13}^{f})$ for (a) down-type quarks
and (b) up-type quarks in $1\sigma$ CL.

#### Lepton mass hierarchies

Although absolute neutrino masses remain unknown, they can be calculated with
the help of the mass-squared difference $\Delta m^{2}_{ij}$ after initializing
the lightest $m^{\nu}_{1}$. Another unknown problem is neutrino mass order. In
the close-to-flat mass structure, flavor symmetry is broken by perturbations
$\delta_{ij}^{f}$, which is in favor of the NH neutrino. For the inverted
hierarchy neutrino, nondiagonal corrections cannot be treated as
perturbations. We treat neutrino mass as a precise value and neglect the
error. Charged lepton mass data are also measured with high precision. This
means that there is a very thin distribution near surface $S1$ in Fig. (1).
Thus, we calculate lepton mass perturbations $\delta_{ij}^{\nu,e}$ from the
second method. The allowed ranges are shown in Fig. (3).

Figure 3: Parameter space for (a) charged lepton and (b) neutrinos

Until now, all quark and lepton masses have been reproduced with successful
physical results. The close-to-flat mass matrix has been proven as a good
mechanism to address the problematic fermion mass hierarchy. It provides a
general flavor structure independent of concrete mass data. In each family,
${\bf M}_{0}^{f}$ are parameterized by 3 $\delta_{ij}^{f}$. The two quantities
$S^{f}$ and $Q^{f}$ are generated as $h_{12}^{f}$ and $h_{23}^{f}$. The role
of the remaining one (chosen as $\delta_{13}^{f}$) will be discussed in flavor
mixing.

##  IV Yukawa phase as a quantum origin of CP violation

In addition to fermion mass, the flavor structure must address fermion
mixings. Experiments have shown that a nonvanishing CPV exists in both CKM and
PMNS mixings. The necessary condition of CPV is the existence of a complex
phase in weak charge current interactions. In terms of eq. (9), it must come
from the left-handed unitary matrix. In this section, we discuss ${\bf
F}_{L}^{f}$, including its parameterization, explanation and verification.

###  IV.1 The parameterization of ${\bf F}_{L,R}^{f}$

Let us consider the number of d.o.f. in flavor structure. Real transformation
${\bf U}_{0}^{f}$ in eq. (8) is determined by 3 nondiagonal vacuum
perturbations $\delta_{ij}^{f}$. Two of the $\delta_{ij}^{f}$ generate family
mass hierarchies, and the remaining $\delta_{ij}^{f}$ remains free in each
family. In the CKM, 4 free parameters are needed to correspond to 3 flavor
mixing angles and 1 Dirac CPV. Thus, in addition to two remaining free
parameters in ${\bf I}_{\Delta}^{u}$ and ${\bf I}_{\Delta}^{d}$, only two
parameters in ${\bf F}_{L}^{u}({\bf F}_{L}^{d})^{\dagger}$ are needed for
flavor mixings. It is noted that a rephasing process still exists in mass
eigenstates. In mass eigenstates labeled by $\psi^{f,m}_{L,R}$, fermion
rephasing can be taken as

$\displaystyle{\psi}^{u,m}_{L,R}\rightarrow{\bf
K}^{u}\psi^{u,m}_{L,R},~{}~{}~{}{\psi}^{d,m}_{L,R}\rightarrow{\bf
K}^{d}\psi^{d,m}_{L,R},~{}~{}~{}$ (29)

with diagonal

$\displaystyle{\bf K}^{u}$ $\displaystyle\equiv$ $\displaystyle{\rm
diag}(e^{i\beta_{1}},e^{i\beta_{2}},e^{i\beta_{3}})$ (30) $\displaystyle{\bf
K}^{d}$ $\displaystyle\equiv$ $\displaystyle{\rm
diag}(1,e^{i\alpha_{1}},e^{i\alpha_{2}})$ (31)

Under the rephasing, the mass terms remain invariant, but the CKM matrix can
be transformed into the standard from

$\displaystyle{\bf U}_{CKM}\rightarrow{\bf K}_{u}^{\dagger}{\bf U}_{CKM}{\bf
K}_{d}.$ (32)

(See Appendix B for the details of calculation of the rephasing matrix.) Using
the rephasing, ${\bf F}^{f}_{L,R}$ can be eliminated in a global phase and
parameterized as follows:

$\displaystyle{\bf{F}}^{d}_{L}={\rm
diag}(1,1,1),~{}~{}~{}{\bf{F}}^{u}_{L}={\rm
diag}(1,e^{i\lambda^{u}_{1}},e^{i\lambda^{u}_{2}}).$ (33)

Substituting them into eq. (6), the quark mass matrix can be written in the
form

$\displaystyle{\bf M}^{d}$ $\displaystyle=$
$\displaystyle\frac{v_{0}}{3\sqrt{2}}y^{d}{\bf
I}_{\Delta}^{d}=\frac{v_{0}}{3\sqrt{2}}y^{d}\left(\begin{array}[]{ccc}1&1+\delta_{12}^{d}&1+\delta_{13}^{d}\\\
1+\delta_{12}^{d}&1&1+\delta_{23}^{d}\\\
1+\delta_{13}^{d}&1+\delta_{23}^{d}&1\end{array}\right),$ (37)
$\displaystyle{\bf M}^{u}$ $\displaystyle=$
$\displaystyle\frac{v_{0}}{3\sqrt{2}}y^{u}({\bf F}_{L}^{u})^{\dagger}{\bf
I}_{\Delta}^{u}=\frac{v_{0}}{3\sqrt{2}}y^{u}\left(\begin{array}[]{ccc}1&1+\delta_{12}^{u}&1+\delta_{13}^{u}\\\
(1+\delta_{12}^{u})e^{-i\lambda^{u}_{1}}&e^{-i\lambda^{u}_{1}}&(1+\delta_{23}^{u})e^{-i\lambda^{u}_{1}}\\\
(1+\delta_{13}^{u})e^{-i\lambda^{u}_{2}}&(1+\delta_{23}^{u})e^{-i\lambda^{u}_{2}}&e^{-i\lambda^{u}_{2}}\end{array}\right).$
(41)

Additionally, the CKM mixing has the form

$\displaystyle{\bf U}_{CKM}={\bf
U}_{0}^{u}\text{~{}diag}\left(1,e^{i\lambda^{u}_{1}},e^{i\lambda^{u}_{2}}\right)({\bf
U}_{0}^{d})^{T}$ (42)

Real orthogonal transformations ${\bf U}_{0}^{u,d}$ are completely determined
by perturbations $\delta_{ij}^{u,d}$ from eq. (7). Up to now, we have
parameterized the fermion mass matrix into ten free parameters, for which
roles in flavor structure are summarized in Tab. (1).

structure |  | parameters |  | observables
---|---|---|---|---
$\begin{array}[]{c}\\\ {\bf I}_{\Delta}^{u},~{}{\bf I}_{\Delta}^{d}\\\ \\\ {\bf F}_{L}^{u}\\\ {\bf F}_{L}^{d}={\bf 1}\\\ \end{array}$ | $\left.\begin{array}[]{c}\left\\{\begin{array}[]{c}\\\ \\\ \\\ \end{array}\right.\\\ \left\\{\begin{array}[]{c}\\\ \\\ \end{array}\right.\end{array}\right.$ | $\begin{array}[]{c}\begin{array}[]{cc}\delta_{12}^{d},&\delta_{12}^{u}\\\ \delta_{23}^{d},&\delta_{23}^{u}\\\ \delta_{13}^{d},&\delta_{13}^{u}\end{array}\\\ \begin{array}[]{c}\lambda^{u}_{1},\lambda^{u}_{2}\\\ -\end{array}\end{array}$ | $\left.\begin{array}[]{c}\left.\begin{array}[]{c}\\\ \\\ \end{array}\right\\}\\\ \left.\begin{array}[]{c}\\\ \\\ \\\ \end{array}\right\\}\end{array}\right.$ | $\left.\begin{array}[]{c}\left.\begin{array}[]{cc}h_{12}^{d},&h_{12}^{u}\\\ h_{23}^{d},&h_{23}^{u}\end{array}\right\\}\text{4 mass hierarchies}\\\ \left.\begin{array}[]{cc}\theta_{12},\\\ \theta_{23},&\delta_{CP},\\\ \theta_{13},\end{array}\right\\}\text{3 mixing angles and 1 CPV}\end{array}\right.$
|  | $y^{u},~{}y^{d}$ |  |  $\sum_{i}m_{i}^{u},\sum_{i}m_{i}^{d}$ : 2 total family masses
Table 1: Parameters in the minimal quark flavor structure

Similarly, lepton unitary matrixes ${\bf F}_{L,R}^{\nu,e}$ are set as

$\displaystyle{\bf{F}}^{\nu}_{L}=diag(1,e^{i\lambda_{1}^{\nu}},e^{i\lambda^{\nu}_{2}}),~{}~{}~{}{\bf{F}}_{L}^{e}=diag(1,1,1).$
(43)

The charged lepton mass structure can be expressed as

$\displaystyle{\bf M}^{e}$ $\displaystyle=$
$\displaystyle\frac{v_{0}}{3\sqrt{2}}y^{e}{\bf
I}_{\Delta}^{e}=\frac{v_{0}}{3\sqrt{2}}y^{e}\left(\begin{array}[]{ccc}1&1+\delta_{12}^{e}&1+\delta_{13}^{e}\\\
1+\delta_{12}^{e}&1&1+\delta_{23}^{e}\\\
1+\delta_{13}^{e}&1+\delta_{23}^{e}&1\end{array}\right).$ (47)

The PMNS mixings are determined by real orthogonal matrix ${\bf
U}_{0}^{e,\nu}$ as

$\displaystyle{\bf U}_{PMNS}$ $\displaystyle=$ $\displaystyle{\bf
U}_{0}^{e}\text{~{}diag}\left(1,e^{-i\lambda^{\nu}_{1}},e^{-i\lambda^{\nu}_{2}}\right)({{\bf
U}_{0}^{\nu}})^{T}.$ (53)

###  IV.2 Yukawa phase

Now, let us analyze the physical means of ${\bf F}_{L,R}^{f}$. In eq. (9),
${\bf{F}}^{u}_{L}({\bf{F}}_{L}^{d})^{\dagger}$ provides the origin of CPV in
flavor mixing. Using eqs. (2) and (6), the quark mass term is

$\displaystyle-\mathcal{L}_{M}^{f}$ $\displaystyle=$
$\displaystyle\bar{d}_{L}({\bf F}_{L}^{d})^{\dagger}{\bf M}_{0}^{d}{\bf
F}_{R}^{d}d_{R}+\bar{u}_{L}({\bf F}_{L}^{u})^{\dagger}{\bf M}_{0}^{u}{\bf
F}_{R}^{u}u_{R}+H.c.$ (54)

Defining Yukawa interaction eigenstates as

$\displaystyle d_{L,R}^{(Y)}={\bf
F}^{d}_{L,R}d_{L,R},~{}~{}~{}u_{L,R}^{(Y)}={\bf
F}^{u}_{L,R}u_{L,R},~{}~{}~{}Q_{L}^{(Y)}=(u_{L}^{(Y)},d_{L}^{(Y)})^{T}$ (55)

the above mass term is written as

$\displaystyle-\mathcal{L}_{M}^{q}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}\bar{d}_{L}^{(Y)}{y}^{d}{\bf
I}_{\Delta}^{d}v_{0}d_{R}^{(Y)}+\frac{1}{\sqrt{2}}\bar{u}_{L}^{(Y)}{y}^{u}{\bf
I}_{\Delta}^{d}v_{0}u_{R}^{(Y)}+H.c.$ (60) $\displaystyle=$
$\displaystyle\bar{Q}_{L}^{(Y)}{y}^{d}{\bf
I}_{\Delta}^{d}\left(\begin{array}[]{c}0\\\
v_{0}/\sqrt{2}\end{array}\right)d_{R}^{(Y)}+\bar{Q}_{L}^{(Y)}{y}^{u}{\bf
I}_{\Delta}^{u}\left(\begin{array}[]{c}v_{0}/\sqrt{2}\\\
0\end{array}\right)u_{R}^{(Y)}+H.c.$

The results can be generalized into the lepton sector. The Dirac neutrino mass
and charged lepton mass can be expressed into Yukawa state
$L_{L}^{(Y)},\nu_{R}^{(Y)},e_{R}^{(Y)}$ as

$\displaystyle-\mathcal{L}_{M}^{l}$ $\displaystyle=$
$\displaystyle\bar{L}_{L}^{(Y)}{y}^{e}{\bf
I}_{\Delta}^{e}\left(\begin{array}[]{c}0\\\
v_{0}/\sqrt{2}\end{array}\right)e_{R}^{(Y)}+\bar{L}_{L}^{(Y)}{y}^{\nu}{\bf
I}_{\Delta}^{\nu}\left(\begin{array}[]{c}v_{0}/\sqrt{2}\\\
0\end{array}\right)\nu_{R}^{(Y)}+H.c.$ (65)

This means that the unitary matrix ${\bf F}_{L,R}^{f}$ can be explained as a
quantum phase between weak eigenstates and Yukawa interaction eigenstates. Due
to fermion Yukawa terms being nongauge interactions, these need not be weak
gauge eigenstates. In general, the phases ${\bf F}_{L,R}^{f}$ are allowed,
which are called Yukawa phases. This means that CPV in flavor mixing arises
from a quantum effect.

###  IV.3 Flavor mixing angles and CPV

In addition to parameters for mass hierarchies, the minimal flavor structure
provides four remaining free parameters for flavor mixings:
$\delta_{13}^{u,d}$ and Yukawa phases $\lambda_{1}^{u},\lambda_{2}^{u}$. The
next mission is to verify the validity of the minimal flavor structure with
respect to quark CKM mixing. The process is operated as follows: (1) choose a
random point $(\delta_{12}^{d},\delta_{23}^{d},\delta_{13}^{d})$ from Fig.
2(a) and another $(\delta_{12}^{u},\delta_{23}^{u},\delta_{13}^{u})$ from Fig.
2(b); (2) calculate $U_{CKM}$ by scanning all possible Yukawa phases
$\lambda^{u}_{1}$ and $\lambda_{2}^{u}$ in terms of eqs. (77, 78) in Appendix
B; (3) if the mixing results agree with the CKM data, record
$(\delta_{12}^{d},\delta_{23}^{d},\delta_{13}^{d})$ and
$(\delta_{12}^{u},\delta_{23}^{u},\delta_{13}^{u})$ as a set of acceptable
results. After tedious numerical computation, a matched result is listed in
Tab. (2). (Alternatively, Wolfenstein parameterization can be defined in
WolfensteinPRL1983 .) Up until now, all current data, quark masses and CKM
mixings have been reproduced successfully. At the matched points, allowed
Yukawa phase space is shown in Fig. 4. It is also effectively checked by
operating the same matching process as for PMNS. Setting
$m^{\nu}_{1}=0.0001~{}eV$ as an example, the matched results are listed in
Tab. 3, and the allowed Yukawa phase is shown in Fig. 5.

| $\delta^{d}_{ij}$ | $\delta^{u}_{ij}$ | $\lambda^{u}_{i}$
---|---|---|---
para. | $\begin{array}[]{c}\delta_{12}^{d}=-0.00723\\\ \delta_{23}^{d}=-0.0644\\\ \delta_{13}^{d}=-0.0377\end{array}$ | $\begin{array}[]{c}\delta_{12}^{u}=-0.0000453\\\ \delta_{23}^{u}=-0.0172\\\ \delta_{13}^{u}=-0.0165\end{array}$ | $\begin{array}[]{c}\lambda_{1}^{u}=-0.00504\\\ \lambda_{2}^{u}=0.0851\end{array}$
| $m^{d}_{i}$ | $m^{u}_{i}$ | $CKM$
results | $\begin{array}[]{c}m^{d}=4.750~{}\textrm{MeV}\\\ m^{s}=98.96~{}\textrm{MeV}\\\ m^{b}=4.176~{}\textrm{GeV}\end{array}$ | $\begin{array}[]{c}m^{u}=2.205~{}\textrm{MeV}\\\ m^{c}=1.303~{}\textrm{GeV}\\\ m^{t}=173.0~{}\textrm{GeV}\end{array}$ | $\begin{array}[]{c}s_{12}=0.2243\\\ s_{23}=0.04141\\\ s_{13}=0.003942\\\ \delta_{CP}=75.07^{\circ}\end{array}$
Table 2: The matched results from quark flavor structure. $v_{0}y^{f}/\sqrt{2}$ is set as the total family mass. | $\delta^{e}_{ij}$ | $\delta^{\nu}_{ij}$ | $\lambda^{\nu}_{i}$
---|---|---|---
para. | $\begin{array}[]{c}\delta_{12}^{e}=-0.00443\\\ \delta_{23}^{e}=-0.146\\\ \delta_{13}^{e}=-0.105\end{array}$ | $\begin{array}[]{c}\delta_{12}^{\nu}=-0.176\\\ \delta_{23}^{\nu}=-0.435\\\ \delta_{13}^{\nu}=-0.073\end{array}$ | $\begin{array}[]{c}\lambda_{1}^{\nu}=-0.0111\\\ \lambda_{2}^{\nu}=1.60\end{array}$
| $m^{e}_{i}$ | $m^{\nu}_{i}$ | $PMNS$
results | $\begin{array}[]{c}m^{e}=0.5108~{}\textrm{MeV}\\\ m^{\mu}=105.55~{}\textrm{MeV}\\\ m^{\tau}=1.777~{}\textrm{GeV}\end{array}$ | $\begin{array}[]{c}m^{\nu}_{1}=0.0001~{}\textrm{eV}\\\ m^{\nu}_{2}=0.008640~{}\textrm{eV}\\\ m^{\nu}_{3}=0.04996~{}\textrm{eV}\end{array}$ | $\begin{array}[]{c}s^{2}_{12}=0.3353\\\ s^{2}_{23}=0.4393\\\ s^{2}_{13}=0.02009\\\ \delta_{CP}=1.486\pi\end{array}$
Table 3: The matched results from leptonic flavor structure

Figure 4: Constraints from three CKM mixing angles (left) at 95% CL and
contour line of CPV (right) [degree] at the matched $\delta_{ij}^{u,d}$ given
in Tab. 2 on the $\lambda^{u}_{1},\lambda^{u}_{2}$ plane.

Figure 5: Constraints from three PMNS mixing angles (left) at $3\sigma$ and
contour line of CPV (right) [radian/$\pi$] at the matched
$\delta_{ij}^{\nu,e}$ given in Tab. 3 on the
$\lambda^{\nu}_{1},\lambda^{\nu}_{2}$ plane.

##  V Flavor Models

The minimal flavor structure must be realized by parameterization of real
${\bf M}_{0}^{f}$ and left-handed unitary phase ${\bf F}_{L,R}^{f}$ in each
family. All fermion masses, flavor mixing angles and CPV have been reproduced
successfully. In this section, we discuss some possible phenomenological
scenarios that can yield the minimal flavor structure. In terms of the origin
of close-to-flat structure, the flavor models have primarily been of three
types:

  * 1.

Quasi-universal interaction model. In this case, close-to-flat flavor
structure comes from Yukawa couplings. This pattern indicates quasi-family-
universal fermion couplings to Higgs. Only slight deviations appear at
nondiagonal element $\delta_{ij}^{f}$. The model is still based on the SM
Higgs doublet, in which FCNC does not appear in the tree level. The quark
Yukawa terms can be expressed by

$\displaystyle-\mathcal{L}^{q}_{Y}=y^{d}({\bf
I}_{\Delta}^{d})_{ij}\bar{Q}_{L}^{(Y),i}Hd_{R}^{(Y),j}+y^{u}({\bf
I}_{\Delta}^{u})_{ij}\bar{Q}_{L}^{(Y),i}\tilde{H}u_{R}^{(Y),j}+H.c.$ (66)

The quasi-democratic model in Sogami1998 ; Miura2000  proposes a similar mass
matrix, in which the pattern was expressed by a unitarity matrix with complex
nondiagonal perturbations. The difference is that we need only three real
symmetric perturbations to replace complex nondiagonal corrections in the
democratic-type model. The origin of CPV in our paper comes from quantum
phases.

  * 2.

Multi-Higgs models. In this case, ${\bf I}_{\Delta}^{f}$ comes not from
couplings but from the vacuum structure. Nondiagonal flavor breakings can be
fulfilled in the context of an extension of the SM, where some Higgs doublets
and/or even more multiplets are introduced Botella2010PLB ; Haba2010EPJC .
Generally, more than two Higgs scalars are needed for CPV in flavor mixing. By
appropriate assignment of VEVs of scalars and Yukawa couplings, it is possible
to obtain close-to-flat vacuum. More generally, in multiscalar-inspired
scenarios, couplings between scalars and fermions can result in special flavor
structures after scalar gain of corresponding VEVs, which is the case in so-
called flavon models Bazzocchi2004PRD ; BazzocchiPRD2004 ; PascoliJHEP2016 .

  * 3.

There is another possible mechanism to provide the flavor structure. When
considering flavor interaction dependence on energy scale, the small broken
flavor effects may result from the flat Higgs vacuum and/or Yukawa couplings
by a renormalization effect ZXingPRD2001 . If the mechanism is confirmed in
future work, the flavor puzzle will be self-determined in the SM.

These possible scenarios provide a wide range of new physics. Regardless of
the scenario, a (quasi-)family universal Yukawa coupling is always hinted as a
favorable result.

##  VI Conclusion

The SM fermion flavor struture has been analyzed by bi-unitary decomposition
of mass matrix. The real matrix ${\bf M}_{0}^{f}$ and unitary ${\bf
F}_{L}^{f}$ play different roles. The former generates fermion mass
hierarchies, and the latter provides the CPV in flavor mixing. The minimal
flavor structure has been realized by close-to-flat real symmetric ${\bf
I}_{\Delta}^{f}$. The number of parameters in the minimal flavor structure is
equal to phenomenological observables, six fermion masses and four mixings,
without any redundancy. The structure successfully reproduces current mass and
mixing data in both quark sector and lepton sector. An important indication
from the minimal flavor structure is that the origin of CPV existing in CKM
and PMNS may be attributed to a quantum effect. The complex phases can arise
from the superposition between Yukawa interaction states and weak gauge
states. This information helps us understand the properties of Yukawa
interactions in a new way. In future research, the possible mechanisms or
models corresponding to ${\bf I}_{\Delta}^{f}$ merit further investigation,
which will reveal the nature of flavor.

## Acknowledgements

I thank my collaborator Prof. Rong Li for helpful discussions on the subject.
This work is partially supported by the Fundamental Research Funds for the
Central Universities.

##  Appendix A Data of Fermion Masses and Mixings

All data of quark and charged lepton masses as well as CKM and PMNS mixings
come from 2018PDG . In Tab. (4), we set $m^{\nu}_{1}=0.0001$ eV, and other
neutrino masses can be calculated from the mass squared differences $\Delta
m^{2}_{ij}$.

Table 4: Quark and lepton masses quark mass
---
$\begin{array}[]{ll}m_{u}=2.2^{+0.5}_{-0.4}~{}\textrm{MeV}&m_{d}=4.7^{+0.5}_{-0.3}~{}\textrm{MeV}\\\
m_{c}=1.275^{+0.025}_{-0.035}~{}\textrm{GeV}&m_{s}=95^{+9}_{-3}~{}\textrm{MeV}\\\
m_{t}=173.0\pm
0.4~{}\textrm{GeV}&m_{b}=4.18^{+0.04}_{-0.03}~{}\textrm{GeV}\end{array}$
lepton mass
$\begin{array}[]{ll}m_{e}=0.5109989461\pm
0.0000000031~{}\textrm{MeV}&m_{1}^{\nu}=0.0001~{}\textrm{eV}(input)\\\
m_{\mu}=105.6583745\pm
0.0000024~{}\textrm{MeV}&m_{2}^{\nu}=0.0086~{}\textrm{eV}\\\
m_{\tau}=1776.86\pm
0.12~{}\textrm{MeV}&m_{3}^{\nu}=0.050~{}\textrm{eV}\end{array}$
Table 5:  CKM and PMNS experimental data CKM | PMNS, $3\sigma$(2018PDG)
---|---
$\begin{array}[]{l}s_{12}=\frac{|V_{us}|}{\sqrt{|V_{ud}|^{2}+|V_{us}|^{2}}}=0.2244\pm 0.0005\\\ s_{23}=\frac{|V_{cb}|}{\sqrt{|V_{ud}|^{2}+|V_{us}|^{2}}}=0.0422\pm 0.0008\\\ s_{13}=|V_{ub}|=0.00394\pm 0.00036\\\ \delta=(73.5^{+4.2}_{-5.1})^{\circ}\end{array}$ | $\begin{array}[]{l}s_{12}^{2}=0.297,~{}~{}0.250-0.354\\\ s_{23}^{2}=0.425,~{}~{}0.381-0.615\\\ s_{13}^{2}=0.0215(NH),~{}0.0190-0.0240\\\ \delta=1.38\pi,~{}~{}2\sigma:(1.0-1.9)\pi\end{array}$

##  Appendix B Calculation of Flavor Mixing Angles and CPV

After bi-unitary transformation eq. (2), the diagonalized quark masses and
weak charge current terms are

$\displaystyle\mathcal{L}_{W}^{q}=\bar{u}^{m}_{L}{\bf
m}^{u}u^{m}_{R}+\bar{d}^{m}_{L}{\bf
m}^{d}d^{m}_{R}+\frac{g}{\sqrt{2}}\bar{u}^{m}_{L}\gamma^{\mu}{\bf
U}_{CKM}d^{m}_{L}W_{\mu}+H.c.$ (67)

There is still a rephasing transformation to keep the diagonal mass invariant:

$\displaystyle{u}^{m}_{L,R}\rightarrow{\bf
K}^{u}u^{m}_{L,R},~{}~{}~{}{d}^{m}_{L,R}\rightarrow{\bf K}^{d}d^{m}_{L,R}$
(68)

with

$\displaystyle{\bf K}_{u}$ $\displaystyle\equiv$ $\displaystyle{\rm
diag}(e^{i\beta_{1}},e^{i\beta_{2}},e^{i\beta_{3}})$ (69) $\displaystyle{\bf
K}_{d}$ $\displaystyle\equiv$ $\displaystyle{\rm
diag}(1,e^{i\alpha_{1}},e^{i\alpha_{2}}).$ (70)

However, the CKM matrix transform is

$\displaystyle{\bf U}_{CKM}\rightarrow{\bf K}_{u}^{\dagger}{\bf U}_{CKM}{\bf
K}_{d}.$ (71)

By choice of rephasing phases, CKM can be expressed in the standard form

$\displaystyle{\bf
U}_{CKM}=\left(\begin{array}[]{ccc}c_{12}c_{13}&s_{12}c_{13}&s_{13}e^{-i\delta_{CP}}\\\
-s_{12}c_{23}-c_{12}s_{23}s_{13}e^{i\delta_{CP}}&c_{12}c_{23}-s_{12}s_{23}s_{13}e^{i\delta_{CP}}&s_{23}c_{13}\\\
s_{12}s_{23}-c_{12}c_{23}s_{13}e^{i\delta_{CP}}&-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\delta_{CP}}&c_{23}c_{13}\end{array}\right).$
(75)

Alternatively, Wolfenstein parameterization can be defined as

$\displaystyle
s_{12}=\lambda,~{}~{}s_{23}=A\lambda^{2},~{}~{}s_{13}e^{i\delta}=\frac{A\lambda^{3}(\bar{\rho}+i\bar{\eta})\sqrt{1-A^{2}\lambda^{4}}}{\sqrt{1-\lambda^{2}}[1-A^{2}\lambda^{4}(\bar{\rho}+i\bar{\eta})]}$
(76)

For the sake of convenience, the non-rephased CKM matrix is labeled as ${\bf
U}$. To make ${\bf U}_{11},{\bf U}_{12},{\bf U}_{22},{\bf U}_{23}$ real,
rephasing phases can be set as follows:

$\displaystyle\beta_{1}$ $\displaystyle=$ $\displaystyle{\rm arg}({\bf
U}_{11})$ $\displaystyle\beta_{2}$ $\displaystyle=$ $\displaystyle{\rm
arg}({\bf U}_{23})+\alpha_{2}$ $\displaystyle\beta_{3}$ $\displaystyle=$
$\displaystyle{\rm arg}({\bf U}_{33})+\alpha_{2}$ $\displaystyle\alpha_{1}$
$\displaystyle=$ $\displaystyle{\rm arg}({\bf U}_{11})-{\rm arg}({\bf
U}_{12})$

$\alpha_{2}$ and $\delta_{CP}$ can be solved from

$\displaystyle\left({\bf K}_{u}{\bf U}{\bf K}_{d}^{\dagger}\right)_{21}={\bf
U}_{CKM}\Big{|}_{21},$ $\displaystyle\left({\bf K}_{u}{\bf U}{\bf
K}_{d}^{\dagger}\right)_{22}={\bf U}_{CKM}\Big{|}_{22}.$

Ignoring these details, mixing angles can be calculated directly from

$\displaystyle
s_{13}=|U_{13}|,~{}~{}~{}s_{23}^{2}=\frac{|U_{23}|^{2}}{1-|U_{13}|^{2}},~{}~{}~{}s_{12}^{2}=\frac{|U_{12}|^{2}}{1-|U_{13}|^{2}}$
(77)

And the CPV can be determined by $J_{CP}$ or another rephasing invariant:

$\displaystyle
c_{12}^{2}c_{23}^{2}+s_{13}^{2}s_{23}^{2}s_{12}^{2}-2c_{12}s_{12}c_{23}s_{23}s_{13}\cos[\delta_{CP}]=|U_{22}|^{2}.$
(78)

## References

  * (1) Zhi-zhong Xing, Phys.Rept. 854 (2020) 1-147 [arXiv: 1909.09610 [hep-ph]]
  * (2) F. Feruglio, Eur. Phys. J. C 75, no.8, 373 (2015) [arXiv:1503.04071 [hep-ph]].
  * (3) M. Raidal, A. van der Schaaf, I. Bigi, M.L. Mangano, Y.K. Semertzidis, S. Abel, S. Albino, S. Antusch, E. Arganda and B. Bajc, et al. Eur. Phys. J. C 57, 13-182 (2008) [arXiv:0801.1826 [hep-ph]].
  * (4) J. Zupan, CERN Yellow Rep.School Proc. 6 (2019) 181-212 [arXiv:1903.05062 [hep-ph]].
  * (5) G. Bhattacharyya, I. de M. Varzielas, P. Leser, Phys. Rev. Lett. 109(2012), 241603 [arXiv:1210.0545 [hep-ph]].
  * (6) S.T. Petcov, Nucl. Phys. B 908 (2016) 279-301.
  * (7) I. Girardi, S.T. Petcov, A.J. Stuart, A.V. Titov, Nucl.Phys. B902 (2016), 1-57 [arXiv:1509.02502 [hep-ph]].
  * (8) Z.-Z. Xing, S. Zhou, Phys.Lett. B 737 (2014) 196-200 [arXiv:1404.7021 [hep-ph]].
  * (9) S.F. King, C. Luhn, Rept. Prog. Phys. 76 (2013) 056201 [arXiv:1301.1340 [hep-ph]].
  * (10) A. G. Dias, J. Leite, D. D. Lopes, C. C. Nishi, Phys. Rev. D 98 (2018) 11, 115017 [arXiv:1810.01893 [hep-ph].
  * (11) C.E. Diaz, S.F. Mantilla, R. Martinez, Phys. Rev. D 98 (2018) 1, 015038 [arXiv:1712.07263 [hep-ph]].
  * (12) P. R. Archer, JHEP. 2012, 095 (2012) [arXiv:1204.4730 [hep-ph]].
  * (13) N. Arkani-Hamed, M. Schmaltz, Phys. Rev. D 61(2000) 033005 [arXiv:hep-ph/9903417 [hep-ph]].
  * (14) G. Altarelli, F. Feruglio, L. Merlo, Fortsch.Phys. 61 (2013) 507-534 [arXiv:1205.5133 [hep-ph]].
  * (15) Z. Zhao, Phys.Rev.D 86 (2012) 096010 [arXiv:1207.2545 [hep-ph]]
  * (16) H. Fritzsch, Z. Xing, D. Zhang, Chin.Phys.C 41 (2017) 9, 093104 [arXiv: 1705.01391 [hep-ph]]
  * (17) K. Fukuura, T. Miura, E. Takasugi, M. Yoshimura, Phys.Rev.D 61 (2000) 073002 [arXiv:hep-ph/9909415 [hep-ph]]
  * (18) I.S. Sogami, K. Nishida, H. Tanaka, T. Shinohara, Prog.Theor.Phys. 99 (1998) 281-292 [arXiv: hep-ph/9801342 [hep-ph]]
  * (19) D. Du, C. Liu, Mod.Phys.Lett.A 8 (1993) 2271-2276.
  * (20) T. Shinohara, H. Tanaka, I.S. Sogami, Prog.Theor.Phys. 100 (1998) 615-625 [arXiv: hep-ph/9807457 [hep-ph]]
  * (21) G.C. Branco, J.I. Silva-Marcos, Phys.Lett.B 359 (1995) 166-174[arXiv: hep-ph/9507299 [hep-ph]]
  * (22) T. Teshima, T. Sakai, Prog.Theor.Phys. 97 (1997) 653-664 [arXiv: hep-ph/9608447 [hep-ph]]
  * (23) G.C. Branco, R.G. Felipe, F.R. Joaquim, Rev.Mod.Phys. 84 (2012) 515-565 [arXiv: 1111.5332 [hep-ph]]
  * (24) L.-L. Chau and W.-Y. Keung, Phys. Rev. Lett. 53, 1802 (1984).
  * (25) S. Weinberg, Phys.Rev.D 101 (2020) 3, 035020 [arXiv: 2001.06582 [hep-th]].
  * (26) L. Wolfenstein, Phys. Rev. Lett. 51, 1945 (1983).
  * (27) I.S. Sogami, K. Nishida, H. Tanaka and T. Shinohara, Prog.Theor.Phys., 99(1998), 281-292 [arXiv:hep-ph/9801342 [hep-ph]].
  * (28) T. Miura, E. Takasugi, M. Yoshimura, Prog.Theor.Phys. 104 (2000) 1173-1187 [arXiv:hep-ph/0007066 [hep-ph]].
  * (29) F.J. Botella, G.C. Branco, M.N. Rebelo, Phys. Lett. B 687 (2010) 194-200 [arXiv:0911.1753 [hep-ph]].
  * (30) N. Haba and M. Hirotsu, Eur.Phys.J. C 69 (2010) 481-492 [arXiv:1005.1372 [hep-ph]].
  * (31) F. Bazzocchi, S. Bertolini, M. Fabbrichesi, M. Piai, Phys.Rev.D 69 (2004) 036002 [arXiv: hep-ph/0309182 [hep-ph]].
  * (32) F. Bazzocchi, Phys.Rev.D 70 (2004) 013002 [arXiv: hep-ph/0401105 [hep-ph]].
  * (33) S. Pascoli, Y. Zhou, JHEP 10 (2016) 145 [arXiv: 1607.05599 [hep-ph]].
  * (34) Z. Xing, Phys.Rev.D 63 (2001) 057301 [arXiv: hep-ph/0011217 [hep-ph].
  * (35) Particle Data Group (M. Tanabashi et al.), Phys. Rev. D 98 (2018) 3, 030001.