# Questions of flavor physics and neutrino mass from a flipped hypercharge

Duong Van Loi<EMAIL_ADDRESS>(corresponding author)
Phenikaa Institute for Advanced Study and Faculty of Basic Science, Phenikaa
University, Yen Nghia, Ha Dong, Hanoi 100000, Vietnam  N. T. Duy
<EMAIL_ADDRESS>Institute of Physics, Vietnam Academy of Science and
Technology, 10 Dao Tan, Ba Dinh, Hanoi 100000, Vietnam  Nguyen Huy Thao
<EMAIL_ADDRESS>Department of Physics, Hanoi Pedagogical University
2, Phuc Yen, Vinh Phuc, Vietnam  Phung Van Dong  dong.phungvan@phenikaa-
uni.edu.vn Phenikaa Institute for Advanced Study and Faculty of Basic Science,
Phenikaa University, Yen Nghia, Ha Dong, Hanoi 100000, Vietnam

###### Abstract

The flavor structure of quarks and leptons is not yet fully understood, but it
hints a more fundamental theory of non-universal generations. We therefore
propose a simple extension of the Standard Model by flipping (i.e., enlarging)
the hypercharge $U(1)_{Y}$ to $U(1)_{X}\otimes U(1)_{N}$ for which both $X$
and $N$ depend on generations of both quark and lepton. By anomaly
cancellation, this extension not only explains the existence of just three
fermion generations as observed but also requires the presence of a right-
handed neutrino per generation, which motivates seesaw neutrino mass
generation. Furthermore, in its minimal version with a scalar doublet and two
scalar singlets, the model naturally generates the measured fermion-mixing
matrices while it successfully accommodates several flavor anomalies observed
in the neutral meson mixings, $B$-meson decays, lepton-flavor-violating decays
of charged leptons, as well as satisfying constraints from particle colliders.

##  I Introduction

Although the Standard Model (SM) has been highly successful in describing
observed phenomena, it leaves many striking features of the physics of our
world unexplained. This work would focus on the issues relating to the number
of fermion generations, the generation of neutrino masses, fermion mass
hierarchies, and flavor mixing profiles [1].

In the SM, the electroweak symmetry reveals a partial unification of weak and
electromagnetic interactions, which is based upon the non-Abelian gauge group
$SU(2)_{L}\otimes U(1)_{Y}$, where $Y$ is an Abelian charge, well known as
hypercharge [2, 3, 4, 5]. The electric charge operator takes the form
$Q=T_{3}+Y$, in which $T_{3}$ is the third component of the $SU(2)_{L}$ weak
isospin. The value of $T_{3}$ is quantized due to the non-Abelian nature of
$SU(2)_{L}$, whereas in contrast the value of $Y$ is entirely arbitrary on the
theoretical ground, because the Abelian $U(1)_{Y}$ algebra is trivial. Indeed,
the hypercharge is often chosen to describe observed electric charges, while
it does not explain them. An interesting question relating to the nature of
the SM hypercharge is whether the conventional choice of generation-universal
hypercharge causes the SM to be unable to address the issues. The present work
does not directly answer this question. Instead of that, we look for an
extension of $U(1)_{Y}$ to generation-dependent Abelian factors, as general,
which naturally solves the issues.

For this aim, we embed $U(1)_{Y}$ in $U(1)_{X}\otimes U(1)_{N}$ for which both
$X$ and $N$ are generation-dependent but determining $Y=X+N$, as observed. It
is clear that $X,N$ may be an intermediate new physics phase resulting from a
GUT and/or string breaking. Additionally, anomaly cancellation fixes both the
number of fermion generations and values of $X,N$. Interestingly, we find for
the first time that both quark and lepton generations are not universal under
a gauge charge as of $X,N$. We investigate the model with a minimal scalar
content in detail, which is responsible for the small, nonzero neutrino masses
[6, 7], the measured fermion-mixing matrices [1, 8], and several flavor-
physics anomalies, such as mass splittings in $K$\- and $B$-meson systems [1,
9], $B$-meson decays [9, 10], and lepton-flavor-violating (LFV) decays of
charged leptons [1, 11, 12, 13].

Let us remind the reader concering the two features of the present work.
First, we argue that the number of fermion generations is precisely three, as
observed, which comes only from anomaly cancellation. This is quite different
from the 3-3-1 model [14, 15, 16, 17, 18, 19, 20, 21, 22] as well as our
previous proposals [23, 24, 25, 26], in which anomaly cancellation implies
that the fermion generation number is an integer multiple of three, and then
it is necessary to add the QCD asymptotic freedom condition to get the number
of fermion generations equal to three. Second, in the present work, we
consider the possibility that the first lepton generation (the third quark
generation) carries Abelian charges different from the remaining lepton
(quark) generations under the new gauge groups, $U(1)_{X}\otimes U(1)_{N}$.
Consequently, the fermion-mixing matrices are recovered, appropriate to
experiment [1, 8], because necessary small mixings arise only from
nonrenormalizable operators. Interestingly enough, flavor-changing neutral
currents (FCNCs) appear at the tree level in both the quark and lepton
sectors.

The rest of this work is organized as follows. We present the new model in
Sec. II. We investigate the fermion mass spectra in Sec. III. We diagonalize
the gauge and scalar sectors in Sec. IV to identify physical fields. We
determine the interactions of fermions and gauge bosons in Sec. V. We examine
flavor physics observables and compare them to experimental results in Sec.
VI. We discuss the collider bounds in Sec. VII. Finally, we summarize our
results and conclude this work in Sec. VIII.

##  II The model

###  II.1 Anomaly cancellation and generation number

As mentioned, the model under consideration is based on gauge symmetry,

$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{X}\otimes U(1)_{N},$ (1)

in which the first two factors are exactly those of the SM, whereas the last
two factors are flipped (i.e., extended) from the weak hypercharge symmetry
$U(1)_{Y}$. The new gauge charges depend on flavors of both quarks and leptons
as

$\displaystyle X$ $\displaystyle=$ $\displaystyle
3z[Bi^{r^{2}(r-1)}+Li^{r(r-1)}],$ (2) $\displaystyle N$ $\displaystyle=$
$\displaystyle Y-X,$ (3)

where $B(L)$ denotes normal baryon (lepton) number, $Y$ labels the
hypercharge, $z$ is an arbitrary non-zero parameter, $i$ is the imaginary
unit, and $r$ is a flavor index, $r=1,2,\cdots,N_{f}$. Notice that $X$ is
Hermitian, since $i^{r(r-1)}=(-1)^{\frac{r(r-1)}{2}}$ is always real.
Additionally, the charges $X$’s of quark and lepton generations determined by
Eq. (2) are either the same or opposite in sign, leading to reduced degrees of
freedom in the model. The electric charge operator is embedded in the gauge
symmetry as

$Q=T_{3}+X+N$ (4)

with $T_{n}\,(n=1,2,3)$ are the $SU(2)_{L}$ generators. The SM fermions
transform under the gauge symmetry as follows:

$\displaystyle l_{rL}$ $\displaystyle=$
$\displaystyle\left(\nu_{rL},e_{rL}\right)^{T}\sim\left({\bf 1},{\bf
2},3zi^{r(r-1)},-1/2-3zi^{r(r-1)}\right),$ (5) $\displaystyle e_{rR}$
$\displaystyle\sim$ $\displaystyle\left({\bf 1},{\bf
1},3zi^{r(r-1)},-1-3zi^{r(r-1)}\right),$ (6) $\displaystyle q_{rL}$
$\displaystyle=$ $\displaystyle\left(u_{rL},d_{rL}\right)^{T}\sim\left({\bf
3},{\bf 2},zi^{r^{2}(r-1)},1/6-zi^{r^{2}(r-1)}\right),$ (7) $\displaystyle
u_{rR}$ $\displaystyle\sim$ $\displaystyle\left({\bf 3},{\bf
1},zi^{r^{2}(r-1)},2/3-zi^{r^{2}(r-1)}\right),$ (8) $\displaystyle d_{rR}$
$\displaystyle\sim$ $\displaystyle\left({\bf 3},{\bf
1},zi^{r^{2}(r-1)},-1/3-zi^{r^{2}(r-1)}\right).$ (9)

It is interesting that the charge $X$ defined by Eq. (2) is periodic in $r$
with period 4, i.e., with $r=1,2,3,4,5,6,7,8,\cdots$, then

$X=z,z,-z,z,z,z,-z,z,\cdots$ (10)

for the quark generations and

$X=3z,-3z,-3z,3z,3z,-3z,-3z,3z,\cdots$ (11)

for the lepton generations. Hence, we express the number of fermion
generations as $N_{f}=4x-y$ with $x=1,2,3,\cdots$ and $y=0,1,2,3$. Take an
example, $N_{f}=5$ then $x=2$ and $y=3$. Considering the anomaly
$[SU(2)_{L}]^{2}U(1)_{X}$, we get

$[SU(2)_{L}]^{2}U(1)_{X}\sim\sum_{\mathrm{doublets}}X_{f_{L}}=\left\\{\begin{array}[]{ll}6z(x-1)&\text{
if }y=1\\\ 6zx&\text{ if }y=0,2,3\end{array},\right.$ (12)

which implies that this anomaly is canceled if and only if $x=y=1$, or
equivalently $N_{f}=3$ as observed.111The result $N_{f}=3$ is unique and
independent of the QCD asymptotic freedom condition. This is quite different
from the 3-3-1 model [14, 15, 16, 17, 18, 19, 20, 21, 22] as well as our
previous works [23, 24, 25, 26]. Because of lepton and quark generation
discrepancies, we conveniently use two kinds of generation indices, such as
$\alpha,\beta=1,2$ for the first two quark generations, while $a,b=2,3$ for
the last two lepton generations; generically, $n,m=1,2,3$ run over $N_{f}=3$.

With $N_{f}=3$ and the fermion content as in Eqs. (5)–(9), two anomalies
$[\text{Gravity}]^{2}U(1)_{X}$ and $[U(1)_{X}]^{3}$ are not canceled yet,
namely

$\displaystyle\left[\text{Gravity}\right]^{2}U(1)_{X}$ $\displaystyle\sim$
$\displaystyle\sum_{\mathrm{fermions}}(X_{f_{L}}-X_{f_{R}})=-3z,$ (13)
$\displaystyle\left[U(1)_{X}\right]^{3}$ $\displaystyle\sim$
$\displaystyle\sum_{\mathrm{fermions}}(X^{3}_{f_{L}}-X^{3}_{f_{R}})=-27z^{3}.$
(14)

To cancel these anomalies, we introduce right-handed neutrinos
$\nu_{pR}\sim({\bf 1},{\bf 1},X_{\nu_{pR}},-X_{\nu_{pR}})$ for
$p=1,2,\cdots,N_{R}$ into the theory as fundamental constituents, satisfying

$\sum_{p=1}^{N_{R}}X_{\nu_{pR}}=-3z,\hskip
14.22636pt\sum_{p=1}^{N_{R}}X^{3}_{\nu_{pR}}=-27z^{3}.$ (15)

Solving the equations in Eq. (15), as well as requiring that at least two
right-handed neutrinos to be identically responsible for neutrino mass
generation, we obtain a unique nontrivial solution, such as

$X_{\nu_{1R}}=3z,\hskip 14.22636ptX_{\nu_{aR}}=-3z,$ (16)

which implies that the resulting right-handed neutrinos have the lepton number
as usual.222The solution as obtained differs from that in the conventional
$U(1)_{B-L}$ extension whose $(B-L)_{\nu_{nR}}=(-1,-1,-1)$ or
$(B-L)_{\nu_{nR}}=(-4,-4,5)$ [27, 28].

With presence of the three right-handed neutrinos, whose $X$ charges obey Eq.
(16), it is easily checked that the remaining anomalies, including
$[SU(3)_{C}]^{2}U(1)_{X}$, $[SU(3)_{C}]^{2}U(1)_{N}$,
$[SU(2)_{L}]^{2}U(1)_{N}$, $[\text{Gravity}]^{2}U(1)_{N}$, $[U(1)_{N}]^{3}$,
$[U(1)_{X}]^{2}U(1)_{N}$, and $U(1)_{X}[U(1)_{N}]^{2}$, are all canceled,
independent of arbitrary $z$.

###  II.2 Minimal particle content and symmetry breaking

The particle content of the model, including fermions and scalars, as well as
their quantum numbers under the gauge symmetry, are listed in Table 1. In
addition to the SM fermions, three right-handed neutrinos must be included as
fundamental fermions to suppress the anomalies, as shown in the previous
subsection. Concerning the scalar sector, we introduce two singlets
$\chi_{1,2}$ and a doublet $\phi$ under $SU(2)_{L}$. The singlets $\chi_{1,2}$
are necessarily presented to break $U(1)_{X}\otimes U(1)_{N}$ down to the weak
hypercharge symmetry $U(1)_{Y}$, provide the Majorana masses for right-handed
neutrinos, and recover the mixing matrices in quark and lepton sectors. Of
course, the scalar doublet $\phi$ that is identified to the SM-Higgs doublet
must be used to break $SU(2)_{L}\otimes U(1)_{Y}$ down to the electromagnetic
symmetry $U(1)_{Q}$ and generate the masses for ordinary charged fermions, as
well as Dirac masses for neutrinos.

The scheme of symmetry breaking is given by

$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{X}\otimes U(1)_{N}$
---
$\downarrow\Lambda_{1,2}$
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}$
$\downarrow v$
$SU(3)_{C}\otimes U(1)_{Q}$

Here, the scalar fields develop the vacuum expectation values (VEVs), such as

$\langle\phi\rangle=\begin{pmatrix}0\\\ \frac{v}{\sqrt{2}}\end{pmatrix},\hskip
14.22636pt\langle\chi_{1}\rangle=\frac{\Lambda_{1}}{\sqrt{2}},\hskip
14.22636pt\langle\chi_{2}\rangle=\frac{\Lambda_{2}}{\sqrt{2}},$ (17)

satisfying $v=246\text{ GeV}$ and $\Lambda_{1,2}\gg v$ for consistency with
the SM.

Multiplets | $SU(3)_{C}$ | $SU(2)_{L}$ | $U(1)_{X}$ | $U(1)_{N}$
---|---|---|---|---
$l_{1L}=(\nu_{1L},e_{1L})^{T}$ | $\bf 1$ | $\bf 2$ | $3z$ | $-1/2-3z$
$\nu_{1R}$ | $\bf 1$ | $\bf 1$ | $3z$ | $-3z$
$e_{1R}$ | $\bf 1$ | $\bf 1$ | $3z$ | $-1-3z$
$l_{aL}=(\nu_{aL},e_{aL})^{T}$ | $\bf 1$ | $\bf 2$ | $-3z$ | $-1/2+3z$
$\nu_{aR}$ | $\bf 1$ | $\bf 1$ | $-3z$ | $3z$
$e_{aR}$ | $\bf 1$ | $\bf 1$ | $-3z$ | $-1+3z$
$q_{\alpha L}=(u_{\alpha L},d_{\alpha L})^{T}$ | $\bf 3$ | $\bf 2$ | $z$ | $1/6-z$
$u_{\alpha R}$ | $\bf 3$ | $\bf 1$ | $z$ | $2/3-z$
$d_{\alpha R}$ | $\bf 3$ | $\bf 1$ | $z$ | $-1/3-z$
$q_{3L}=(u_{3L},d_{3L})^{T}$ | $\bf 3$ | $\bf 2$ | $-z$ | $1/6+z$
$u_{3R}$ | $\bf 3$ | $\bf 1$ | $-z$ | $2/3+z$
$d_{3R}$ | $\bf 3$ | $\bf 1$ | $-z$ | $-1/3+z$
$\phi=(\phi_{1}^{+},\phi_{2}^{0})^{T}$ | $\bf 1$ | $\bf 2$ | $0$ | $1/2$
$\chi_{1}$ | $\bf 1$ | $\bf 1$ | $2z$ | $-2z$
$\chi_{2}$ | $\bf 1$ | $\bf 1$ | $6z$ | $-6z$
Table 1: Matter content in the model, where $\alpha=1,2$ and $a=2,3$ are
generation indices, while $z$ is an arbitrarily nonzero parameter.

Notice that the scalar content introduced above is minimal. Alternatively, a
generic model can be constructed by introducing two new scalar doublets,
namely $\phi^{\prime}\sim({\bf 1},{\bf 2},2z,1/2-2z)$ and
$\phi^{\prime\prime}\sim({\bf 1},{\bf 2},6z,1/2-6z)$, in addition to the usual
doublet $\phi$. This would produce a complete mixing in the quark and lepton
sectors at tree level. Additionally, such a model presents phenomenological
aspects of interest to be differently from the current model and should be
published elsewhere.

##  III Fermion mass

The Yukawa Lagrangian for quarks and leptons in the current model is given by

$\displaystyle\mathcal{L}_{\mathrm{Yukawa}}$ $\displaystyle=$ $\displaystyle
y^{d}_{\alpha\beta}\bar{q}_{\alpha L}\phi d_{\beta
R}+y^{d}_{33}\bar{q}_{3L}\phi d_{3R}+\frac{y^{d}_{\alpha 3}}{M}\bar{q}_{\alpha
L}\phi\chi_{1}d_{3R}+\frac{y^{d}_{3\beta}}{M}\bar{q}_{3L}\phi\chi_{1}^{*}d_{\beta
R}$ (18) $\displaystyle+y^{u}_{\alpha\beta}\bar{q}_{\alpha
L}\tilde{\phi}u_{\beta
R}+y^{u}_{33}\bar{q}_{3L}\tilde{\phi}u_{3R}+\frac{y^{u}_{\alpha
3}}{M}\bar{q}_{\alpha
L}\tilde{\phi}\chi_{1}u_{3R}+\frac{y^{u}_{3\beta}}{M}\bar{q}_{3L}\tilde{\phi}\chi_{1}^{*}u_{\beta
R}$ $\displaystyle+y^{e}_{11}\bar{l}_{1L}\phi
e_{1R}+y^{e}_{ab}\bar{l}_{aL}\phi
e_{bR}+\frac{y^{e}_{1b}}{M}\bar{l}_{1L}\phi\chi_{2}e_{bR}+\frac{y^{e}_{a1}}{M}\bar{l}_{aL}\phi\chi_{2}^{*}e_{1R}$
$\displaystyle+y^{\nu}_{11}\bar{l}_{1L}\tilde{\phi}\nu_{1R}+y^{\nu}_{ab}\bar{l}_{aL}\tilde{\phi}\nu_{bR}+\frac{y^{\nu}_{1b}}{M}\bar{l}_{1L}\tilde{\phi}\chi_{2}\nu_{bR}+\frac{y^{\nu}_{a1}}{M}\bar{l}_{aL}\tilde{\phi}\chi_{2}^{*}\nu_{1R}$
$\displaystyle+\frac{1}{2}f^{\nu}_{11}\bar{\nu}^{c}_{1R}\chi_{2}^{*}\nu_{1R}+\frac{1}{2}f^{\nu}_{ab}\bar{\nu}^{c}_{aR}\chi_{2}\nu_{bR}+F^{\nu}_{1b}\bar{\nu}^{c}_{1R}\nu_{bR}+\mathrm{H.c.},$

where $\tilde{\phi}=i\sigma_{2}\phi^{*}$ with $\sigma_{2}$ is the second Pauli
matrix, $M$ is a new physics scale that defines the effective interactions,
and the couplings $y$ and $f$ are dimensionless. The bare mass $F$ connects
$\nu_{1R}$ and $\nu_{2,3R}$, possibly obtaining a value ranging from zero to
$M$.

###  III.1 Charged fermion mass

From terms in the first three lines of Eq. (18), we obtain the mass matrices
for charged fermions, which are given by

$\displaystyle[M_{q}]_{\alpha\beta}$ $\displaystyle=$
$\displaystyle-y^{q}_{\alpha\beta}\frac{v}{\sqrt{2}},\hskip
14.22636pt[M_{q}]_{33}=-y^{q}_{33}\frac{v}{\sqrt{2}},$ (19)
$\displaystyle\left[M_{q}\right]_{\alpha 3}$ $\displaystyle=$
$\displaystyle-y^{q}_{\alpha 3}\frac{v\Lambda_{1}}{2M},\hskip
14.22636pt[M_{q}]_{3\beta}=-y^{q}_{3\beta}\frac{v\Lambda_{1}}{2M},$ (20)
$\displaystyle\left[M_{e}\right]_{11}$ $\displaystyle=$
$\displaystyle-y^{e}_{11}\frac{v}{\sqrt{2}},\hskip
14.22636pt[M_{e}]_{ab}=-y^{e}_{ab}\frac{v}{\sqrt{2}},$ (21)
$\displaystyle\left[M_{e}\right]_{1b}$ $\displaystyle=$
$\displaystyle-y^{e}_{1b}\frac{v\Lambda_{2}}{2M},\hskip
14.22636pt[M_{e}]_{a1}=-y^{e}_{a1}\frac{v\Lambda_{2}}{2M},$ (22)

where $q=u,d$. Notice that the small mixing between the first two and third
quark generations can be induced by either $y_{\alpha
3}^{q},y_{3\beta}^{q}<y_{\alpha\beta}^{q},y_{33}^{q}$ or $\Lambda_{1}<M$,
while between the first and last two lepton generations can be understood by
either $y_{1b}^{e},y_{a1}^{e}<y_{11}^{q},y_{ab}^{e}$ or $\Lambda_{2}<M$. By
applying biunitary transformations, we can diagonalize these mass matrices
separately, and then get the realistic masses of the up quarks $u,c,t$, the
down quarks $d,s,b$, as well as the charged leptons $e,\mu,\tau$, such as

$\displaystyle V_{u_{L}}^{\dagger}M_{u}V_{u_{R}}$ $\displaystyle=$
$\displaystyle M_{u}^{D}=\text{diag}(m_{u},m_{c},m_{t}),$ (23) $\displaystyle
V_{d_{L}}^{\dagger}M_{d}V_{d_{R}}$ $\displaystyle=$ $\displaystyle
M_{d}^{D}=\text{diag}(m_{d},m_{s},m_{b}),$ (24) $\displaystyle
V_{e_{L}}^{\dagger}M_{e}V_{e_{R}}$ $\displaystyle=$ $\displaystyle
M_{e}^{D}=\text{diag}(m_{e},m_{\mu},m_{\tau}),$ (25)

where $V_{u_{L,R}}$, $V_{d_{L,R}}$ and $V_{e_{L,R}}$ are unitary matrices,
linking gauge states, $u=(u_{1},u_{2},u_{3})^{T}$,
$d=(d_{1},d_{2},d_{3})^{T}$, $e=(e_{1},e_{2},e_{3})^{T}$, to mass eigenstates,
$u^{\prime}=(u,c,t)^{T}$, $d^{\prime}=(d,s,b)^{T}$,
$e^{\prime}=(e,\mu,\tau)^{T}$, respectively,

$u_{L,R}=V_{u_{L,R}}u^{\prime}_{L,R},\hskip
14.22636ptd_{L,R}=V_{d_{L,R}}d^{\prime}_{L,R},\hskip
14.22636pte_{L,R}=V_{e_{L,R}}e^{\prime}_{L,R}.$ (26)

The Cabibbo-Kobayashi-Maskawa (CKM) matrix is then given by
$V=V_{u_{L}}^{\dagger}V_{d_{L}}$.

###  III.2 Neutrino mass

In the current model, neutrinos have both Dirac and Majorana mass terms, and
their total mass matrix takes a specific form,

$\mathcal{L}_{\text{Yukawa}}\supset-\frac{1}{2}(\begin{array}[]{cc}\bar{\nu}_{L}^{c}&\bar{\nu}_{R}\end{array})\left(\begin{array}[]{cc}0&M_{D}\\\
M_{D}^{T}&M_{M}\end{array}\right)\left(\begin{array}[]{cc}\nu_{L}\\\
\nu_{R}^{c}\end{array}\right)+\mathrm{H.c.},$ (27)

where $\nu_{L,R}=(\nu_{1},\nu_{2},\nu_{3})_{L,R}^{T}$ are related to gauge
states, $M_{D}$ and $M_{M}$ are respectively the Dirac and Majorana mass
matrices,

$\displaystyle\left[M_{D}\right]_{11}$ $\displaystyle=$
$\displaystyle-y^{\nu}_{11}\frac{v}{\sqrt{2}},\hskip
14.22636pt[M_{D}]_{ab}=-y^{\nu}_{ab}\frac{v}{\sqrt{2}},$ (28)
$\displaystyle\left[M_{D}\right]_{1b}$ $\displaystyle=$
$\displaystyle-y^{\nu}_{1b}\frac{v\Lambda_{2}}{2M},\hskip
14.22636pt[M_{D}]_{a1}=-y^{\nu}_{a1}\frac{v\Lambda_{2}}{2M},$ (29)
$\displaystyle\left[M_{M}\right]_{11}$ $\displaystyle=$
$\displaystyle-f^{\nu}_{11}\frac{\Lambda_{2}}{\sqrt{2}},\hskip
14.22636pt[M_{M}]_{ab}=-f^{\nu}_{ab}\frac{\Lambda_{2}}{\sqrt{2}},$ (30)
$\displaystyle\left[M_{M}\right]_{1b}$ $\displaystyle=$
$\displaystyle-F^{\nu}_{1b},\hskip 14.22636pt[M_{M}]_{a1}=-F^{\nu}_{1a}.$ (31)

Supposing $M>\Lambda_{2}\gg v$, i.e. $M_{M}\gg M_{D}$, the total mass matrix
of neutrinos in Eq. (27) can be diagonalized via a transformation as

$\left(\begin{array}[]{cc}\nu_{L}\\\
\nu_{R}^{c}\end{array}\right)\simeq\left(\begin{array}[]{cc}1&\kappa^{*}\\\
-\kappa^{T}&1\end{array}\right)\left(\begin{array}[]{cc}V_{\nu_{L}}&0\\\
0&V_{\nu_{R}}^{*}\end{array}\right)\left(\begin{array}[]{cc}\nu^{\prime}_{L}\\\
{\nu^{\prime}}_{R}^{c}\end{array}\right),$ (32)

where $\kappa$ is the $\nu_{L}$-$\nu_{R}$ mixing element,
$\kappa=M_{D}M_{M}^{-1}\sim v/(\Lambda_{2},M)$, while
$\nu^{\prime}_{L,R}=(\nu^{\prime}_{1},\nu^{\prime}_{2},\nu^{\prime}_{3})_{L,R}^{T}$
are related to mass eigenstates, connecting to $\nu_{L,R}$ via unitary
matrices $V_{\nu_{L,R}}$ as

$\nu_{L}\simeq V_{\nu_{L}}\nu^{\prime}_{L},\hskip 14.22636pt\nu_{R}\simeq
V_{\nu_{R}}\nu^{\prime}_{R}.$ (33)

Then, the mass eigenvalues are approximately given by

$\displaystyle\text{diag}(m_{1},m_{2},m_{3})$ $\displaystyle\simeq$
$\displaystyle-V_{\nu_{L}}^{T}M_{D}M_{M}^{-1}M_{D}^{T}V_{\nu_{L}},$ (34)
$\displaystyle\text{diag}(M_{1},M_{2},M_{3})$ $\displaystyle\simeq$
$\displaystyle V_{\nu_{R}}^{\dagger}M_{M}V_{\nu_{R}}^{*},$ (35)

in which $m_{1,2,3}\sim v^{2}/\Lambda_{2}$ are appropriately small, identified
with the observed neutrino masses, whereas $M_{1,2,3}\sim\Lambda_{2}$ are the
sterile neutrino masses, being at the new physics scale. Note that the
Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix can be written as
$U=V_{e_{L}}^{\dagger}V_{\nu_{L}}$. Note also that $F$ only contributes to
right-handed neutrino mixing but does not set the seesaw scale.

##  IV Gauge and scalar sectors

###  IV.1 Gauge sector

The gauge bosons acquire masses via the scalar kinetic term
$\sum_{S=\phi,\chi_{1},\chi_{2}}(D^{\mu}\langle
S\rangle)^{\dagger}(D_{\mu}\langle S\rangle)$ when the gauge symmetry breaking
occurs. The covariant derivative takes the form

$D_{\mu}=\partial_{\mu}+ig_{s}t_{p}G_{p\mu}+igT_{n}A_{n\mu}+ig_{X}XB_{\mu}+ig_{N}NC_{\mu},$
(36)

where $(g_{s},g,g_{X},g_{N})$, $(t_{p},T_{n},X,N)$, and $(G_{p},A_{n},B,C)$
are coupling constants, generators, and gauge bosons of the ($SU(3)_{C}$,
$SU(2)_{L}$, $U(1)_{X}$, $U(1)_{N}$) groups, respectively. Identifying the
charged gauge bosons as $W^{\pm}_{\mu}=(A_{1\mu}\mp iA_{2\mu})/\sqrt{2}$, we
obtain

$\mathcal{L}\supset\frac{g^{2}v^{2}}{4}W^{\mu+}W_{\mu}^{-}+\frac{1}{2}(A^{\mu}_{3}\,B^{\mu}\,C^{\mu})M_{0}^{2}(A_{3\mu}\,B_{\mu}\,C_{\mu})^{T},$
(37)

where

$M_{0}^{2}=\left(\begin{array}[]{ccc}\frac{1}{4}g^{2}v^{2}&0&-\frac{1}{4}gg_{N}v^{2}\\\
0&4g_{X}^{2}z^{2}(\Lambda_{1}^{2}+9\Lambda_{2}^{2})&-4g_{X}g_{N}z^{2}(\Lambda_{1}^{2}+9\Lambda_{2}^{2})\\\
-\frac{1}{4}gg_{N}v^{2}&-4g_{X}g_{N}z^{2}(\Lambda_{1}^{2}+9\Lambda_{2}^{2})&\frac{1}{4}g_{N}^{2}[16z^{2}(\Lambda_{1}^{2}+9\Lambda_{2}^{2})+v^{2}]\end{array}\right).$
(38)

Hence, the boson $W$ is a physical field by itself with mass
$m_{W}^{2}=g^{2}v^{2}/4$, which is identified to the SM $W$ boson, thus
$v=246\text{ GeV}$, as expected.

Concerning the neutral gauge bosons, the mass-squared matrix $M_{0}^{2}$
always has a zero eigenvalue (i.e. photon mass) with corresponding eigenstate
(i.e. photon field),

$A=\frac{g_{X}g_{N}A_{3}+gg_{N}B+gg_{X}C}{\sqrt{g_{X}^{2}g_{N}^{2}+g^{2}g_{N}^{2}+g^{2}g_{X}^{2}}}.$
(39)

From here, the interaction of the photon with fermions can be calculated [29].
Identifying the coefficient of these interaction vertices with the
electromagnetic coupling constant, we get the sine of the Weinberg’s angle as
$s_{W}=g_{X}g_{N}/\sqrt{g_{X}^{2}g_{N}^{2}+g^{2}g_{N}^{2}+g^{2}g_{X}^{2}}$,
and thus the hypercharge coupling to be
$g_{Y}=g_{X}g_{N}/\sqrt{g_{X}^{2}+g_{N}^{2}}=g_{X}s_{\theta}=g_{N}c_{\theta}$,
where the angle $\theta$ is defined by $t_{\theta}=g_{N}/g_{X}$. We rewrite
the photon field,

$A=s_{W}A_{3}+c_{W}(s_{\theta}B+c_{\theta}C).$ (40)

Hence, we define the SM $Z$ boson orthogonal to the photon $A$ and a new gauge
boson $Z^{\prime}$ orthogonal to both $A$ and $Z$, such as

$\displaystyle Z$ $\displaystyle=$ $\displaystyle
c_{W}A_{3}-s_{W}(s_{\theta}B+c_{\theta}C),$ (41) $\displaystyle Z^{\prime}$
$\displaystyle=$ $\displaystyle c_{\theta}B-s_{\theta}C.$ (42)

In the new basis ($A,Z,Z^{\prime}$), the photon $A$ is decoupled as a physical
field, whereas two states $Z$ and $Z^{\prime}$ still mix by themselves via a
$2\times 2$ symmetric submatrix with the elements

$\displaystyle m_{Z}^{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}v^{2}}{4c_{W}^{2}},\hskip
14.22636ptm_{ZZ^{\prime}}^{2}=\frac{g^{2}v^{2}}{4c_{W}^{2}}s_{W}t_{\theta},$
(43) $\displaystyle m_{Z^{\prime}}^{2}$ $\displaystyle=$
$\displaystyle\frac{g^{2}t_{W}^{2}}{s_{2\theta}^{2}}[16z^{2}(\Lambda_{1}^{2}+9\Lambda_{2}^{2})+s_{\theta}^{4}v^{2}].$
(44)

Diagonalizing this submatrix, we get two physical fields,

$Z_{1}=c_{\varphi}Z-s_{\varphi}Z^{\prime},\hskip
14.22636ptZ_{2}=s_{\varphi}Z+c_{\varphi}Z^{\prime},$ (45)

and two corresponding masses,

$\displaystyle m^{2}_{Z_{1}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[m^{2}_{Z}+m^{2}_{Z^{\prime}}-\sqrt{(m^{2}_{Z}-m^{2}_{Z^{\prime}})^{2}+4m^{4}_{ZZ^{\prime}}}\right]\simeq
m^{2}_{Z}-\frac{m^{4}_{ZZ^{\prime}}}{m^{2}_{Z^{\prime}}},$ (46) $\displaystyle
m^{2}_{Z_{2}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left[m^{2}_{Z}+m^{2}_{Z^{\prime}}+\sqrt{(m^{2}_{Z}-m^{2}_{Z^{\prime}})^{2}+4m^{4}_{ZZ^{\prime}}}\right]\simeq
m^{2}_{Z^{\prime}},$ (47)

where the approximations apply due to $v\ll\Lambda_{1,2}$. Also, the mixing
angle $\varphi$ in Eq. (45) is given by

$t_{2\varphi}=\frac{2m^{2}_{ZZ^{\prime}}}{m^{2}_{Z^{\prime}}-m^{2}_{Z}}\simeq\frac{s_{\theta}^{3}c_{\theta}v^{2}}{8s_{W}z^{2}(\Lambda_{1}^{2}+9\Lambda_{2}^{2})}.$
(48)

It is easy to see that the $Z$-$Z^{\prime}$ mixing is small as suppressed by
$v^{2}/\Lambda_{1,2}^{2}$. Additionally, the field $Z_{1}$ has a mass
approximating that of the SM, and thus, it is called the SM $Z$-like boson,
whereas the field $Z_{2}$ is a new heavy gauge boson with mass at
$\Lambda_{1,2}$ scale.

###  IV.2 Scalar sector

The current model’s scalar sector contains a doublet $\phi$ and two singlets
$\chi_{1,2}$ under $SU(2)_{L}$. Thus, the scalar potential has a simple form
as

$\displaystyle V$ $\displaystyle=$
$\displaystyle\mu_{1}^{2}\phi^{\dagger}\phi+\mu_{2}^{2}\chi_{1}^{*}\chi_{1}+\mu_{3}^{2}\chi_{2}^{*}\chi_{2}+(\lambda\chi_{1}^{3}\chi_{2}^{*}+\mathrm{H.c.})$
(49)
$\displaystyle+\lambda_{1}(\phi^{\dagger}\phi)^{2}+\lambda_{2}(\chi_{1}^{*}\chi_{1})^{2}+\lambda_{3}(\chi_{2}^{*}\chi_{2})^{2}$
$\displaystyle+\lambda_{4}(\phi^{\dagger}\phi)(\chi_{1}^{*}\chi_{1})+\lambda_{5}(\phi^{\dagger}\phi)(\chi_{2}^{*}\chi_{2})+\lambda_{6}(\chi_{1}^{*}\chi_{1})(\chi_{2}^{*}\chi_{2}),$

where the couplings $\lambda$’s are dimensionless, whereas $\mu$’s have a mass
dimension. The necessary conditions for this scalar potential to be bounded
from below and yielding a desirable vacuum structure are

$\displaystyle\mu_{1,2,3}^{2}$ $\displaystyle<$ $\displaystyle 0,\hskip
14.22636pt|\mu_{1}|\ll|\mu_{2,3}|,\hskip 14.22636pt\lambda_{1,2,3}>0,$ (50)
$\displaystyle\lambda_{4}$ $\displaystyle>$
$\displaystyle-2\sqrt{\lambda_{1}\lambda_{2}},\hskip
14.22636pt\lambda_{5}>-2\sqrt{\lambda_{1}\lambda_{3}},\hskip
14.22636pt\lambda_{6}>-2\sqrt{\lambda_{2}\lambda_{3}}.$ (51)

To obtain the physical scalar spectrum, we expand the scalar fields around
their VEVs, such as

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\left(\begin{array}[]{c}\phi^{+}_{1}\\\
\frac{1}{\sqrt{2}}(v+S_{1}+iA_{1})\end{array}\right),$ (54)
$\displaystyle\chi_{1}$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2}}(\Lambda_{1}+S_{2}+iA_{2}),\hskip
14.22636pt\chi_{2}=\frac{1}{\sqrt{2}}(\Lambda_{2}+S_{3}+iA_{3}),$ (55)

and then substitute them into the scalar potential. By using the potential
minimum conditions given by

$\displaystyle
2\lambda_{1}v^{2}+\lambda_{4}\Lambda_{1}^{2}+\lambda_{5}\Lambda_{2}^{2}+2\mu_{1}^{2}$
$\displaystyle=$ $\displaystyle 0,$ (56) $\displaystyle
2\lambda_{2}\Lambda_{1}^{2}+\lambda_{4}v^{2}+\lambda_{6}\Lambda_{2}^{2}+3\lambda\Lambda_{1}\Lambda_{2}+2\mu_{2}^{2}$
$\displaystyle=$ $\displaystyle 0,$ (57)
$\displaystyle\lambda\Lambda_{1}^{3}+(2\lambda_{3}\Lambda_{2}^{2}+\lambda_{5}v^{2}+\lambda_{6}\Lambda_{1}^{2}+2\mu_{3}^{2})\Lambda_{2}$
$\displaystyle=$ $\displaystyle 0,$ (58)

we get the mass-squared matrix for $CP$-even scalar sector as

$M_{S}^{2}=\left(\begin{array}[]{ccc}2\lambda_{1}v^{2}&\lambda_{4}v\Lambda_{1}&\lambda_{5}v\Lambda_{2}\\\
\lambda_{4}v\Lambda_{1}&\frac{1}{2}(4\lambda_{2}\Lambda_{1}+3\lambda\Lambda_{2})\Lambda_{1}&\frac{1}{2}(2\lambda_{6}\Lambda_{1}\Lambda_{2}+3\lambda\Lambda_{1}^{2})\\\
\lambda_{5}v\Lambda_{2}&\frac{1}{2}(2\lambda_{6}\Lambda_{1}\Lambda_{2}+3\lambda\Lambda_{1}^{2})&\frac{1}{2\Lambda_{2}}(4\lambda_{3}\Lambda_{2}^{3}-\lambda\Lambda_{1}^{3})\end{array}\right).$
(59)

Because of the condition, $v\ll\Lambda_{1,2}$, the first row and first column
of $M_{S}^{2}$ consist of elements much smaller than those of the rest.
Therefore, the matrix $M_{S}^{2}$ can be diagonalized by using the seesaw
approximation to separate the light state ($S_{1}$) from the heavy states
($S_{2,3}$). Labeling the new basis as ($H,H_{1},H_{2}$), for which $H$ is
decoupled as a physical field, we have

$H\simeq S_{1}-\epsilon_{1}S_{2}-\epsilon_{2}S_{3}$ (60)

with a corresponding mass

$m^{2}_{H}\simeq
2\lambda_{1}v^{2}-(\epsilon_{1}\lambda_{4}\Lambda_{1}+\epsilon_{2}\lambda_{5}\Lambda_{2})v,$
(61)

while the remaining states $H_{1}\simeq\epsilon_{1}S_{1}+S_{2}$ and
$H_{2}\simeq\epsilon_{2}S_{1}+S_{3}$ mix by themselves via a submatrix as

$\mathcal{M}^{2}\simeq\frac{1}{2}\left(\begin{array}[]{cc}(4\lambda_{2}\Lambda_{1}+3\lambda\Lambda_{2})\Lambda_{1}&(2\lambda_{6}\Lambda_{2}+3\lambda\Lambda_{1})\Lambda_{1}\\\
(2\lambda_{6}\Lambda_{2}+3\lambda\Lambda_{1})\Lambda_{1}&\frac{1}{\Lambda_{2}}(4\lambda_{3}\Lambda_{2}^{3}-\lambda\Lambda_{1}^{3})\end{array}\right).$
(62)

Above, the mixing parameters are given by

$\displaystyle\epsilon_{1}$ $\displaystyle=$
$\displaystyle\frac{[\lambda(\lambda_{4}\Lambda_{1}^{3}+3\lambda_{5}\Lambda_{1}\Lambda_{2}^{2})-2(2\lambda_{3}\lambda_{4}-\lambda_{5}\lambda_{6})\Lambda_{2}^{3}]v}{2[3\lambda^{2}\Lambda_{1}^{3}\Lambda_{2}+\lambda(\lambda_{2}\Lambda_{1}^{4}-3\lambda_{3}\Lambda_{2}^{4}+3\lambda_{6}\Lambda_{1}^{2}\Lambda_{2}^{2})-(4\lambda_{2}\lambda_{3}-\lambda_{6}^{2})\Lambda_{1}\Lambda_{2}^{3}]},$
(63) $\displaystyle\epsilon_{2}$ $\displaystyle=$
$\displaystyle\frac{[3\lambda(\lambda_{4}\Lambda_{1}^{2}-\lambda_{5}\Lambda_{2}^{2})-2(2\lambda_{2}\lambda_{5}-\lambda_{4}\lambda_{6})\Lambda_{1}\Lambda_{2}]v\Lambda_{2}}{2[3\lambda^{2}\Lambda_{1}^{3}\Lambda_{2}+\lambda(\lambda_{2}\Lambda_{1}^{4}-3\lambda_{3}\Lambda_{2}^{4}+3\lambda_{6}\Lambda_{1}^{2}\Lambda_{2}^{2})-(4\lambda_{2}\lambda_{3}-\lambda_{6}^{2})\Lambda_{1}\Lambda_{2}^{3}]},$
(64)

which are small as suppressed by $v/\Lambda_{1,2}$.

Diagonalizing the submatrix $\mathcal{M}^{2}$, we get two physical fields,

$\mathcal{H}_{1}=c_{\xi}H_{1}-s_{\xi}H_{2},\hskip
14.22636pt\mathcal{H}_{2}=s_{\xi}H_{1}+c_{\xi}H_{2},$ (65)

with corresponding masses

$\displaystyle m^{2}_{\mathcal{H}_{1,2}}$ $\displaystyle=$
$\displaystyle\frac{1}{4\Lambda_{2}}\left\\{4\lambda_{3}\Lambda_{2}^{3}-\lambda\Lambda_{1}^{3}+(4\lambda_{2}\Lambda_{1}+3\lambda\Lambda_{2})\Lambda_{1}\Lambda_{2}\right.$
(66)
$\displaystyle\left.\mp\sqrt{[4\lambda_{3}\Lambda_{2}^{3}-\lambda\Lambda_{1}^{3}-(4\lambda_{2}\Lambda_{1}+3\lambda\Lambda_{2})\Lambda_{1}\Lambda_{2}]^{2}+4(2\lambda_{6}\Lambda_{2}+3\lambda\Lambda_{1})^{2}\Lambda_{1}^{2}\Lambda_{2}^{2}}\right\\},$

where the mixing angle $\xi$ is given by

$t_{2\xi}=\frac{2(2\lambda_{6}\Lambda_{2}+3\lambda\Lambda_{1})\Lambda_{1}\Lambda_{2}}{4\lambda_{3}\Lambda_{2}^{3}-\lambda\Lambda_{1}^{3}-(4\lambda_{2}\Lambda_{1}+3\lambda\Lambda_{2})\Lambda_{1}\Lambda_{2}}.$
(67)

The Higgs boson $H$ has a mass in weak scale like the SM Higgs boson, so $H$
is called the SM-like Higgs boson, whereas $\mathcal{H}_{1,2}$ are the new
Higgs bosons, heavy in $\Lambda_{1,2}$ scale.

The $CP$-odd scalars, $A_{1,2,3}$, mix by themselves via a mass-squared matrix

$M_{A}^{2}=\frac{\lambda}{2}\left(\begin{array}[]{ccc}0&0&0\\\
0&-9\Lambda_{1}\Lambda_{2}&3\Lambda_{1}^{2}\\\
0&3\Lambda_{1}^{2}&-\Lambda_{1}^{3}/\Lambda_{2}\end{array}\right).$ (68)

This matrix has exactly two zero eigenvalues corresponding to two eigenstates,

$G_{Z_{1}}=A_{1},\hskip
14.22636ptG_{Z_{2}}=\frac{\Lambda_{1}A_{2}+3\Lambda_{2}A_{3}}{\sqrt{\Lambda_{1}^{2}+9\Lambda_{2}^{2}}},$
(69)

which are the Goldstone bosons associated with the neutral gauge bosons,
$Z_{1}$ and $Z_{2}$, respectively. The remaining eigenstate labeled
$\mathcal{A}$ is a physical pseudoscalar orthogonal to $G_{Z_{2}}$, heavy at
the $\Lambda_{1,2}$ scale, namely

$\mathcal{A}=\frac{3\Lambda_{2}A_{2}-\Lambda_{1}A_{3}}{\sqrt{\Lambda_{1}^{2}+9\Lambda_{2}^{2}}},\hskip
14.22636ptm^{2}_{\mathcal{A}}=-\frac{\lambda(\Lambda_{1}^{2}+9\Lambda_{2}^{2})\Lambda_{1}}{2\Lambda_{2}}.$
(70)

Here, the requirement of positive squared mass implies the parameter $\lambda$
to be negative.

Concerning the charged scalars, we obtain a massless eigenstate,
$G_{W}^{\pm}\equiv\phi_{1}^{\pm}$, identical to the Goldstone boson eaten by
the SM $W$ boson.

##  V  Fermion-gauge boson interaction

We now consider the interaction of gauge bosons with fermions, which results
from the fermion kinetic term, i.e., $\sum_{F}\bar{F}i\gamma^{\mu}D_{\mu}F$,
where $F$ runs over fermion multiplets in the model. For convenience, we
rewrite the covariant derivative in Eq. (36) in the new form of

$\displaystyle D_{\mu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}+ig_{s}t_{p}G_{p\mu}+igs_{W}QA_{\mu}+ig(T_{+}W^{+}_{\mu}+\mathrm{H.c.})$
(71)
$\displaystyle+\frac{ig}{c_{W}}\left[c_{\varphi}(T_{3}-s^{2}_{W}Q)-s_{\varphi}\frac{s_{W}}{s_{\theta}c_{\theta}}(X-s^{2}_{\theta}Y)\right]Z_{1\mu}$
$\displaystyle+\frac{ig}{c_{W}}\left[s_{\varphi}(T_{3}-s^{2}_{W}Q)+c_{\varphi}\frac{s_{W}}{s_{\theta}c_{\theta}}(X-s^{2}_{\theta}Y)\right]Z_{2\mu},$

where $T_{\pm}=(T_{1}\pm iT_{2})/\sqrt{2}$ are the weight-raising and lowering
operators of the $SU(2)_{L}$ group. Notice that $Q$, $T_{3}$, and $Y$ are
universal for every flavor of neutrinos, charged leptons, up-type quarks, and
down-type quarks, but $X$ is not. Consequently, both $Z_{1}$ and $Z_{2}$
flavor-change when interacting with fermions, in which the flavor-changing
effect associated with $Z_{1}$ results from the $Z$–$Z^{\prime}$ mixing to be
small, whereas the flavor change associated with $Z_{2}$ is dominant, even for
$\varphi=0$.

It is easily checked that the interaction of gluons and photon with fermions
is similar to the SM, while the interaction of the $W$ boson with fermions is
modified by the PMNS matrix,

$\mathcal{L}\supset-\frac{g}{\sqrt{2}}(\bar{\nu}_{iL}\gamma^{\mu}U_{ij}^{\dagger}e_{jL}+\bar{u}_{iL}\gamma^{\mu}V_{ij}d_{jL})W^{+}_{\mu}+\mathrm{H.c.},$
(72)

where $i,j=1,2,3$ are mass eigenstate indexes, i.e.,
$\nu_{iL}=\\{\nu^{\prime}_{1L},\nu^{\prime}_{2L},\nu^{\prime}_{3L}\\}$,
$e_{i}=\\{e,\mu,\tau\\}$, $u_{i}=\\{u,c,t\\}$, and $d_{i}=\\{d,s,b\\}$.

For the interaction of $Z_{1,2}$ with fermions, using the unitary condition of
mixing matrices,

$V_{\nu_{L,R}}^{\dagger}V_{\nu_{L,R}}=V_{e_{L,R}}^{\dagger}V_{e_{L,R}}=V_{u_{L,R}}^{\dagger}V_{u_{L,R}}=V_{d_{L,R}}^{\dagger}V_{d_{L,R}}=1,$
(73)

we obtain a flavor-conserving part, given in the form of

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle-\frac{g}{2c_{W}}\left\\{C_{1L}^{Z_{I}}\bar{\nu}^{\prime}_{1L}\gamma^{\mu}\nu^{\prime}_{1L}+C_{2L}^{Z_{I}}(\bar{\nu}^{\prime}_{2L}\gamma^{\mu}\nu^{\prime}_{2L}+\bar{\nu}^{\prime}_{3L}\gamma^{\mu}\nu^{\prime}_{3L})\right.$
(74)
$\displaystyle\left.+C_{R}^{Z_{I}}(\bar{\nu}^{\prime}_{1R}\gamma^{\mu}\nu^{\prime}_{1R}-\bar{\nu}^{\prime}_{2R}\gamma^{\mu}\nu^{\prime}_{2R}-\bar{\nu}^{\prime}_{3R}\gamma^{\mu}\nu^{\prime}_{3R})+\bar{f}\gamma^{\mu}[g^{Z_{I}}_{V}(f)-g^{Z_{I}}_{A}(f)\gamma_{5}]f\right\\}Z_{I\mu},$

where $I=1,2$, and $f$ denotes the physical charged fermions in the model.
Additionally, the flavor-conserving couplings are given by

$\displaystyle C_{1L}^{Z_{1}}$ $\displaystyle=$ $\displaystyle
c_{\varphi}-s_{\varphi}\frac{s_{W}}{s_{\theta}c_{\theta}}(6z+s_{\theta}^{2}),\hskip
14.22636ptC_{2L}^{Z_{1}}=c_{\varphi}+s_{\varphi}\frac{s_{W}}{s_{\theta}c_{\theta}}(6z-s_{\theta}^{2}),$
(75) $\displaystyle C_{R}^{Z_{1}}$ $\displaystyle=$ $\displaystyle-
s_{\varphi}\frac{6s_{W}z}{s_{\theta}c_{\theta}},$ (76) $\displaystyle
g_{V}^{Z_{1}}(f)$ $\displaystyle=$ $\displaystyle
c_{\varphi}[T_{3}(f_{L})-2Q(f)s_{W}^{2}]-s_{\varphi}\frac{s_{W}}{s_{\theta}c_{\theta}}\\{[T_{3}(f_{L})-2Q(f)]s_{\theta}^{2}+2X(f)\\},$
(77) $\displaystyle g_{A}^{Z_{1}}(f)$ $\displaystyle=$ $\displaystyle
T_{3}(f_{L})(c_{\varphi}-s_{\varphi}s_{W}t_{\theta}),$ (78) $\displaystyle
C_{1L,2L}^{Z_{2}}$ $\displaystyle=$ $\displaystyle
C_{1L,2L}^{Z_{1}}|_{c_{\varphi}\to s_{\varphi},s_{\varphi}\to-
c_{\varphi}},\hskip 14.22636ptC_{R}^{Z_{2}}=C_{R}^{Z_{1}}|_{s_{\varphi}\to-
c_{\varphi}},$ (79) $\displaystyle g_{V,A}^{Z_{2}}(f)$ $\displaystyle=$
$\displaystyle g_{V,A}^{Z_{1}}(f)|_{c_{\varphi}\to s_{\varphi},s_{\varphi}\to-
c_{\varphi}}.$ (80)

More specifically, we show the flavor-conserving couplings of $Z_{1,2}$ with
the charged fermions in Tables 2 and 3, respectively. It is easy to see that
the $Z_{1}$ couplings with to the fermions are identical to those of the SM
$Z$ boson in the limit $\varphi\to 0$.

$f$ | $g^{Z_{1}}_{V}(f)$ | $g^{Z_{1}}_{A}(f)$
---|---|---
$e$ | $c_{\varphi}\left(2s^{2}_{W}-\frac{1}{2}\right)-3s_{\varphi}s_{W}\left(\frac{1}{2}t_{\theta}+\frac{2z}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(s_{\varphi}s_{W}t_{\theta}-c_{\varphi}\right)$
$\mu,\tau$ | $c_{\varphi}\left(2s^{2}_{W}-\frac{1}{2}\right)-3s_{\varphi}s_{W}\left(\frac{1}{2}t_{\theta}-\frac{2z}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(s_{\varphi}s_{W}t_{\theta}-c_{\varphi}\right)$
$u,c$ | $c_{\varphi}\left(\frac{1}{2}-\frac{4}{3}s^{2}_{W}\right)+s_{\varphi}s_{W}\left(\frac{5}{6}t_{\theta}-\frac{2z}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(c_{\varphi}-s_{\varphi}s_{W}t_{\theta}\right)$
$t$ | $c_{\varphi}\left(\frac{1}{2}-\frac{4}{3}s^{2}_{W}\right)+s_{\varphi}s_{W}\left(\frac{5}{6}t_{\theta}+\frac{2z}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(c_{\varphi}-s_{\varphi}s_{W}t_{\theta}\right)$
$d,s$ | $c_{\varphi}\left(\frac{2}{3}s^{2}_{W}-\frac{1}{2}\right)-s_{\varphi}s_{W}\left(\frac{1}{6}t_{\theta}+\frac{2z}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(s_{\varphi}s_{W}t_{\theta}-c_{\varphi}\right)$
$b$ | $c_{\varphi}\left(\frac{2}{3}s^{2}_{W}-\frac{1}{2}\right)-s_{\varphi}s_{W}\left(\frac{1}{6}t_{\theta}-\frac{2z}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(s_{\varphi}s_{W}t_{\theta}-c_{\varphi}\right)$
Table 2:  Flavor-conserving couplings of $Z_{1}$ with the charged fermions. $f$ | $g^{Z_{2}}_{V}(f)$ | $g^{Z_{2}}_{A}(f)$
---|---|---
$e$ | $s_{\varphi}\left(2s^{2}_{W}-\frac{1}{2}\right)+3c_{\varphi}s_{W}\left(\frac{1}{2}t_{\theta}+\frac{2z}{s_{\theta}c_{\theta}}\right)$ | $-\frac{1}{2}\left(c_{\varphi}s_{W}t_{\theta}+s_{\varphi}\right)$
$\mu,\tau$ | $s_{\varphi}\left(2s^{2}_{W}-\frac{1}{2}\right)+3c_{\varphi}s_{W}\left(\frac{1}{2}t_{\theta}-\frac{2z}{s_{\theta}c_{\theta}}\right)$ | $-\frac{1}{2}\left(c_{\varphi}s_{W}t_{\theta}+s_{\varphi}\right)$
$u,c$ | $s_{\varphi}\left(\frac{1}{2}-\frac{4}{3}s^{2}_{W}\right)-c_{\varphi}s_{W}\left(\frac{5}{6}t_{\theta}-\frac{2z}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(c_{\varphi}s_{W}t_{\theta}+s_{\varphi}\right)$
$t$ | $s_{\varphi}\left(\frac{1}{2}-\frac{4}{3}s^{2}_{W}\right)-c_{\varphi}s_{W}\left(\frac{5}{6}t_{\theta}+\frac{2z}{s_{\theta}c_{\theta}}\right)$ | $\frac{1}{2}\left(c_{\varphi}s_{W}t_{\theta}+s_{\varphi}\right)$
$d,s$ | $s_{\varphi}\left(\frac{2}{3}s^{2}_{W}-\frac{1}{2}\right)+c_{\varphi}s_{W}\left(\frac{1}{6}t_{\theta}+\frac{2z}{s_{\theta}c_{\theta}}\right)$ | $-\frac{1}{2}\left(c_{\varphi}s_{W}t_{\theta}+s_{\varphi}\right)$
$b$ | $s_{\varphi}\left(\frac{2}{3}s^{2}_{W}-\frac{1}{2}\right)+c_{\varphi}s_{W}\left(\frac{1}{6}t_{\theta}-\frac{2z}{s_{\theta}c_{\theta}}\right)$ | $-\frac{1}{2}\left(c_{\varphi}s_{W}t_{\theta}+s_{\varphi}\right)$
Table 3:  Flavor-conserving couplings of $Z_{2}$ with the charged fermions.

To obtain flavor-changing part, we look at fermion-$Z_{1,2}$ interactions
induced by $X$-charge, namely

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\frac{zgt_{W}}{s_{\theta}c_{\theta}}(\bar{l}_{L,R}\gamma^{\mu}T_{l}l_{L,R}+\bar{q}_{L,R}\gamma^{\mu}T_{q}q_{L,R})(s_{\varphi}Z_{1\mu}-c_{\varphi}Z_{2\mu}),$
(81)

where we have denoted $T_{l}=\text{diag}(3,-3,-3)$,
$T_{q}=\text{diag}(1,1,-1)$, and $l=\nu,e$, while $q=u,d$. Changing to the
mass basis via transformations $l_{L,R}=V_{l_{L,R}}l^{\prime}_{L,R}$ and
$q_{L,R}=V_{q_{L,R}}q^{\prime}_{L,R}$, we obtain

$\displaystyle\mathcal{L}$ $\displaystyle\supset$
$\displaystyle\frac{zgt_{W}}{s_{\theta}c_{\theta}}(\bar{l}^{\prime}_{L,R}\gamma^{\mu}V_{l_{L,R}}^{\dagger}T_{l}V_{l_{L,R}}l^{\prime}_{L,R}+\bar{q}^{\prime}_{L,R}\gamma^{\mu}V_{q_{L,R}}^{\dagger}T_{q}V_{q_{L,R}}q^{\prime}_{L,R})(s_{\varphi}Z_{1\mu}-c_{\varphi}Z_{2\mu})$
(82) $\displaystyle\supset$
$\displaystyle\frac{2zgt_{W}}{s_{\theta}c_{\theta}}(3[V_{l_{L}}^{*}]_{1i}[V_{l_{L}}]_{1j}\bar{l}_{iL}\gamma^{\mu}l_{jL}-[V_{q_{L}}^{*}]_{3i}[V_{q_{L}}]_{3j}\bar{q}_{iL}\gamma^{\mu}q_{jL})(s_{\varphi}Z_{1\mu}-c_{\varphi}Z_{2\mu})+(L\to
R)$ $\displaystyle\equiv$
$\displaystyle(\Gamma_{ij}^{l_{L}}\bar{l}_{iL}\gamma^{\mu}l_{jL}+\Gamma_{ij}^{q_{L}}\bar{q}_{iL}\gamma^{\mu}q_{jL})(s_{\varphi}Z_{1\mu}-c_{\varphi}Z_{2\mu})+(L\to
R),$

which give rise to flavor-changing interactions for $i\neq j$. Here, we have
labeled

$\Gamma_{ij}^{l_{L}}=\frac{6gzt_{W}}{s_{\theta}c_{\theta}}[V_{l_{L}}^{*}]_{1i}[V_{l_{L}}]_{1j},\hskip
14.22636pt\Gamma_{ij}^{q_{L}}=-\frac{2gzt_{W}}{s_{\theta}c_{\theta}}[V_{q_{L}}^{*}]_{3i}[V_{q_{L}}]_{3j}.$
(83)

##  VI  Flavor phenomenologies

To explain some flavor anomalies based on flavor-changing interactions in the
current model, we first perform some assumptions for related parameters. It
has been previously mentioned that the CKM and PMNS matrices are determined as
$V=V_{u_{L}}^{\dagger}V_{d_{L}}$ and $U=V_{e_{L}}^{\dagger}V_{\nu_{L}}$,
respectively. For the sake of simplicity, in this section, we disregard the
effect of neutrino mixing and set the mixing matrix of left-handed charged
leptons $V_{e_{L}}^{\dagger}$ to the PMNS matrix $U$. For the quark sector, we
focus solely on studying the flavor-changing of down quarks. Hence, we
identify the mixing matrix of left-handed down-type quarks $V_{d_{L}}$ with
the CKM matrix $V$, while the up-type quark-mixing matrices $V_{u_{L,R}}$ are
unit matrices. For the mixing matrix of right-handed down-type quarks
$V_{d_{R}}$, we parameterize it through three mixing angles
$\theta^{d_{R}}_{ij}$ and a CP-violating phase $\delta^{d_{R}}$, similar to
the parametrization of the CKM and PMNS matrices, i.e.,

$\displaystyle
V_{d_{R}}=\left(\begin{array}[]{ccc}c^{d_{R}}_{12}c^{d_{R}}_{13}&s^{d_{R}}_{12}c^{d_{R}}_{13}&s^{d_{R}}_{13}e^{-i\delta^{d_{R}}}\\\
-s^{d_{R}}_{12}c_{23}^{d_{R}}-c_{12}^{d_{R}}s^{d_{R}}_{13}s_{23}^{d_{R}}e^{i\delta^{d_{R}}}&c^{d_{R}}_{12}c^{d_{R}}_{23}-s_{12}^{d_{R}}s^{d_{R}}_{13}s_{23}^{d_{R}}e^{i\delta^{d_{R}}}&c_{13}^{d_{R}}s_{23}^{d_{R}}\\\
s^{d_{R}}_{12}s_{23}^{d_{R}}-c_{12}^{d_{R}}s^{d_{R}}_{13}c_{23}^{d_{R}}e^{i\delta^{d_{R}}}&-c^{d_{R}}_{12}s_{23}^{d_{R}}-s_{12}^{d_{R}}s^{d_{R}}_{13}c_{23}^{d_{R}}e^{i\delta^{d_{R}}}&c^{d_{R}}_{13}c^{d_{R}}_{23}\end{array}\right),$
(87)

where $c^{d_{R}}_{ij}\equiv\cos\theta^{d_{R}}_{ij}$ and
$s^{d_{R}}_{ij}\equiv\sin\theta^{d_{R}}_{ij}$ [30]. Additionally, we assume
that there are four different scenarios of the hierarchy of these mixing
angles as

$\displaystyle\frac{s_{13}^{d_{R}}}{s_{12}^{d_{R}}}$ $\displaystyle=$
$\displaystyle\frac{s^{\text{CKM}}_{13}}{s^{\text{CKM}}_{12}},\hskip
14.22636pt\frac{s_{23}^{d_{R}}}{s_{12}^{d_{R}}}=\frac{s^{\text{CKM}}_{23}}{s^{\text{CKM}}_{12}},\hskip
14.22636pt\text{Normal hierarchy-QNH},$ (88)
$\displaystyle\frac{s_{13}^{d_{R}}}{s_{12}^{d_{R}}}$ $\displaystyle=$
$\displaystyle\frac{s^{\text{CKM}}_{12}}{s^{\text{CKM}}_{13}},\hskip
14.22636pt\frac{s_{23}^{d_{R}}}{s_{12}^{d_{R}}}=\frac{s^{\text{CKM}}_{12}}{s^{\text{CKM}}_{23}},\hskip
14.22636pt\text{Inverted hierarchy-QIH},$ (89)
$\displaystyle\frac{s_{13}^{d_{R}}}{s_{12}^{d_{R}}}$ $\displaystyle=$
$\displaystyle\frac{s^{\text{CKM}}_{12}}{s^{\text{CKM}}_{13}},\hskip
14.22636pt\frac{s_{23}^{d_{R}}}{s_{12}^{d_{R}}}=\frac{s^{\text{CKM}}_{23}}{s^{\text{CKM}}_{12}},\hskip
14.22636pt\text{Mixed hierarchy-QMH1},$ (90)
$\displaystyle\frac{s_{13}^{d_{R}}}{s_{12}^{d_{R}}}$ $\displaystyle=$
$\displaystyle\frac{s^{\text{CKM}}_{13}}{s^{\text{CKM}}_{12}},\hskip
14.22636pt\frac{s_{23}^{d_{R}}}{s_{12}^{d_{R}}}=\frac{s^{\text{CKM}}_{12}}{s^{\text{CKM}}_{23}},\hskip
14.22636pt\text{Mixed hierarchy-QMH2},$ (91)

in which $s^{\text{CKM}}_{ij}\equiv\sin\theta^{\text{CKM}}_{ij}$ with
$\theta^{\text{CKM}}_{ij}$ to be the mixing angles in the CKM matrix. Hence,
only $s_{12}^{d_{R}}$ and $\delta^{d_{R}}$ are free parameters in $V_{d_{R}}$.
Notice that the mixing angles of the CKM matrix can be defined via the
Wolfenstein parameters $\lambda,A,\bar{\rho},\bar{\eta}$ [31, 32, 33], i.e.,

$\displaystyle s^{\text{CKM}}_{12}=\lambda,\hskip
14.22636pts^{\text{CKM}}_{23}=A\lambda^{2},\hskip
14.22636pts^{\text{CKM}}_{13}=A\lambda^{3}\sqrt{\bar{\rho}^{2}+\bar{\eta}^{2}}/(1-\lambda^{2}/2).$
(92)

Similarly, we parameterize the mixing matrix of right-handed charged leptons
$V_{e_{R}}$ via three mixing angles $\theta^{e_{R}}_{ij}$ and also assume that
there are four hierarchy scenarios, such as

$\displaystyle\frac{s_{12}^{e_{R}}}{s_{23}^{e_{R}}}$ $\displaystyle=$
$\displaystyle\frac{s^{\text{PMNS}}_{12}}{s^{\text{PMNS}}_{23}},\hskip
14.22636pt\frac{s_{13}^{e_{R}}}{s_{23}^{e_{R}}}=\frac{s^{\text{PMNS}}_{13}}{s^{\text{PMNS}}_{23}},\hskip
14.22636pt\text{Normal hierarchy-LNH},$ (93)
$\displaystyle\frac{s_{12}^{e_{R}}}{s_{23}^{e_{R}}}$ $\displaystyle=$
$\displaystyle\frac{s^{\text{PMNS}}_{23}}{s^{\text{PMNS}}_{12}},\hskip
14.22636pt\frac{s_{13}^{e_{R}}}{s_{23}^{e_{R}}}=\frac{s^{\text{PMNS}}_{23}}{s^{\text{PMNS}}_{13}},\hskip
14.22636pt\text{Inverted hierarchy-LIH},$ (94)
$\displaystyle\frac{s_{12}^{e_{R}}}{s_{23}^{e_{R}}}$ $\displaystyle=$
$\displaystyle\frac{s^{\text{PMNS}}_{23}}{s^{\text{PMNS}}_{12}},\hskip
14.22636pt\frac{s_{13}^{e_{R}}}{s_{23}^{e_{R}}}=\frac{s^{\text{PMNS}}_{13}}{s^{\text{PMNS}}_{23}},\hskip
14.22636pt\text{Mixed hierarchy-LMH1},$ (95)
$\displaystyle\frac{s_{12}^{e_{R}}}{s_{23}^{e_{R}}}$ $\displaystyle=$
$\displaystyle\frac{s^{\text{PMNS}}_{12}}{s^{\text{PMNS}}_{23}},\hskip
14.22636pt\frac{s_{13}^{e_{R}}}{s_{23}^{e_{R}}}=\frac{s^{\text{PMNS}}_{23}}{s^{\text{PMNS}}_{13}},\hskip
14.22636pt\text{Mixed hierarchy-LMH2},$ (96)

where $s^{e_{R}}_{ij}\equiv\sin\theta^{e_{R}}_{ij}$ and
$s^{\text{PMNS}}_{ij}\equiv\sin\theta^{\text{PMNS}}_{ij}$ with
$\theta^{\text{PMNS}}_{ij}$ to be the mixing angles in the PMNS matrix. In
this work, we take the best-fit values of neutrino oscillation data with
normal ordering hierarchy, given in Ref. [8]. The matrix $V_{e_{R}}$ therefore
contains only $s_{23}^{e_{R}}$ and $\delta^{e_{R}}$ as free parameters.
Furthermore, in the limit $v\ll\Lambda_{1,2}$, we have $t_{2\varphi}\sim
v^{2}/\Lambda_{1,2}^{2}\ll 1$, hence we can neglect the $Z$–$Z^{\prime}$
mixing. For the VEVs $\Lambda_{1,2}$, we assume that
$\Lambda_{1}=k\Lambda_{2}$ where $k\sim\mathcal{O}(1)$. Consequently, our
model leaves six free parameters $z$, $k,\Lambda_{2}$,
$s^{d_{R}}_{12},s^{e_{R}}_{23}$, and $\theta$. Numerical values of the
relevant common SM parameters are listed in Table 4, while those of known
input parameters associated with quark and lepton flavors are listed in Tables
5 and 6, respectively.

We would like to note that the new scalars $\mathcal{H}_{1,2}$ and
$\mathcal{A}$ also induce flavor-violating interactions, in addition to the
new gauge boson $Z^{\prime}$. However, these flavor-violating interactions are
proportional to $m_{u,d,e}/\Lambda_{1,2}\ll 1$, and thus, significantly
smaller compared to those caused by the $Z^{\prime}$ gauge boson. Therefore,
the following analysis will only focus on flavor phenomenologies from the
$Z^{\prime}$ gauge boson.

Parameters | Values
---|---
$\alpha_{\text{em}}$ |  $1/137$ [1]
$m_{W}$ |  $80.377\ \text{GeV}$ [1]
$m_{Z}$ |  $91.1876\ \text{GeV}$ [1]
$G_{F}$ |  $1.1663788\times 10^{-5}\ \text{GeV}^{-2}$ [1]
$s_{W}^{2}$ |  $0.23121$ [1]
Table 4: Common SM parameters.  Parameters | Values | Parameters | Values
---|---|---|---
$f_{K}$ |  $155.7(3)\ \text{MeV}$ [34] | $m_{K}$ |  $497.611(13)\ \text{MeV}$ [1]
$f_{B_{s}}$ |  $230.3(1.3)\ \text{MeV}$ [34] | $m_{B_{s}}$ |  $5366.88(11)\ \text{MeV}$ [1]
$f_{B_{d}}$ |  $190.0(1.3)\ \text{MeV}$ [34] | $m_{B_{d}}$ |  $5279.65(12)\ \text{MeV}$ [1]
$m_{u}$ |  $2.14(8)\ \text{MeV}$ [34] | $m_{d}$ |  $4.70(5)\ \text{MeV}$ [34]
$\bar{m}_{c}\ (3\ \text{GeV})$ |  $0.988(11)\ \text{GeV}$ [34] | $m_{s}$ |  $93.40(57)\ \text{MeV}$ [34]
$m_{t}$ |  $172.69(30)\ \text{GeV}$ [1] | $\bar{m}_{b}(\bar{m}_{b})$ |  $4.203(11)\ \text{GeV}$ [35]
$N(E_{\gamma})$ |  $3.3\times 10^{-3}$ [36] | $C_{7}^{\text{SM}}(\mu_{b}=2.0\ \text{GeV})$ |  $-0.3636$ [37, 38, 36]
$C_{9}^{\text{SM}}(\mu_{b}=5.0\ \text{GeV})$ |  $4.344$ [39] | $C_{10}^{\text{SM}}(\mu_{b}=5.0\ \text{GeV})$ |  $-4.198$ [39]
$y_{s}$ |  $0.0645(3)$ [9] | $\lambda$ |  $0.22548^{+0.00068}_{-0.00034}$ [40]
$A$ |  $0.810^{+0.018}_{-0.024}$ [40] | $\bar{\rho}$ |  $0.145^{+0.0013}_{-0.007}$ [40]
$\bar{\eta}$ |  $0.343^{+0.0011}_{-0.012}$ [40] |  |
Table 5: Numerical values of known input parameters for quark flavors. Parameters | Values | Parameters | Values
---|---|---|---
$m_{e}$ |  $5\times 10^{-4}\ \text{GeV}$ [1] | $(s_{12}^{\text{PMNS}})^{2}$ |  $0.304^{+0.012}_{-0.012}$ [8]
$m_{\mu}$ |  $0.105\ \text{GeV}$ [1] | $(s_{23}^{\text{PMNS}})^{2}$ |  $0.573^{+0.016}_{-0.020}$ [8]
$m_{\tau}$ |  $1.776\ \text{GeV}$ [34] | $(s_{13}^{\text{PMNS}})^{2}$ |  $0.02219^{+0.00062}_{-0.00063}$ [8]
$\Gamma_{\mu}$ |  $3\times 10^{-19}\ \text{GeV}$ [1] | $\delta_{\text{CP}}^{\text{PMNS}}(^{\circ})$ |  $197^{+27}_{-24}$ [8]
$\Gamma_{\tau}$ |  $2.27\times 10^{-12}\ \text{GeV}$ [1] |  |
Table 6: Numerical values of known input parameters for lepton flavors.

###  VI.1 Quark flavor phenomenologies

This subsection focuses on flavor phenomenologies in the quark sector with
controllable theoretical uncertainties. Because the quark generations are not
universal under $U(1)_{X}\otimes U(1)_{N}$, the model predicts flavor-changing
processes in the quark sector associated with the new gauge boson
$Z^{\prime}$. These processes occur at the tree level for $K,B_{s}$, and
$B_{d}$ meson oscillations or at both tree and loop levels for the quark
transitions $b\to se_{I}^{+}e_{I}^{-}$ with
$e_{I}=\\{e_{1},e_{2}\\}=\\{e,\mu\\}$, such as branching ratio of
$B_{s}\to\mu^{+}\mu^{-}$, branching ratio of inclusive decay BR$(\bar{B}\to
X_{s}\gamma)$, and ratios $R_{K,K^{*}}=\text{BR}(B^{+,0}\to
K^{+,0*}\mu^{+}\mu^{-})/\text{BR}(B^{+,0}\to K^{+,0*}e^{+}e^{-})$.

The effective Hamiltonian relevant for the above processes can be written as
[41]

$\displaystyle\mathcal{H}^{\text{quark}}_{\text{eff}}$ $\displaystyle=$
$\displaystyle\sum_{X=K,B_{s},B_{d}}(C_{X}^{2}\mathcal{O}_{X}^{2}+C^{{}^{\prime}2}_{X}\mathcal{O}_{X}^{{}^{\prime}2}+2C_{X}C^{{}^{\prime}}_{X}\mathcal{O}_{X}\mathcal{O}^{{}^{\prime}}_{X})$
(97)
$\displaystyle-\frac{4G_{F}}{\sqrt{2}}V^{*}_{ts}V_{tb}\sum_{Y=7,8,9,10}(C_{Y}\mathcal{O}_{Y}+C_{Y}^{{}^{\prime}}\mathcal{O}_{Y}^{{}^{\prime}}),$

where $G_{F}$ is the Fermi constant and $V_{ts,tb}$ are the CKM matrix
elements. The first summation contains contributions to meson mixing systems
$\bar{d}_{i}d_{j}\to d_{j}\bar{d_{j}}$ with
$d_{i,j}=\\{d_{1},d_{2},d_{3}\\}=\\{d,s,b\\}$ and $d_{i}\neq d_{j}$, while the
second summation relevant to the $b\to se_{I}^{+}e_{I}^{-}$ observables. The
primed operators $\mathcal{O}^{{}^{\prime}}_{X,Y}$ are chirally flipped
counterparts $P_{L}\leftrightarrow P_{R}$ of unprimed operators
$\mathcal{O}_{X,Y}$, they are defined as

$\displaystyle\mathcal{O}^{(^{\prime})}_{K}$ $\displaystyle=$
$\displaystyle\bar{s}\gamma^{\mu}P_{L(R)}d,\hskip
14.22636pt\mathcal{O}^{(^{\prime})}_{B_{s}}=\bar{s}\gamma^{\mu}P_{L(R)}b,\hskip
14.22636pt\mathcal{O}^{(^{\prime})}_{B_{d}}=\bar{d}\gamma^{\mu}P_{L(R)}b,$
(98) $\displaystyle\mathcal{O}^{(^{\prime})}_{7}$ $\displaystyle=$
$\displaystyle\frac{e}{16\pi^{2}}m_{b}(\bar{s}\sigma^{\mu\nu}P_{R(L)}b)F_{\mu\nu},\hskip
14.22636pt\mathcal{O}^{(^{\prime})}_{8}=\frac{g_{s}}{16\pi^{2}}m_{b}(\bar{s}\sigma^{\mu\nu}T^{a}P_{R(L)}b)G^{a}_{\mu\nu},$
(99) $\displaystyle\mathcal{O}^{(^{\prime})}_{9}$ $\displaystyle=$
$\displaystyle\frac{e^{2}}{16\pi^{2}}(\bar{s}\gamma^{\mu}P_{L(R)}b)(\bar{e}_{I}\gamma_{\mu}e_{I}),\hskip
14.22636pt\mathcal{O}^{(^{\prime})}_{10}=\frac{e^{2}}{16\pi^{2}}(\bar{s}\gamma^{\mu}P_{L(R)}b)(\bar{e}_{I}\gamma_{\mu}\gamma_{5}e_{I}),$
(100)

where $P_{L,R}=\frac{1}{2}(1\mp\gamma_{5})$. The operators
$\mathcal{O}^{(^{\prime})}_{7,8}$ contribute mainly to BR$(\bar{B}\to
X_{s}\gamma)$, whereas $\mathcal{O}^{(^{\prime})}_{9,10}$ dominate the
BR$(B_{s}\to e_{I}^{+}e_{I}^{-})$ and the ratios $R_{K,K^{*}}$. The new
physics contributions to the Wilson coefficients (WCs)
$C^{(^{\prime})\text{NP}}_{X,Y}$ can come from either the tree level or the
quantum level (loop, penguin, and box diagrams) or both ones. Generally, we
can decompose the new physics contributions as
$C^{(^{\prime})\text{NP}}_{X,Y}=C^{(^{\prime})\text{tree}}_{X,Y}+C^{(^{\prime})\text{loop}}_{X,Y}+C^{(^{\prime})\text{penguin}}_{X,Y}+C^{(^{\prime})\text{box}}_{X,Y}$,
where the contribution of each style of diagrams is indicted by the
superscripts. For the tree-level contributions as described by Feynman
diagrams in Fig. 1, we obtain

$\displaystyle C_{K}^{\text{tree}}$ $\displaystyle=$
$\displaystyle-\frac{2gzt_{W}}{s_{\theta}c_{\theta}m_{Z^{\prime}}}[V^{*}_{d_{L}}]_{31}[V_{d_{L}}]_{32},\hskip
14.22636ptC_{K}^{{}^{\prime}\text{tree}}=-\frac{2gzt_{W}}{s_{\theta}c_{\theta}m_{Z^{\prime}}}[V^{*}_{d_{R}}]_{31}[V_{d_{R}}]_{32},$
(101) $\displaystyle C_{B_{s}}^{\text{tree}}$ $\displaystyle=$
$\displaystyle-\frac{2gzt_{W}}{s_{\theta}c_{\theta}m_{Z^{\prime}}}[V^{*}_{d_{L}}]_{32}[V_{d_{L}}]_{33},\hskip
14.22636ptC_{B_{s}}^{{}^{\prime}\text{tree}}=-\frac{2gzt_{W}}{s_{\theta}c_{\theta}m_{Z^{\prime}}}[V^{*}_{d_{R}}]_{32}[V_{d_{R}}]_{33},$
(102) $\displaystyle C_{B_{d}}^{\text{tree}}$ $\displaystyle=$
$\displaystyle-\frac{2gzt_{W}}{s_{\theta}c_{\theta}m_{Z^{\prime}}}[V^{*}_{d_{L}}]_{31}[V_{d_{L}}]_{33},\hskip
14.22636ptC_{B_{d}}^{{}^{\prime}\text{tree}}=-\frac{2gzt_{W}}{s_{\theta}c_{\theta}m_{Z^{\prime}}}[V^{*}_{d_{R}}]_{31}[V_{d_{R}}]_{33},$
(103) $\displaystyle C^{\text{tree},e_{I}}_{9}$ $\displaystyle=$
$\displaystyle\Gamma^{d_{L}}_{23}\frac{m_{W}^{2}}{c_{W}V^{*}_{ts}V_{tb}}\frac{1}{g}\frac{(4\pi)^{2}}{e^{2}}\frac{g_{V}^{Z_{2}}(e_{I})}{m_{Z^{\prime}}^{2}},\hskip
14.22636ptC^{{}^{\prime}\text{tree},e_{I}}_{9}=\Gamma^{d_{R}}_{23}\frac{m_{W}^{2}}{c_{W}V^{*}_{ts}V_{tb}}\frac{1}{g}\frac{(4\pi)^{2}}{e^{2}}\frac{g_{V}^{Z_{2}}(e_{I})}{m_{Z^{\prime}}^{2}},$
(104) $\displaystyle C^{\text{tree},e_{I}}_{10}$ $\displaystyle=$
$\displaystyle-\Gamma^{d_{L}}_{23}\frac{m_{W}^{2}}{c_{W}V^{*}_{ts}V_{tb}}\frac{1}{g}\frac{(4\pi)^{2}}{e^{2}}\frac{g_{A}^{Z_{2}}(e_{I})}{m_{Z^{\prime}}^{2}},\hskip
14.22636ptC^{{}^{\prime}\text{tree},e_{I}}_{10}=-\Gamma^{d_{R}}_{23}\frac{m_{W}^{2}}{c_{W}V^{*}_{ts}V_{tb}}\frac{1}{g}\frac{(4\pi)^{2}}{e^{2}}\frac{g_{A}^{Z_{2}}(e_{I})}{m_{Z^{\prime}}^{2}}.$
(105)

For the quantum-level contributions, they are obtained from the one-loop,
penguin, and box diagrams that contain gauge boson $Z^{\prime}$, down quarks
$f=d,s,b$, and charged leptons $k=e,\mu,\tau$ to be internal lines, given in
Fig. 2. We use t Hooft gauge $\zeta=1$ for calculating these diagrams. With
the diagrams (a) and (b), we calculate on-shell, i.e., $q^{2}=0$,
$p_{s}^{2}=m_{s}^{2}$, and $p_{b}^{2}=m_{b}^{2}$. Because $m_{s}\ll m_{b}$, we
set the $s$ quark mass to be zero, $m_{s}=0$, and keep the mass of $b$ quark
at the linear order, i.e. $m_{b}^{2}=0$. Additionally, we calculate in the
limit $m_{f}^{2}/m_{Z^{\prime}}^{2},m_{k}^{2}/m_{Z^{\prime}}^{2}\ll 1$ since
$m_{f,k}\sim\mathcal{O}(1)\ \text{GeV}\ll m_{Z^{\prime}}\sim\mathcal{O}(1)$
TeV, for simplicity. It is important to note that under this limit other loop
diagrams with unphysical Goldstone boson $\phi_{Z^{\prime}}$ are suppressed by
factors $m_{f}^{2}/m_{Z^{\prime}}^{2},m_{k}^{2}/m_{Z^{\prime}}^{2}\ll 1$,
hence we can safely ignore the box diagrams with the Goldstone boson
$\phi_{Z^{\prime}}$ and keep only the ones with physical gauge boson
$Z^{\prime}$. We have the expressions for these contributions at the scale
$\mu_{Z^{\prime}}=m_{Z^{\prime}}$ as

$\displaystyle C^{\text{loop}}_{7}(\mu_{Z^{\prime}})$ $\displaystyle\simeq$
$\displaystyle-\frac{2m_{W}^{2}}{9m_{Z^{\prime}}^{2}V^{*}_{ts}V_{tb}}\frac{1}{g^{2}}\sum_{i=1,2,3}\left(\Gamma^{*d_{L}}_{i2}\Gamma^{d_{L}}_{i3}-3\frac{m_{d_{i}}}{m_{b}}\Gamma^{*d_{L}}_{i2}\Gamma^{d_{R}}_{i3}\right),$
(106) $\displaystyle C^{{}^{\prime}\text{loop}}_{7}(\mu_{Z^{\prime}})$
$\displaystyle\simeq$
$\displaystyle-\frac{2m_{W}^{2}}{9m_{Z^{\prime}}^{2}V^{*}_{ts}V_{tb}}\frac{1}{g^{2}}\sum_{i=1,2,3}\left(\Gamma^{*d_{R}}_{i2}\Gamma^{d_{R}}_{i3}-3\frac{m_{d_{i}}}{m_{b}}\Gamma^{*d_{R}}_{i2}\Gamma^{d_{L}}_{i3}\right),$
(107) $\displaystyle C^{\text{loop}}_{8}(\mu_{Z^{\prime}})$
$\displaystyle\simeq$
$\displaystyle-3C^{\text{loop}}_{7}(\mu_{Z^{\prime}}),\hskip
14.22636ptC^{{}^{\prime}\text{loop}}_{8}(\mu_{Z^{\prime}})\simeq-3C^{{}^{\prime}\text{loop}}_{7}(\mu_{Z^{\prime}}),$
(108) $\displaystyle C_{9}^{\text{penguin},e_{I},\gamma}(\mu_{Z^{\prime}})$
$\displaystyle\simeq$
$\displaystyle-\frac{m_{W}^{2}}{m_{Z^{\prime}}^{2}}\frac{1}{g^{2}}\sum_{i=1,2,3}\frac{\Gamma^{*d_{L}}_{i2}\Gamma^{d_{L}}_{i3}}{V_{ts}^{*}V_{tb}}\left(-\frac{1}{27}+\frac{2}{9}\ln{\frac{m^{2}_{d_{i}}}{m^{2}_{Z^{\prime}}}}\right),$
(109) $\displaystyle
C_{9}^{{}^{\prime}\text{penguin},e_{I},\gamma}(\mu_{Z^{\prime}})$
$\displaystyle\simeq$
$\displaystyle-\frac{m_{W}^{2}}{m_{Z^{\prime}}^{2}}\frac{1}{g^{2}}\sum_{i=1,2,3}\frac{\Gamma^{*d_{R}}_{i2}\Gamma^{d_{R}}_{i3}}{V_{ts}^{*}V_{tb}}\left(-\frac{1}{27}+\frac{2}{9}\ln{\frac{m^{2}_{d_{i}}}{m^{2}_{Z^{\prime}}}}\right),$
(110) $\displaystyle C^{\text{box},e_{I}}_{9}(\mu_{Z^{\prime}})$
$\displaystyle\simeq$
$\displaystyle-\frac{m_{W}^{2}}{4s_{W}^{2}m_{Z^{\prime}}^{2}}\frac{1}{g^{4}}\sum_{i=1,2,3}\sum_{j=1,2,3}\frac{\Gamma^{*d_{L}}_{i2}\Gamma^{d_{L}}_{i3}(|\Gamma^{e_{L}}_{jI}|^{2}+|\Gamma^{e_{R}}_{jI}|^{2})}{V^{*}_{ts}V_{tb}},$
(111) $\displaystyle C^{{}^{\prime}\text{box},e_{I}}_{9}(\mu_{Z^{\prime}})$
$\displaystyle\simeq$
$\displaystyle-\frac{m_{W}^{2}}{4s_{W}^{2}m_{Z^{\prime}}^{2}}\frac{1}{g^{4}}\sum_{i=1,2,3}\sum_{j=1,2,3}\frac{\Gamma^{*d_{R}}_{i2}\Gamma^{d_{R}}_{i3}(|\Gamma^{e_{L}}_{jI}|^{2}+|\Gamma^{e_{R}}_{jI}|^{2})}{V^{*}_{ts}V_{tb}},$
(112) $\displaystyle C^{\text{box},e_{I}}_{10}(\mu_{Z^{\prime}})$
$\displaystyle\simeq$
$\displaystyle-\frac{m_{W}^{2}}{4s_{W}^{2}m_{Z^{\prime}}^{2}}\frac{1}{g^{4}}\sum_{i=1,2,3}\sum_{j=1,2,3}\frac{\Gamma^{*d_{L}}_{i2}\Gamma^{d_{L}}_{i3}(|\Gamma^{e_{L}}_{jI}|^{2}-|\Gamma^{e_{R}}_{jI}|^{2})}{V^{*}_{ts}V_{tb}},$
(113) $\displaystyle C^{{}^{\prime}\text{box},e_{I}}_{10}(\mu_{Z^{\prime}})$
$\displaystyle\simeq$
$\displaystyle-\frac{m_{W}^{2}}{4s_{W}^{2}m_{Z^{\prime}}^{2}}\frac{1}{g^{4}}\sum_{i=1,2,3}\sum_{j=1,2,3}\frac{\Gamma^{*d_{R}}_{i2}\Gamma^{d_{R}}_{i3}(|\Gamma^{e_{L}}_{jI}|^{2}-|\Gamma^{e_{R}}_{jI}|^{2})}{V^{*}_{ts}V_{tb}}.$
(114)

It should be noted that the penguin diagrams with off-shell SM $Z$-like boson
do not give the contributions to WCs in the limit
$m_{d_{i}}^{2}/m_{Z^{\prime}}^{2}\to 0$; thus, we do not include these
diagrams in our calculation.

---
Figure 1: Tree-level diagrams induced by new gauge boson $Z^{\prime}$ for
meson mixings (left) and $b\to se_{I}^{+}e_{I}^{-}$ transitions (right), where
$d_{i,j}=\\{d,s,b\\}$ and $d_{i}\neq d_{j}$, $e_{I}=\\{e,\mu\\}$.
---
Figure 2: Radiative contributions to $C^{(^{\prime})}_{7,8,9,10}$, where
$k=e,\mu,\tau$ and $f=d,s,b$.

Next, we determine the new physics contributions to each observable in terms
of WCs. Firstly, for the meson mixings, we can decompose the contributions to
meson mass differences as $\Delta m_{K,B_{s},B_{d}}=\Delta
m^{\text{SM}}_{K,B_{s},B_{d}}+\Delta m^{\text{NP}}_{K,B_{s},B_{d}}$, where the
SM contributions $\Delta m^{\text{SM}}_{K,B_{s},B_{d}}$ are shown in the
second column in Table 7, and the new physics contributions $\Delta
m^{\text{NP}}_{K,B_{s},B_{d}}$ are estimated by [42, 43]

$\displaystyle\Delta m_{K}^{\text{NP}}$ $\displaystyle\simeq$
$\displaystyle\frac{2}{3}\mathrm{Re}\left\\{(C^{\text{tree}}_{K})^{2}-\left[\frac{3}{2}+\left(\frac{m_{K}}{m_{d}+m_{s}}\right)^{2}\right]C^{\text{tree}}_{K}C^{{}^{\prime}\text{tree}}_{K}+(C^{{}^{\prime}\text{tree}}_{K})^{2}\right\\}m_{K}f^{2}_{K},$
(115) $\displaystyle\Delta m_{B_{s}}^{\text{NP}}$ $\displaystyle\simeq$
$\displaystyle\frac{2}{3}\mathrm{Re}\left\\{(C^{\text{tree}}_{B_{s}})^{2}-\left[\frac{3}{2}+\left(\frac{m_{B_{s}}}{m_{s}+m_{b}}\right)^{2}\right]C^{\text{tree}}_{B_{s}}C^{{}^{\prime}\text{tree}}_{B_{s}}+(C^{{}^{\prime}\text{tree}}_{B_{s}})^{2}\right\\}m_{B_{s}}f^{2}_{B_{s}},$
(116) $\displaystyle\Delta m_{B_{d}}^{\text{NP}}$ $\displaystyle\simeq$
$\displaystyle\frac{2}{3}\mathrm{Re}\left\\{(C^{\text{tree}}_{B_{d}})^{2}-\left[\frac{3}{2}+\left(\frac{m_{B_{d}}}{m_{d}+m_{b}}\right)^{2}\right]C^{\text{tree}}_{B_{d}}C^{{}^{\prime}\text{tree}}_{B_{d}}+(C^{{}^{\prime}\text{tree}}_{B_{d}})^{2}\right\\}m_{B_{d}}f^{2}_{B_{d}}.$
(117)

Note that the SM $Z$-like boson also contributes to meson mass differences due
to the mixing of $Z$–$Z^{\prime}$. However, these contributions are
proportional with $s_{\varphi}^{4}$, so we ignore them.

Observables | SM predictions | Experimental values
---|---|---
$\Delta m_{K}$ |  $0.467\times 10^{-2}\ \text{ps}^{-1}$ [1] |  $0.5293(9)\times 10^{-2}\ \text{ps}^{-1}$ [1]
$\Delta m_{B_{s}}$ |  $18.77(86)\ \text{ps}^{-1}$ [44] |  $17.765(6)\ \text{ps}^{-1}$ [9]
$\Delta m_{B_{d}}$ |  $0.543(29)\ \text{ps}^{-1}$ [44] |  $0.5065(19)\ \text{ps}^{-1}$ [9]
$\text{BR}(B_{s}\to\mu^{+}\mu^{-})$ |  $(3.66\pm 0.14)\times 10^{-9}$ [45] |  $(3.45\pm 0.29)\times 10^{-9}$ [10]
$\text{BR}(\bar{B}\to X_{s}\gamma)$ |  $(3.40\pm 0.17)\times 10^{-4}$ [36] |  $(3.49\pm 0.19)\times 10^{-4}$ [9]
$R_{K}$ |  $1.00\pm 0.01$ [46] |  $0.949^{+0.042}_{-0.041}\pm 0.022$ [10]
$R_{K^{*}}$ |  $1.00\pm 0.01$ [46] |  $1.027^{+0.072+0.027}_{-0.068-0.026}$ [10]
Table 7:  The SM predictions and experimental values for flavor-changing
observables related to quark sectors.

For the branching ratio BR$(B_{s}\to\mu^{+}\mu^{-})$, we have the following
formula [47],

$\mathrm{BR}(B_{s}\to\mu^{+}\mu^{-})=\frac{\tau_{B_{s}}}{16\pi^{3}}\alpha_{\text{em}}^{2}G_{F}^{2}f_{B_{s}}^{2}|V_{tb}V^{*}_{ts}|^{2}m_{B_{s}}m_{\mu}^{2}\sqrt{1-\frac{4m_{\mu}^{2}}{m^{2}_{B_{s}}}}|C_{10}^{\mu}-C^{{}^{\prime}\mu}_{10}|^{2},$
(118)

where $\tau_{B_{s}}$ is the lifetime of $B_{s}$ meson, $\alpha_{\text{em}}$ is
the fine-structure constant, and the WCs are defined as
$C_{10}^{\mu}=C_{10}^{\text{SM}}+C_{10}^{\text{NP},\mu}$,
$C^{{}^{\prime}\mu}_{10}=C_{10}^{{}^{\prime}\text{NP},\mu}$ with
$C_{10}^{\text{SM}}$ is the SM WC and given in Table 5 and
$C_{10}^{(^{\prime})\text{NP},\mu}=C_{10}^{(^{\prime})\text{tree},\mu}+C_{10}^{(^{\prime})\text{box},\mu}$.
Due to the effect of $B_{s}$–$\bar{B}_{s}$ oscillations, the available
experimental value relates to theoretical prediction as [48]

$\mathrm{BR}(B_{s}\to\mu^{+}\mu^{-})_{\text{exp}}\simeq\frac{1}{1-y_{s}}\mathrm{BR}(B_{s}\to\mu^{+}\mu^{-}),$
(119)

where $y_{s}=\frac{\Delta\Gamma_{B_{s}}}{2\Gamma_{B_{s}}}$ and the value of
$y_{s}$ is presented in Table 5.

The branching ratio for the decay $\bar{B}\to X_{s}\gamma$ is given as [49,
50]

$\text{BR}(\bar{B}\to X_{s}\gamma)=\frac{6\alpha_{\text{em}}}{\pi
C}\left|\frac{V_{ts}^{*}V_{tb}}{V_{cb}}\right|^{2}\left[|C_{7}(\mu_{b})|^{2}+|C^{{}^{\prime}}_{7}(\mu_{b})|^{2}+N(E_{\gamma})\right]\text{BR}(\bar{B}\rightarrow
X_{c}e\bar{\nu}),$ (120)

where $N(E_{\gamma})$ is a non-perturbative contribution which amounts around
$4\%$ of the branching ratio. We compute the leading order contribution to
$N(E_{\gamma})$ followed the Eq. (3.8) in Ref. [36] and then obtain
$N(E_{\gamma})\simeq 3.3\times 10^{-3}$. Additionally, $C$ is the semileptonic
phase-space factor, $C=|V_{ub}/V_{cb}|^{2}\Gamma(\bar{B}\rightarrow
X_{c}e\bar{\nu}_{e})/\Gamma(\bar{B}\rightarrow X_{u}e\bar{\nu}_{e})$, and
BR$(\bar{B}\rightarrow X_{c}e\bar{\nu})$ is the branching ratio for semi-
leptonic decay. It is necessary to consider the QCD corrections to complete
the calculation for this branching ratio. The WCs
$C_{7}^{(^{\prime})}(\mu_{b})$ are evaluated at the matching scale $\mu_{b}=2$
GeV by running down from the higher scale $\mu_{Z^{\prime}}$ via the
renormalization group equations. Its expression can be split as

$\displaystyle
C_{7}(\mu_{b})=C_{7}^{\text{SM}}(\mu_{b})+C_{7}^{\text{NP}}(\mu_{b}),\hskip
14.22636ptC^{{}^{\prime}}_{7}(\mu_{b})=C_{7}^{{}^{\prime}\text{NP}}(\mu_{b}),$
(121)

where $C_{7}^{\text{SM}}(\mu_{b})$ is the SM WC and have been calculated up to
next-to-next-leading order of QCD corrections with the result shown in Table
5. Otherwise, for NP contribution, we have the result at leading order [50] as

$\displaystyle
C^{(^{\prime})\text{NP}}_{7}(\mu_{b})=\kappa_{7}C^{(^{\prime})\text{loop}}_{7}(\mu_{Z^{\prime}})+\kappa_{8}C_{8}^{(^{\prime})\text{loop}}(\mu_{Z^{\prime}})+\Delta_{Z^{\prime}}^{(^{\prime})\text{current}}(\mu_{b}),$
(122)

where the last term stems from the mixing of neutral current-current operators
generated by $Z^{\prime}$ and the dipole operators $\mathcal{O}_{7,8}$.
Besides, the coefficients $\kappa_{7,8}$ are called NP magic numbers, and
their numerical values are given in Ref. [50].

Lepton flavor universality violating (LFUV) observables $R_{K,K^{*}}$ in the
range of squared dilepton mass $q^{2}=[1.1,6.0]\ \text{GeV}^{2}$ are defined
in terms of new physics WCs $C^{(^{\prime})\text{NP},e_{I}}_{9,10}$, given in
[51],

$\displaystyle\frac{R_{K}}{R_{K}^{\text{SM}}}$ $\displaystyle=$
$\displaystyle\left\\{1+0.24\text{Re}[C^{\text{NP},\mu}_{9}+C^{{}^{\prime}\text{NP},\mu}_{9}]-0.26\text{Re}[C^{\text{NP},\mu}_{10}+C^{{}^{\prime}\text{NP},\mu}_{10}]+0.03\left(|C^{\text{NP},\mu}_{9}+C^{{}^{\prime}\text{NP},\mu}_{9}|^{2}\right.\right.$
(123)
$\displaystyle\left.\left.+|C^{\text{NP},\mu}_{10}+C^{{}^{\prime}\text{NP},\mu}_{10}|^{2}\right)\right\\}\left\\{1+0.24\text{Re}[C^{\text{NP},e}_{9}+C^{{}^{\prime}\text{NP},e}_{9}]-0.26\text{Re}[C^{\text{NP},e}_{10}+C^{{}^{\prime}\text{NP},e}_{10}]\right.$
$\displaystyle\left.+0.03\left(|C^{\text{NP},e}_{9}+C^{{}^{\prime}\text{NP},e}_{9}|^{2}+|C^{\text{NP},e}_{10}+C^{{}^{\prime}\text{NP},e}_{10}|^{2}\right)\right\\}^{-1},$
$\displaystyle\frac{R_{K^{*}}}{R_{K^{*}}^{\text{SM}}}$ $\displaystyle=$
$\displaystyle\left\\{1+0.18\text{Re}[C^{\text{NP},\mu}_{9}+C^{{}^{\prime}\text{NP},\mu}_{9}]-0.29\text{Re}[C^{\text{NP},\mu}_{10}+C^{{}^{\prime}\text{NP},\mu}_{10}]+0.03\left(|C^{\text{NP},\mu}_{9}+C^{{}^{\prime}\text{NP},\mu}_{9}|^{2}\right.\right.$
(124)
$\displaystyle\left.\left.+|C^{\text{NP},\mu}_{10}+C^{{}^{\prime}\text{NP},\mu}_{10}|^{2}\right)\right\\}\left\\{1+0.18\text{Re}[C^{\text{NP},e}_{9}+C^{{}^{\prime}\text{NP},e}_{9}]-0.29\text{Re}[C^{\text{NP},e}_{10}+C^{{}^{\prime}\text{NP},e}_{10}]\right.$
$\displaystyle\left.+0.03\left(|C^{\text{NP},e}_{9}+C^{{}^{\prime}\text{NP},e}_{9}|^{2}+|C^{\text{NP},e}_{10}+C^{{}^{\prime}\text{NP},e}_{10}|^{2}\right)\right\\}^{-1}.$

We also need to take into account QCD corrections here. At the leading order,
the $C^{(^{\prime}),e_{I}}_{9,10}$ are shifted by
$\epsilon\simeq\frac{\alpha_{s}}{4\pi}\ln{(m_{Z^{\prime}}/m_{b})}$ where
$\alpha_{s}$ is the strong coupling at scale $m_{Z^{\prime}}$. This effect of
QCD corrections modifies the value of WCs by around a few percent with
$m_{Z^{\prime}}\sim\mathcal{O}(1)\ \text{TeV}\gg m_{b}$. However, the effect
of QCD correction is insignificant in the ratios $R_{K,K^{*}}$ because they
are small and canceled between the numerator and the denominator of these
ratios. Therefore, in this work, we ignore the effect of QCD corrections in
$R_{K,K^{*}}$ and BR$(B_{s}\to\mu^{+}\mu^{-})$.

All observables mentioned above should be compared with the experimental
values in the last column in Table 7. It is important to note that the central
values of SM prediction and the measurement results of these observables are
very close. However, the uncertainties in SM prediction are quite large,
especially in meson mass differences, compared to experimental ones.
Therefore, it is better to consider the ratio between SM and respective
experimental values on each observable since the uncertainties can be canceled
via the numerator and the denominator of these ratios. Hence, we obtain
constraints for $B^{0}_{s,d}$–$\bar{B}^{0}_{s,d}$ meson systems as

$\frac{(\Delta m_{B_{d}})_{\mathrm{SM}}}{(\Delta
m_{B_{d}})_{\mathrm{exp}}}=1.0721(1\pm 0.0535),\hskip 14.22636pt\frac{(\Delta
m_{B_{s}})_{\mathrm{SM}}}{(\Delta m_{B_{s}})_{\mathrm{exp}}}=1.0566(1\pm
0.0458),$ (125)

which are equivalent

$\displaystyle\frac{(\Delta m_{B_{d}})_{\text{NP}}}{(\Delta
m_{B_{d}})_{\mathrm{exp}}}\in[-0.1295,-0.0147],\hskip 14.22636pt\frac{(\Delta
m_{B_{s}})_{\text{NP}}}{(\Delta
m_{B_{s}})_{\mathrm{exp}}}\in[-0.105,-0.0082].$ (126)

However, in $K^{0}$–$\bar{K}^{0}$ meson system, the lattice QCD calculations
for long-distance effect are not well-controlled. Therefore, we assume the
present theory contributes about 30% to $\Delta m_{K}$, it reads

$\displaystyle\frac{(\Delta m_{K})_{\text{SM}}}{(\Delta
m_{K})_{\text{exp}}}=1(1\pm 0.3),$ (127)

and then translates to the following constraint

$\displaystyle\frac{(\Delta m_{K})_{\text{NP}}}{(\Delta
m_{K})_{\text{exp}}}\in[-0.3,0.3],$ (128)

in agreement with [52]. For the branching ratios BR$(B_{s}\to\mu^{+}\mu^{-})$
and BR$(\bar{B}\to X_{s}\gamma)$, we have constraints as

$\displaystyle\frac{\text{BR}(B_{s}\to\mu^{+}\mu^{-})_{\text{exp}}}{\text{BR}(B_{s}\to\mu^{+}\mu^{-})_{\text{SM}}}$
$\displaystyle=$
$\displaystyle\frac{1}{1-y_{s}}\frac{|C^{\mu}_{10}-C^{{}^{\prime}\mu}_{10}|^{2}}{|C_{10}^{\text{SM}}|^{2}}$
(129) $\displaystyle=$ $\displaystyle 0.9426(1\pm 0.0924),$
$\displaystyle\frac{\text{BR}(\bar{B}\to
X_{s}\gamma)_{\text{exp}}}{\text{BR}(\bar{B}\to X_{s}\gamma)_{\text{SM}}}$
$\displaystyle=$ $\displaystyle
1+\frac{|C_{7}^{\text{NP}}|^{2}+|C_{7}^{{}^{\prime}\text{NP}}|^{2}+2C_{7}^{\text{SM}}\text{Re}[C^{\text{NP}}_{7}]}{|C_{7}^{\text{SM}}|^{2}+N(E_{\gamma})}$
(130) $\displaystyle=$ $\displaystyle 1.0265(1\pm 0.0739).$

###  VI.2 Lepton flavor phenomenologies

For the lepton flavor violating (LFV) decays $e_{j}\to e_{i}\gamma$ with
$e_{i,j}=\\{e_{1},e_{2},e_{3}\\}=\\{e,\mu,\tau\\}$ and $e_{i}\neq e_{j}$, we
have the following the effective Hamiltonian contributing by new neutral gauge
boson $Z^{\prime}$ at the one-loop level

$\displaystyle\mathcal{H}^{\text{lepton}}_{\text{eff}}=C^{ij}_{L}\bar{e}_{i}\sigma_{\mu\nu}P_{L}e_{j}F^{\mu\nu}+(L\to
R),$ (131)

where the coefficients $C^{ij}_{L,R}$ are obtained by calculating one-loop
diagrams containing the SM charged leptons
$e_{k}=\\{e_{1},e_{2},e_{3}\\}=\\{e,\mu,\tau\\}$ and new neutral gauge boson
$Z^{\prime}$ as internal lines, see subfigure (b) of Fig. 3. Here we calculate
these diagrams in the limit $m_{e_{k}}^{2}/m_{Z^{\prime}}^{2}\ll 1$ and
keeping the masses of external leptons $m_{e_{i,j}}$, similar to the quark
flavor section. We obtain the expressions for these coefficients as

$\displaystyle C_{L}^{ij}$ $\displaystyle=$
$\displaystyle\frac{e}{3(4\pi)^{2}m_{Z^{\prime}}^{2}}\sum_{k=1}^{3}(m_{e_{j}}\Gamma^{*e_{R}}_{ki}\Gamma^{e_{R}}_{kj}-3m_{e_{k}}\Gamma^{*e_{R}}_{ki}\Gamma^{e_{L}}_{kj}+m_{e_{i}}\Gamma^{*e_{L}}_{ki}\Gamma^{e_{L}}_{kj}),$
(132) $\displaystyle C_{R}^{ij}$ $\displaystyle=$
$\displaystyle\frac{e}{3(4\pi)^{2}m_{Z^{\prime}}^{2}}\sum_{k=1}^{3}(m_{e_{j}}\Gamma^{*e_{L}}_{ki}\Gamma^{e_{L}}_{kj}-3m_{e_{k}}\Gamma^{*e_{L}}_{ki}\Gamma^{e_{R}}_{kj}+m_{e_{i}}\Gamma^{*e_{R}}_{ki}\Gamma^{e_{R}}_{kj}),$
(133)

where $\Gamma_{ij}^{e_{L,R}}$ are the LFV couplings given in Eq. (83). The
branching ratios of the LFV decays are determined by [53]

$\text{BR}(e_{j}\to e_{j}\gamma)=\frac{(m_{e_{j}}^{2}-m_{e_{i}}^{2})^{3}}{4\pi
m_{e_{j}}^{3}\Gamma_{e_{j}}}(|C^{ij}_{L}|^{2}+|C^{ij}_{R}|^{2}),$ (134)

where $\Gamma_{e_{j}}$ is the total decay width of decaying lepton $e_{j}$.

Besides, the effective Hamiltonian in Eq. (131) also contributes to branching
ratios of three-body leptonic decays such as $\tau\to 3\mu(3e)$, $\tau\to
e\mu\mu(ee\mu)$, and $\mu\to 3e$. There are three contributions to these
observables, including the tree-level shown in subfigure (a) of Fig. 3 with
the following operators

$\displaystyle\mathcal{O}_{ab,\alpha\beta}^{L(R)L(R)}=(\bar{e}_{i}\gamma_{\mu}P_{L(R)}e_{j})(\bar{e}_{\rho}\gamma^{\mu}P_{L(R)}e_{\delta}),$
(135)

where $e_{i,j}=\\{e_{2},e_{3}\\}=\\{\mu,\tau\\}$, $i\neq j$, and
$e_{\rho,\delta}=\\{e_{1},e_{2}\\}=\\{e,\mu\\}$. Note that these operators are
also generated by the SM $Z$-like boson but suppressed due to small
$Z$–$Z^{\prime}$ mixing. This setup also does not allow the LFV decays of $Z$
boson, namely $Z\to e_{i}e_{j}$. Besides the tree level, the dipole operators
in Eq. (131) also generate the three-body decays via penguin diagrams, as
shown in subfigures (c) and (d) of Fig. 3. Furthermore, there are one-loop
contributions that arise from the mixing of tree-level operators defined in
Eq. (135) with “hidden” operators that do not trigger flavor violating decays
at the tree level but do so in QED penguin diagrams, such as
$\mathcal{O}^{L(R),L(R)}_{e\mu,\mu\mu(\tau\tau)},\mathcal{O}^{L(R),L(R)}_{e\tau,\tau\mu(\tau\tau)},\mathcal{O}^{L(R),L(R)}_{\mu\tau,\tau\tau}$
[54]. The branching ratios of three-body leptonic decays, including all
mentioned contributions, were explicitly given in Ref. [54].

On the other hand, for the lepton flavor conversing (LFC) observables
including the electron and muon anomalous magnetic moments $\Delta a_{e,\mu}$
and the electric dipole moments $d_{e,\mu}$, we have the following formulas
[53],

$\displaystyle\Delta a_{e_{I}}$ $\displaystyle=$
$\displaystyle-\frac{4m_{e_{I}}}{e}\text{Re}[C_{R}^{II}],$ (136)
$\displaystyle d_{e_{I}}$ $\displaystyle=$
$\displaystyle-2\text{Im}[C_{R}^{II}],\hskip 14.22636ptI=1,2.$ (137)

---
Figure 3: Feynman diagrams for three-body leptonic and LFV decays.

All predicted observables above should be compared with experimental results
listed in Table 8. It is straightforward to recognize that the contribution of
the $Z^{\prime}$ gauge boson with a mass at several TeVs implied by the
collider searches (discussed below) to the anomalous magnetic moments,
especially for $\Delta a_{\mu}$, be quite suppressed in comparison with other
leptonic observables. Indeed, from Eqs. (133) and (136), it is easy to see
that $\Delta a_{\mu}$ is proportional with a factor,
$-\frac{4m_{\mu}}{e}\frac{e}{48\pi^{2}m_{Z^{\prime}}^{2}}\sim\mathcal{O}(10^{-11}-10^{-10})$,
while the internal terms,
Re$[m_{e_{k}}\Gamma^{e_{L(R)}}_{k2}\Gamma^{*e_{L(R)}}_{k2}]\sim\mathcal{O}(10^{-1}-10^{0})$.
Therefore, our model predicts $\Delta
a_{\mu}\sim\mathcal{O}(10^{-12}-10^{-11})$, remarkably smaller than
experimental result $\Delta a_{\mu}^{\text{exp}}\sim\mathcal{O}(10^{-9})$
[55]. In the following numerical analysis, we will investigate branching
ratios of LFV, three-body leptonic decays, and electric dipole moments.

LFV Observables | Experimental limits | LFC Observables | Experimental limits
---|---|---|---
BR$(\mu\to e\gamma)$ |  $\leq 4.2\times 10^{-13}$ [11, 12, 13] | $\Delta a_{e}^{\text{Cs}}$ |  $-0.88(36)\times 10^{-12}$ [1]
BR$(\tau\to e\gamma)$ |  $\leq 3.3\times 10^{-8}$ [11, 12, 13] | $\Delta a_{e}^{\text{Rb}}$ |  $0.48(30)\times 10^{-12}$ [1]
BR$(\tau\to\mu\gamma)$ |  $\leq 4.4\times 10^{-8}$ [11, 12, 13] | $\Delta a_{\mu}$ |  $249(48)\times 10^{-11}$ [55]
BR$(\mu^{-}\to e^{-}e^{+}e^{-})$ |  $\leq 1.0\times 10^{-12}$ [1] | $|d_{e}|$ |  $<1.1\times 10^{-29}e$ cm [56]
BR$(\tau^{-}\to e^{-}e^{+}e^{-})$ |  $\leq 1.4\times 10^{-8}$ [1] | $|d_{\mu}|$ |  $<1.9\times 10^{-19}e$ cm [57]
BR$(\tau^{-}\to e^{-}\mu^{+}\mu^{-})$ |  $\leq 1.6\times 10^{-8}$[1] |  |
BR$(\tau^{-}\to\mu^{-}e^{+}\mu^{-})$ |  $\leq 9.8\times 10^{-9}$ [1] |  |
BR$(\tau^{-}\to e^{-}\mu^{+}e^{-})$ |  $\leq 8.4\times 10^{-9}$ [1] |  |
BR$(\tau^{-}\to\mu^{-}\mu^{+}\mu^{-})$ |  $\leq 1.1\times 10^{-8}$ [1] |  |
Table 8:  Experimental results for leptonic flavor observables.

###  VI.3 Numerical results

In this subsection, we will use the values of known input parameters from
Tables 4, 5, and 6 in our numerical study. For the lepton flavor
phenomenologies, we randomly seed the free parameters $z,s_{23}^{e_{R}},k$,
$\Lambda_{2}$, and $\theta$ in ranges as

$z\in[-1,1],\hskip 14.22636pts_{23}^{e_{R}}\in[0,1],\hskip
14.22636pt\theta\in[0,\pi/2],\hskip 14.22636ptk\in[1,10],\hskip
14.22636pt\Lambda_{2}\in[1,50]\ \text{TeV}.$ (138)

Besides, we also compare the results of four hierarchy scenarios of lepton
mixing angles $\theta_{ij}^{e_{R}}$ shown in Eqs. (93)–(96).

We first obtain the correlation between mixing angle $s_{23}^{e_{R}}$ and
charge parameter $z$ satisfying all constraints of leptonic observables within
four hierarchy scenarios of lepton mixing angles $\theta_{ij}^{e_{R}}$ as in
Fig. 4. It is easy to see that all the hierarchy scenarios potentially fulfill
the constraints, and the obtained values of $s_{23}^{e_{R}}$ are mostly
independent of values of $z$. This can be understood because the coefficients
$C_{L,R}^{ij}$ are free of $z$ in the limit $v\ll\Lambda_{1,2}$. In addition,
the whole range $s_{23}^{e_{R}}$ pleases the constraints in the LNH scenario.
In contrast, the remaining scenarios accept only a partial range of
$s_{23}^{e_{R}}$, namely, $s_{23}^{e_{R}}\in[0,0.75]$ for the LMH1 scenario
and $s_{23}^{e_{R}}\in[0,0.3]$ for the LIH and LMH2 scenarios. It is
noteworthy that the LIH and LMH2 scenarios have
$s_{13}^{e_{R}}=s_{23}^{e_{R}}s_{23}^{\text{PMNS}}/s_{13}^{\text{PMNS}}\simeq
5.082s_{23}^{e_{R}}$, hence in order to ensure $s_{13}^{e_{R}}\leq 1$ then
$s_{23}^{e_{R}}\leq 0.197$. Thus, the viable range for $s_{23}^{e_{R}}$ in the
LIH and LMH2 scenarios is $s_{23}^{e_{R}}\in[0,0.197]$. Furthermore, the LIH
and LMH2 scenarios have an inverse relationship between $s_{12}^{e_{R}}$ and
$s_{23}^{e_{R}}$. Therefore, the nearly identical panels of these scenarios
also illustrate that the leptonic observables do not significantly rely on
$s_{12}^{e_{R}}$, but primarily on $s_{23}^{e_{R}}$. This behavior is also
applied to the LNH and LMH1 scenarios since they have the same
$s_{13}^{e_{R}}$ whereas $s_{12}^{e_{R}}$ is changed, but the result is not
modified remarkably.

Figure 4: The correlations between mixing angle $s_{23}^{e_{R}}$ with charge
parameter $z$ in four hierarchy scenarios of lepton mixing angles
$\theta_{ij}^{e_{R}}$.

Furthermore, we obtain the correlation between the ratio
$k=\Lambda_{1}/\Lambda_{2}$ and VEV $\Lambda_{2}$ within four hierarchy
scenarios of lepton mixing angles $\theta_{ij}^{e_{R}}$, as respectively shown
in four panels of Fig. 5. We see that the whole ranges of $k$ and
$\Lambda_{2}$ in four panels generally satisfy the leptonic constraints,
although these points distribute mainly with high $k$ and $\Lambda_{2}$
values, namely $k\gtrsim 7.4$ and $\Lambda_{2}\gtrsim 39$ TeV. Besides, the
panels of LIH and LMH2 scenarios are almost similar. This result also occurred
in Fig. 4. Hence, we comment that the LIH and LMH2 scenarios give the same
results, while the LNH and LMH1 scenarios do not change considerably.
Therefore, in the following, we consider the model under only the LNH and LIH
scenarios that satisfy the constraints from the lepton flavor violation
process.

Figure 5: The correlations between the ratio $k=\Lambda_{1}/\Lambda_{2}$ with
VEV $\Lambda_{2}$ for the four hierarchy scenarios of lepton mixing angles
$\theta_{ij}^{e_{R}}$.

Now, we turn to the quark flavor phenomenologies. We randomly generate the
free parameters $z,k$, $\Lambda_{2}$, and $\theta$ similarly in the studies of
leptonic flavor phenomenologies. Besides, the parameters $s_{12}^{d_{R}}$ and
$\delta^{d_{R}}$ are randomly extracted from ranges as

$\displaystyle s_{12}^{d_{R}}\in[0,1],\hskip
14.22636pt\delta^{d_{R}}\in[-2\pi,2\pi],$ (139)

whereas the lepton mixing angles $\theta_{ij}^{e_{R}}$ are chosen in the LNH
and LIH scenarios.

In Fig. 6, we show the correlation points between the $\Delta m_{B_{s}}$ and
$\Delta m_{B_{d}}$ in four hierarchy scenarios of quark mixing angles
$\theta_{ij}^{d_{R}}$ determined by Eqs. (88)–(91), which satisfy the
constraints of $\Delta m_{K}$, BR$(B_{s}\to\mu^{+}\mu^{-})$, and
BR$(\bar{B}\to X_{s}\gamma)$ respectively expressed in Eqs. (128)–(130), and
lepton mixing angle $\theta_{ij}^{e_{R}}$ is taken in the LNH scenario. The
dot-dashed red and green lines are correspondingly the most current
experimental limits of $\Delta m_{B_{s}}$ and $\Delta m_{B_{d}}$ [9]. It is
important that the model with QNH scenario supplies points simultaneously
fulfilling the constraints of both $\Delta m_{B_{s}}$ and $\Delta m_{B_{d}}$,
whereas the QIH and QMH2 scenarios supply points adapt the constraint of only
$\Delta m_{B_{d}}$. In addition, the QMH1 scenario provides points satisfying
the constraint of either $\Delta m_{B_{s}}$ or $\Delta m_{B_{d}}$. A similar
result is also shown in Fig. 7 plotted with the LIH scenario. These two
figures imply that the model under QNH and LNH scenarios is preferred.
Therefore, the following numerical studies will focus on these hierarchy
scenarios.

Figure 6: The correlations between predicted values of $\Delta m_{B_{s}}$ and
$\Delta m_{B_{d}}$ for four hierarchy scenarios of quark mixing angles
$\theta_{ij}^{d_{R}}$ with the LNH scenario of $\theta_{ij}^{e_{R}}$. The dot-
dashed red and green lines present the experimental upper and lower limits of
$\Delta m_{B_{s}}$ and $\Delta m_{B_{d}}$ [9].

Figure 7: The correlations between predicted values of $\Delta m_{B_{s}}$ and
$\Delta m_{B_{d}}$ in four hierarchy scenarios of quark mixing angles
$\theta_{ij}^{d_{R}}$ with the LIH scenario of $\theta_{ij}^{e_{R}}$. The dot-
dashed red and green lines present the experimental upper and lower limits of
$\Delta m_{B_{s}}$ and $\Delta m_{B_{d}}$ [9].

Now, we focus particularly on four correlations, $s_{12}^{d_{R}}-z$,
$s_{12}^{d_{R}}-s_{23}^{e_{R}}$, $s_{12}^{d_{R}}-\delta^{d_{R}}$, and
$k-\Lambda_{2}$, which are respectively shown in four panels of Fig. 8. For
the first panel, we see that the mixing angle $s_{12}^{d_{R}}$ is limited in a
range $s_{12}^{d_{R}}\in[0.1,0.2]$ for $|z|\in[0.1,1]$. However, when $|z|$
decreases to less than 0.1, there are many points distributed in a wider range
of $s_{12}^{d_{R}}$, i.e., $s_{12}^{d_{R}}\in[0.1,0.4]$. This can be
interpreted by the following reason: when $|z|\sim 1$, the electroweak term
proportional $v$ in $m_{Z^{\prime}}$ will be much smaller than one relevant to
the new physics contribution and can be ignored. As a result, $m_{Z^{\prime}}$
now depends linearly on $z$, and then, several of the WCs are free of $z$
since it is canceled between numerator and denominator, such as the WCs in
Eqs. (101)-(103) and (106)-(110). Therefore, the quark flavor observable is
approximately independent of $z$. On the other hand, when $|z|$ is
sufficiently small, the electroweak term significantly affects quark flavor
processes. We would like to note that the upper limit of $s_{12}^{d_{R}}\sim
0.4$ is larger than the center value of $s_{12}^{\text{CKM}}=\lambda$ given in
the Table 5. The second panel demonstrates that the mixing parameter
$s_{12}^{d_{R}}$ of $V_{d_{R}}$ is independent of mixing parameter
$s_{23}^{e_{R}}$ of $V_{e_{R}}$. The range of $s_{23}^{e_{R}}$ is not
constrained tightly as the $s_{12}^{d_{R}}$ and whole range of
$s_{23}^{e_{R}}$ satisfying the mentioned constraints. The correlation between
$s_{12}^{d_{R}}$ with CP violation phase $\delta^{d_{R}}$ is displayed in the
third panel. Here, the total range of $\delta^{d_{R}}$ fulfills the
constraints from Eqs. (126) and (128)–(130). This also implies that the effect
of $\delta^{d_{R}}$ on the quark flavor observables is negligible and can be
ignored. The last panel focuses on the correlation between $k$ and
$\Lambda_{2}$. It is easy to see that the behavior of $k$ and $\Lambda_{2}$ is
inverse; if the value of $k$ increases then $\Lambda_{2}$ decreases.
Additionally, when $k$ becomes much larger than $10$, $\Lambda_{2}$ tends to
the regime of $\mathcal{O}(10^{2})$ GeV , which is inconsistent with the
condition $\Lambda_{1,2}\gg v=246$ GeV. Therefore, to the new physics scale in
the TeV scale, the relevant values of $k$ should be $k\leq 10$.

Figure 8: The correlation between parameter pairs, $s_{12}^{d_{R}}-z$,
$s_{12}^{d_{R}}-s_{23}^{e_{R}}$, $s_{12}^{d_{R}}-\delta^{d_{R}}$, and
$k-\Lambda_{2}$.

In Fig. 9, we show two correlations, $m_{Z^{\prime}}-z$ (left panel) and
$m_{Z^{\prime}}-\theta$ (right panel). From here, we comment that the viable
range of $z$ is $[-0.5,0.1]$, whereas the whole range of $\theta$ is
available. However, if $\theta\in[5\pi/16,\pi/2)$ then $m_{Z^{\prime}}\gtrsim
6$ TeV. This is consistent with the collider bounds (discussed below).
Comparing with Fig. 8, we obtain the viable ranges for several parameters as

$\displaystyle s_{12}^{d_{R}}\in[0.1,0.22],\hskip
14.22636ptz\in[-0.5,0.1],\hskip 14.22636pt\theta\in[5\pi/16,\pi/2),\hskip
14.22636ptk\leq 10,\hskip 14.22636pts_{23}^{e_{R}}\in[0,1],$ (140)

while the effect of $\delta^{d_{R}}$ is insignificantly and it can be chosen
arbitrarily.

Figure 9: The correlation between the mass of new gauge boson $m_{Z^{\prime}}$
with the charge parameter $z$ (left panel), and with the mixing angle $\theta$
(right panel).

Last but not least, we consider the LFUV ratios $R_{K}$ and $R_{K^{*}}$; their
results are shown in Fig. 10. The blue points are plotted in which parameters
satisfy all constraints given in Eqs. (126) and (128)–(130). We realize that
the figure shows points concurrently meeting the measured results of both
$R_{K}$ and $R_{K^{*}}$ given in the last column in Table 7. Therefore, the
model with the QNH hierarchy scenario can explain several of quark flavor
observables, including the meson oscillations $\Delta m_{K,B_{s},B_{d}}$,
BR$(\bar{B}\to X_{s}\gamma)$, BR$(B_{s}\to\mu^{+}\mu^{-})$, and $R_{K^{(*)}}$.

---
Figure 10: The correlation between the predicted $R_{K}$ and $R_{K^{*}}$. The
dot-dashed red and green lines are correspondingly the current experimental
limits of $R_{K}^{\text{exp}}$ and $R_{K^{*}}^{\text{exp}}$ [10].

##  VII  Collider bounds

The $Z^{\prime}$ gauge boson in our model directly interacts with both
ordinary quarks ($q$) and charged leptons ($l$), so it can be produced at the
large electron-positron (LEP) experiments even the large hadron collider
(LHC). In this section, we take the current negative search results reported
by these experiments to impose a lower bound on the mass of $Z^{\prime}$ boson
[58, 59, 60, 61, 62, 63].

###  VII.1 LEP

One of the processes searched at the LEP experiments is $e^{+}e^{-}\to
f\bar{f}$, which generates a pair of ordinary charged leptons ($f=e,\mu,\tau$)
through the exchange of $Z^{\prime}$ boson. This process can be described by
the following effective Lagrangian,

$\mathcal{L}_{\text{eff}}=\frac{1}{1+\delta_{ef}}\frac{g^{2}}{c^{2}_{W}m^{2}_{Z^{\prime}}}\sum_{i,j=L,R}C_{i}^{Z^{\prime}}(e)C_{j}^{Z^{\prime}}(f)(\bar{e}\gamma_{\mu}P_{i}e)(\bar{f}\gamma^{\mu}P_{j}f),$
(141)

where $\delta_{ef}=1(0)$ for $f=e(\neq e)$, and the chiral gauge couplings are
given by $C_{L,R}^{Z^{\prime}}(f)=\frac{1}{2}[g_{V}^{Z_{2}}(f)\pm
g_{A}^{Z_{2}}(f)]$. LEP-II has probed all such effective contact interactions,
and no significant evidence has been found for the existence of a $Z^{\prime}$
boson. LEP-II also provided the lower limits of the scale of the contact
interactions, $\Lambda$, for all possible chiral structures and for various
combinations of fermions [60]. Consequently, the mass of $Z^{\prime}$ boson is
bounded by

$m^{2}_{Z^{\prime}}\gtrsim\frac{g^{2}}{4\pi
c^{2}_{W}}|C_{i}^{Z^{\prime}}(e)C_{j}^{Z^{\prime}}(f)|[\Lambda_{ij}^{\pm}(f)]^{2},$
(142)

where $\Lambda^{+}$ for $C_{i}^{Z^{\prime}}(e)C_{j}^{Z^{\prime}}(f)>0$ and
$\Lambda^{-}$ for $C_{i}^{Z^{\prime}}(e)C_{j}^{Z^{\prime}}(f)<0$.

The strongest constraint for our model comes from the
$e^{+}e^{-}\to\mu^{+}\mu^{-},\tau^{+}\tau^{-}$ channel with
$\Lambda^{+}_{VV}=24.6$ TeV. It results $m_{Z^{\prime}}\gtrsim 5.6$ TeV for
$z\simeq-0.1$ and $\theta\simeq 3\pi/8$.

###  VII.2 LHC

At the LHC experiment, the $Z^{\prime}$ neutral gauge boson can be resonantly
produced in the new physics processes $pp\to Z^{\prime}\to f\bar{f}$ for
$f=q,l$. Additionally, the most significant decay channel of $Z^{\prime}$ is
given by $Z^{\prime}\to l\bar{l}$ because of well-understood backgrounds [61,
63] and that it signifies a boson $Z^{\prime}$ having both couplings to lepton
and quark like ours. The cross-section for the relevant process, in the narrow
width approximation, takes the form [64]

$\sigma(pp\to Z^{\prime}\to
l\bar{l})\simeq\frac{1}{3}\sum_{q}\frac{dL_{q\bar{q}}}{dm_{Z^{\prime}}^{2}}\hat{\sigma}(q\bar{q}\to
Z^{\prime})\text{BR}(Z^{\prime}\to l\bar{l}),$ (143)

where the parton luminosities $\frac{dL_{q\bar{q}}}{dm_{Z^{\prime}}^{2}}$ can
be found in Ref. [65], while the peak cross-section is given by

$\hat{\sigma}(q\bar{q}\to Z^{\prime})\simeq\frac{\pi
g^{2}}{12c_{W}^{2}}[(g_{V}^{Z_{2}}(q))^{2}+(g_{A}^{Z_{2}}(q))^{2}].$ (144)

The branching ratio of $Z^{\prime}$ decaying into the lepton pairs is
$\text{BR}(Z^{\prime}\to l\bar{l})=\Gamma(Z^{\prime}\to
l\bar{l})/\Gamma_{Z^{\prime}}$, where the partial and total decay widths are
respectively given by

$\displaystyle\Gamma(Z^{\prime}\to l\bar{l})$ $\displaystyle\simeq$
$\displaystyle\frac{g^{2}m_{Z^{\prime}}}{48\pi
c_{W}^{2}}[(g_{V}^{Z_{2}}(l))^{2}+(g_{A}^{Z_{2}}(l))^{2}],$ (145)
$\displaystyle\Gamma_{Z^{\prime}}$ $\displaystyle\simeq$
$\displaystyle\frac{g^{2}m_{Z^{\prime}}}{48\pi
c_{W}^{2}}\left\\{\sum_{f}N_{C}(f)[(g_{V}^{Z_{2}}(f))^{2}+(g_{A}^{Z_{2}}(f))^{2}]+\frac{(C_{1L}^{Z_{2}})^{2}}{2}+(C_{2L}^{Z_{2}})^{2}\right\\}$
(146) $\displaystyle+\frac{g^{2}m_{Z^{\prime}}}{96\pi
c_{W}^{2}}(C_{R}^{Z_{2}})^{2}\sum_{i=1}^{3}\left(1-\frac{4M_{i}^{2}}{m_{Z^{\prime}}^{2}}\right)^{3/2}\Theta\left(\frac{m_{Z^{\prime}}}{2}-M_{i}\right),$

assuming that $Z^{\prime}$ is lighter than new Higgs bosons
$\mathcal{H}_{1,2}$ and $\mathcal{A}$. Here, $f$ denotes the SM charged
fermions, $N_{C}(f)$ is the color number of the fermion $f$, and $\Theta$ is
the step function.

Setting center-of-mass energy of $\sqrt{s}=13$ TeV and assuming
$M_{1,2,3}=m_{Z^{\prime}}/3$, in Fig. 11, we plot the cross section for the
relevant processes as a function of the $Z^{\prime}$ boson mass, given that
$z=-0.1$ and $\theta=3\pi/8$. Here, we also include the upper limits on the
cross-section of these processes reported by ATLAS [61] and CMS [63]
experiments. We obtain a lower bound on the $Z^{\prime}$ boson mass of $5.9$
TeV under the $ee$ channel, while the $\mu\mu~{}(\tau\tau)$ channel even
implies a more strict constraint. Significantly, these signal strengths are
separated, which can be used to approve or rule out the model under
consideration.

---
Figure 11: Dilepton production cross-section as a function of the new gauge
boson mass $m_{Z^{\prime}}$ for $z=-0.1$ and $\theta=3\pi/8$. The black (gray)
curve shows the upper bound on the cross-section obtained by the ATLAS 2019
results for $\Gamma/m=3\%$ [61] (the CMS 2019 results for $\Gamma/m=0.6\%$
[63]).

We would like to note that the dijet signals also can provide a lower bound
for the $Z^{\prime}$ boson mass [62]. However, in the present model, the
coupling strengths between $Z^{\prime}$ and the charged leptons are
approximately equal to those of $Z^{\prime}$ with the quarks, while the
current bound on dijet signals is less sensitive than the lepton one, so the
lower bounds implied by the dijet search are quite smaller than those result
from the dilepton. In other words, in the present model, the dijet bounds for
the $Z^{\prime}$ boson mass are not significant.

##  VIII Conclusion

In this work, we have proposed a model that is based on the gauge symmetry
$SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{X}\otimes U(1)_{N}$. This model is
general and flavor-dependent but constructed minimally. The new charges $X$
and $N$ of the light fermions differ from those of the remaining fermions but
determine the hypercharge $Y$ as $Y=X+N$, which conforms to the observables.
We have shown that our model can provide a possible solution to several
puzzles of the SM, including the observed number of fermion generations, the
neutrino mass generation mechanism, and the flavor anomalies in both the quark
and lepton sectors. The new physics effects at the LEP and LHC experiments
have also been examined.

With the relevant assumptions, the model leaves only six free parameters,
including the charge parameter $z$, the new physics scale $\Lambda_{2}$, the
ratio $k=\Lambda_{1}/\Lambda_{2}$, the mixing angles $\theta_{12}^{d_{R}}$,
$\theta_{23}^{e_{R}}$ and $\theta$. We have identified the allowed parameter
space of the model, which is consistent with the experimental constraints,
namely, $\sin[\theta_{12}^{d_{R}}]\in[0.1,0.22],\hskip
14.22636ptz\in[-0.5,0.1],\hskip 14.22636pt\theta\in[5\pi/16,\pi/2),\hskip
14.22636ptk\in[1,10],\hskip 14.22636pt\sin[\theta_{23}^{e_{R}}]\in[0,1]$, and
the new physics scale $\Lambda_{2}$ should generally be above several TeVs.

## Acknowledgement

N.T. Duy was funded by the Postdoctoral Scholarship Programme of Vingroup
Innovation Foundation (VINIF), code VINIF.2023.STS.65.

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