# The flavor-dependent $U(1)_{F}$ model

Jin-Lei Yang1,2,3111jlyang@hbu.edu.cn,Hai-Bin
Zhang1,2,3222hbzhang@hbu.edu.cn,Tai-Fu<EMAIL_ADDRESS>Department of Physics, Hebei University, Baoding, 071002, China1
Key Laboratory of High-precision Computation and Application of Quantum Field
Theory of Hebei Province, Baoding, 071002, China2
Research Center for Computational Physics of Hebei Province, Baoding, 071002,
China3

###### Abstract

A flavor-dependent model (FDM) is proposed in this work. The model extends the
Standard Model by an extra $U(1)_{F}$ local gauge group, two scalar doublets,
one scalar singlet and two right-handed neutrinos, where the additional
$U(1)_{F}$ charges are related to the particles’ flavor. The new fermion
sector in the FDM can explain the flavor mixings puzzle and the mass hierarchy
puzzle simultaneously, and the nonzero Majorana neutrino masses can be
obtained naturally by the Type I see-saw mechanism. In addition, the $B$ meson
rare decay processes $\bar{B}\to X_{s}\gamma$, $B_{s}^{0}\to\mu^{+}\mu^{-}$,
the top quark rare decay processes $t\to ch$, $t\to uh$ and the $\tau$ lepton
flavor violation processes $\tau\to 3e$, $\tau\to 3\mu$, $\mu\to 3e$ predicted
in the FDM are analyzed.

##  I Introduction

The Standard Model (SM) achieves great success in describing the interactions
of fundamental particles, and most of the observations coincide well with the
SM predictions. However, the Yukawa couplings in the SM are still enigmatic,
because the flavor mixings of quarks described by the Cabibbo-Kobayashi-
Maskawa (CKM) matrix Cabibbo:1963yz ; Kobayashi:1973fv  are not predicted from
first principles in the SM, which is the so-called flavor puzzle. And the
fermions of the three families have largely distinct masses

$\displaystyle\frac{m_{t}}{m_{c}}\approx 104,\;\frac{m_{c}}{m_{u}}\approx
773,\;\frac{m_{b}}{m_{s}}\approx 51,\;\frac{m_{s}}{m_{d}}\approx
20,\;\frac{m_{\tau}}{m_{\mu}}\approx 17,\;\frac{m_{\mu}}{m_{e}}\approx 207.$
(1)

It exhibits the large hierarchical structure of masses across the three
families while they share common SM gauge group quantum numbers, which is the
so-called mass hierarchy puzzle. Meanwhile, the nonzero neutrino masses and
mixings observed at the neutrino oscillation experiments
ParticleDataGroup:2022pth  make the so-called flavor puzzle in the SM more
acutely. All of these indicate that explaining the observed fermionic mass
spectrum and mixings is one of the most enigmatic questions in particle
physics, which may help understand the flavor nature and seek possible new
physics (NP).

In literatures, there are some attempts to explain the fermionic mass
hierarchies and flavor mixings. For example, the authors of Refs.
Froggatt:1978nt ; Koide:1982ax ; Leurer:1992wg ; Ibanez:1994ig ; Babu:1995hr
try to explain the fermionic flavor mixings by imposing the family symmetries,
some proposals to explain the fermionic mass hierarchies can be found in Refs.
Berezhiani:1990wn ; Berezhiani:1990jj ; Sakharov:1994pr ; Randall:1999ee ;
Kaplan:2001ga ; Chen:2008tc ; Buras:2011ph ; King:2013eh ; King:2014nza ;
King:2015aea ; King:2017guk ; Weinberg:2020zba ; Feruglio:2019ybq ;
Abbas:2018lga ; Mohanta:2022seo ; Mohanta:2023soi ; Abbas:2022zfb . The three-
Higgs doublet model is motivated to account for the flavor structure with
three generations of fermions Weinberg:1976hu ; Lavoura:2007dw ; Ivanov:2012ry
; GonzalezFelipe:2013xok ; GonzalezFelipe:2013yhh ; Keus:2013hya ;
Ivanov:2014doa ; Buskin:2021eig ; Izawa:2022viu , in which the number of Higgs
doublets is three and has a flavor symmetry. A flavor-dependent $U(1)$
extension of the SM is proposed in Ref. VanLoi:2023utt , the authors explained
the small mixing at quark sector by introducing a NP cut-off scale parameter.
Weinberg proposed a new mechanism in which only the third generation of quarks
and leptons achieve the masses at the tree level, while the masses for the
second and first generations are produced by one-loop and two-loop radiative
corrections respectively Weinberg:2020zba . However, his analysis indicated
that the ratios of the masses of the second and third generations fermions are
independent of various masses of the third generation, i.e.

$\frac{m_{c}}{m_{t}}=\frac{m_{s}}{m_{b}}=\frac{m_{\mu}}{m_{\tau}}\;\;{\rm
or}\;\;\frac{m_{c}}{m_{t}^{3}}=\frac{m_{s}}{m_{b}^{3}}=\frac{m_{\mu}}{m_{\tau}^{3}},\\\
$

which is not true of observed masses as shown in Eq. (1). Hence, he pointed
out in his work that “this kind of NP models are not realistic for some
reasons” Weinberg:2020zba . In addition, Weinberg was the first to propose the
Type I see-saw mechanism to give the tiny neutrino masses naturally
Weinberg:1979sa , which provides one of the most popular mechanisms so far to
give the tiny Majorana neutrino masses.

In this work, we adopt the Weinberg’s idea “only the third generation of
fermions achieve the masses at the tree level”, but abandon “the masses for
the second and first generations are produced by one-loop and two-loop
radiative corrections respectively”. Instead, we propose to obtain the masses
for the second and first generations by the “see-saw mechanism” (which is
proposed by Weinberg to give the tiny Majorana neutrino masses as mentioned
above), i.e. the masses for the second and first generations are produced by
mixing with the third generation. In this case, the ratios of the masses of
the second and third generations fermions depend on the tree-level mixing
parameters of fermions, and we propose a flavor-dependent model (FDM) to
realize this mechanism. The model extends the SM by an extra $U(1)_{F}$ local
gauge group which relates to the particles’ flavor, two scalar doublets, one
scalar singlet and two right-handed neutrinos. The new fermion sector in the
FDM relates the fermionic flavor mixings to the fermionic mass hierarchies,
which provides a new understanding about the mass hierarchy puzzle and the
flavor mixing puzzle. In addition, the nonzero neutrino masses can be obtained
naturally in the FDM by the Type I see-saw mechanism.

The paper is organized as follows: The structure of the FDM including particle
content, scalar sector, fermion masses and gauge sector are collected in Sec.
II. The numerical results of CKM matrix and Higgs masses predicted in the FDM
are presented in Sec. III. The processes mediated by the flavor changed
neutral currents (FCNCs) in the FDM are analyzed in Sec. IV. The summary is
made in Sec. V.

##  II The flavor-dependent model

Multiplets | $SU(3)_{C}$ | $SU(2)_{L}$ | $U(1)_{Y}$ | $U(1)_{F}$
---|---|---|---|---
$l_{1}=(\nu_{1L},e_{1L})^{T}$ | 1 | 2 | $-\frac{1}{2}$ | $z$
$l_{2}=(\nu_{2L},e_{2L})^{T}$ | 1 | 2 | $-\frac{1}{2}$ | $-z$
$l_{3}=(\nu_{3L},e_{3L})^{T}$ | 1 | 2 | $-\frac{1}{2}$ | $0$
$\nu_{1R}$ | 1 | 1 | $0$ | $-z$
$\nu_{2R}$ | 1 | 1 | $0$ | $z$
$e_{1R}$ | 1 | 1 | $-1$ | $-z$
$e_{2R}$ | 1 | 1 | $-1$ | $z$
$e_{3R}$ | 1 | 1 | $-1$ | $0$
$q_{1}=(u_{1L},d_{1L})^{T}$ | 3 | 2 | $\frac{1}{6}$ | $z$
$q_{2}=(u_{2L},d_{2L})^{T}$ | 3 | 2 | $\frac{1}{6}$ | $-z$
$q_{3}=(u_{3L},d_{3L})^{T}$ | 3 | 2 | $\frac{1}{6}$ | $0$
$d_{1R}$ | 3 | 1 | -$\frac{1}{3}$ | $-z$
$d_{2R}$ | 3 | 1 | -$\frac{1}{3}$ | $z$
$d_{3R}$ | 3 | 1 | -$\frac{1}{3}$ | $0$
$u_{1R}$ | 3 | 1 | $\frac{2}{3}$ | $-z$
$u_{2R}$ | 3 | 1 | $\frac{2}{3}$ | $z$
$u_{3R}$ | 3 | 1 | $\frac{2}{3}$ | $0$
$\Phi_{1}=(\phi_{1}^{+},\phi_{1}^{0})^{T}$ | 1 | 2 | $\frac{1}{2}$ | $z$
$\Phi_{2}=(\phi_{2}^{+},\phi_{2}^{0})^{T}$ | 1 | 2 | $\frac{1}{2}$ | $-z$
$\Phi_{3}=(\phi_{3}^{+},\phi_{3}^{0})^{T}$ | 1 | 2 | $\frac{1}{2}$ | 0
$\chi$ | 1 | 1 | 0 | $2z$
Table 1: Matter content in the FDM, where the nonzero constant $z$ denotes the
extra $U(1)_{F}$ charge.

The gauge group of the FDM is $SU(3)_{C}\otimes SU(2)_{L}\otimes
U(1)_{Y}\otimes U(1)_{F}$, where the extra $U(1)_{F}$ local gauge group is
related to the particles’ flavor. In the FDM, the third generation of fermions
obtain masses through the tree-level couplings with the SM scalar doublet, and
the first two generations of fermions achieve masses through the tree-level
mixings with the third generation as mentioned above. Hence, two additional
scalar doublets are introduced in the FDM to realize the tree-level mixings of
the first two generations and the third generation. In addition, to coincide
with the observed neutrino oscillations, two right-handed neutrinos and one
scalar singlet are introduced. Then the right-handed neutrinos obtain large
Majorana masses after the scalar singlet achieving large vacuum expectation
value (VEV), and the tiny neutrino masses and neutrino flavor mixings can be
obtained by the Type I see-saw mechanism.

All fields in the FDM and the corresponding gauge symmetry charges are
presented in Tab. 1, where $\Phi_{3}$ corresponds to the SM Higgs doublet, the
nonzero constant $z$ denotes the extra $U(1)_{F}$ charge. It can be noted in
Tab. 1 that there are only two generations of right-handed neutrinos in the
FDM, because both $U(1)_{F}$ and $U(1)_{Y}$ charges of the third generation of
right-handed neutrinos $\nu_{R_{3}}$ are zero, which is trivial. In addition,
it is obvious that the chiral anomaly cancellation can be guaranteed for the
fermionic charges presented in Tab. 1.

###  II.1 The scalar sector of the FDM

The scalar potential in the FDM can be written as

$\displaystyle
V=-M_{\Phi_{1}}^{2}\Phi_{1}^{\dagger}\Phi_{1}-M_{\Phi_{2}}^{2}\Phi_{2}^{\dagger}\Phi_{2}-M_{\Phi_{3}}^{2}\Phi_{3}^{\dagger}\Phi_{3}-M_{\chi}^{2}\chi^{*}\chi+\lambda_{\chi}(\chi^{*}\chi)^{2}+\lambda_{1}(\Phi_{1}^{\dagger}\Phi_{1})^{2}$
$\displaystyle\qquad+\lambda_{2}(\Phi_{2}^{\dagger}\Phi_{2})^{2}+\lambda_{3}(\Phi_{3}^{\dagger}\Phi_{3})^{2}+\lambda^{\prime}_{4}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{2}^{\dagger}\Phi_{2})+\lambda_{4}^{\prime\prime}(\Phi_{1}^{\dagger}\Phi_{2})(\Phi_{2}^{\dagger}\Phi_{1})$
$\displaystyle\qquad+\lambda_{5}^{\prime}(\Phi_{1}^{\dagger}\Phi_{1})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{5}^{\prime\prime}(\Phi_{1}^{\dagger}\Phi_{3})(\Phi_{3}^{\dagger}\Phi_{1})+\lambda_{6}^{\prime}(\Phi_{2}^{\dagger}\Phi_{2})(\Phi_{3}^{\dagger}\Phi_{3})+\lambda_{6}^{\prime\prime}(\Phi_{2}^{\dagger}\Phi_{3})(\Phi_{3}^{\dagger}\Phi_{2})$
$\displaystyle\qquad+\lambda_{7}(\Phi_{1}^{\dagger}\Phi_{1})(\chi^{*}\chi)+\lambda_{8}(\Phi_{2}^{\dagger}\Phi_{2})(\chi^{*}\chi)+\lambda_{9}(\Phi_{3}^{\dagger}\Phi_{3})(\chi^{*}\chi)+[\lambda_{10}(\Phi_{3}^{\dagger}\Phi_{1})(\Phi_{3}^{\dagger}\Phi_{2})$
$\displaystyle\qquad+\kappa(\Phi_{1}^{\dagger}\Phi_{2})\chi+h.c.],$ (2)

where

$\displaystyle\Phi_{1}=\left(\begin{array}[]{c}\phi_{1}^{+}\\\
\frac{1}{\sqrt{2}}(iA_{1}+S_{1}+v_{1})\end{array}\right),\Phi_{2}=\left(\begin{array}[]{c}\phi_{2}^{+}\\\
\frac{1}{\sqrt{2}}(iA_{2}+S_{2}+v_{2})\end{array}\right),\Phi_{3}=\left(\begin{array}[]{c}\phi_{3}^{+}\\\
\frac{1}{\sqrt{2}}(iA_{3}+S_{3}+v_{3})\end{array}\right),$ (9)
$\displaystyle\chi=\frac{1}{\sqrt{2}}(iA_{\chi}+S_{\chi}+v_{\chi}),$ (10)

and $v_{i}\;(i=1,\;2,\;3),\;v_{\chi}$ are the VEVs of $\Phi_{i},\;\chi$
respectively.

Based on the scalar potential in Eq. (2), the tadpole equations in the FDM can
be written as444Calculating the exact vacuum stability conditions for any new
physics model is difficult generally, and the obtained tadpole equations can
be used to calculate the stationary points. In this case, we apply tadpole
equations in the calculations, and guarantee the stability of vacuum
numerically by keeping the scalar potential at the input
$v_{1},\;v_{2},\;v_{3},\;v_{\chi}$ are smaller than all the other stationary
points.

$\displaystyle
M_{\Phi_{1}}^{2}=\lambda_{1}v_{1}^{2}+\frac{1}{2}\Big{[}(\lambda_{4}^{\prime}+\lambda_{4}^{\prime\prime})v_{2}^{2}+(\lambda_{5}^{\prime}+\lambda_{5}^{\prime\prime})v_{3}^{2}+\frac{v_{2}}{v_{1}}v_{3}^{2}{\rm
Re}(\lambda_{10})+\sqrt{2}\frac{v_{2}}{v_{1}}v_{\chi}{\rm
Re}(\kappa)+\lambda_{7}v_{\chi}^{2}\Big{]},$ $\displaystyle
M_{\Phi_{2}}^{2}=\lambda_{2}v_{2}^{2}+\frac{1}{2}\Big{[}(\lambda_{4}^{\prime}+\lambda_{4}^{\prime\prime})v_{1}^{2}+(\lambda_{6}^{\prime}+\lambda_{6}^{\prime\prime})v_{3}^{2}+\frac{v_{1}}{v_{2}}v_{3}^{2}{\rm
Re}(\lambda_{10})+\sqrt{2}\frac{v_{1}}{v_{2}}v_{\chi}{\rm
Re}(\kappa)+\lambda_{8}v_{\chi}^{2}\Big{]},$ $\displaystyle
M_{\Phi_{3}}^{2}=\lambda_{3}v_{3}^{2}+{\rm
Re}(\lambda_{10})v_{1}v_{2}+\frac{1}{2}[(\lambda_{5}^{\prime}+\lambda_{5}^{\prime\prime})v_{1}^{2}+(\lambda_{6}^{\prime}+\lambda_{6}^{\prime\prime})v_{2}^{2}+\lambda_{9}v_{c}^{2}],$
$\displaystyle
M_{\chi}^{2}=\lambda_{\chi}v_{\chi}^{2}+\frac{1}{2}\Big{[}\lambda_{7}v_{1}^{2}+\lambda_{8}v_{2}^{2}+\lambda_{9}v_{3}^{2}+\sqrt{2}\frac{v_{1}v_{2}}{v_{\chi}}{\rm
Re}(\kappa)\Big{]}.$ (11)

On the basis $(S_{1},\;S_{2},\;S_{3},\;S_{\chi})$, the CP-even Higgs squared
mass matrix in the FDM is

$\displaystyle
M_{h}^{2}=\left(\begin{array}[]{*{20}{cccc}}M_{h,11}^{2}&M_{h,12}^{2}&M_{h,13}^{2}&M_{h,14}^{2}\\\\[6.0pt]
M_{h,12}^{2}&M_{h,22}^{2}&M_{h,23}^{2}&M_{h,24}^{2}\\\\[6.0pt]
M_{h,13}^{2}&M_{h,23}^{2}&M_{h,33}^{2}&M_{h,34}^{2}\\\\[6.0pt]
M_{h,14}^{2}&M_{h,24}^{2}&M_{h,34}^{2}&M_{h,44}^{2}\\\\[6.0pt]
\end{array}\right),$ (16)

where

$\displaystyle
M_{h,11}^{2}=2\lambda_{1}v_{1}^{2}-\frac{v_{2}}{2v_{1}}\Big{[}v_{3}^{2}{\rm
Re}(\lambda_{10})+\sqrt{2}v_{\chi}{\rm Re}(\kappa)\Big{]},$ $\displaystyle
M_{h,22}^{2}=2\lambda_{2}v_{2}^{2}-\frac{v_{1}}{2v_{2}}\Big{[}v_{3}^{2}{\rm
Re}(\lambda_{10})+\sqrt{2}v_{\chi}{\rm Re}(\kappa)\Big{]},$ $\displaystyle
M_{h,33}^{2}=2\lambda_{3}v_{3}^{2},\;\;M_{h,44}^{2}=2\lambda_{\chi}v_{\chi}^{2}-\frac{\sqrt{2}v_{1}v_{2}}{2v_{\chi}}{\rm
Re}(\kappa),$ $\displaystyle
M_{h,12}^{2}=(\lambda_{4}^{\prime}+\lambda_{4}^{\prime\prime})v_{1}v_{2}+\frac{1}{2}{\rm
Re}(\lambda_{10})v_{3}^{2}+\frac{\sqrt{2}}{2}{\rm Re}(\kappa)v_{\chi},$
$\displaystyle
M_{h,13}^{2}=(\lambda_{5}^{\prime}+\lambda_{5}^{\prime\prime})v_{1}v_{3}+{\rm
Re}(\lambda_{10})v_{2}v_{3},\;\;M_{h,14}^{2}=\lambda_{7}v_{1}v_{\chi}+\frac{\sqrt{2}}{2}{\rm
Re}(\kappa)v_{2},$ $\displaystyle
M_{h,23}^{2}=(\lambda_{6}^{\prime}+\lambda_{6}^{\prime\prime})v_{2}v_{3}+{\rm
Re}(\lambda_{10})v_{1}v_{3},\;\;M_{h,24}^{2}=\lambda_{8}v_{2}v_{\chi}+\frac{\sqrt{2}}{2}{\rm
Re}(\kappa)v_{1},$ $\displaystyle M_{h,34}^{2}=\lambda_{9}v_{3}v_{\chi}.$ (17)

The tadpole equations in Eq. (11) are used to obtain the matrix elements
above.

Then, on the basis $(A_{1},\;A_{2},\;A_{3},\;A_{\chi})$, the squared mass
matrix of CP-odd Higgs in the FDM can be written as

$\displaystyle
M_{A}^{2}=\left(\begin{array}[]{*{20}{cccc}}M_{A,11}^{2}&M_{A,12}^{2}&M_{A,13}^{2}&M_{A,14}^{2}\\\\[6.0pt]
M_{A,12}^{2}&M_{A,22}^{2}&M_{A,23}^{2}&M_{A,24}^{2}\\\\[6.0pt]
M_{A,13}^{2}&M_{A,23}^{2}&M_{A,33}^{2}&M_{A,34}^{2}\\\\[6.0pt]
M_{A,14}^{2}&M_{A,24}^{2}&M_{A,34}^{2}&M_{A,44}^{2}\\\\[6.0pt]
\end{array}\right),$ (22)

where

$\displaystyle M_{A,11}^{2}=-\frac{v_{2}}{2v_{1}}[{\rm
Re}(\lambda_{10})v_{3}^{2}+\sqrt{2}v_{\chi}{\rm
Re}(\kappa)],\;\;M_{A,33}^{2}=-2{\rm Re}(\lambda_{10})v_{1}v_{2},$
$\displaystyle M_{A,22}^{2}=-\frac{v_{1}}{2v_{2}}[{\rm
Re}(\lambda_{10})v_{3}^{2}+\sqrt{2}v_{\chi}{\rm
Re}(\kappa)],\;\;M_{A,44}^{2}=-\frac{\sqrt{2}v_{1}v_{2}}{2v_{\chi}}{\rm
Re}(\kappa),$ $\displaystyle M_{A,12}^{2}=\frac{\sqrt{2}}{2}v_{\chi}{\rm
Re}(\kappa)-\frac{1}{2}{\rm Re}(\lambda_{10})v_{3}^{2},\;\;M_{A,13}^{2}={\rm
Re}(\lambda_{10})v_{2}v_{3},$ $\displaystyle
M_{A,14}^{2}=\frac{\sqrt{2}}{2}v_{2}{\rm Re}(\kappa),\;\;M_{A,23}^{2}={\rm
Re}(\lambda_{10})v_{1}v_{3},\;\;M_{A,24}^{2}=-\frac{\sqrt{2}}{2}v_{1}{\rm
Re}(\kappa),$ $\displaystyle M_{A,34}^{2}=0.$ (23)

On the basis $(\phi_{1}^{+},\;\phi_{2}^{+},\;\phi_{3}^{+})$ and
$(\phi_{1}^{-},\;\phi_{2}^{-},\;\phi_{3}^{-})^{T}$, the squared mass matrix of
singly charged Higgs in the FDM can be written as

$\displaystyle
M_{H^{\pm}}^{2}=\left(\begin{array}[]{*{20}{ccc}}M_{H^{\pm},11}^{2}&M_{H^{\pm},12}^{2}&M_{H^{\pm},13}^{2}\\\\[6.0pt]
(M_{H^{\pm},12}^{2})^{*}&M_{H^{\pm},22}^{2}&M_{H^{\pm},23}^{2}\\\\[6.0pt]
(M_{H^{\pm},13}^{2})^{*}&(M_{H^{\pm},23}^{2})^{*}&M_{H^{\pm},33}^{2}\\\\[6.0pt]
\end{array}\right),$ (27)

where

$\displaystyle M_{H^{\pm},11}^{2}=-\frac{v_{2}}{2v_{1}}[{\rm
Re}(\lambda_{10})v_{3}^{2}+\sqrt{2}v_{\chi}{\rm
Re}(\kappa)]-\frac{1}{2}(\lambda_{4}^{\prime\prime}v_{2}^{2}+\lambda_{5}^{\prime\prime}v_{3}^{2}),$
$\displaystyle M_{H^{\pm},22}^{2}=-\frac{v_{1}}{2v_{2}}[{\rm
Re}(\lambda_{10})v_{3}^{2}+\sqrt{2}v_{\chi}{\rm
Re}(\kappa)]-\frac{1}{2}(\lambda_{4}^{\prime\prime}v_{1}^{2}+\lambda_{6}^{\prime\prime}v_{3}^{2}),$
$\displaystyle M_{H^{\pm},33}^{2}=-{\rm
Re}(\lambda_{10})v_{1}v_{2}-\frac{1}{2}(\lambda_{5}^{\prime\prime}v_{1}^{2}+\lambda_{6}^{\prime\prime}v_{2}^{2}),$
$\displaystyle
M_{H^{\pm},12}^{2}=\frac{\sqrt{2}}{2}v_{\chi}\kappa+\frac{1}{2}\lambda_{4}^{\prime\prime}v_{1}v_{2},\;\;M_{H^{\pm},13}^{2}=\frac{1}{2}v_{3}(\lambda_{5}^{\prime\prime}v_{1}+\lambda_{10}^{*}v_{2}),$
$\displaystyle
M_{H^{\pm},23}^{2}=\frac{1}{2}v_{3}(\lambda_{6}^{\prime\prime}v_{2}+\lambda_{10}^{*}v_{1}).$
(28)

It is easy to verify that there are two neutral Goldstones and one singly
charged Goldstone in the FDM.

###  II.2 The fermion masses in the FDM

Based on the matter content listed in Tab. 1, the Yukawa couplings in the FDM
can be written as

$\displaystyle\mathcal{L}_{Y}=Y_{u}^{33}\bar{q}_{3}\tilde{\Phi}_{3}u_{R_{3}}+Y_{d}^{33}\bar{q}_{3}\Phi_{3}d_{R_{3}}+Y_{u}^{32}\bar{q}_{3}\tilde{\Phi}_{1}u_{R_{2}}+Y_{u}^{23}\bar{q}_{2}\tilde{\Phi}_{1}u_{R_{3}}+Y_{d}^{32}\bar{q}_{3}\Phi_{2}d_{R_{2}}$
$\displaystyle\qquad\;+Y_{d}^{23}\bar{q}_{2}\Phi_{2}d_{R_{3}}+Y_{u}^{21}\bar{q}_{2}\tilde{\Phi}_{3}u_{R_{1}}+Y_{u}^{12}\bar{q}_{1}\tilde{\Phi}_{3}u_{R_{2}}+Y_{d}^{21}\bar{q}_{2}\Phi_{3}d_{R_{1}}+Y_{d}^{12}\bar{q}_{1}\Phi_{3}d_{R_{2}}$
$\displaystyle\qquad\;+Y_{u}^{31}\bar{q}_{3}\tilde{\Phi}_{2}u_{R_{1}}+Y_{u}^{13}\bar{q}_{1}\tilde{\Phi}_{2}u_{R_{3}}+Y_{d}^{31}\bar{q}_{3}\Phi_{1}d_{R_{1}}+Y_{d}^{13}\bar{q}_{1}\Phi_{1}d_{R_{3}}$
$\displaystyle\qquad\;+Y_{e}^{33}\bar{l}_{3}\Phi_{3}e_{R_{3}}+Y_{e}^{32}\bar{l}_{3}\Phi_{2}e_{R_{2}}+Y_{e}^{23}\bar{l}_{2}\Phi_{2}e_{R_{3}}+Y_{e}^{21}\bar{l}_{2}\Phi_{3}e_{R_{1}}+Y_{e}^{12}\bar{l}_{1}\Phi_{3}e_{R_{2}}$
$\displaystyle\qquad\;+Y_{e}^{31}\bar{l}_{3}\Phi_{1}e_{R_{1}}+Y_{e}^{13}\bar{l}_{1}\Phi_{1}e_{R_{3}}+Y_{R}^{11}\bar{\nu}^{c}_{R_{1}}\nu_{R_{1}}\chi+Y_{R}^{22}\bar{\nu}^{c}_{R_{2}}\nu_{R_{2}}\chi^{*}+Y_{D}^{21}\bar{l}_{2}\tilde{\Phi}_{3}\nu_{R_{1}}$
$\displaystyle\qquad\;+Y_{D}^{12}\bar{l}_{1}\tilde{\Phi}_{3}\nu_{R_{2}}+Y_{D}^{31}\bar{l}_{3}\tilde{\Phi}_{2}\nu_{R_{1}}+Y_{D}^{32}\bar{l}_{3}\tilde{\Phi}_{1}\nu_{R_{2}}+h.c.,$
(29)

Then the mass matrices of quarks and leptons can be written as

$\displaystyle m_{q}=\left(\begin{array}[]{ccc}0&m_{q,12}&m_{q,13}\\\
m_{q,12}^{*}&0&m_{q,23}\\\
m_{q,13}^{*}&m_{q,23}^{*}&m_{q,33}\end{array}\right),m_{e}=\left(\begin{array}[]{ccc}0&m_{e,12}&m_{e,13}\\\
m_{e,12}^{*}&0&m_{e,23}\\\
m_{e,13}^{*}&m_{e,23}^{*}&m_{e,33}\end{array}\right),m_{\nu}=\left(\begin{array}[]{cc}0&M_{D}^{T}\\\
M_{D}&M_{R}\end{array}\right),$ (38)

where $q=u,d$, the parameters $m_{q,33}$ and $m_{e,33}$ are real, $M_{D}$ is
$2\times 3$ Dirac mass matrix and $M_{R}$ is $2\times 2$ Majorana mass matrix
(the nonzero neutrino masses are obtained by the Type I see-saw mechanism).
The elements of the matrices in Eq. (38) are

$\displaystyle
m_{u,11}=m_{u,22}=0,\;m_{u,33}=\frac{1}{\sqrt{2}}Y_{u}^{33}v_{3},\;m_{u,12}=\frac{1}{\sqrt{2}}Y_{u}^{12}v_{3},\;m_{u,13}=\frac{1}{\sqrt{2}}Y_{u}^{13}v_{1},$
$\displaystyle m_{u,23}=\frac{1}{\sqrt{2}}Y_{u}^{23}v_{2},$ (39)
$\displaystyle
m_{d,11}=m_{d,22}=0,\;m_{d,33}=\frac{1}{\sqrt{2}}Y_{d}^{33}v_{3},\;m_{d,12}=\frac{1}{\sqrt{2}}Y_{d}^{12}v_{3},\;m_{d,13}=\frac{1}{\sqrt{2}}Y_{d}^{13}v_{1},$
$\displaystyle m_{d,23}=\frac{1}{\sqrt{2}}Y_{d}^{23}v_{2},$ (40)
$\displaystyle
m_{e,11}=m_{e,22}=0,\;m_{e,33}=\frac{1}{\sqrt{2}}Y_{e}^{33}v_{3},\;m_{e,12}=\frac{1}{\sqrt{2}}Y_{e}^{12}v_{3},\;m_{e,13}=\frac{1}{\sqrt{2}}Y_{e}^{13}v_{1},$
$\displaystyle m_{e,23}=\frac{1}{\sqrt{2}}Y_{e}^{23}v_{2},$ (41)
$\displaystyle
M_{D,11}=M_{D,22}=0,\;\;M_{D,12}=\frac{1}{\sqrt{2}}Y_{D}^{12}v_{3},\;M_{D,31}=\frac{1}{\sqrt{2}}Y_{D}^{31}v_{1},$
$\displaystyle
M_{D,32}=\frac{1}{\sqrt{2}}Y_{D}^{32}v_{2},\;M_{R,12}=M_{R,21}=0,\;M_{R,11}=\frac{1}{\sqrt{2}}Y_{R}^{11}v_{\chi},\;M_{R,22}=\frac{1}{\sqrt{2}}Y_{R}^{22}v_{\chi}.$
(42)

###  II.3 The gauge sector of the FDM

Due to the introducing of an extra $U(1)_{F}$ local gauge group in the FDM,
the covariant derivative corresponding to $SU(2)_{L}\otimes U(1)_{Y}\otimes
U(1)_{F}$ is defined as

$\displaystyle
D_{\mu}=\partial_{\mu}+ig_{2}T_{j}A_{j\mu}+ig_{1}YB_{\mu}+ig_{{}_{F}}FB^{\prime}_{\mu}+ig_{{}_{YF}}YB^{\prime}_{\mu},\;(j=1,\;2,\;3),$
(43)

where $(g_{2},\;g_{1},\;g_{{}_{F}})$, $(T_{j},\;Y,\;F)$,
$(A_{j\mu},\;B_{\mu},\;B^{\prime}_{\mu})$ denote the gauge coupling constants,
generators and gauge bosons of groups $(SU(2)_{L},\;U(1)_{Y},\;U(1)_{F})$
respectively, $g_{{}_{YF}}$ is the gauge coupling constant arises from the
gauge kinetic mixing effect which presents in the models with two Abelian
groups. Then the $W$ boson mass can be written as

$\displaystyle M_{W}=\frac{1}{2}g_{2}(v_{1}^{2}+v_{2}^{2}+v_{3}^{2})^{1/2},$
(44)

where $(v_{1}^{2}+v_{2}^{2}+v_{3}^{2})^{1/2}=v\approx 246\;{\rm GeV}$ and we
have $v_{1},\;v_{2}<v_{3}$ in the FDM. The $\gamma$, $Z$ and $Z^{\prime}$
boson masses in the FDM can be written as

$\displaystyle
M_{\gamma}=0,\;M_{Z}\approx\frac{1}{2}(g_{1}^{2}+g_{2}^{2})^{1/2}v,\;M_{Z^{\prime}}\approx
2|zg_{{}_{F}}|v_{\chi},$ (45)

and

$\displaystyle\gamma=c_{W}B+s_{W}A_{3},\;Z=-s_{W}B+c_{W}A_{3}+s^{\prime}_{W}B^{\prime},\;Z^{\prime}=s_{W}^{\prime}(s_{W}B-c_{W}A_{3})+c^{\prime}_{W}B^{\prime},$
(46)

where $\gamma,\;Z,\;Z^{\prime}$ are the mass eigenstates,
$c_{W}\equiv\cos\theta_{W},\;s_{W}\equiv\sin\theta_{W}$ with $\theta_{W}$
denoting the Weinberg angle, $s_{W}^{\prime}\equiv\sin\theta^{\prime}_{W}$,
$c_{W}^{\prime}\equiv\cos\theta^{\prime}_{W}$ with $\theta_{W}^{\prime}$
representing the $Z-Z^{\prime}$ mixing effect.

##  III CKM matrix and Higgs masses in the FDM

The quark sector in the FDM are redefined and the additional two scalar
doublets, one scalar singlet modify the scalar potential of the model, hence
we focus on the quark sector and scalar sector of the model in this section.

###  III.1 CKM matrix

The analysis in our previous work Yang:2024kfs  shows that the quark mass
matrices obtained in the FDM can fit the measured CKM matrix well. In this
work, we perform a $\chi^{2}$ test to explore the best fit describing the
quark masses and CKM matrix in the model. Generally, the $\chi^{2}$ function
can be constructed as

$\displaystyle\chi^{2}=\sum_{1}\Big{(}\frac{O_{i}^{\rm th}-O_{i}^{\rm
exp}}{\sigma_{i}^{\rm exp}}\Big{)}^{2},$ (47)

where $O_{i}^{\rm th}$ denotes the $i-$th observable computed theoretically,
$O_{i}^{\rm exp}$ is the corresponding experimental value and $\sigma_{i}^{\rm
exp}$ is the uncertainty in $O_{i}^{\rm exp}$. Taking into account $10$
observables including $6$ quark masses, $3$ mixing angles and a phase in the
CKM matrix, we scan the parameter space

$\displaystyle|m_{u,13}|=(0.0\sim 1.0)\;{\rm GeV},\;|m_{d,13}|=(0.0\sim
0.1)\;{\rm GeV},\;\theta_{q,ij}=(-\pi,\pi)$ (48)

with $m_{q,ij}=|m_{q,ij}|e^{i\theta_{q,ij}}$, $q=u,d$, $ij=12,13,23$. Then we
obtain the best fit solution corresponding to $\chi^{2}=0.0072$, the results
of this fit are listed in Tab. 2 where various $O_{i}^{\rm exp}$ we use are
listed in the third column and $\sigma_{i}^{\rm exp}$ are take from PDG
ParticleDataGroup:2022pth .

Observables | $O_{i}^{\rm th}$ | $O_{i}^{\rm exp}$ | Deviations in $\%$
---|---|---|---
$m_{u}$[MeV] | 2.15 | 2.16 | 0.47
$m_{c}$[GeV] | 1.67 | 1.67 | 0
$m_{t}$[GeV] | 172.5 | 172.5 | 0
$m_{d}$[MeV] | 4.67 | 4.67 | 0
$m_{s}$[MeV] | 93.4 | 93.4 | 0
$m_{b}$[GeV] | 4.78 | 4.78 | 0
$|v_{us}|$ | 0.2253 | 0.2253 | 0
$|v_{ub}|$ | 0.3616 | 0.003616 | 0
$|v_{cb}|$ | 0.4149 | 0.04149 | 0
Table 2: The results obtained for the best fit corresponding to
$\chi^{2}=\mathbf{0.0072}$.

The parameters for the best fit listed in Tab. 2 are

$\displaystyle|m_{u,13}|=0.3152\;{\rm
GeV},\;\theta_{u,12}=-0.8248\pi,\;\theta_{u,13}=0.1231\pi,\;\theta_{u,23}=-0.1129\pi,$
$\displaystyle|m_{d,13}|=7.235\;{\rm
MeV},\;\theta_{d,12}=0.6705\pi,\;\theta_{d,13}=-0.3622\pi,\;\theta_{d,23}=-0.06389\pi,$
(49)

and $|m_{q,12}|,\;|m_{q,23}|,\;m_{q,33}\;(q=u,\;d)$ can be obtained by555Eq.
(50) is obtained with the approximation
$|m_{q,12}|,\;|m_{q,13}|,\;|m_{q,23}|\ll m_{q,33}$, and the terms of
$\mathcal{O}(m_{q,ij}/m_{q,33})^{3}$ are neglected. For down type quark,
higher order corrections to Eq. (50) are also important, we calculated them
numerically and the corrections to $m_{q,33}$, $|m_{q,23}|$, $|m_{q,12}|$ are
$9.32$ MeV, $-27.03$ MeV, $0.196$ MeV respectively.

$\displaystyle m_{q,33}=m_{q_{3}}-m_{q_{1}}-m_{q_{2}},$
$\displaystyle|m_{q,23}|=[(m_{q_{1}}+m_{q_{2}})m_{q,33}-|m_{q,13}|^{2}]^{1/2},$
$\displaystyle|m_{q,12}|=\frac{|m_{q,13}||m_{q,23}|}{|m_{q,33}|}\cos(\theta_{q,12}+\theta_{q,23}-\theta_{q,13})$
$\displaystyle\qquad\qquad+\Big{\\{}[\frac{|m_{q,13}||m_{q,23}|}{|m_{q,33}|^{2}}\cos(\theta_{q,12}+\theta_{q,23}-\theta_{q,13})]^{2}+\frac{m_{q_{1}}m_{q_{2}}}{|m_{q,33}|^{2}}\Big{\\}}^{1/2}|m_{q,33}|,$
(50)

with $m_{q_{k}}$ being the $k-$generation quark $q$ mass.

It is obvious that there are additional CP phases in the obtained CKM matrix
by taking the parameters in Eq. (49) as inputs, but we do not list the
observed CP phase in the CKM matrix in Tab. 2. Because the CKM matrix is
defined as

$\displaystyle V_{\rm CKM}=(U_{L}^{u}U_{L}^{d\dagger})^{*},$ (51)

where $U_{L}^{u}$, $U_{L}^{d}$ are the unitary matrices which diagonalize the
quark matrices in Eq. (38)

$\displaystyle m_{u}^{\rm diag}=U_{L}^{u*}m_{u}U_{R}^{u\dagger},\;m_{d}^{\rm
diag}=U_{L}^{d*}m_{d}U_{R}^{d\dagger}.$ (52)

Eq. (52) is invariant under

$\displaystyle U_{L}^{u}\to{\rm
diag}(e^{-i\theta_{u}},e^{-i\theta_{c}},e^{-i\theta_{t}})\cdot
U_{L}^{u},\;U_{R}^{u}\to{\rm
diag}(e^{i\theta_{u}},e^{i\theta_{c}},e^{i\theta_{t}})\cdot U_{R}^{u},$
$\displaystyle U_{L}^{d}\to{\rm
diag}(e^{i\theta_{d}},e^{i\theta_{s}},e^{i\theta_{b}})\cdot
U_{L}^{d},\;U_{R}^{d}\to{\rm
diag}(e^{-i\theta_{d}},e^{-i\theta_{s}},e^{-i\theta_{b}})\cdot U_{R}^{d},$
(53)

so there are six free parameters
$\theta_{u},\;\theta_{c},\;\theta_{t},\;\theta_{d},\;\theta_{s},\;\theta_{b}$
which can absorb the extra CP phases in the obtained CKM matrix, these
parameters are not observable. As a result, the observed CP phase in the CKM
matrix can be obtained by defining appropriate
$\theta_{u},\;\theta_{c},\;\theta_{t},\;\theta_{d},\;\theta_{s},\;\theta_{b}$.
This fact is always valid because there are also six possible CP violation
parameters at the quark sector in the model.

###  III.2 Perturbative unitary bounds

The scalar sector of the FDM is extended by two scalar doublets and one scalar
singlet, additional new couplings are also introduced correspondingly.
Therefore, the tree-level perturbative unitary should be applied to the scalar
elastic scattering processes in this model Lee:1977eg . The zero partial wave
amplitude

$\displaystyle a_{0}=\frac{1}{32\pi}\sqrt{\frac{4p_{f}^{\rm CM}p_{i}^{\rm
CM}}{s}}\int_{-1}^{+1}T_{2\to 2}d\cos\theta$ (54)

must satisfy the condition $|{\rm Re}(a_{0})|\leq\frac{1}{2}$, where $s$ is
the centre of mass (CM) energy, $T_{2\to 2}$ denotes the matrix element for
$2\to 2$ processes, $\theta$ is the incident angle between two incoming
particles, $p_{i}^{\rm CM}$ and $p_{f}^{\rm CM}$ are the initial and final
momenta in the CM system respectively.

The possible two particle states of $2\to 2$ scattering processes at the
scalar sector in the FDM are $S_{1}S_{1}$, $S_{2}S_{2}$, $S_{3}S_{3}$,
$S_{\chi}S_{\chi}$, $S_{1}S_{2}$, $S_{1}S_{3}$, $S_{1}S_{\chi}$, $S_{2}S_{3}$,
$S_{2}S_{\chi}$, $S_{3}S_{\chi}$. Considering the limit $s\gg
M_{S_{1}}^{2},\;M_{S_{2}}^{2},\;M_{S_{3}}^{2},\;M_{S_{\chi}}^{2}$, we have

$\displaystyle a_{0}^{S_{1}S_{1}\to
S_{1}S_{1}}\approx\frac{1}{16\pi}(6\lambda_{1}),\;a_{0}^{S_{2}S_{2}\to
S_{2}S_{2}}\approx\frac{1}{16\pi}(6\lambda_{2}),$ $\displaystyle
a_{0}^{S_{3}S_{3}\to
S_{3}S_{3}}\approx\frac{1}{16\pi}(6\lambda_{3}),\;a_{0}^{S_{\chi}S_{\chi}\to
S_{\chi}S_{\chi}}\approx\frac{1}{16\pi}(6\lambda_{\chi}),$ $\displaystyle
a_{0}^{S_{1}S_{1}\to
S_{2}S_{2}}\approx\frac{1}{16\pi}(\lambda_{4}^{\prime}+\lambda_{4}^{\prime\prime}),\;a_{0}^{S_{1}S_{2}\to
S_{1}S_{2}}\approx\frac{1}{16\pi}(\lambda_{4}^{\prime}+\lambda_{4}^{\prime\prime}),$
$\displaystyle a_{0}^{S_{1}S_{1}\to
S_{3}S_{3}}\approx\frac{1}{16\pi}(\lambda_{5}^{\prime}+\lambda_{5}^{\prime\prime}),\;a_{0}^{S_{1}S_{3}\to
S_{1}S_{3}}\approx\frac{1}{16\pi}(\lambda_{5}^{\prime}+\lambda_{5}^{\prime\prime}),$
$\displaystyle a_{0}^{S_{2}S_{2}\to
S_{3}S_{3}}\approx\frac{1}{16\pi}(\lambda_{6}^{\prime}+\lambda_{6}^{\prime\prime}),\;a_{0}^{S_{2}S_{3}\to
S_{2}S_{3}}\approx\frac{1}{16\pi}(\lambda_{6}^{\prime}+\lambda_{6}^{\prime\prime}),$
$\displaystyle a_{0}^{S_{1}S_{1}\to
S_{\chi}S_{\chi}}\approx\frac{1}{16\pi}(\lambda_{7}),\;a_{0}^{S_{1}S_{\chi}\to
S_{1}S_{\chi}}\approx\frac{1}{16\pi}(\lambda_{7}),$ $\displaystyle
a_{0}^{S_{2}S_{2}\to
S_{\chi}S_{\chi}}\approx\frac{1}{16\pi}(\lambda_{8}),\;a_{0}^{S_{2}S_{\chi}\to
S_{2}S_{\chi}}\approx\frac{1}{16\pi}(\lambda_{8}),$ $\displaystyle
a_{0}^{S_{3}S_{3}\to
S_{\chi}S_{\chi}}\approx\frac{1}{16\pi}(\lambda_{9}),\;a_{0}^{S_{3}S_{\chi}\to
S_{3}S_{\chi}}\approx\frac{1}{16\pi}(\lambda_{9}),$ $\displaystyle
a_{0}^{S_{3}S_{3}\to S_{1}S_{2}}\approx\frac{1}{16\pi}[2{\rm
Re}(\lambda_{10})],\;a_{0}^{S_{1}S_{3}\to
S_{2}S_{3}}\approx\frac{1}{16\pi}[2{\rm Re}(\lambda_{10})].$ (55)

Requiring $|{\rm Re}(a_{0})|\leq\frac{1}{2}$ for each individual process, we
have

$\displaystyle|\lambda_{1}|,\;|\lambda_{2}|,\;|\lambda_{3}|,\;|\lambda_{\chi}|\leq\frac{4\pi}{3},\;|{\rm
Re}(\lambda_{10})|\leq 4\pi,$
$\displaystyle|\lambda_{4}^{\prime}+\lambda_{4}^{\prime\prime}|,\;|\lambda_{5}^{\prime}+\lambda_{5}^{\prime\prime}|,\;|\lambda_{6}^{\prime}+\lambda_{6}^{\prime\prime}|,\;|\lambda_{7}|,\;|\lambda_{8}|,\;|\lambda_{9}|\leq
8\pi.$ (56)

###  III.3 Higgs masses in the FDM

For the free parameters in the scalar sector of the FDM, we take
$v_{1}=v_{2}$,
$\lambda_{4}^{\prime}=\lambda_{4}^{\prime\prime}=\lambda_{4}/2$,
$\lambda_{5}^{\prime}=\lambda_{5}^{\prime\prime}=\lambda_{5}/2$,
$\lambda_{6}^{\prime}=\lambda_{6}^{\prime\prime}=\lambda_{6}/2$ and all
parameters to be real for simplicity. In addition, $v_{\chi}$ is limited by
the new $Z^{\prime}$ boson mass as shown in Eq. (45), hence we take
$v_{\chi}\geq 5\;{\rm TeV}$ in the following analysis. Considering the
perturbative unitary bounds presented in Eq. (56), we scan the following
parameter space

$\displaystyle v_{1}=(0,\;40)\;{\rm GeV},\;\lambda_{i}=(0,\;4)\;\;{\rm
with}\;\;(i=1,...,9,\chi),\;\lambda_{10}=(-4,\;0),$ $\displaystyle
v_{\chi}=(5,\;40)\;{\rm TeV},\;\kappa=(-3,\;-0.1)\;{\rm TeV},$ (57)

to explore the Higgs mass spectrum in the FDM. In the scanning, we keep the
next-to-lightest CP-even Higgs mass $M_{H_{2}}$ in the range $124\;{\rm
GeV}<M_{H_{2}}<126\;{\rm GeV}$ and the scalar potential at the input
$v_{1},\;v_{2},\;v_{3},\;v_{\chi}$ are smaller than all the other stationary
points to guarantee the stability of vacuum.

####  III.3.1 CP-even Higgs masses

Figure 1: The results of CP-even Higgs masses $M_{H_{1}}$ (a), $M_{H_{3}}$
(b), $M_{H_{4}}$ (c) versus $v_{1}$, $\kappa$, $v_{\chi}$ are plotted
respectively by scanning the parameter space in Eq. (57).

The results of CP-even Higgs masses $M_{H_{1}}$ versus $v_{1}$, $M_{H_{3}}$
versus $\kappa$, $M_{H_{4}}$ versus $v_{\chi}$ are plotted in Fig. 1 (a), Fig.
1 (b), Fig. 1 (c) respectively. We do not present the results of the next-to-
lightest CP-even Higgs mass $M_{H_{2}}$ in Fig. 1 because $M_{H_{2}}$ is
limited in the range $124\;{\rm GeV}<M_{H_{2}}<126\;{\rm GeV}$ in the
plotting. Fig. 1 (a) shows that the lightest Higgs mass $M_{H_{1}}$ in the FDM
mainly depends on the chosen value of $v_{1}$, and $M_{H_{1}}$ can reach
$95\;{\rm GeV}$ for $v_{1}\gtrsim 30\;{\rm GeV}$ which will be explored in
detail in our next work. As shown in Fig. 1 (b) and (c), $M_{H_{3}}$,
$M_{H_{4}}$ are dominated by $\kappa$, $v_{\chi}$, where $M_{H_{4}}$ is about
$110\;{\rm TeV}$ for large $v_{\chi}$ while $M_{H_{3}}$ is about $13\;{\rm
TeV}$ for large $|\kappa|$.

Figure 2: The results of $M_{H_{2}}$ versus $\lambda_{3}$ (a), $\lambda_{9}$
(b), where the solid, dashed, dotted curves denotes the results for
$\lambda_{\chi}=2,\;3,\;4$ respectively, and the gray areas denote the range
$124\;{\rm GeV}<M_{H_{2}}<126\;{\rm GeV}$.

As mentioned above, $\Phi_{3}$ corresponds to the SM Higgs doublet, hence
$\lambda_{3}$ may affect the $125\;{\rm GeV}$ Higgs boson mass $M_{H_{2}}$
significantly. In addition, Eq. (17) shows that there are large mixing effects
between $S_{3}$ and $S_{\chi}$, it indicates $\lambda_{9}$, $\lambda_{\chi}$
can also affect $M_{H_{2}}$. And to verify numerically that $M_{H_{2}}$ is
mainly affected by $\lambda_{3}$, $\lambda_{9}$, $\lambda_{\chi}$, we take
$v_{1}=20\;{\rm
GeV},\;\lambda_{1}=\lambda_{2}=\lambda_{4}=\lambda_{5}=\lambda_{6}=\lambda_{7}=\lambda_{8}=2,\;\lambda_{10}=-2,\;v_{\chi}=10\;{\rm
TeV},\;\kappa=-1\;{\rm TeV}$ to explore the effects of $\lambda_{3}$,
$\lambda_{9}$, $\lambda_{\chi}$ on $M_{H_{2}}$. Then $M_{H_{2}}$ versus
$\lambda_{3}$, $\lambda_{9}$ are plotted in Fig. 2 (a) and Fig. 2 (b)
respectively, where the solid, dashed, dotted curves denotes the results for
$\lambda_{\chi}=2,\;3,\;4$ respectively, and the gray areas denote the range
$124\;{\rm GeV}<M_{H_{2}}<126\;{\rm GeV}$. The picture shows $M_{H_{2}}$
increases with increasing $\lambda_{3}$, $\lambda_{\chi}$ and decreases with
increasing $\lambda_{9}$, and $\lambda_{3}$, $\lambda_{9}$, $\lambda_{\chi}$
affect the theoretical predictions on $M_{H_{2}}$ significantly.

####  III.3.2 CP-odd Higgs masses

Figure 3: The results of CP-even Higgs masses $M_{A_{1}}$ (a), $M_{A_{2}}$
(b), $M_{A_{2}}$ (c) versus $\lambda_{10}$, $\kappa$, $v_{\chi}$ are plotted
respectively by scanning the parameter space in Eq. (57).

There are two physical CP-odd Higgs in the FDM, and the results of CP-odd
Higgs masses $M_{A_{1}}$ versus $\lambda_{10}$, $M_{A_{2}}$ versus $\kappa$,
$M_{A_{2}}$ versus $v_{\chi}$ are plotted in Fig. 3 (a), Fig. 3 (b), Fig. 3
(c) respectively. It is obvious in Fig. 3 (a) that $M_{A_{1}}$ is dominated by
the value of $\lambda_{10}$ completely, and $M_{A_{1}}$ increases with
increasing $|\lambda_{10}|$. In addition, $M_{A_{2}}$ is dominated by
$\kappa$, $v_{\chi}$ as shown in Fig. 3 (b) and Fig. 3 (c), where $M_{A_{2}}$
can be large when $|\kappa|$ and $v_{\chi}$ are large.

####  III.3.3 Charged Higgs masses

Figure 4: The results of charged Higgs masses $M_{H^{\pm}_{1}}$ (a),
$M_{H^{\pm}_{2}}$ (b), $M_{H^{\pm}_{2}}$ (c) versus $\lambda_{10}$, $\kappa$,
$v_{\chi}$ are plotted respectively by scanning the parameter space in Eq.
(57).

Similar to the case of CP-odd Higgs, there are also two physical charged Higgs
in the FDM. The results of charged Higgs masses $M_{H^{\pm}_{1}}$ versus
$\lambda_{10}$, $M_{H^{\pm}_{2}}$ versus $\kappa$, $M_{H^{\pm}_{2}}$ versus
$v_{\chi}$ are plotted in Fig. 4 (a), Fig. 4 (b), Fig. 4 (c) respectively.
Fig. 4 (a) indicates that $M_{H^{\pm}_{1}}$ is mainly affected by
$\lambda_{10}$, but the other scanning parameter can also influence the
predicted $M_{H^{\pm}_{1}}$ especially for small $|\lambda_{10}|$. Similar to
the case of $M_{A_{2}}$, Fig. 4 (b) and Fig. 4 (c) show that $M_{H^{\pm}_{2}}$
is also dominated by $\kappa$, $v_{\chi}$.

##  IV The flavor changed neutral currents in the FDM

As shown in Eq. (29), the different generations of fermions couple to
different Higgs bosons while $\Phi_{3}$ corresponding to the SM-like Higgs,
which are quite different from the ones in the SM. In addition, the new
defined $Z$ and $Z^{\prime}$ gauge bosons can also mediate the FCNCs. Hence,
observing the FCNCs in the FDM may be effective to test the model. In this
section, we focus on the $B$ meson rare decay processes $\bar{B}\to
X_{s}\gamma$, $B_{s}^{0}\to\mu^{+}\mu^{-}$, the top quark rare decay processes
$t\to ch$, $t\to uh$ and the charged lepton flavor violation processes
$\tau\to 3e$, $\tau\to 3\mu$, $\mu\to 3e$ predicted in the FDM. And for
simplicity, we take the nonzero $U_{F}(1)$ charge $z=1$ in the following
analysis.

###  IV.1 $B$ meson rare decay processes $\bar{B}\to X_{s}\gamma$ and
$B_{s}^{0}\to\mu^{+}\mu^{-}$ in the FDM

The $B$ meson rare decay processes $\bar{B}\to X_{s}\gamma$,
$B_{s}^{0}\to\mu^{+}\mu^{-}$ are related closely to the NP contributions, and
the average experimental data on the branching ratios of $\bar{B}\to
X_{s}\gamma$, $B_{s}^{0}\to\mu^{+}\mu^{-}$ are ParticleDataGroup:2022pth

$\displaystyle{\rm Br}(\bar{B}\to X_{s}\gamma)=(3.49\pm 0.19)\times 10^{-4},$
$\displaystyle{\rm Br}(B_{s}^{0}\to\mu^{+}\mu^{-})=(3.01\pm 0.35)\times
10^{-9}.$ (58)

The newly introduced scalars in the FDM including CP-even Higgs, CP-odd Higgs
and charged Higgs can make contributions to these two processes, the
analytical calculations of the contributions are collected in the appendix
Yang:2018fvw .

Scanning the parameter spaces in Eq. (48), Eq. (57) and the parameters
$|m_{e,13}|$, $\theta_{e,ij},\;(ij=12,\;13,\;23)$ in the following range

$\displaystyle|m_{e,13}|=(0.0\sim 0.3)\;{\rm
GeV},\;\theta_{e,ij}=(-\pi\sim\pi),$ (59)

keeping $0.2243<|V_{us}|<0.2263$, $0.003516<|V_{ub}|<0.003716$,
$0.4139<|V_{cb}|<0.4159$ and $M_{H_{2}}$ in the range $124\;{\rm
GeV}<M_{H_{2}}<126\;{\rm GeV}$, we plot $M_{H^{\pm}_{1}}-M_{A_{1}}$,
$v_{1}-\lambda_{10}$ in Fig. 5 (a), Fig. 5 (b) respectively, where the black
points, green points denote the results for ${\rm Br}(\bar{B}\to X_{s}\gamma)$
and ${\rm Br}(B_{s}^{0}\to\mu^{+}\mu^{-})$ in the experimental $2\sigma$,
$1\sigma$ intervals respectively, the ‘red star’ denotes the best fit with the
B meson rare decay branching ratios in Eq. (58) corresponding to
$\chi^{2}=0.022$. The results of this best fit are listed in Tab. 3.

Figure 5: Scanning the parameter spaces in Eq. (48), Eq. (57), Eq. (59) and keeping $0.2243<|V_{us}|<0.2263$, $0.003516<|V_{ub}|<0.003716$, $0.4139<|V_{cb}|<0.4159$, $M_{H_{2}}$ in the range $124\;{\rm GeV}<M_{H_{2}}<126\;{\rm GeV}$, the allowed ranges of $M_{H^{\pm}_{1}}-M_{A_{1}}$ (a) and $v_{1}-\lambda_{10}$ (b) are plotted, where the black points, green points denote the results for ${\rm Br}(\bar{B}\to X_{s}\gamma)$ and ${\rm Br}(B_{s}^{0}\to\mu^{+}\mu^{-})$ in the experimental $2\sigma$, $1\sigma$ intervals respectively, the ‘red star’ denotes the best fit with the B meson rare decay branching ratios in Eq. (58) corresponding to $\chi^{2}=\mathbf{0.022}$. Observables | $O_{i}^{\rm th}$ | $O_{i}^{\rm exp}$ | Deviations in $\%$
---|---|---|---
$m_{u}$[MeV] | 2.154 | 2.16 | 0.28
$m_{c}$[GeV] | 1.658 | 1.67 | 0.72
$m_{t}$[GeV] | 172.5 | 172.5 | 0
$m_{d}$[MeV] | 4.67 | 4.67 | 0
$m_{s}$[MeV] | 93.4 | 93.4 | 0
$m_{b}$[GeV] | 4.78 | 4.78 | 0
$|v_{us}|$ | 0.2252 | 0.2253 | 0.044
$|v_{ub}|$ | 0.003617 | 0.003616 | 0.028
$|v_{cb}|$ | 0.4138 | 0.04149 | 0
${\rm Br}(\bar{B}\to X_{s}\gamma)$ | $3.49\times 10^{-4}$ | $3.49\times 10^{-4}$ | 0
${\rm Br}(B_{s}^{0}\to\mu^{+}\mu^{-})$ | $3.03\times 10^{-9}$ | $3.01\times 10^{-9}$ | 0.66
Table 3: Fit with the B meson rare decays: the results obtained for the best
fit corresponding to $\chi^{2}=\mathbf{0.022}$.

For the parameters not shown in Fig. 5 such as
$\lambda_{1},\;\lambda_{2},...$, they affect the predicted ${\rm
Br}(\bar{B}\to X_{s}\gamma)$ and ${\rm Br}(B_{s}^{0}\to\mu^{+}\mu^{-})$
mildly. The results presented in Fig. 5 (a) indicate that $M_{H^{\pm}_{1}}$ is
correlated strongly to $M_{A_{1}}$ because both of them mainly depend on
$\lambda_{10}$ as shown in Fig. 3 (a) and Fig. 4 (a). Eq. (39) and Eq. (40)
show that the Yukawa couplings increase with decreasing $v_{1}$, i.e. the
scalars in the FDM can make significant contributions to the $B$ meson rare
decay processes $\bar{B}\to X_{s}\gamma$, $B_{s}^{0}\to\mu^{+}\mu^{-}$ for
small $v_{1}$, which leads to the experimental observations of ${\rm
Br}(\bar{B}\to X_{s}\gamma)$ and ${\rm Br}(B_{s}^{0}\to\mu^{+}\mu^{-})$ prefer
large $v_{1}$, and Fig. 5 (b) shows that $v_{1}$ is limited in the range
$v_{1}\gtrsim 15\;{\rm GeV}$.

###  IV.2 top quark rare decay processes $t\to ch$ and $t\to uh$

The branching ratios of the top quark rare decay processes $t\to ch$ and $t\to
uh$ can be written as YANG2018

$\displaystyle{\rm Br}(t\rightarrow
q_{u}h)=\frac{|\mathcal{M}_{tq_{u}h}|^{2}\sqrt{((m_{t}+m_{h})^{2}-m_{q_{u}}^{2})((m_{t}-m_{h})^{2}-m_{q_{u}}^{2})}}{32\pi
m_{t}^{3}\Gamma^{t}_{{\rm total}}},$ (60)

where $q_{u}=u,\;c$, the amplitude $\mathcal{M}_{tq_{u}h}$ can be read
directly from the Yukawa couplings in Eq. (29), and $\Gamma^{t}_{{\rm
total}}=1.42\;$GeV ParticleDataGroup:2022pth  is the total decay width of top
quark. The measured quark masses, CKM matrix and $B$ meson rare decay
processes $\bar{B}\to X_{s}\gamma$, $B_{s}^{0}\to\mu^{+}\mu^{-}$ should be
considered in the calculations of top quark rare decay processes $t\to ch$ and
$t\to uh$, hence we take the points obtained in Fig. 5 as inputs.

Figure 6: Taking the points obtained in Fig. 5 as inputs, the results of ${\rm
Br}(t\to ch)$ versus $|m_{u,13}|$ (a), $\theta_{u,12}$ (b) and the results of
${\rm Br}(t\to uh)$ versus $|m_{u,13}|$ (c), $\theta_{u,12}$ (d) are plotted,
where red and blue lines denote the upper bounds on ${\rm Br}(t\to ch)$ and
${\rm Br}(t\to uh)$ from Particle Data Group ParticleDataGroup:2022pth
respectively.

Then we plot the results of ${\rm Br}(t\to ch)$ versus $v_{1}$ in Fig. 6 (a),
${\rm Br}(t\to uh)$ versus $v_{1}$ in Fig. 6 (b), and ${\rm Br}(t\to ch)$
versus ${\rm Br}(t\to uh)$ in Fig. 6 (c). The red and blue lines in Fig. 6
denote the upper bounds on ${\rm Br}(t\to ch)$ and ${\rm Br}(t\to uh)$ from
Particle Data Group ParticleDataGroup:2022pth  respectively. The picture
illustrates that the results of ${\rm Br}(t\to ch)$, ${\rm Br}(t\to uh)$
obtained in the FDM can be large, which indicates that the processes $t\to
ch$, $t\to uh$ have great opportunities to be observed experimentally. In
addition, the parameter space of the model suffers constraints from the
experimental upper bounds on ${\rm Br}(t\to ch)$ and ${\rm Br}(t\to uh)$.

###  IV.3 Lepton flavor violation processes $\tau\to 3e$, $\tau\to 3\mu$ and
$\mu\to 3e$

Finally, we focus on the lepton flavor violation processes $\tau\to 3e$,
$\tau\to 3\mu$, $\mu\to 3e$ predicted in the FDM. The corresponding amplitude
can be written as Hisano:1995cp

$\displaystyle\mathcal{M}(e_{j}\rightarrow
e_{i}e_{i}\bar{e}_{i})=C_{1}^{L}\bar{u}_{e_{i}}(p_{2})\gamma_{\mu}P_{L}u_{e_{j}}(p_{1})u_{e_{i}}(p_{3})\gamma^{\mu}P_{L}\nu_{e_{i}}(p_{4})$
$\displaystyle\qquad\quad+C_{1}^{R}\bar{u}_{e_{i}}(p_{2})\gamma_{\mu}P_{R}u_{e_{j}}(p_{1})u_{e_{i}}(p_{3})\gamma^{\mu}P_{R}\nu_{e_{i}}(p_{4})$
$\displaystyle\qquad\quad+[C_{2}^{L}\bar{u}_{e_{i}}(p_{2})\gamma_{\mu}P_{L}u_{e_{j}}(p_{1})u_{e_{i}}(p_{3})\gamma^{\mu}P_{R}\nu_{e_{i}}(p_{4})$
$\displaystyle\qquad\quad+C_{2}^{R}\bar{u}_{e_{i}}(p_{2})\gamma_{\mu}P_{R}u_{e_{j}}(p_{1})u_{e_{i}}(p_{3})\gamma^{\mu}P_{L}\nu_{e_{i}}(p_{4})-(p_{2}\leftrightarrow
p_{3})]$
$\displaystyle\qquad\quad+[C_{3}^{L}\bar{u}_{e_{i}}(p_{2})P_{L}u_{e_{j}}(p_{1})u_{e_{i}}(p_{3})P_{L}\nu_{e_{i}}(p_{4})$
$\displaystyle\qquad\quad+C_{3}^{R}\bar{u}_{e_{i}}(p_{2})P_{R}u_{e_{j}}(p_{1})u_{e_{i}}(p_{3})P_{R}\nu_{e_{i}}(p_{4})-(p_{2}\leftrightarrow
p_{3})],$ (61)

where $i=1,\;2$ for $j=3$, $i=1$ for $j=2$, $u_{e_{i}}$ denotes the spinor of
lepton, $\nu_{e_{i}}$ denotes the spinor of antilepton,
$P_{L}=(1-\gamma_{5})/2$, $P_{R}=(1+\gamma_{5})/2$, and $p_{k}$ denotes the
momentum of charged lepton with $k=1,2,3,4$. The coefficients
$C_{1,2,3}^{L,R}$ from the contributions of Higgs bosons and $Z,\;Z^{\prime}$
gauge bosons, can be obtained through the Yukawa couplings in Eq. (29) and the
definition of covariant derivative in Eq. (43). Then we can calculate the
decay rate Hisano:1995cp

$\displaystyle\Gamma(e_{j}\rightarrow
e_{i}e_{i}\bar{e}_{i})=\frac{m_{e_{j}}^{5}}{1536\pi^{3}}\Big{[}\frac{1}{2}(|C_{1}^{L}|^{2}+|C_{1}^{R}|^{2})+|C_{2}^{L}|^{2}+|C_{2}^{R}|^{2}+\frac{1}{8}(|C_{3}^{L}|^{2}+|C_{3}^{R}|^{2})\Big{]}.$
(62)

The total decay widthes of $\mu,\;\tau$ are taken as $\Gamma^{\mu}_{{\rm
total}}=2.996\times 10^{-19}\;$GeV, $\Gamma^{\tau}_{{\rm total}}=2.265\times
10^{-12}\;$GeV ParticleDataGroup:2022pth .

Scanning the free parameter space in Eq. (57), Eq. (59) and

$\displaystyle g_{F}=(0,\;0.8),\;g_{F}=(-0.8,\;0.8),$ (63)

we plot the results of ${\rm Br}(\tau\to 3e)$ versus $|m_{e,13}|$, ${\rm
Br}(\tau\to 3\mu)$ versus $\theta_{e,23}$, ${\rm Br}(\mu\to 3e)$ versus
$\theta_{e,12}$ in Fig. 7 (a), (b), (c) respectively by keeping $M_{H_{2}}$ in
the range $124\;{\rm GeV}<M_{H_{2}}<126\;{\rm GeV}$ and ${\rm Br}(\tau\to
3e)<10^{-7}$, ${\rm Br}(\tau\to 3\mu)<10^{-7}$, ${\rm Br}(\mu\to 3e)<2\times
10^{-12}$, then the allowed ranges of $v_{\chi}-v_{1}$, $g_{F}-g_{YF}$,
$M_{Z^{\prime}}-g_{F}$ are plotted in Fig. 7 (d), (e), (f) respectively. The
gray points in Fig. 7 are excluded by the present limits, and the green points
denote the results which can reach future experimental sensitivities, where
the present limits and future sensitivities for the branching ratios of these
LFV processes are listed in Tab. 4.

Figure 7: Scanning the parameter space Eq. (57), Eq. (59), Eq. (63) and keeping $M_{H_{2}}$ in the range $124\;{\rm GeV}<M_{H_{2}}<126\;{\rm GeV}$ and ${\rm Br}(\tau\to 3e)<10^{-7}$, ${\rm Br}(\tau\to 3\mu)<10^{-7}$, ${\rm Br}(\mu\to 3e)<2\times 10^{-12}$, the results of ${\rm Br}(\tau\to 3e)$ versus $|m_{e,13}|$ (a), ${\rm Br}(\tau\to 3\mu)$ versus $\theta_{e,23}$ (b), ${\rm Br}(\mu\to 3e)$ versus $\theta_{e,12}$ (c) and the allowed ranges of $v_{\chi}-v_{1}$ (d), $g_{F}-g_{YF}$ (e), $M_{Z^{\prime}}-g_{F}$ (f) are plotted, where the gray points are excluded by the present limits, the green points denote the results which can reach future experimental sensitivities. Branching ratios | Present limit | Future sensitivity
---|---|---
${\rm Br}(\tau\to 3e)$ | $<2.7\times 10^{-8}$ | $\sim 10^{-10}$
${\rm Br}(\tau\to 3\mu)$ | $<2.1\times 10^{-8}$ | $\sim 10^{-10}$
${\rm Br}(\mu\to 3e)$ | $<10^{-12}$ | $\sim 10^{-16}$
Table 4: Present limits and future sensitivities for the branching ratios for
the LFV processes $\tau\to 3e$, $\tau\to 3\mu$ and $\mu\to 3e$
ParticleDataGroup:2022pth ; Blondel:2013ia ; Hayasaka:2013dsa .

Fig. 7 (a), (b), (c) show that the experimental upper bounds on ${\rm
Br}(\tau\to 3\mu)$, ${\rm Br}(\mu\to 3e)$ limit the parameter space strictly
while the predicted ${\rm Br}(\tau\to 3e)$ is less than about $10^{-12}$ which
is hard to be observed in near future. In addition, the model predicts ${\rm
Br}(\tau\to 3\mu)\gtrsim 2\times 10^{-10}$, ${\rm Br}(\mu\to 3e)\gtrsim
5\times 10^{-14}$, which indicates observing the LFV processes $\tau\to 3\mu$
and $\mu\to 3e$ is also effective to test the FDM. It is obvious in Fig. 7 (d)
that the experimental upper bounds on the branching ratios of these LFV
processes limit $v_{1}\gtrsim 23\;{\rm GeV}$, $v_{\chi}\gtrsim 20\;{\rm TeV}$.
Fig. 7 (e) indicates experimental constraints prefer $-0.2\lesssim
g_{YF}\lesssim 0.2$ and $g_{YF}\lesssim-0.2$ is excluded completely. From Fig.
7 (f), it can be seen explicitly that the allowed range of $M_{Z^{\prime}}$ is
related closely with the chosen value of $g_{F}$ and $M_{Z^{\prime}}\gtrsim
5\;{\rm TeV}$.

##  V Summary

Motivated by the hierarchical structure of fermionic masses puzzle and
fermionic flavor mixings puzzle, we propose a flavor-dependent model (FDM) to
relate these two puzzles, i.e. the proposed FDM can explain the flavor mixings
puzzle and mass hierarchy puzzle simultaneously. The model extends the SM by
an extra $U(1)_{F}$ local gauge group, two scalar doublets, one scalar singlet
and two right-handed neutrinos, where the new $U(1)_{F}$ charges are related
to the particles’ flavor. In the FDM, only the third generation of quarks and
charged leptons achieve the masses at the tree level, the first two
generations achieve masses through the mixings with the third generation, and
the neutrinos obtain tiny Majorana masses through the so-called Type I see-saw
mechanism. In addition, the $B$ meson rare decay processes $\bar{B}\to
X_{s}\gamma$, $B_{s}^{0}\to\mu^{+}\mu^{-}$, the top quark rare decay processes
$t\to ch$, $t\to uh$ and the $\tau$ LFV processes $\tau\to 3e$, $\tau\to
3\mu$, $\mu\to 3e$ predicted in the FDM are analyzed. It is found that
observing the top quark rare decay processes $t\to ch$, $t\to uh$ and the
$\tau$ LFV decays $\tau\to 3\mu$, $\mu\to 3e$ is effective to test the FDM,
while $\tau$ LFV decay $\tau\to 3e$ is hard to be observed experimentally. In
addition, the model can fit the observed quark masses, CKM matrix, ${\rm
Br}(\bar{B}\to X_{s}\gamma)$, ${\rm Br}(B_{s}^{0}\to\mu^{+}\mu^{-})$ well, and
the VEVs of the two extra scalar doublets are limited to be larger than about
$23\;{\rm GeV}$, new $Z^{\prime}$ gauge boson is heavier than about $5\;{\rm
TeV}$ and gauge kinetic mixing constant $g_{YF}$ is lager than $-0.2$ by
considering the experimental upper bounds on the branching ratios of LFV
decays $\tau\to 3\mu$, $\mu\to 3e$.

##  Appendix A Contributions to $\bar{B}\to X_{s}\gamma$ and
$B_{s}^{0}\to\mu^{+}\mu^{-}$ in the FDM.

Generally, the effective Hamilton for the transition $b\rightarrow s$ at
hadronic scale can be written as

$\displaystyle
H_{eff}=-\frac{4G_{F}}{\sqrt{2}}V_{ts}^{\ast}V_{tb}\Big{[}C_{1}\mathcal{O}^{c}_{1}+C_{2}\mathcal{O}_{2}^{c}+\sum_{i=3}^{6}\mathcal{O}_{i}+\sum_{i=7}^{10}(C_{i}\mathcal{O}_{i}+C^{\prime}_{i}\mathcal{O}^{\prime}_{i})$
$\displaystyle\qquad\;\quad\;+\sum_{i=S,P}(C_{i}\mathcal{O}_{i}+C^{\prime}_{i}\mathcal{O}^{\prime}_{i})\Big{]},$
(64)

where O1 ; O2 ; Altmannshofer:2008dz ; O3 ; O4 ; O6

$\displaystyle{\cal
O}_{{}_{1}}^{u}=(\bar{s}_{{}_{L}}\gamma_{\mu}T^{a}u_{{}_{L}})(\bar{u}_{{}_{L}}\gamma^{\mu}T^{a}b_{{}_{L}})\;,\;\;{\cal
O}_{{}_{2}}^{u}=(\bar{s}_{{}_{L}}\gamma_{\mu}u_{{}_{L}})(\bar{u}_{{}_{L}}\gamma^{\mu}b_{{}_{L}})\;,$
$\displaystyle{\cal
O}_{{}_{3}}=(\bar{s}_{{}_{L}}\gamma_{\mu}b_{{}_{L}})\sum\limits_{q}(\bar{q}\gamma^{\mu}q)\;,\;\;{\cal
O}_{{}_{4}}=(\bar{s}_{{}_{L}}\gamma_{\mu}T^{a}b_{{}_{L}})\sum\limits_{q}(\bar{q}\gamma^{\mu}T^{a}q)\;,$
$\displaystyle{\cal
O}_{{}_{5}}=(\bar{s}_{{}_{L}}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}b_{{}_{L}})\sum\limits_{q}(\bar{q}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}q)\;,\;\;{\cal
O}_{{}_{6}}=(\bar{s}_{{}_{L}}\gamma_{\mu}\gamma_{\nu}\gamma_{\rho}T^{a}b_{{}_{L}})\sum\limits_{q}(\bar{q}\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}T^{a}q)\;,$
$\displaystyle{\cal O}_{{}_{7}}={e\over
16\pi^{2}}m_{{}_{b}}(\bar{s}_{{}_{L}}\sigma_{{}_{\mu\nu}}b_{{}_{R}})F^{\mu\nu}\;,\;\;{\cal
O}_{{}_{7}}^{\prime}={e\over
16\pi^{2}}m_{{}_{b}}(\bar{s}_{{}_{R}}\sigma_{{}_{\mu\nu}}b_{{}_{L}})F^{\mu\nu}\;,\;\;$
$\displaystyle{\cal O}_{{}_{8}}={g_{{}_{s}}\over
16\pi^{2}}m_{{}_{b}}(\bar{s}_{{}_{L}}\sigma_{{}_{\mu\nu}}T^{a}b_{{}_{R}})G^{a,\mu\nu}\;,\;\;{\cal
O}_{{}_{8}}^{\prime}={g_{{}_{s}}\over
16\pi^{2}}m_{{}_{b}}(\bar{s}_{{}_{R}}\sigma_{{}_{\mu\nu}}T^{a}b_{{}_{L}})G^{a,\mu\nu}\;,\;\;$
$\displaystyle{\cal O}_{{}_{9}}={e^{2}\over
g_{{}_{s}}^{2}}(\bar{s}_{{}_{L}}\gamma_{\mu}b_{{}_{L}})\bar{l}\gamma^{\mu}l\;,\;\;{\cal
O}_{{}_{9}}^{\prime}={e^{2}\over
g_{{}_{s}}^{2}}(\bar{s}_{{}_{R}}\gamma_{\mu}b_{{}_{R}})\bar{l}\gamma^{\mu}l\;,\;\;$
$\displaystyle{\cal O}_{{}_{10}}={e^{2}\over
g_{{}_{s}}^{2}}(\bar{s}_{{}_{L}}\gamma_{\mu}b_{{}_{L}})\bar{l}\gamma^{\mu}\gamma_{5}l\;,\;\;{\cal
O}_{{}_{10}}^{\prime}={e^{2}\over
g_{{}_{s}}^{2}}(\bar{s}_{{}_{R}}\gamma_{\mu}b_{{}_{R}})\bar{l}\gamma^{\mu}\gamma_{5}l\;,\;\;$
$\displaystyle{\cal O}_{{}_{S}}={e^{2}\over
16\pi^{2}}m_{{}_{b}}(\bar{s}_{{}_{L}}b_{{}_{R}})\bar{l}l\;,\;\;{\cal
O}_{{}_{S}}^{\prime}={e^{2}\over
16\pi^{2}}m_{{}_{b}}(\bar{s}_{{}_{R}}b_{{}_{L}})\bar{l}l\;,\;\;$
$\displaystyle{\cal O}_{{}_{P}}={e^{2}\over
16\pi^{2}}m_{{}_{b}}(\bar{s}_{{}_{L}}b_{{}_{R}})\bar{l}\gamma_{5}l\;,\;\;{\cal
O}_{{}_{P}}^{\prime}={e^{2}\over
16\pi^{2}}m_{{}_{b}}(\bar{s}_{{}_{R}}b_{{}_{L}})\bar{l}\gamma_{5}l.$ (65)

In the definitions above, $g_{s}$ is the strong coupling constant,
$T^{a}\,(a=1,...,8)$ are $SU(3)$ generators, $F^{\mu\nu}$ and $G^{\mu\nu}$ are
the electromagnetic and gluon field strength tensors respectively.

Figure 8: The one loop Feynman diagrams contributing to $\bar{B}\rightarrow
X_{s}\gamma$ from charged Higgs in the FDM.

The dominant contributions to $b\rightarrow s\gamma$ come from the charged
Higgs in the FDM, and the leading-order Feynman diagrams are plotted in Fig.
8. Then the branching ratio of $\bar{B}\rightarrow X_{s}\gamma$ can be written
as

$\displaystyle Br(\bar{B}\rightarrow
X_{s}\gamma)=R\Big{(}|C_{7\gamma}(\mu_{b})|^{2}+N(E_{\gamma})\Big{)}\;,$ (66)

where the overall factor $R=2.47\times 10^{-3}$, and the nonperturbative
contribution $N(E_{\gamma})=(3.6\pm 0.6)\times 10^{-3}$H5 .
$C_{7\gamma}(\mu_{b})$ can be written as

$\displaystyle
C_{7\gamma}(\mu_{b})=C_{7\gamma,SM}(\mu_{b})+C_{7,NP}(\mu_{b}),$ (67)

where the hadron scale $\mu_{b}=2.5$ GeV and $C_{7\gamma,SM}(\mu_{b})=-0.3689$
for the SM contribution at NNLO level H5 ; H6 ; H7 ; H8 . In new physics
models, the corresponding Wilson coefficients at the bottom quark scale are H9
; H10

$\displaystyle C_{7,NP}(\mu_{b})\approx
0.5696C_{7,NP}(\mu_{EW})+0.1107C_{8,NP}(\mu_{EW}),$ (68)

where

$\displaystyle
C_{7,NP}^{NP}(\mu_{EW})=C_{7,NP}^{(1)}(\mu_{EW})+C_{7,NP}^{(2)}(\mu_{EW})+C_{7,NP}^{\prime(1)}(\mu_{EW})+C_{7,NP}^{\prime(2)}(\mu_{EW}),$
$\displaystyle
C_{8,NP}(\mu_{EW})=C_{8g,NP}(\mu_{EW})+C_{8g,NP}^{\prime}(\mu_{EW}),$ (69)

The coefficients $C_{7,NP}^{(1,2)}(\mu_{EW})$ are Wilson coefficients of the
process $b\rightarrow s\gamma$ and can be calculated from the diagrams in Fig.
8 (1), (2) respectively, the results read

$\displaystyle
C_{7,NP}^{(1)}(\mu_{EW})=\sum_{H^{-}_{i},u_{j}}\frac{s_{W}^{2}}{2e^{2}V^{*}_{ts}V_{tb}}\Big{\\{}\frac{1}{2}C_{H^{-}_{i}\bar{s}u_{j}}^{R}C_{H^{-}_{i}b\bar{u}_{j}}^{L}[-I_{3}(x_{u_{j}},x_{H^{-}_{i}})+I_{4}(x_{u_{j}},x_{H^{-}_{i}})]+$
$\displaystyle\qquad\qquad\qquad\quad\frac{m_{u_{j}}}{m_{b}}C_{H^{-}_{i}\bar{s}u_{j}}^{L}C_{H^{-}_{i}b\bar{u}_{j}}^{L}[-I_{1}(x_{u_{j}},x_{H^{-}_{i}})+I_{3}(x_{u_{j}},x_{H^{-}_{i}})]\Big{\\}},$
$\displaystyle
C_{7,NP}^{(2)}(\mu_{EW})=\sum_{H^{-}_{j},u_{i}}\frac{s_{W}^{2}}{3e^{2}V^{*}_{ts}V_{tb}}\Big{\\{}\frac{1}{2}C_{H^{-}_{j}\bar{s}u_{i}}^{R}C_{H^{-}_{j}b\bar{u}_{i}}^{L}[-I_{1}(x_{u_{i}},x_{H^{-}_{j}})+2I_{3}(x_{u_{i}},x_{H^{-}_{j}})$
$\displaystyle\qquad\qquad\qquad\quad-
I_{4}(x_{u_{i}},x_{H^{-}_{j}})]+\frac{m_{u_{i}}}{m_{b}}C_{H^{-}_{j}\bar{s}u_{i}}^{L}C_{H^{-}_{j}b\bar{u}_{i}}^{L}[I_{1}(x_{u_{i}},x_{H^{-}_{j}})-I_{2}(x_{u_{i}},x_{H^{-}_{j}})$
$\displaystyle\qquad\qquad\qquad\quad-
I_{3}(x_{u_{i}},x_{H^{-}_{j}})]\Big{\\}},$ $\displaystyle C_{7}^{\prime
NP(a)}(\mu_{EW})=C_{7}^{\prime NP(a)}(\mu_{EW})(L\leftrightarrow R),(a=1,2),$
(70)

where $x_{i}=\frac{m_{i}^{2}}{m_{W}^{2}}$, $C_{abc}^{L,R}$ denotes the scalar
parts of the interaction vertex about $abc$ with $a,b,c$ denoting the
interactional particles, and the loop integral functions $I_{1,...,4}$ can be
found in our previous work Yang:2018fvw . In addition, $C_{8g,NP}(\mu_{EW})$
and $C_{8g,NP}^{\prime}(\mu_{EW})$ at electroweak scale are

$\displaystyle
C_{8g,NP}(\mu_{EW})=[C_{7,NP}^{(2)}(\mu_{EW})+C_{7,NP}^{(3)}(\mu_{EW})]/Q_{u},$
$\displaystyle
C_{8g,NP}^{\prime}(\mu_{EW})=C_{8g,NP}(\mu_{EW})(L\leftrightarrow R),$ (71)

where $Q_{u}=2/3$.

The main Feynman diagrams contributing to $B_{s}^{0}\to\mu^{+}\mu^{-}$ are
plotted in Fig. 9.

Figure 9: The Feynman diagrams contributing to the decay
$B_{s}^{0}\rightarrow\mu^{+}\mu^{-}$ in the B-LSSM

At the electroweak energy scale $\mu_{EW}$, the corresponding Wilson
coefficients can be written as

$\displaystyle C_{{}_{S,NP}}(\mu_{{}_{\rm
EW}})=\frac{\sqrt{2}s_{{}_{W}}c_{{}_{W}}}{4m_{b}e^{3}V_{ts}^{*}V_{tb}}\Big{[}C_{{}_{S,NP}}^{(1)}(\mu_{{}_{\rm
EW}})+C_{{}_{S,NP}}^{(2)}(\mu_{{}_{\rm EW}})+C_{{}_{S,NP}}^{(3)}(\mu_{{}_{\rm
EW}})+C_{{}_{S,NP}}^{(4)}(\mu_{{}_{\rm EW}})$
$\displaystyle\qquad\;\qquad\;\qquad+C_{{}_{S,NP}}^{(6)}(\mu_{{}_{\rm
EW}})\Big{]},$ $\displaystyle C_{{}_{S,NP}}^{\prime}(\mu_{{}_{\rm
EW}})=C_{{}_{S,NP}}(\mu_{{}_{\rm EW}})(L\leftrightarrow R),$ $\displaystyle
C_{{}_{P,NP}}(\mu_{{}_{\rm
EW}})=\frac{\sqrt{2}s_{{}_{W}}c_{{}_{W}}}{4m_{b}e^{3}V_{ts}^{*}V_{tb}}\Big{[}C_{{}_{P,NP}}^{(1)}(\mu_{{}_{\rm
EW}})+C_{{}_{P,NP}}^{(2)}(\mu_{{}_{\rm EW}})+C_{{}_{P,NP}}^{(3)}(\mu_{{}_{\rm
EW}})+C_{{}_{P,NP}}^{(4)}(\mu_{{}_{\rm EW}})$
$\displaystyle\qquad\;\qquad\;\qquad+C_{{}_{P,NP}}^{(6)}(\mu_{{}_{\rm
EW}})\Big{]},$ $\displaystyle C_{{}_{P,NP}}^{\prime}(\mu_{{}_{\rm
EW}})=-C_{{}_{P,NP}}(\mu_{{}_{\rm EW}})(L\leftrightarrow R),$ $\displaystyle
C_{{}_{9,NP}}(\mu_{{}_{\rm
EW}})=\frac{\sqrt{2}s_{{}_{W}}c_{{}_{W}}g_{{}_{s}}^{2}}{64\pi^{2}e^{3}V_{ts}^{*}V_{tb}}\Big{[}C_{{}_{9,NP}}^{(5)}(\mu_{{}_{\rm
EW}})+C_{{}_{9,NP}}^{(6)}(\mu_{{}_{\rm EW}})+C_{{}_{9,NP}}^{(7)}(\mu_{{}_{\rm
EW}})+C_{{}_{9,NP}}^{(8)}(\mu_{{}_{\rm EW}})\Big{]}\;,$ $\displaystyle
C_{{}_{9,NP}}^{\prime}(\mu_{{}_{\rm EW}})=C_{{}_{9,NP}}(\mu_{{}_{\rm
EW}})(L\leftrightarrow R),$ $\displaystyle C_{{}_{10,NP}}(\mu_{{}_{\rm
EW}})=\frac{\sqrt{2}s_{{}_{W}}c_{{}_{W}}g_{{}_{s}}^{2}}{64\pi^{2}e^{3}V_{ts}^{*}V_{tb}}\Big{[}C_{{}_{10,NP}}^{(5)}(\mu_{{}_{\rm
EW}})+C_{{}_{10,NP}}^{(6)}(\mu_{{}_{\rm
EW}})+C_{{}_{10,NP}}^{(7)}(\mu_{{}_{\rm
EW}})+C_{{}_{10,NP}}^{(8)}(\mu_{{}_{\rm EW}})\Big{]}\;,$ $\displaystyle
C_{{}_{10,NP}}^{\prime}(\mu_{{}_{\rm EW}})=-C_{{}_{10,NP}}(\mu_{{}_{\rm
EW}})(L\leftrightarrow R).$ (72)

The superscripts $(1,...,8)$ corresponding to the contributions in Fig. 9
(1,…,8) respectively and the results can be written as

$\displaystyle C_{{}_{S,NP}}^{(1)}(\mu_{{}_{\rm
EW}})=\sum_{H^{-}_{i},u_{j},u_{k}}^{S=H_{l},A_{l}}\frac{C_{\bar{\mu}S\mu}^{L}+C_{\bar{\mu}S\mu}^{R}}{2(m_{b}^{2}-m_{S}^{2})}\Big{[}C_{H^{-}_{i}\bar{s}u_{j}}^{R}C_{\bar{u}_{j}Su_{k}}^{L}C_{\bar{u}_{k}bH^{-}_{i}}^{R}G_{2}(x_{\tilde{H}^{\pm}_{i}},x_{u_{j}},x_{u_{k}})$
$\displaystyle\qquad\qquad\qquad+m_{u_{j}}m_{u_{k}}C_{H^{-}_{i}\bar{s}u_{j}}^{R}C_{\bar{u}_{j}Su_{k}}^{R}C_{\bar{u}_{k}bH^{-}_{i}}^{R}G_{1}(x_{\tilde{H}^{\pm}_{i}},x_{u_{j}},x_{u_{k}})\Big{]},$
$\displaystyle C_{{}_{P,NP}}^{(1)}(\mu_{{}_{\rm
EW}})=\sum_{H^{-}_{i},u_{j},u_{k}}^{S=H_{l},A_{l}}\frac{-C_{\bar{\mu}S\mu}^{L}+C_{\bar{\mu}S\mu}^{R}}{2(m_{b}^{2}-m_{S}^{2})}\Big{[}C_{H^{-}_{i}\bar{s}u_{j}}^{R}C_{\bar{u}_{j}Su_{k}}^{L}C_{\bar{u}_{k}bH^{-}_{i}}^{R}G_{2}(x_{\tilde{H}^{\pm}_{i}},x_{u_{j}},x_{u_{k}})$
$\displaystyle\qquad\qquad\qquad+m_{u_{j}}m_{u_{k}}C_{H^{-}_{i}\bar{s}u_{j}}^{R}C_{\bar{u}_{j}Su_{k}}^{R}C_{\bar{u}_{k}bH^{-}_{i}}^{R}G_{1}(x_{\tilde{H}^{\pm}_{i}},x_{u_{j}},x_{u_{k}})\Big{]},$
(73) $\displaystyle C_{{}_{S,NP}}^{(2)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{j},H^{\pm}_{k}}^{S=H_{l},A_{l}}\frac{1}{2(m_{b}^{2}-m_{S}^{2})}m_{u_{i}}C_{\bar{s}u_{i}H^{\pm}_{j}}^{R}C_{\bar{u}_{i}bH^{\pm}_{k}}^{R}C_{SH^{\pm}_{j}H^{\pm}_{k}}G_{1}(x_{u_{i}},x_{H^{\pm}_{j}},x_{H^{\pm}_{k}})$
$\displaystyle\qquad\qquad\qquad\times(C_{\bar{\mu}S\mu}^{L}+C_{\bar{\mu}S\mu}^{R}),$
$\displaystyle C_{{}_{p,NP}}^{(2)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{j},H^{\pm}_{k}}^{S=H_{l},A_{l}}\frac{1}{2(m_{b}^{2}-m_{S}^{2})}m_{u_{i}}C_{\bar{s}u_{i}H^{\pm}_{j}}^{R}C_{\bar{u}_{i}bH^{\pm}_{k}}^{R}C_{SH^{\pm}_{j}H^{\pm}_{k}}G_{1}(x_{u_{i}},x_{H^{\pm}_{j}},x_{H^{\pm}_{k}})$
$\displaystyle\qquad\qquad\qquad\times(-C_{\bar{\mu}S\mu}^{L}+C_{\bar{\mu}S\mu}^{R}),$
(74) $\displaystyle C_{{}_{S,NP}}^{(3)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{k}}^{S=H_{l},A_{l}}\frac{-C_{W^{\pm}SH^{\pm}_{k}}}{2(m_{b}^{2}-m_{S}^{2})}\Big{[}C_{\bar{s}W^{\pm}u_{i}}^{L}C_{\bar{u}_{i}H^{\pm}_{k}b}^{R}G_{2}(x_{u_{i}},1,x_{H^{\pm}_{k}})-2m_{b}m_{u_{i}}C_{\bar{s}W^{\pm}u_{i}}^{L}$
$\displaystyle\qquad\qquad\qquad\times
C_{\bar{u}_{i}H^{\pm}_{k}b}^{L}G_{1}(x_{u_{i}},1,x_{H^{\pm}_{k}})\Big{]}(C_{\bar{\mu}S\mu}^{L}+C_{\bar{\mu}S\mu}^{R}),$
$\displaystyle C_{{}_{P,NP}}^{(3)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{k}}^{S=H_{l},A_{l}}\frac{-C_{W^{\pm}SH^{\pm}_{k}}}{2(m_{b}^{2}-m_{S}^{2})}\Big{[}C_{\bar{s}W^{\pm}u_{i}}^{L}C_{\bar{u}_{i}H^{\pm}_{k}b}^{R}G_{2}(x_{u_{i}},1,x_{H^{\pm}_{k}})-2m_{b}m_{u_{i}}C_{\bar{s}W^{\pm}u_{i}}^{L}$
$\displaystyle\qquad\qquad\qquad\times
C_{\bar{u}_{i}H^{\pm}_{k}b}^{L}G_{1}(x_{u_{i}},1,x_{H^{\pm}_{k}})\Big{]}(-C_{\bar{\mu}S\mu}^{L}+C_{\bar{\mu}S\mu}^{R}),$
(75) $\displaystyle C_{{}_{S,NP}}^{(4)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{j}}^{S=H_{l},A_{l}}\frac{-C_{W^{\pm}SH^{\pm}_{j}}}{2(m_{b}^{2}-m_{S}^{2})}C_{\bar{s}H^{\pm}_{j}u_{i}}^{R}C_{\bar{u}_{i}W^{\pm}b}^{R}G_{2}(x_{u_{i}},x_{H^{\pm}_{j}},1)(C_{\bar{\mu}S\mu}^{L}+C_{\bar{\mu}S\mu}^{R}),$
$\displaystyle C_{{}_{S,NP}}^{(4)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{j}}^{S=H_{l},A_{l}}\frac{-C_{W^{\pm}SH^{\pm}_{j}}}{2(m_{b}^{2}-m_{S}^{2})}C_{\bar{s}H^{\pm}_{j}u_{i}}^{R}C_{\bar{u}_{i}W^{\pm}b}^{R}G_{2}(x_{u_{i}},x_{H^{\pm}_{j}},1)(-C_{\bar{\mu}S\mu}^{L}+C_{\bar{\mu}S\mu}^{R}),$
$\displaystyle C_{{}_{9,NP}}^{(5)}(\mu_{{}_{\rm
EW}})=\sum_{\tilde{H}^{\pm}_{i},u_{j},u_{k}}^{V}\frac{C_{\bar{\mu}V\mu}^{L}+C_{\bar{\mu}V\mu}^{R}}{-2(m_{b}^{2}-m_{V}^{2})}\Big{[}-\frac{1}{2}C_{H^{\pm}_{i}\bar{s}u_{j}}^{R}C_{\bar{u}_{j}Vu_{k}}^{R}C_{u_{k}sH^{\pm}_{i}}^{L}G_{2}(x_{H^{\pm}_{i}},x_{u_{j}},x_{u_{k}})$
$\displaystyle\qquad\qquad\qquad+m_{u_{j}}m_{u_{k}}C_{H^{\pm}_{i}\bar{s}u_{j}}^{R}C_{\bar{u}_{j}Vu_{k}}^{L}C_{\bar{u}_{k}sH^{\pm}_{i}}^{L}G_{1}(x_{H^{\pm}_{i}},x_{u_{j}},x_{u_{k}})\Big{]},$
$\displaystyle C_{{}_{10,NP}}^{(5)}(\mu_{{}_{\rm
EW}})=\sum_{\tilde{H}^{\pm}_{i},u_{j},u_{k}}^{V}\frac{-C_{\bar{\mu}V\mu}^{L}+C_{\bar{\mu}V\mu}^{R}}{-2(m_{b}^{2}-m_{V}^{2})}\Big{[}-\frac{1}{2}C_{H^{\pm}_{i}\bar{s}u_{j}}^{R}C_{\bar{u}_{j}Vu_{k}}^{R}C_{u_{k}sH^{\pm}_{i}}^{L}G_{2}(x_{H^{\pm}_{i}},x_{u_{j}},x_{u_{k}})$
$\displaystyle\qquad\qquad\qquad+m_{u_{j}}m_{u_{k}}C_{H^{\pm}_{i}\bar{s}u_{j}}^{R}C_{\bar{u}_{j}Vu_{k}}^{L}C_{\bar{u}_{k}sH^{\pm}_{i}}^{L}G_{1}(x_{H^{\pm}_{i}},x_{u_{j}},x_{u_{k}})\Big{]},$
(77) $\displaystyle C_{{}_{9,NP}}^{(6)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{j},H^{\pm}_{k}}^{V}\frac{C_{\bar{\mu}V\mu}^{L}+C_{\bar{\mu}V\mu}^{R}}{4(m_{b}^{2}-m_{V}^{2})}C_{\bar{s}u_{i}H^{\pm}_{j}}^{R}C_{\bar{u}_{i}bH^{\pm}_{k}}^{L}C_{VH^{\pm}_{j}H^{\pm}_{k}}G_{2}(x_{u_{i}},x_{H^{\pm}_{j}},x_{H^{\pm}_{k}}),$
$\displaystyle C_{{}_{10,NP}}^{(6)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{j},H^{\pm}_{k}}^{V}\frac{-C_{\bar{\mu}V\mu}^{L}+C_{\bar{\mu}V\mu}^{R}}{4(m_{b}^{2}-m_{V}^{2})}C_{\bar{s}u_{i}H^{\pm}_{j}}^{R}C_{\bar{u}_{i}bH^{\pm}_{k}}^{L}C_{VH^{\pm}_{j}H^{\pm}_{k}}G_{2}(x_{u_{i}},x_{H^{\pm}_{j}},x_{H^{\pm}_{k}}),$
$\displaystyle C_{{}_{S,NP}}^{(6)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{j},H^{\pm}_{k}}^{V}\frac{C_{\bar{\mu}V\mu}^{L}+C_{\bar{\mu}V\mu}^{R}}{-2(m_{b}^{2}-m_{V}^{2})}m_{b}m_{u_{i}}C_{\bar{s}u_{i}H^{\pm}_{j}}^{R}C_{\bar{u}_{i}bH^{\pm}_{k}}^{R}C_{VH^{\pm}_{j}H^{\pm}_{k}}G_{1}(x_{u_{i}},x_{H^{\pm}_{j}},x_{H^{\pm}_{k}}),$
$\displaystyle C_{{}_{P,NP}}^{(6)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{j},H^{\pm}_{k}}^{V}\frac{C_{\bar{\mu}V\mu}^{L}-C_{\bar{\mu}V\mu}^{R}}{-2(m_{b}^{2}-m_{V}^{2})}m_{b}m_{u_{i}}C_{\bar{s}u_{i}H^{\pm}_{j}}^{R}C_{\bar{u}_{i}bH^{\pm}_{k}}^{R}C_{VH^{\pm}_{j}H^{\pm}_{k}}G_{1}(x_{u_{i}},x_{H^{\pm}_{j}},x_{H^{\pm}_{k}}),$
$\displaystyle C_{{}_{9,NP}}^{(7)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{k}}^{V}\frac{C_{\bar{\mu}V\mu}^{L}+C_{\bar{\mu}V\mu}^{R}}{2(m_{b}^{2}-m_{V}^{2})}m_{u_{i}}C_{\bar{s}u_{i}W^{\pm}}^{L}C_{\bar{u}_{i}bH^{\pm}_{k}}^{L}C_{VW^{\pm}H^{\pm}_{k}}G_{1}(x_{u_{i}},x_{W},x_{H^{\pm}_{k}}),$
$\displaystyle C_{{}_{10,NP}}^{(7)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{k}}^{V}\frac{-C_{\bar{\mu}V\mu}^{L}+C_{\bar{\mu}V\mu}^{R}}{2(m_{b}^{2}-m_{V}^{2})}m_{u_{i}}C_{\bar{s}u_{i}W^{\pm}}^{L}C_{\bar{u}_{i}bH^{\pm}_{k}}^{L}C_{VW^{\pm}H^{\pm}_{k}}G_{1}(x_{u_{i}},x_{W},x_{H^{\pm}_{k}}),$
$\displaystyle C_{{}_{9,NP}}^{(8)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{j}}^{V}\frac{C_{\bar{\mu}V\mu}^{L}+C_{\bar{\mu}V\mu}^{R}}{2(m_{b}^{2}-m_{V}^{2})}m_{u_{i}}C_{\bar{s}u_{i}H^{\pm}_{j}}^{R}C_{\bar{u}_{i}bW^{\pm}}^{L}C_{VW^{\pm}H^{\pm}_{k}}G_{1}(x_{u_{i}},x_{H^{\pm}_{j}},x_{W}),$
$\displaystyle C_{{}_{10,NP}}^{(8)}(\mu_{{}_{\rm
EW}})=\sum_{u_{i},H^{\pm}_{j}}^{V}\frac{-C_{\bar{\mu}V\mu}^{L}+C_{\bar{\mu}V\mu}^{R}}{2(m_{b}^{2}-m_{V}^{2})}m_{u_{i}}C_{\bar{s}u_{i}H^{\pm}_{j}}^{R}C_{\bar{u}_{i}bW^{\pm}}^{L}C_{VW^{\pm}H^{\pm}_{k}}G_{1}(x_{u_{i}},x_{H^{\pm}_{j}},x_{W}),$

where $V$ denotes the vector bosons $\gamma$, $Z$, $Z^{\prime}$, and $C_{aVb}$
denote the scalar parts of the corresponding interaction vertex $aVb$. The
concrete expressions for loop integral $G_{k}(k=1,...,4)$ can be found in the
Appendix B of our previous work Yang:2018fvw . The Wilson coefficients at
hadronic energy scale from the SM to next-to-next-to-logarithmic accuracy are
shown in Table I.

$C_{{}_{7}}^{eff,SM}$ | $C_{{}_{8}}^{eff,SM}$ | $C_{{}_{9}}^{eff,SM}$ | $C_{{}_{10}}^{eff,SM}$
---|---|---|---
$-0.304$ | $-0.167$ | $4.211$ | $-4.103$
Table 5: At hadronic scale $\mu=m_{{}_{b}}$, SM Wilson coefficients to next-
to-next-to-logarithmic accuracy.

In addition, the Wilson coefficients in Eq. (72) should be evolved down to
hadronic scale $\mu\sim m_{b}$ by the renormalization group equations:

$\displaystyle\overrightarrow{C}_{{}_{NP}}(\mu)=\widehat{U}(\mu,\mu_{0})\overrightarrow{C}_{{}_{NP}}(\mu_{0})\;,$
$\displaystyle\overrightarrow{C^{\prime}}_{{}_{NP}}(\mu)=\widehat{U^{\prime}}(\mu,\mu_{0})\overrightarrow{C^{\prime}}_{{}_{NP}}(\mu_{0})$
(81)

with

$\displaystyle\overrightarrow{C}_{{}_{NP}}^{T}=\Big{(}C_{{}_{1,NP}},\;\cdots,\;C_{{}_{6,NP}},C_{{}_{7,NP}}^{eff},\;C_{{}_{8,NP}}^{eff},\;C_{{}_{9,NP}}^{eff}-Y(q^{2}),\;C_{{}_{10,NP}}^{eff}\Big{)}\;,$
$\displaystyle\overrightarrow{C}_{{}_{NP}}^{\prime,\;T}=\Big{(}C_{{}_{7,NP}}^{\prime,\;eff},\;C_{{}_{8,NP}}^{\prime,\;eff},\;C_{{}_{9,NP}}^{\prime,\;eff},\;C_{{}_{10,NP}}^{\prime,\;eff}\Big{)}\;.$
(82)

Correspondingly, the evolving matrices
$\widehat{U}(\mu,\mu_{0}),\;\widehat{U^{\prime}}(\mu,\mu_{0})$ can be found in
our previous work Yang:2018fvw .

Then, the squared amplitude can be written as

$\displaystyle|\mathcal{M}_{s}|^{2}=16G_{F}^{2}|V_{tb}V_{ts}^{*}|^{2}M_{B_{s}^{0}}^{2}\Big{[}|F_{S}^{s}|^{2}+|F_{P}^{s}+2m_{\mu}F_{A}^{s}|^{2}\Big{]},$
(83)

and

$\displaystyle
F_{S}^{s}=\frac{\alpha_{EW}(\mu_{b})}{8\pi}\frac{m_{b}M_{B_{s}^{0}}^{2}}{m_{b}+m_{s}}f_{B_{s}^{0}}(C_{S}-C_{S}^{\prime}),$
(84) $\displaystyle
F_{P}^{s}=\frac{\alpha_{EW}(\mu_{b})}{8\pi}\frac{m_{b}M_{B_{s}^{0}}^{2}}{m_{b}+m_{s}}f_{B_{s}^{0}}(C_{P}-C_{P}^{\prime}),$
(85) $\displaystyle
F_{A}^{s}=\frac{\alpha_{EW}(\mu_{b})}{8\pi}f_{B_{s}^{0}}\Big{[}C_{10}^{eff}(\mu_{b})-C_{10}^{\prime
eff}(\mu_{b})\Big{]},$ (86)

where $f_{B_{s}^{0}}=(227\pm 8)\;{\rm MeV}$ denote the decay constants,
$M_{B_{s}^{0}}=5.367\;{\rm GeV}$ denote the masses of neutral meson
$B_{s}^{0}$. The branching ratio of $B_{s}^{0}\rightarrow\mu^{+}\mu^{-}$ can
be written as

$\displaystyle
Br(B_{s}^{0}\rightarrow\mu^{+}\mu^{-})=\frac{\tau_{B_{s}^{0}}}{16\pi}\frac{|\mathcal{M}_{s}|^{2}}{M_{B_{s}^{0}}}\sqrt{1-\frac{4m_{\mu}^{2}}{M_{B_{s}^{0}}^{2}}},$
(87)

with $\tau_{B_{s}^{0}}=1.466(31)\;{\rm ps}$ denoting the life time of meson.

###### Acknowledgements.

The work has been supported by the National Natural Science Foundation of
China (NNSFC) with Grants No. 12075074, No. 12235008, Hebei Natural Science
Foundation with Grant No. A2022201017, No. A2023201041, Natural Science
Foundation of Guangxi Autonomous Region with Grant No. 2022GXNSFDA035068, the
youth top-notch talent support program of the Hebei Province.

## References

  * (1) N. Cabibbo, Phys. Rev. Lett. 10, 531-533 (1963).
  * (2) M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652-657 (1973).
  * (3) R. L. Workman et al. [Particle Data Group], PTEP 2022, 083C01 (2022).
  * (4) C. D. Froggatt and H. B. Nielsen, Nucl. Phys. B 147, 277-298 (1979).
  * (5) Y. Koide, Phys. Lett. B 120, 161-165 (1983).
  * (6) M. Leurer, Y. Nir and N. Seiberg, Nucl. Phys. B 398, 319-342 (1993).
  * (7) L. E. Ibanez and G. G. Ross, Phys. Lett. B 332, 100-110 (1994).
  * (8) K. S. Babu and S. M. Barr, Phys. Lett. B 381, 202-208 (1996).
  * (9) Z. G. Berezhiani and M. Y. Khlopov, Sov. J. Nucl. Phys. 51, 739-746 (1990)
  * (10) Z. G. Berezhiani and M. Y. Khlopov, Sov. J. Nucl. Phys. 51, 935-942 (1990)
  * (11) A. S. Sakharov and M. Y. Khlopov, Phys. Atom. Nucl. 57, 651-658 (1994)
  * (12) L. Randall and R. Sundrum, Phys. Rev. Lett. 83, 3370-3373 (1999).
  * (13) D. E. Kaplan and T. M. P. Tait, JHEP 11, 051 (2001).
  * (14) M. C. Chen, D. R. T. Jones, A. Rajaraman and H. B. Yu, Phys. Rev. D 78, 015019 (2008).
  * (15) A. J. Buras, C. Grojean, S. Pokorski and R. Ziegler, JHEP 08, 028 (2011).
  * (16) S. F. King and C. Luhn, Rept. Prog. Phys. 76, 056201 (2013).
  * (17) S. F. King, A. Merle, S. Morisi, Y. Shimizu and M. Tanimoto, New J. Phys. 16, 045018 (2014).
  * (18) S. F. King, J. Phys. G 42, 123001 (2015).
  * (19) S. F. King, Prog. Part. Nucl. Phys. 94, 217-256 (2017).
  * (20) S. Weinberg, Phys. Rev. D 101, no.3, 035020 (2020).
  * (21) F. Feruglio and A. Romanino, Rev. Mod. Phys. 93, no.1, 015007 (2021).
  * (22) G. Abbas, Int. J. Mod. Phys. A 36, no.18, 2150090 (2021).
  * (23) G. Mohanta and K. M. Patel, Phys. Rev. D 106, no.7, 075020 (2022).
  * (24) G. Mohanta and K. M. Patel, JHEP 10, 128 (2023).
  * (25) G. Abbas, V. Singh, N. Singh and R. Sain, Eur. Phys. J. C 83, no.4, 305 (2023).
  * (26) S. Weinberg, Phys. Rev. Lett. 37, 657 (1976).
  * (27) L. Lavoura and H. Kuhbock, Eur. Phys. J. C 55, 303-308 (2008).
  * (28) I. P. Ivanov and E. Vdovin, Phys. Rev. D 86, 095030 (2012).
  * (29) R. González Felipe, H. Serôdio and J. P. Silva, Phys. Rev. D 87, 055010 (2013).
  * (30) R. Gonzalez Felipe, H. Serodio and J. P. Silva, Phys. Rev. D 88, 015015 (2013).
  * (31) V. Keus, S. F. King and S. Moretti, JHEP 01, 052 (2014).
  * (32) I. P. Ivanov and C. C. Nishi, JHEP 01, 021 (2015).
  * (33) N. Buskin and I. P. Ivanov, J. Phys. A 54, 325401 (2021).
  * (34) Y. Izawa, Y. Shimizu and H. Takei, PTEP 2023, 063B04 (2023).
  * (35) D. Van Loi and P. Van Dong, Eur. Phys. J. C 83, no.11, 1048 (2023).
  * (36) S. Weinberg, Phys. Rev. Lett. 43, 1566-1570 (1979).
  * (37) J. L. Yang, H. B. Zhang and T. F. Feng, Phys. Lett. B 853, 138677 (2024).
  * (38) B. W. Lee, C. Quigg and H. B. Thacker, Phys. Rev. D 16, 1519 (1977)
  * (39) J. L. Yang, T. F. Feng, S. M. Zhao, R. F. Zhu, X. Y. Yang and H. B. Zhang, Eur. Phys. J. C 78, 714 (2018).
  * (40) J. L. Yang, T. F. Feng, H. B. Zhang, G. Z. Ning and X. Y. Yang, Eur. Phys. J. C 78, 438 (2018).
  * (41) J. Hisano, T. Moroi, K. Tobe and M. Yamaguchi, Phys. Rev. D 53, 2442-2459 (1996).
  * (42) A. Blondel, A. Bravar, M. Pohl, S. Bachmann, N. Berger, M. Kiehn, A. Schoning, D. Wiedner, B. Windelband and P. Eckert, et al. [arXiv:1301.6113 [physics.ins-det]].
  * (43) K. Hayasaka [Belle and Belle-II], J. Phys. Conf. Ser. 408, 012069 (2013).
  * (44) R.Grigjanis, P. J. O Donnell,M. Sutherland and H. Navelet, Phys. Rep. 22, 93 (1993).
  * (45) G. Buchalla, A. J. Buras and M. E. Lautenbacher, Rev. Mod. Phys. 68, 1125 (1996).
  * (46) W. Altmannshofer, P. Ball, A. Bharucha, A. J. Buras, D. M. Straub and M. Wick, JHEP 0901, 019 (2009) [arXiv:0811.1214 [hep-ph]].
  * (47) L. Lin, T. F. Feng and F. Sun, Mod. Phys. Lett. A 24, 2181-2186 (2009).
  * (48) X. Y. Yang and T. F. Feng, JHEP 1005, 059 (2010).
  * (49) P. Goertz and T. Pfoh, Phys. Rev. D 84, 095016 (2011).
  * (50) A. J. Buras, L. Merlo, E. Stamou, JHEP 1108 124 (2011).
  * (51) F. Goertz, T. Pfoh, Phys. Rev. D 84 095016 (2011).
  * (52) P. Gambino, M. Misiak, Nucl. Phys. B 611 338 (2001).
  * (53) M. Czakon, U. Haisch, M. Misiak, JHEP 0703 008 (2007).
  * (54) A.J. Buras, M. Misiak, M. Müunz and S. Pokorski, Nucl. Phys. B 424 374 (1994).
  * (55) T. J. Gao, T. F. Feng, J. B. Chen, Mod. Phys. Lett. A 7 1250011 (2012).