# New Contributions to $b\rightarrow s\gamma$ in Minimal G2HDM

Che Hao Liu<EMAIL_ADDRESS>Institute of Physics, Academia Sinica,
Nangang, Taipei 11529, Taiwan Van Que Tran  0000-0003-4643-4050
<EMAIL_ADDRESS>Tsung-Dao Lee Institute $\&$ School of Physics and
Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China  Phenikaa
Institute for Advanced Study, Phenikaa University, Yen Nghia, Ha Dong, Hanoi
100000, Vietnam Qiaoyi Wen  0000-0001-5544-7123 Department of Physics, College
of Physics $\&$ Optoelectronic Engineering,
Jinan University, Guangzhou 510632, P.R. China
Fanrong Xu  0000-0003-0986-8028<EMAIL_ADDRESS>Department of Physics,
College of Physics $\&$ Optoelectronic Engineering,
Jinan University, Guangzhou 510632, P.R. China  Tzu-Chiang Yuan
0000-0001-8546-5031<EMAIL_ADDRESS>Institute of Physics, Academia
Sinica, Nangang, Taipei 11529, Taiwan

(August 31, 2024)

###### Abstract

We study the flavor-changing bottom quark radiative decay $b\rightarrow
s\gamma$ induced at one-loop level within the minimal gauged two-Higgs-doublet
model (G2HDM). Among the three new contributions to this rare process in
G2HDM, we find that only the charged Higgs $\mathcal{H^{\pm}}$ contribution
can be constrained by the current global fit data in $B$-physics. Other two
contributions from the complex vectorial dark matter $\mathcal{W}$ and dark
Higgs $\mathcal{D}$ are not sensitive to the current data. Combining with
theoretical constraints imposed on the scalar potential and electroweak
precision data for the oblique parameters, we exclude mass regions
$m_{\mathcal{H}^{\pm}}\lesssim 250$ GeV and $m_{\mathcal{D}}\lesssim 100$ GeV
at the 95% confidence level.

##  I Introduction

The discovery of a Higgs boson near the vicinity of 125 GeV ATLAS:2012yve ;
CMS:2012qbp  at the year 2012 completes the building blocks set up in the
Standard Model (SM). Well-known unanswered questions in SM like the neutrino
masses for neutrino oscillations, dark matter and dark energy problem for the
cosmic energy reserve in the standard $\Lambda$CDM cosmology, gauge hierarchy
problem concerning the stability of the electroweak scale under quantum
fluctuations, etc. must be faced by new physics (NP) beyond the SM (BSM). With
the direct search limits of new particles at the Large Hadron Collider (LHC)
reach multi-TeV, many simple extensions of SM are either under severely
constrained or completely ruled out. At this stage, it is upmost important to
scrutinize a plethora of all available experimental data to explore where NP
may still be hiding from us. Indirect probes of NP from loop-induced rare
processes thus provide an unique opportunity in this endeavour. Rare $B$-meson
decays can play a crucial role as both low-$p_{T}$ and high-$p_{T}$ searches
at the LHCb and LHC respectively are accumulating more and more precise and
complementary data for the indirect probes.

In this paper, we focus on the one-loop process $b\rightarrow s\gamma$ in the
minimal gauged two-Higgs-doublet model (G2HDM) advocated by some of us
Ramos:2021omo ; Ramos:2021txu . The original model Huang:2015wts  was
motivated by gauging the popular inert two-Higgs-doublet model (I2HDM)
Barbieri:2006dq ; LopezHonorez:2006gr ; Arhrib:2013ela ; Belyaev:2016lok ;
Tsai:2019eqi ; Fan:2022dck  for scalar dark matter, augmented by an extended
gauge-Higgs sector of $SU(2)_{H}\times U(1)_{X}$ with a hidden doublet and a
hidden triplet. Thus the complete Higgs sector of the original model is quite
rich but rather complicated to analyze. Nonetheless, various refinements
Arhrib:2018sbz ; Huang:2019obt  and collider implications Huang:2015rkj ;
Huang:2017bto ; Chen:2018wjl ; Chen:2019pnt ; Dirgantara:2020lqy  were pursued
with the same particle content as the original model. As demonstrated in
Ramos:2021omo ; Ramos:2021txu ; Tran:2022yrh , one can drop the hidden triplet
field of the extra $SU(2)_{H}$ without jeopardizing the symmetry breaking
pattern and the mass spectra. Furthermore omitting the hidden triplet vastly
simplifies the scalar potential by getting rid of 6 parameters. We will refer
this as minimal G2HDM, or simply G2HDM, in this work.

Since the two Higgs doublets $H_{1}$ and $H_{2}$ in I2HDM are lumped into an
irreducible representation of the hidden $SU(2)_{H}$ in G2HDM, there are new
Yukawa couplings between the SM and hidden heavy fermions with the inert Higgs
doublet $H_{2}$. In fact, both the charged and neutral components of $H_{2}$
can couple one SM fermion in one generation and one hidden heavy fermion in
another generation. The latter one gives rise to flavor changing neutral
current (FCNC) Higgs interaction between one SM fermion and one hidden heavy
fermion in different generations. Furthermore, unlike the extra gauge boson
$W^{\prime\,\pm}$ in left-right symmetric models Mohapatra:1979ia ;
Senjanovic:1975rk , the $SU(2)_{H}$ gauge boson $\mathcal{W}^{(p,m)}$, one of
the dark matter candidate in G2HDM, carries no electric charge and hence does
not mix with the SM $W^{\pm}$. $\mathcal{W}^{(p,m)}$ also give rise to FCNC
gauge interaction via a right-handed current formed by one SM fermion and one
hidden heavy fermion. All other neutral particles in G2HDM like the photon,
$Z$, SM Higgs along with its hidden sibling as well as the dark photon and
dark $Z$ couple diagonally in flavors with the SM fermion pairs and heavy
hidden fermion pairs. Thus the naturalness of neutral current interactions
proposed by Glashow and Weinberg Glashow:1976nt  can be fulfilled in G2HDM as
far as the SM sector is concerned. Regarding this we note the following fine
point: In Glashow:1976nt , a discrete $Z_{2}$ symmetry was imposed by hand in
the scalar potential of the general 2HDM to forbid the unwanted FCNC Higgs
interactions with SM fermions at tree level. In G2HDM, however, there is an
accidental $h$-parity Chen:2019pnt  in the model to guarantee the absence of
SM particles couple to odd number of new particles with odd $h$-parity coming
from the hidden sector.

All low energy FCNC processes must then be induced by quantum loops in G2HDM.
This motivates our interests in rare $B$ meson decays, in particular
$b\rightarrow s\gamma$ in this study. We will focus on $b\rightarrow s\gamma$
in this work as a warm up and reserve the more complicated penguin process
$b\rightarrow sl^{+}l^{-}$ with $l=e$ or $\mu$ in our future effort.

The organization of this paper is as follows. In Section II, we give a
succinct review of the minimal G2HDM. The relevant G2HDM interaction
Lagrangians for the loop computations are given in Section III, followed by a
discussion of the Wilson coefficients that govern the amplitudes of
$b\rightarrow s(\gamma,g)$ in Section IV. Relevant flavor phenomenology
including renormalization group running effects is discussed in Section V. In
Section VI, after a brief discussion of the scanning methodology we present
our numerical results. We draw our conclusions in VII. Some analytical
formulas are relegated to two appendices. Appendix A gives the detailed
expressions of the loop amplitudes entered in the Wilson coefficients, while
Appendix B lists the Feynman parameterized loop integrals with all internal
and external masses retained.

##  II Minimal G2HDM - A Succinct Review

In this Section, we will briefly review the minimal G2HDM. The quantum numbers
of the matter particles in G2HDM under $SU(3)_{C}\times SU(2)_{L}\times
SU(2)_{H}\times U(1)_{Y}\times U(1)_{X}$ are 111The last two entries in the
tuples are the $Y$ hypercharge and $X$ charge of the two $U(1)$ factors. Note
that the $Q_{X}$ charges of $\pm 1$ of the some fields in our earlier works
Huang:2015wts ; Arhrib:2018sbz ; Huang:2019obt ; Chen:2018wjl ; Huang:2017bto
; Chen:2019pnt ; Huang:2015rkj ; Dirgantara:2020lqy  had been changed to $\pm
1/2$ here. This makes the interaction terms for the hidden $X$ gauge field
look similar to those of the $B$ gauge field associated with the hypercharge.
The anomaly cancellation remains intact with these changes.

Scalars:

$H=\left(H_{1}\;\;H_{2}\right)^{\rm T}\sim\left({\bf 1},{\bf 2},{\bf
2},\frac{1}{2},\frac{1}{2}\right)\;,$
$\Phi_{H}=\left(G^{p}_{H}\;\;\Phi^{0}_{H}\right)^{\rm T}\sim\left({\bf 1},{\bf
1},{\bf 2},0,{\frac{1}{2}}\right)\;.$

We note that the two $SU(2)_{L}$ doublets $H_{1}$ and $H_{2}$ are grouped
together as $H=\left(H_{1}\;\;H_{2}\right)^{\rm T}$ to form a doublet of
$SU(2)_{H}$ with $U(1)_{X}$ charge $+1/2$.

Spin 1/2 Fermions:

Quarks

$Q_{L}=\left(u_{L}\;\;d_{L}\right)^{\rm T}\sim\left({\bf 3},{\bf 2},{\bf
1},\frac{1}{6},0\right)\;,$ $U_{R}=\left(u_{R}\;\;u^{H}_{R}\right)^{\rm
T}\sim\left({\bf 3},{\bf 1},{\bf 2},\frac{2}{3},{\frac{1}{2}}\right)\;,$
$D_{R}=\left(d^{H}_{R}\;\;d_{R}\right)^{\rm T}\sim\left({\bf 3},{\bf 1},{\bf
2},-\frac{1}{3},{-\frac{1}{2}}\right)\;,$ $u_{L}^{H}\sim\left({\bf 3},{\bf
1},{\bf 1},\frac{2}{3},0\right)\;,\;d_{L}^{H}\sim\left({\bf 3},{\bf 1},{\bf
1},-\frac{1}{3},0\right)\;.$

Even though the lepton sector is not relevant in this work, it is shown below
for completeness.

Leptons

$L_{L}=\left(\nu_{L}\;\;e_{L}\right)^{\rm T}\sim\left({\bf 1},{\bf 2},{\bf
1},-\frac{1}{2},0\right)\;,$ $N_{R}=\left(\nu_{R}\;\;\nu^{H}_{R}\right)^{\rm
T}\sim\left({\bf 1},{\bf 1},{\bf 2},0,{\frac{1}{2}}\right)\;,$
$E_{R}=\left(e^{H}_{R}\;\;e_{R}\right)^{\rm T}\sim\left({\bf 1},{\bf 1},{\bf
2},-1,{-\frac{1}{2}}\right)\;,$ $\nu_{L}^{H}\sim\left({\bf 1},{\bf 1},{\bf
1},0,0\right)\;,\;e_{L}^{H}\sim\left({\bf 1},{\bf 1},{\bf 1},-1,0\right)\;.$

The most general renormalizable Higgs potential which is invariant under both
$SU(2)_{L}\times U(1)_{Y}$ and $SU(2)_{H}\times U(1)_{X}$ can be written down
as follows

$\displaystyle V={}$ $\displaystyle-\mu^{2}_{H}\left(H^{\alpha i}H_{\alpha
i}\right)-\mu^{2}_{\Phi}\Phi_{H}^{\dagger}\Phi_{H}+\lambda_{H}\left(H^{\alpha
i}H_{\alpha
i}\right)^{2}+\lambda_{\Phi}\left(\Phi_{H}^{\dagger}\Phi_{H}\right)^{2}$
$\displaystyle+\frac{1}{2}\lambda^{\prime}_{H}\epsilon_{\alpha\beta}\epsilon^{\gamma\delta}\left(H^{\alpha
i}H_{\gamma i}\right)\left(H^{\beta j}H_{\delta j}\right)$ (1)
$\displaystyle+\lambda_{H\Phi}\left(H^{\dagger}H\right)\left(\Phi_{H}^{\dagger}\Phi_{H}\right)+\lambda^{\prime}_{H\Phi}\left(H^{\dagger}\Phi_{H}\right)\left(\Phi_{H}^{\dagger}H\right),$

where ($i$, $j$) and ($\alpha$, $\beta$, $\gamma$, $\delta$) refer to the
$SU(2)_{L}$ and $SU(2)_{H}$ indices respectively, all of which run from one to
two. We denote $H^{\alpha i}=H^{*}_{\alpha i}$, so $H^{\dagger}H=H^{\alpha
i}H_{\alpha i}$ and
$\left(H^{\dagger}\Phi_{H}\right)\left(\Phi_{H}^{\dagger}H\right)=\left(H^{\alpha
i}\Phi_{H\alpha}\right)\left(\Phi^{*}_{H\beta}H_{\beta i}\right)$.

To study spontaneous symmetry breaking (SSB) in the model, we parameterize the
Higgs fields linearly according to standard lore

$\displaystyle H_{1}=\begin{pmatrix}G^{+}\\\ \frac{v+h_{\rm
SM}}{\sqrt{2}}+i\frac{G^{0}}{\sqrt{2}}\end{pmatrix}\;,\;\;\;H_{2}=\begin{pmatrix}\mathcal{H}^{+}\\\
\mathcal{H}_{2}^{0}\end{pmatrix}\;,$ (2)
$\displaystyle\Phi_{H}=\begin{pmatrix}G_{H}^{p}\\\
\frac{v_{\Phi}+\phi_{H}}{\sqrt{2}}+i\frac{G_{H}^{0}}{\sqrt{2}}\end{pmatrix}\;,$
(3)

where $v$ and $v_{\Phi}$ are the only non-vanishing vacuum expectation values
(VEVs) in the SM doublet $H_{1}$ and the hidden doublet $\Phi_{H}$ fields
respectively, with $v=246$ GeV is the SM VEV and $v_{\Phi}$ a hidden VEV at
the TeV scale. $H_{2}$ is the inert doublet with $\langle H_{2}\rangle=0$. In
essence, the scalar sector of minimal G2HDM is a special tailored 3HDM.

##  III G2HDM Interactions

In this Section, we provide the relevant interaction Lagrangians and other
information for the computation of $b\rightarrow s(\gamma,g)$ at one-loop in
minimal G2HDM. We will mainly follow the convention in Peskin and Schroeder
222M. E. Peskin and D. V. Schroeder, “An Introduction to quantum field
theory,” Addison-Wesley, 1995, ISBN 978-0-201-50397-5.

Besides introducing the CKM unitary mixing matrix

$V_{\rm
CKM}\equiv\left(U_{u}^{L}\right)^{\dagger}U^{L}_{d}=\begin{pmatrix}V_{ud}&V_{us}&V_{ub}\\\
V_{cd}&V_{cs}&V_{cb}\\\ V_{td}&V_{ts}&V_{tb}\end{pmatrix}\;,$ (4)

while one diagonalizes the mass matrices of SM quarks, we also need to
introduce the following two unitary mixing matrices while one diagonalizes the
mass matrices of heavy new quarks in G2HDM,

$\displaystyle V^{H}_{u}$
$\displaystyle\equiv\left(U^{R}_{u}\right)^{\dagger}U^{R}_{u^{H}}\,,$ (5)
$\displaystyle V_{d}^{H}$
$\displaystyle\equiv\left(U^{R}_{d}\right)^{\dagger}U^{R}_{d^{H}}\;.$ (6)

###  III.1 Photon, Gluon and $W^{\pm}$ Interactions

For the photon, the relevant interaction Lagrangian is

$\displaystyle\mathcal{L}^{\gamma}$ $\displaystyle\supset$ $\displaystyle-
ie\left(\mathcal{H}^{+}\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}\mathcal{H}^{-}\right)A^{\mu}$
(7)
$\displaystyle+\,e\biggl{[}Q_{u}\sum_{q=u,c,t}\left(\bar{q}\gamma_{\mu}q+\overline{q^{H}}\gamma_{\mu}q^{H}\right)+Q_{d}\sum_{q=d,s,b}\left(\bar{q}\gamma_{\mu}q+\overline{q^{H}}\gamma_{\mu}q^{H}\right)\biggr{]}A^{\mu}$
$\displaystyle+\,ie\biggl{[}\left(\partial_{\mu}W^{+}_{\nu}-\partial_{\nu}W^{+}_{\mu}\right)W^{\mu-}A^{\nu}-\left(\partial_{\mu}W^{-}_{\nu}-\partial_{\nu}W^{-}_{\mu}\right)W^{\mu+}A^{\nu}\biggr{.}$
$\displaystyle\;\;\;\;\biggl{.}+\frac{1}{2}\left(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}\right)\left(W^{\mu+}W^{-\nu}-W^{-\mu}W^{+\nu}\right)\biggr{]}\;,$

where $(a\stackrel{{\scriptstyle\leftrightarrow}}{{\partial}}_{\mu}b)\equiv
a\partial_{\mu}b-b\partial_{\mu}a$, $Q_{u}=2/3$ and $Q_{d}=-1/3$.

For the gluon, we have

$\displaystyle\mathcal{L}^{g}$ $\displaystyle\supset$ $\displaystyle
g_{s}\sum_{q=u,d,s,c,b,t}\left(\bar{q}T^{a}\gamma_{\mu}q+\overline{q^{H}}T^{a}\gamma_{\mu}q^{H}\right)G^{\mu}_{a}\;,$
(8)

where $T^{a}$ are the generators of the color group $SU(3)_{C}$ associated
with the gluon fields $G^{\mu}_{a}$ for $a=1,\cdots,8$.

Figure 1:  Contributions to $b\rightarrow s\gamma$ from the SM $W^{\pm}$ loop
in the unitary gauge.

The SM charged current interaction for the quarks is

$\mathcal{L}^{W}\supset\frac{g}{2\sqrt{2}}\sum_{i,j}\bar{u}_{j}\left(V_{\rm
CKM}\right)_{ji}\gamma^{\mu}\left(1-\gamma_{5}\right)d_{i}W^{+}_{\mu}+{\rm
h.c.}$ (9)

where $V_{\rm CKM}$ is defined in (4) with $i,j$ being the generation indices.
Since the effective Lagrangian describing the rare FCNC decays $b\rightarrow
s(\gamma,g)$ is given by the chirality flipped transition dipole operators,
the chiral $V-A$ structure of SM interaction (9) implies the loop amplitudes
can enjoy the enhancement by two internal top quark mass insertions, besides
the mass insertion from either side of the external lines due to equation of
motion. This is to be compared with the similar processes $t\rightarrow
c(\gamma,g)$ in which SM contribution arises from the hermitian conjugate of
(9), and hence involves two bottom quark mass insertions instead. This
distinctive feature is reflected in the SM branching ratios of $b\rightarrow
s\gamma$ and $t\rightarrow c\gamma$ which are about $\sim 3\times 10^{-4}$
Misiak:2006zs ; Belle:2014nmp  and $\sim 10^{-14}$ Aguilar-Saavedra:2002lwv
respectively.

###  III.2 G2HDM Interactions

There are three new charged (electric charge or dark charge) current
interactions in G2HDM mediated by the dark Higgs $\mathcal{D}$, charged Higgs
$\mathcal{H}^{\pm}$, and $\mathcal{W}^{(p,m)}$ that can give rise to
$b\rightarrow s(\gamma,g)$ at one-loop. The first contribution is from the
dark Higgs $\mathcal{D}$ which is a linear combination of two odd $h$-parity
components $\mathcal{H}_{2}^{0}$ and $G_{H}^{m}$ 333The other orthogonal
combination is
$\tilde{G}=-\sin\theta_{2}\mathcal{H}_{2}^{0}+\cos\theta_{2}G_{H}^{m}$, which
together with its complex conjugate, are the Goldstone bosons absorbed by the
longitudinal components of $\mathcal{W}^{(m,p)}$.

$\mathcal{D}=\cos\theta_{2}\mathcal{H}_{2}^{0}+\sin\theta_{2}G_{H}^{m}\;,$
(10)

where $\theta_{2}$ is a mixing angle giving by

$\tan 2\theta_{2}=\frac{2vv_{\Phi}}{v_{\Phi}^{2}-v^{2}}\;.$ (11)

The mass of $\mathcal{D}$ is

$m_{\mathcal{D}}^{2}=\frac{1}{2}\lambda^{\prime}_{H\Phi}\left(v^{2}+v_{\Phi}^{2}\right)\;.$
(12)

Figure 2:  Contribution to $b\rightarrow s\gamma$ from the $\mathcal{D}$ loop.

The relevant interaction Lagrangian for the dark boson $\mathcal{D}$ interacts
with the SM down-type quarks $d_{i}$ and new heavy down-type quarks
$d^{H}_{j}$ in G2HDM is given by

$\displaystyle{\mathcal{L}}^{\mathcal{D}}$ $\displaystyle\supset$
$\displaystyle\sum_{i,j}\overline{d^{H}_{j}}\left[\left(S^{\mathcal{D}}_{d}\right)_{ji}+\left(P^{\mathcal{D}}_{d}\right)_{ji}\gamma_{5}\right]d_{i}\,\mathcal{D}^{*}+\;{\rm
h.c.}$ (13)

where the Yukawa couplings matrices $S^{\mathcal{D}}_{d}$ and
$P^{\mathcal{D}}_{d}$ are given by

$\displaystyle(S^{\mathcal{D}}_{d})_{ji}$ $\displaystyle=$
$\displaystyle\frac{\sqrt{2}}{2v}\cos\theta_{2}\left(V_{d}^{H\,\dagger}M_{d}\right)_{ji}+\frac{\sqrt{2}}{2v_{\Phi}}\sin\theta_{2}\left(M_{d^{H}}V_{d}^{H\,\dagger}\right)_{ji}\;,$
(14) $\displaystyle(P^{\mathcal{D}}_{d})_{ji}$ $\displaystyle=$
$\displaystyle-\frac{\sqrt{2}}{2v}\cos\theta_{2}\left(V_{d}^{H\,\dagger}M_{d}\right)_{ji}+\frac{\sqrt{2}}{2v_{\Phi}}\sin\theta_{2}\left(M_{d^{H}}V_{d}^{H\,\dagger}\right)_{ji}\;,$
(15)

with $V^{H}_{d}$ defined in (6) and

$\displaystyle M_{d}$ $\displaystyle=$ $\displaystyle{\rm
diag}\left(m_{d},m_{s},m_{b}\right)\;,$ (16) $\displaystyle M_{d^{H}}$
$\displaystyle=$ $\displaystyle{\rm
diag}\left(m_{d^{H}},m_{s^{H}},m_{b^{H}}\right)\;.$ (17)

Note that the ordering of the mass matrices and the mixing matrices are
important in the Yukawa couplings (14) and (15). Also, fixing $v$, $v_{\Phi}$
and $V_{d}^{H}$, for small (large) mixing angle $\theta_{2}$, these Yukawa
couplings are suppressed (enhanced) by the down-type quark (heavy quark) mass
$M_{d}$ ($M_{d^{H}}$). For $v_{\Phi}\gg v$, the contributions from
$\mathcal{D}$ are expected to be minuscule. Similar small effects from
$\mathcal{D}$ was found in $t\rightarrow c(\gamma,g)$ as well tcgamma .

Figure 3:  Contributions to $b\rightarrow s\gamma$ from the
$\mathcal{H^{\pm}}$ loop.

The second contribution to $b\rightarrow s(\gamma,g)$ is from the dark charged
Higgs $\mathcal{H}^{\pm}$ which is quite peculiar in G2HDM as compared with
other multi-Higgs doublet model since it has odd $h$-parity. Thus the
following vertices $W^{\pm}\mathcal{H}^{\mp}\gamma$,
$W^{\pm}\mathcal{H}^{\mp}Z$ and $W^{\pm}\mathcal{H}^{\mp}h$ are all nil in the
model. The mass of the charged Higgs is given by

$m_{\mathcal{H}^{\pm}}^{2}=\frac{1}{2}\left(\lambda^{\prime}_{H\Phi}v_{\Phi}^{2}-\lambda^{\prime}_{H}v^{2}\right)\;.$
(18)

The relevant interaction Lagrangian for the charged Higgs exchange is

${\mathcal{L}}^{\mathcal{H}}\supset\sum_{i,j}\overline{u^{H}_{j}}\left[\left(y^{\mathcal{H}}_{u}\right)_{ji}\left(1-\gamma_{5}\right)\right]d_{i}\,\mathcal{H}^{+}+\;{\rm
h.c.}$ (19)

where the Yukawa coupling matrix $y^{\mathcal{H}}_{u}$ is given by

$\displaystyle(y^{\mathcal{H}}_{u})_{ji}=\frac{\sqrt{2}}{2v}\left(V_{u}^{H\,\dagger}M_{u}V_{\rm
CKM}\right)_{ji}\;,$ (20)

with $V_{\rm CKM}$ and $V^{H}_{u}$ defined in (4) and (5) respectively, and

$M_{u}={\rm diag}\left(m_{u},m_{c},m_{t}\right)\;.$ (21)

Since the Yukawa coupling $y^{\mathcal{H}}_{u}$ is proportional to the up-type
quark mass matrix $M_{u}$, we expect charged Higgs contribution to
$b\rightarrow s(\gamma,g)$ from the third generation heavy fermions is more
relevant than the $\mathcal{D}$ contribution. This is to be compared with the
charged Higgs contribution to $t\rightarrow c(\gamma,g)$ where the
corresponding Yukawa coupling $y^{\mathcal{H}}_{d}=\sqrt{2}(V_{\rm
CKM}M_{d}V^{H}_{d})/2v$ is proportional to the down-type quark mass matrix
$M_{d}$ and therefore has smaller impact tcgamma .

The third contribution to $b\rightarrow s(\gamma,g)$ is from the vector dark
matter $\mathcal{W}^{(p,m)}$ assumed to be the lightest $h$-parity odd
particle in minimal G2HDM with mass given by

$m_{\mathcal{W}}=\frac{1}{2}g_{H}\sqrt{v^{2}+v_{\Phi}^{2}}\;.$ (22)

The relevant interaction Lagrangian for $\mathcal{W}$ is given by

${\mathcal{L}}^{\mathcal{W}}\supset\frac{g_{H}}{2\sqrt{2}}\sum_{i,j}\overline{d^{H}_{j}}\left[\left(V^{H\,\dagger}_{d}\right)_{ji}\gamma^{\mu}(1+\gamma_{5})\right]d_{i}\mathcal{W}^{p}_{\mu}+{\rm
h.c.}$ (23)

It is interesting to note that the dark matter gauge boson $\mathcal{W}$
couples to a right-handed current formed by one SM fermion and one hidden
heavy fermion. However from our previous works, we know the hidden gauge
coupling $g_{H}$ is constrained to be small, of order one percent or less, we
expect the contribution to the processes $b\rightarrow s(\gamma,g)$ from the
dark matter $\mathcal{W}$ is not significant too. Similar situation is found
in $t\rightarrow c(\gamma,g)$ tcgamma .

Figure 4:  Contribution to $b\rightarrow s\gamma$ from the
$\mathcal{W}^{(p,m)}$ loop in the unitary gauge.

The interaction Lagrangians $\mathcal{L}^{\mathcal{D}}$,
$\mathcal{L}^{\mathcal{H}}$ and $\mathcal{L}^{\mathcal{W}}$ given by (13),
(19) and (23) respectively, are the three new contributions from minimal G2HDM
that can induce one-loop FCNC $b\rightarrow s(\gamma,g)$ decays competed with
those from the SM $W^{\pm}$ boson contributions from $\mathcal{L}^{W}$ given
by (9). Note that the mediation of $\mathcal{D}$, $\mathcal{H}^{\pm}$ and
$\mathcal{W}$ are always involved a SM fermion and a new hidden heavy fermion
in G2HDM. Feynman diagrams contributing to $b\rightarrow s\gamma$ from
$W^{\pm}$, $\mathcal{D}$, $\mathcal{H}^{\pm}$ and $\mathcal{W}^{(p,m)}$ are
depicted in Figs. 1, 2, 3 and 4 respectively. Needless to say, for the gluon
case $b\rightarrow sg$, one simply replace the photon line attached to the
colored quarks in these diagrams by the gluon appropriately and hence we will
not bother to depict again here.

##  IV Wilson Coefficients for $b\rightarrow s(\gamma,g)$

As mentioned before, the processes $b\rightarrow s(\gamma,g)$ can be described
by the following effective Lagrangian

$\displaystyle\mathscr{L}_{\rm eff}=$ $\displaystyle-$
$\displaystyle\frac{1}{32\pi^{2}}e\,m_{b}\,\overline{s}\sigma_{\mu\nu}\left(A^{M}+i\gamma_{5}A^{E}\right)b\,F^{\mu\nu}$
(24) $\displaystyle+$
$\displaystyle\frac{1}{32\pi^{2}}g_{s}\,m_{b}\,\overline{s}\sigma_{\mu\nu}T^{a}\left(C^{M}+i\gamma_{5}C^{E}\right)b\,G_{a}^{\mu\nu}\;,$

where $A^{M}(C^{M})$ and $A^{E}(C^{E})$ are the transition (chromo)magnetic
and (chromo)electric dipole form factors respectively, and
$F^{\mu\nu}(G_{a}^{\mu\nu})$ is the electromagnetic (gluon) field strength.
Our task is to compute and evaluate these form factors for the on-shell photon
at one-loop from the SM $W$ boson loop as well as the three new contributions
in minimal G2HDM, as described in previous Section III.

The computation is similar to the charged lepton flavor violation process
$l_{i}\rightarrow l_{j}\gamma$ as was presented in Tran:2022cwh , so we can
simply recycle our previous formulas. The total contribution for $b\rightarrow
s\gamma$ in minimal G2HDM is given by

$A^{(M,E)}=A^{(M,E)}(W)+\Delta A^{(M,E)}\;,$ (25)

where

$\displaystyle\Delta
A^{(M,E)}=A^{(M,E)}(\mathcal{D})+\left(A_{1}^{(M,E)}(\mathcal{H})+A_{2}^{(M,E)}(\mathcal{H})\right)+A^{(M,E)}(\mathcal{W})\;.$
(26)

The SM contributions $A^{(M,E)}(W)$ and new contributions
$A^{(M,E)}(\tilde{D})$, $A_{1,2}^{(M,E)}(\mathcal{H})$, and
$A^{(M,E)}(\mathcal{W})$ can be found in Appendix A.

Similarly the total contribution to $b\rightarrow sg$ is given by

$C^{(M,E)}=C^{(M,E)}(W)+\Delta C^{(M,E)}\;,$ (27)

where

$\displaystyle\Delta
C^{(M,E)}=C^{(M,E)}(\mathcal{D})+C^{(M,E)}(\mathcal{H})+C^{(M,E)}(\mathcal{W})\;,$
(28)

with

$\displaystyle C^{(M,E)}(W)$ $\displaystyle=$ $\displaystyle
A^{(M,E)}_{2}(W)/Q_{u}\;,$ (29) $\displaystyle C^{(M,E)}(\mathcal{D})$
$\displaystyle=$ $\displaystyle A^{(M,E)}(\mathcal{D})/Q_{d}\;,$ (30)
$\displaystyle C^{(M,E)}(\mathcal{H})$ $\displaystyle=$ $\displaystyle
A^{(M,E)}_{2}(\mathcal{H})/Q_{u}\;,$ (31) $\displaystyle
C^{(M,E)}(\mathcal{W})$ $\displaystyle=$ $\displaystyle
A^{(M,E)}(\mathcal{W})/Q_{d}\;.$ (32)

In the $B$-physics community, the processes $b\rightarrow s(\gamma,g)$ are
usually described by the effective Hamiltonian as Wen:2023pfq

$\displaystyle\mathscr{H}_{\rm eff}$ $\displaystyle=$
$\displaystyle-\frac{4G_{F}}{\sqrt{2}}\frac{e^{2}}{16\pi^{2}}V_{tb}V_{ts}^{*}\left[C_{7}(\mu)\mathcal{O}_{7}(\mu)+C_{7}^{\prime}(\mu)\mathcal{O}_{7}^{\prime}(\mu)\right.$
(33)
$\displaystyle\quad\quad\quad\;\;\;\;\;\;\;\;\;\;\;\;\;\left.+\;C_{8}(\mu)\mathcal{O}_{8}(\mu)+C_{8}^{\prime}(\mu)\mathcal{O}_{8}^{\prime}(\mu)\right]\,+\rm{h.c.}\;,$

with the operators

$\displaystyle\mathcal{O}_{7}$ $\displaystyle=$
$\displaystyle\frac{m_{b}}{e}\overline{s}\sigma_{\mu\nu}P_{R}bF^{\mu\nu}\;,\;\;\;\;\mathcal{O}_{7}^{\prime}=\frac{m_{b}}{e}\overline{s}\sigma_{\mu\nu}P_{L}bF^{\mu\nu}\;,$
(34) $\displaystyle\mathcal{O}_{8}$ $\displaystyle=$ $\displaystyle
g_{s}\frac{m_{b}}{e^{2}}\overline{s}\sigma_{\mu\nu}T^{a}P_{R}bG^{\mu\nu}_{a}\;,\;\;\;\;\mathcal{O}_{8}^{\prime}=g_{s}\frac{m_{b}}{e^{2}}\overline{s}\sigma_{\mu\nu}T^{a}P_{L}bG^{\mu\nu}_{a}\;,$
(35)

where $P_{L,R}=(1\mp\gamma_{5})/2$ are the chiral projection operators,
$G_{F}$ is the Fermi constant and $C_{7,8}^{(\prime)}(\mu)$ are the
dimensionless Wilson coefficients at the scale $\mu$. These Wilson
coefficients contain two parts:

$C_{7,8}^{(\prime)}(\mu)=C_{7,8\;\rm{SM}}^{(\prime)}(\mu)+\Delta
C_{7,8}^{(\prime)}(\mu)\;.$ (36)

Comparing the effective Hamiltonian $\mathscr{H}_{\rm eff}$ in (33) with
$(-\mathscr{L}_{\rm eff})$ in (24), we can read off the Wilson coefficients
$C_{7,8}^{(\prime)}(M)$ at a high mass scale $M$ where the heavy particles in
G2HDM are integrated out. Explicitly, we found for the SM $W$ contribution

$\displaystyle\left(C_{7\;\rm{SM}}(M),C^{\prime}_{7\;{\rm SM}}(M)\right)$
$\displaystyle=$
$\displaystyle-\left(\frac{8G_{F}}{\sqrt{2}}\right)^{-1}\left(V_{tb}V_{ts}^{*}\right)^{-1}$
(37)
$\displaystyle\times\left(A^{M}(W)+iA^{E}(W),A^{M}(W)-iA^{E}(W)\right)\;,$
$\displaystyle\left(C_{8\;\rm{SM}}(M),C^{\prime}_{8\;{\rm SM}}(M)\right)$
$\displaystyle=$
$\displaystyle+\left(\frac{8G_{F}}{\sqrt{2}}\right)^{-1}\left(V_{tb}V_{ts}^{*}\right)^{-1}$
(38)
$\displaystyle\times\left(C^{M}(W)+iC^{E}(W),C^{M}(W)-iC^{E}(W)\right)\;,$

and the new contributions from G2HDM

$\displaystyle\left(\Delta C_{7}(M),\Delta C^{\prime}_{7}(M)\right)$
$\displaystyle=$
$\displaystyle-\left(\frac{8G_{F}}{\sqrt{2}}\right)^{-1}\left(V_{tb}V_{ts}^{*}\right)^{-1}$
(39) $\displaystyle\times\left(\Delta A^{M}+i\Delta A^{E},\Delta A^{M}-i\Delta
A^{E}\right)\;,$ $\displaystyle\left(\Delta C_{8}(M),\Delta
C^{\prime}_{8}(M)\right)$ $\displaystyle=$
$\displaystyle+\left(\frac{8G_{F}}{\sqrt{2}}\right)^{-1}\left(V_{tb}V_{ts}^{*}\right)^{-1}$
(40) $\displaystyle\times\left(\Delta C^{M}+i\Delta C^{E},\Delta C^{M}-i\Delta
C^{E}\right)\;,$

where $\Delta A^{(M,E)}$ and $\Delta C^{(M,E)}$ are defined in (26) and (28)
respectively.

One can then use QCD renormalization group equations to evolve the Wilson
coefficients down to $\mu=m_{b}$ for evaluation of the hadronic matrix
elements for $b\rightarrow s$ transitions.

##  V Flavor Phenomenology

Both inclusive and exclusive decays will be taken into account in the
following analysis. For the inclusive decay, the branching fraction is

$\displaystyle\mathcal{B}\left(B_{q}\rightarrow X_{s}\gamma\right)$
$\displaystyle=\\!\mathcal{B}\left(\bar{B}\rightarrow
X_{c}e\bar{\nu}\right)_{\text{exp}}\Big{|}\frac{V_{ts}^{\ast}V_{tb}}{V_{cb}}\Big{|}^{2}\frac{6\alpha_{e}}{\pi
C}\Delta(E_{0};C^{(\prime)\text{eff}}_{7,8})\,,$ (41)
$\displaystyle\Delta(E_{0};C^{(\prime)}_{7,8})$
$\displaystyle=\\!\left[P(E_{0};C^{(\prime)\text{eff}}_{7,8})+N(E_{0};C^{(\prime)\text{eff}}_{7,8})+\epsilon_{\text{EM}}(C^{(\prime)\text{eff}}_{7,8})\right]\,,$

where $C,\leavevmode\nobreak\ P(E_{0}),\leavevmode\nobreak\ N(E_{0})$ and
$\epsilon_{\text{EM}}$ are functions of $C_{7,8}^{(\prime){\rm eff}}$ and
their expressions are given in Gambino:2013rza ; Misiak:2006ab ;
Gambino:2001ew ; Belanger:2004yn . The unprimed effective coefficients can be
evaluated via the RGE evolution

$\mu\frac{d\vec{C}^{\text{eff}}(\mu)}{d\mu}=\gamma^{T}\vec{C}^{\text{eff}}(\mu)\,,$
(42)

where the anomalous dimension matrix is defined as
$\gamma=\sum\limits_{n=0}\gamma^{n}\left(\frac{\alpha_{s}(\mu)}{4\pi}\right)^{n+1}$
with $\gamma^{n}$ of NLO Chetyrkin:1996vx ; Gambino:2003zm  and NNLO
Bobeth:2003at ; Huber:2005ig ; Czakon:2006ss . For the effective coefficients
Buras:1993xp ; Chetyrkin:1996vx ; Greub:1996tg  at EW scale, here we adopt the
convention
$C_{7,8}^{\text{eff}}=C_{7,8}+\sum\limits_{j=1}^{6}y^{(7,8)}_{j}C_{j}$ with
$y_{j}^{(7)}=\left(0,0,-\frac{1}{3},-\frac{4}{9},-\frac{20}{3},-\frac{80}{9}\right)$
and $y_{j}^{(8)}=\left(0,0,1,-\frac{1}{6},20,-\frac{10}{3}\right)$. Notice in
the practical calculation we have neglected new physics contributions to four-
quark operators $O_{1,\ldots,6}$. Since new particles in G2HDM are supposed
not to emerge between $\mu_{W}$ and $\mu_{t}$, the primed coefficients
$C_{7,8}^{\prime\,\rm{eff}}$ share the common evolution equations with their
chiral-flipped counterparts Everett:2001yy ; Eberl:2021ulg ; Borzumati:1999qt
.

The branching fraction of exclusive radiative decay $B\rightarrow V\gamma$ can
be generally written Paul:2016urs  as

$\displaystyle\mathcal{B}\left(B_{q}\rightarrow
V\gamma\right)=\tau_{B_{q}}\frac{\alpha_{e}G_{F}^{2}m^{3}_{B_{q}}m_{b}^{2}}{32\pi^{4}}\left(1-\frac{m_{V}^{2}}{m_{B}^{2}}\right)^{3}|\lambda_{t}|^{2}\left(|C_{7}^{\text{eff}}|^{2}+|C_{7}^{\prime\text{eff}}|^{2}\right)T_{1}(0)\,,$
(43)

where the final state dependent form factors $T_{1}(0)$ are taken from
Paul:2016urs  based on a combination with light-cone sum rule and Lattice QCD.

In the normalized CP asymmetry for $B_{s}\rightarrow V\gamma$, assuming its
parametrization obeying generic time dependent form 444In the parametrization
$\mathcal{A}_{\text{CP}}(B_{s}\rightarrow V\gamma)[t]=\frac{S\sin(\Delta
m_{s}t)-C\cos(\Delta
m_{s}t)}{\cosh(\frac{1}{2}\Delta\Gamma_{s}t)-H\cosh(\frac{1}{2}\Delta\Gamma_{s}t)}$,
we adopt the convention $C_{V\gamma}=C$, $S_{V\gamma}=S$,
$A^{\Delta}_{V\gamma}=H$ in this work. , the observables are defined
Muheim:2008vu  as

$\displaystyle C_{V\gamma}$
$\displaystyle=\frac{|\mathcal{A}_{L}|^{2}+|\mathcal{A}_{R}|^{2}-|\bar{\mathcal{A}}_{R}|^{2}-|\bar{\mathcal{A}}_{L}|^{2}}{|\mathcal{A}_{L}|^{2}+|\bar{\mathcal{A}}_{L}|^{2}+|\mathcal{A}_{R}|^{2}+|\bar{\mathcal{A}}_{R}|^{2}}\;,$
(44) $\displaystyle S_{V\gamma}$
$\displaystyle=2\Im\left[\frac{\frac{q}{p}(\bar{\mathcal{A}}_{L}\mathcal{A}_{L}^{\ast}+\bar{\mathcal{A}}_{R}\mathcal{A}_{R}^{\ast})}{|\mathcal{A}_{L}|^{2}+|\bar{\mathcal{A}}_{L}|^{2}+|\mathcal{A}_{R}|^{2}+|\bar{\mathcal{A}}_{R}|^{2}}\right]\,,$
$\displaystyle A^{\Delta}_{V\gamma}$
$\displaystyle=2\Re\left[\frac{\frac{q}{p}(\bar{\mathcal{A}}_{L}\mathcal{A}_{L}^{\ast}+\bar{\mathcal{A}}_{R}\mathcal{A}_{R}^{\ast})}{|\mathcal{A}_{L}|^{2}+|\bar{\mathcal{A}}_{L}|^{2}+|\mathcal{A}_{R}|^{2}+|\bar{\mathcal{A}}_{R}|^{2}}\right]\,,$

in terms of the amplitudes
$\mathcal{A}_{L(R)}=\mathcal{N}C_{7}^{(\prime)\text{eff}}T_{1}(0)$ and
$\bar{\mathcal{A}}_{L(R)}\equiv\mathcal{A}(\bar{B}_{s}\rightarrow
V\gamma_{L(R)})$ with
$\mathcal{N}=\lambda_{t}\sqrt{\frac{G_{F}^{2}\alpha_{e}m_{B}^{3}}{32\pi^{4}}\left(1-\frac{m_{V}^{2}}{m_{B}^{2}}\right)^{3}}$.
In particular, $\bar{\mathcal{A}}_{L(R)}$ can be derived straightforwardly
from ${\mathcal{A}}_{L(R)}$ by taking weak phase conjugated while keeping
strong phase unchanged. The defined ratio is
$\left(\frac{q}{p}\right)_{s}=\left|\frac{q}{p}\right|_{s}e^{-i\phi_{s}}$ and
$\left|\frac{q}{p}\right|_{s}=1$ has been utilized to derive Eq. (44). 555
Here we simply adopt the experimental average value $\phi_{s}=-0.008\pm 0.019$
HFLAV:2022esi  in the following numerical analysis.

To date, the branching fractions of $B^{(0,+)}\rightarrow
K^{\ast(0,+)}\gamma$, $B\rightarrow\phi\gamma$ and CP asymmetry parameters of
$B_{s}\rightarrow\phi\gamma$ have been measured, which can be taken as input
in the following numerical analysis.

##  VI Numerical Analysis

In this section, we present the numerical results including the new
contributions to the Wilson Coefficients $C_{7}^{(\prime)}$ and
$C_{8}^{(\prime)}$ and the preferred regions on the model parameter space for
data from various low-energy flavor observables as well as the constraints
derived from theoretical conditions on the scalar potential and oblique
parameters. In this analysis, we assume the new mixing matrices
$V_{u}^{H}\equiv V_{d}^{H}\equiv V_{\rm CKM}$ and fix the hidden quark masses
as $m_{s^{H}}=m_{d^{H}}$, $m_{b^{H}}=m_{d^{H}}+\Delta m_{d^{H}}$ for down-type
quarks, and $m_{c^{H}}=m_{u^{H}}$, $m_{t^{H}}=m_{u^{H}}+\Delta m_{u^{H}}$ for
up-type quarks.

Figure 5: Real part of the new contributions to the Wilson coefficients
$\Delta C_{7}$ and $\Delta C_{8}$. Top panels: Contribution from the
$\mathcal{D}$ loop as a function of $m_{\mathcal{D}}$ (left) and from the
$\mathcal{W}$ loop as a function of $m_{\mathcal{W}}$ (right). We set
$V_{d}^{H}=V_{\rm CKM}$, $m_{d^{H}}=500$ GeV, and $\Delta m_{d^{H}}=200$ GeV.
The solid (dashed) blue and red lines indicate Re[$\Delta C_{7}$] (Re[$\Delta
C_{8}$]) with fixed $\theta_{2}=0.05$ rad and $\theta_{2}=0.15$ rad,
respectively. Bottom panels: Contribution from the ${\mathcal{H}}^{\pm}$ loop
as a function of $m_{\mathcal{H}^{\pm}}$, with $V_{u}^{H}=V_{\rm CKM}$ and
$\Delta m_{u^{H}}=0$ GeV for the left panel, and $\Delta m_{u^{H}}=300$ GeV
for the right panel. The solid (dashed) purple and green lines represent
Re[$\Delta C_{7}$] (Re[$\Delta C_{8}$]) with fixed $m_{u^{H}}=300$ GeV and
$m_{u^{H}}=800$ GeV, respectively.

Figure 6: Similar to Fig. 5 but for $\Delta C_{7}^{\prime}$ and $\Delta
C_{8}^{\prime}$.

###  VI.1 New Contributions to the Wilson Coefficients $C^{(\prime)}_{7}$ and
$C_{8}^{(\prime)}$

Before delving into an examination of the parameter space within the model in
light of observations from low-energy flavor experiments, we would like to
know what are the relative magnitudes of the new contributions to the Wilson
coefficients $\Delta C_{7}^{(\prime)}$ and $\Delta C_{8}^{(\prime)}$.

Fig. 5 illustrates the real part of $\Delta C_{7}$ and $\Delta C_{8}$ 666The
imaginary parts of $\Delta C_{7}^{(\prime)}$ and $\Delta C_{8}^{(\prime)}$ are
significantly smaller, typically by at least two orders of magnitude, compared
to their real parts.. The top panels present the contributions from
$\mathcal{D}$ loop and $\mathcal{W}$ loop diagrams. We illustrate the results
with fixed values of $\theta_{2}=0.05$ rad (red lines) and $\theta_{2}=0.15$
rad (blue lines)777The choice of $\theta_{2}$ values adheres to the current
constraints for the light mass DM candidate ($\theta_{2}\leq 0.15$ rad), as
investigated in Ref. Tran:2022yrh ..

We find that $\Delta C_{7}$ and $\Delta C_{8}$ are highly dependent on the
values of the mixing angle $\theta_{2}$ and the masses involved in the loop. A
smaller $\theta_{2}$ and lower $m_{\mathcal{D}}$ or $m_{\mathcal{W}}$ result
in diminished contributions to both Re[$\Delta C_{7}$] and Re[$\Delta C_{8}$].
Here, we set $V_{d}^{H}=V_{\rm CKM}$, $m_{d^{H}}=500$ GeV, and $\Delta
m_{d^{H}}=200$ GeV. It’s worth noting that if $\Delta m_{d^{H}}=0$, indicating
degenerate masses of heavy down-type quarks, both the contributions from
$\mathcal{D}$ loop and $\mathcal{W}$ loop diagrams vanish.

Conversely, the contribution from ${\mathcal{H}}^{\pm}$ loop diagrams doesn’t
vanish when the masses of heavy up-type quarks are degenerate (i.e., $\Delta
m_{u^{H}}=0$), but it is enhanced compared to the non-degenerate case, as
depicted in the bottom panels of Fig. 5. Furthermore, the contribution is more
significant in regions of lighter $m_{{\mathcal{H}}^{\pm}}$ and $m_{u^{H}}$.
Unlike the $\mathcal{D}$ loop and $\mathcal{W}$ loop diagrams, which provide
positive contributions to both Re[$\Delta C_{7}$] and Re[$\Delta C_{8}$],
${\mathcal{H}}^{\pm}$ loop diagrams yield a positive value for Re[$\Delta
C_{7}$] and a negative value for Re[$\Delta C_{8}$] in the parameter space of
interest. Ultimately, we find that the contribution from ${\mathcal{H}}^{\pm}$
loop diagrams dominates over the $\mathcal{D}$ loop and $\mathcal{W}$ loop
diagrams. This implies that the relic density constraint from the dark matter
candidate $\mathcal{W}$ is not relevant in our analysis.

We note that if one also takes the up-type SM quark masses (in additional to
degenerate up-type hidden heavy quark masses) to be the same, the charged
Higgs $\mathcal{H^{\pm}}$ loop diagram also vanish too, just like the SM
$W^{\pm}$ loop. All these null results for the $W^{\pm}$, $\mathcal{W}$,
$\mathcal{D}$ and $\mathcal{H}^{\pm}$ loop contributions in the degenerate
mass scenarios are just manifestation of a generalized version of GIM
mechanism Glashow:1970gm  in G2HDM.

Figure 6 presents the real part of $\Delta C_{7}^{\prime}$ and $\Delta
C_{8}^{\prime}$, with the parameter space setup identical to that of Figure 5.
We observe a similar dependence of $\Delta C_{7}^{\prime}$ and $\Delta
C_{8}^{\prime}$ on the relevant parameter space as seen from $\Delta C_{7}$
and $\Delta C_{8}$. Moreover, for the $\mathcal{D}$ loop and $\mathcal{W}$
loop diagrams, $\Delta C_{7}^{\prime}$ ($\Delta C_{8}^{\prime}$) can be
approximately two orders of magnitude larger than $\Delta C_{7}$ ($\Delta
C_{8}$) within the same parameter space of interest. This discrepancy
primarily arises from the presence of right-handed currents in the
$\mathcal{D}$ loop and $\mathcal{W}$ loop diagrams. On the other hand, the
contribution from $\mathcal{H}^{\pm}$ loop diagrams to $\Delta C_{7}^{\prime}$
($\Delta C_{8}^{\prime}$) is approximately two orders of magnitude smaller
compared to its contribution to $\Delta C_{7}$ ($\Delta C_{8}$).

###  VI.2 Analysis Strategy and Inputs

In Bayesian analysis, the posterior function is proportional to the product of
likelihood function and prior, giving

$\displaystyle\mathcal{P}(\vec{\theta}|\mathcal{O}_{\text{expt.}})\propto\mathcal{L}(\mathcal{O}|\vec{\theta})\pi(\vec{\theta})\,.$
(45)

The Negative Log Likelihood (NLL) function is further defined via $\chi^{2}$
as

$\displaystyle-2\log\mathcal{L}(\mathcal{O}|\vec{\theta})$
$\displaystyle=\chi^{2}(\vec{\theta})$ (46)
$\displaystyle=(\mathcal{O}_{\text{theo.}}(\vec{\theta})-\mathcal{O}_{\text{expt.}})^{\top}(V_{\text{expt.}}+V_{\text{theo.}})^{-1}(\mathcal{O}_{\text{theo.}}(\vec{\theta})-\mathcal{O}_{\text{expt.}})\,,$

where $\mathcal{O}_{\text{theo.}}$ and $\mathcal{O}_{\text{expt.}}$ denote the
theoretical predictions and experimental values, respectively, of observables
of interest. Additionally, the covariance matrices $V_{\text{theo.}}$ and
$V_{\text{expt.}}$ incorporate their respective errors. The set of
observables, including pseudo-observables $C_{8}^{(\prime)}(\mu_{b})$ 888
Although they are not actual observables, the information extracted from the
model-independent global fit Wen:2023pfq  is essential for imposing additional
constraints on a detailed model., are summarized as

$\mathcal{O}^{\top}=\begin{aligned}
\bigl{[}\mathcal{B}(B\rightarrow\phi\gamma),\leavevmode\nobreak\
\bigr{.}\mathcal{B}(B\rightarrow K^{\ast 0}\gamma),\leavevmode\nobreak\
&\mathcal{B}(B\rightarrow K^{\ast+}\gamma),\leavevmode\nobreak\
\mathcal{B}(B\rightarrow X_{s}\gamma),\\\
S_{K^{\ast}\gamma},S_{\phi\gamma},C_{\phi\gamma},&A^{\Delta}_{\phi\gamma},C_{8},\bigl{.}C^{\prime}_{8}\bigr{]}\,,\end{aligned}$
(47)

with corresponding experimental values collected in Table 1. For our keen
readers, other inputs entered implicitly in the eight observables in (47) for
carrying out the theoretical calculations are summarized in Table 2 as well.

Table 1: Experimental values of observables related to $C_{7}/C_{7}^{\prime}$.

| $10^{5}\mathcal{B}$ | $S$ | $C$ | $A^{\Delta}$
---|---|---|---|---
$B\rightarrow X_{s}\gamma$ |  $37.5\pm 1.8\pm 3.5$Belle:2014nmp  | - | - | -
$B\rightarrow\phi\gamma$ |  $3.6\pm 0.5\pm 0.3\pm 0.6$Belle:2014sac  |  $0.43\pm 0.30\pm 0.11$LHCb:2019vks  |  $0.11\pm 0.29\pm 0.11$LHCb:2019vks  |  $-0.67^{+0.37}_{-0.41}\pm 0.17$LHCb:2019vks
$B\rightarrow K^{\ast 0}\gamma$ |  $4.5\pm 0.3\pm 0.2$BelleII:2021tzi  |  $-0.16\pm 0.22$HFLAV:2022esi  | - | -
$B\rightarrow K^{\ast+}\gamma$ |  $5.2\pm 0.4\pm 0.3$BelleII:2021tzi  | - | - | -

Table 2:  Input parameters adopted in theoretical calculations of observables of interest.  Parameters | Values | Parameters | Values
---|---|---|---
$\mathcal{Y}_{b}/10^{-2}$ | 1.646($8.2$)Deppisch:2018flu  | $\mathcal{Y}_{t}$ | 0.9897(86)Deppisch:2018flu
$\mathcal{Y}_{c}/10^{-3}$ | 3.646$(91)$Deppisch:2018flu  | $\mathcal{Y}_{s}/10^{-4}$ | 3.104$(36)$Deppisch:2018flu
$\mathcal{Y}_{d}/10^{-5}$ | 1.663$(64)$Deppisch:2018flu  | $\mathcal{Y}_{u}/10^{-6}$ | 7.80$(86)$Deppisch:2018flu
$m_{B_{d}}$ | 5279.65(12) MeVPDG2022  | $m_{B_{s}}$ | 5366.92(10) MeVPDG2022
$m_{B_{u}}$ | 5279.34(12) MeVPDG2022  | $m_{\phi}$ | 1019.461(16) MeVPDG2022
$m_{K^{\ast\pm}}$ | 891.67(26) MeVPDG2022  | $m_{K^{\ast 0}}$ | 895.55(20) MeVPDG2022
$\tau_{B_{s}}$ | 1.520(5) psPDG2022  | $\tau_{B_{u}}$ | 1.638(4) psPDG2022
$\tau_{B_{d}}$ | 1.519$\left(4\right)$ psPDG2022  | $G_{F}$ | 1.1663787(6) GeV-2PDG2022
$\alpha_{s}(m_{Z})$ | 0.1179(9)PDG2022  | $\alpha_{\text{e}}(m_{Z})$ | 1/127.944(14)PDG2022
$y_{s}$ | 0.064(4)PDG2022  | $y_{d}$ | 0.0005(50)PDG2022
$\sin^{2}\theta_{W}$ | 0.23121(4)PDG2022  | $\mu^{2}_{G}$ | 0.336$(64)$Gambino:2013rza
$\rho_{D}^{3}$ |  $0.153(45)$Gambino:2013rza  | $\rho_{LS}^{3}$ |  $-0.145(98)$Gambino:2013rza
$\mathcal{B}(B\rightarrow X_{c}e\bar{\nu})_{\text{exp}}$ |  $0.0997(41)$Belle-II:2021jlu  | $\phi_{s}$ |  $0.0046(12)$HFLAV:2022esi
$\sin\theta_{12}$ | 0.22500(67)PDG2022  | $\sin\theta_{13}$ | 0.00369$(11)$PDG2022
$\sin\theta_{23}$ |  $0.04182(^{+85}_{-74})$PDG2022  | $\delta_{CP}$ |  $1.144(27)$PDG2022

###  VI.3 Scanning Results

Figure 7:  The parameter space associated with $\mathcal{H}^{\pm}$ constrained
by low-energy flavor experiments. The solid (dashed) boundary delineates the
region of interest, with darker (lighter) shading indicating areas within a
$68\%$ ($95\%$) confidence level.

To explore the remaining related parameters, we implement the affine-invariant
ensemble sampler for Markov chain Monte Carlo (MCMC) emcee Foreman-
Mackey:2012any . Utilizing this method enables the posterior function (45) to
efficiently converge towards solutions with higher probabilities in the
parameter space.

We specify the following priors for the remaining parameters in the model:

$\displaystyle\vec{\theta}=\left\\{\begin{array}[]{rcl}m_{d^{H}}\in[200,1000]\leavevmode\nobreak\
\text{GeV},&\leavevmode\nobreak\ \Delta
m_{d^{H}}\in[0,500]\leavevmode\nobreak\ \text{GeV},\\\
m_{u^{H}}\in[200,1000]\leavevmode\nobreak\ \text{GeV},&\leavevmode\nobreak\
\Delta m_{u^{H}}\in[0,500]\leavevmode\nobreak\ \text{GeV},\\\
m_{\mathcal{W}}\in[0.01,100]\leavevmode\nobreak\
\text{GeV},&\leavevmode\nobreak\
m_{\mathcal{D}}\in[100,1000]\leavevmode\nobreak\ \text{GeV},\\\
m_{\mathcal{H}^{\pm}}\in[100,1000]\leavevmode\nobreak\
\text{GeV},&\leavevmode\nobreak\
\theta_{2}\in[-\frac{\pi}{2},\frac{\pi}{2}]\,.\end{array}\right.$ (48)

These priors assume that each parameter follows a flat prior probability
(uniform distribution), which assists in defining the search intervals.

Fig. 7 illustrates the favored region from low-energy flavor experiments
spanned on sensitive parameters. We found that the dominant contribution to
the $b\rightarrow s\gamma$ process arises from charged Higgs diagrams, leading
to significant constraints on related parameters such as
$m_{\mathcal{H}^{\pm}}$, $m_{u^{H}}$, and $\Delta m_{u^{H}}$, as illustrated
in Fig. 7. In particular, one can put a lower bound on the charged Higgs mass
depending upon both the hidden up-type quark mass and the mass splitting
$\Delta m_{u^{H}}$. Within the $2\sigma$ confidence interval, we find
$m_{\mathcal{H}^{\pm}}\gtrsim 180$ GeV when $m_{u^{H}}\simeq 700$ GeV and
$\Delta m_{u^{H}}\simeq 350$ GeV. This constraint can become even more
stringent in the lower mass range of the hidden up-type quark and for smaller
values of the mass splitting.

Figure 8: The $68\%$ and $95\%$ confidence level contours from low-energy
flavor experiments (lighter blue and darker blue regions), and exclusion
regions (purple shaded regions) from a combination of theoretical and oblique
parameters constraints projected on the
($m_{\mathcal{H}^{\pm}},m_{\mathcal{D}}$) plane.

In Fig. 8, we present the favored region delineated by low-energy flavor
experiments on the ($m_{\mathcal{H}^{\pm}},m_{\mathcal{D}}$) plane. Owing to
the negligible influence of the $\mathcal{D}$ diagram, the bounds derived from
these experiments show minimal dependence on $m_{\mathcal{D}}$. Within the
same figure, we also show exclusion regions (purple regions) from theoretical
constraints of the scalar potential, which includes the criteria for vacuum
stability and perturbative unitarity Ramos:2021omo ; Ramos:2021txu , alongside
constraints Tran:2022yrh  from oblique parameters Peskin:1991sw . These
constraints introduce a significant correlation between
$m_{\mathcal{H}^{\pm}}$ and $m_{\mathcal{D}}$. When these are combined with
the low-energy flavor experiment constraints, it becomes possible to establish
lower bounds on both $m_{\mathcal{H}^{\pm}}$ and $m_{\mathcal{D}}$. Notably,
the regions where $m_{\mathcal{H}^{\pm}}\lesssim 250$ GeV and
$m_{\mathcal{D}}\lesssim 100$ GeV are excluded at the $95\%$ confidence level
through a synergy of constraints from low-energy flavor experiments, oblique
parameters, and theoretical constraints imposed on the scalar potential.

We note that because of the minor contributions from $\mathcal{D}$ and
$\mathcal{W}$ diagrams, parameters like $m_{d^{H}}$, $\Delta m_{d^{H}}$, and
$m_{\mathcal{W}}$ remain relatively unconstrained and are not depicted here.

##  VII Conclusions

In this study, we have performed computations of the one-loop radiative decay
processes for the flavor-changing bottom quark transitions $b\rightarrow
s\gamma$ and $b\rightarrow sg$ within the framework of a minimal G2HDM. Our
analysis extends beyond the SM contributions, traditionally mediated by the
$W$ boson, to include one-loop flavor-changing processes in the minimal G2HDM
facilitated by new charged current interactions. These interactions are
mediated by the dark Higgs ($\cal D$), the charged Higgs (${\cal H}^{\pm}$),
and the complex dark gauge boson (${\cal W}^{p,m}$), the latter of which is a
candidate for dark matter, involving both SM quarks and new heavy quarks in
the loops.

We have derived new contributions to the Wilson coefficients $\Delta
C_{7}^{(\prime)}$ and $\Delta C_{8}^{(\prime)}$, with our numerical results
illustrated in Fig. 5 and Fig. 6. Within our parameters of interest, we found
that the contributions from ${\cal H}^{\pm}$ loop diagrams significantly
dominate over those from $\cal D$ and ${\cal W}^{p,m}$ loop diagrams. This
dominance is due to the explicit mass factor of SM up-type quarks ( mainly top
quark) in the Yukawa coupling involving the charged Higgs, SM down-type
quarks, and up-type new heavy quarks, as shown in (20), plus the requirement
of mass insertions for internal new heavy quark lines to induce the chirality
flipped magnetic and electric dipole operators in (34) and (35). Additionally,
the impact of the ${\cal H}^{\pm}$ loop diagrams is more significant in
regions with lighter masses for $m_{{\mathcal{H}}^{\pm}}$ and the new heavy
up-type quarks.

Interestingly, we observed that the contributions from $\cal D$ loop and
${\cal W}^{p,m}$ loop diagrams diminish when the masses of the three
generations of new heavy quarks running in the loop are degenerate. In
contrast, the contributions from ${\cal H}^{\pm}$ loop diagrams persist under
such conditions but also vanish should the masses of the three generations of
SM up-type quarks are set to be degenerate as well.

Through an exhaustive parameter space scan, constrained by data from various
low-energy flavor observables, we have showcased our main results in Fig. 7.
Owing to the predominant contribution of the charged Higgs loop diagrams on
flavor-changing bottom quark processes, stringent lower bounds have been
placed on the masses of the particles involved in the loop, including the
charged Higgs and new heavy up-type quarks. Notably, the lower bound on the
charged Higgs mass is more restrictive in regions with smaller masses of the
new heavy up-type quarks.

By integrating these constraints from low-energy flavor experiments with those
from theoretical conditions on the scalar potential and oblique parameters, we
have established lower bounds on the masses of both the charged and dark
Higgs. Specifically, regions where $m_{\mathcal{H}^{\pm}}\lesssim 250$ GeV and
$m_{\mathcal{D}}\lesssim 100$ GeV are excluded at the $95\%$ confidence level
based on our analysis.

Due to the peculiar embedding the two Higgs doublets into a two dimensional
irreducible representation of the hidden $SU(2)_{H}$ in G2HDM, the Yukawa
couplings of charged Higgs are highly correlated with the SM Higgs Yukawa
couplings. Thus flavor physics is quite interesting and rich in G2HDM, as
demonstrated in this work for $b\rightarrow s(\gamma,g)$ in $B$ physics, as
well as in the analogous leptonic process of $l_{i}\rightarrow l_{j}\gamma$ in
Tran:2022cwh . Many other low energy flavor physics can be explored further.
For instance, new contributions from G2HDM to the Wilson’s coefficients
$\Delta C_{9}^{(\prime)}$ and $\Delta C_{10}^{(\prime)}$, which are relevant
to various observables for the semi-leptonic processes
$B\rightarrow(X_{s},K^{(*)})l^{+}l^{-},K^{(*)}\nu\bar{\nu}$ and may have some
rooms for new physics according to the recent global fit studies in LEFT
Wen:2023pfq  and in SMEFT Chen:2024jlj , may be of potentially significance in
providing additional constraints. Work on this direction is now in progress
and will be reported elsewhere.

## Acknowledgments

This work was supported in part by the National Natural Science Foundation of
China, grant Nos. 19Z103010239 and 12350410369 (VQT) and U1932104 (FRX), and
the NSTC grant No. 111-2112-M-001-035 (TCY). VQT would like to thank the
Medium and High Energy Physics group at the Institute of Physics, Academia
Sinica, Taiwan for their hospitality during the course of this work. TCY would
like to thank Khiem Hong Phan for the hospitality he received at Duy Tân
University, HCMC, Việt Nam, where the final phase of this work was completed.

##  Appendix A One-loop Induced Amplitudes for $b\rightarrow s\gamma$

For the SM $W$ boson, there are only two diagrams in the unitary gauge as
depicted in Fig. 1. One obtains

$A^{(M,E)}\left(W\right)=A^{(M,E)}_{1}\left(W\right)+A^{(M,E)}_{2}\left(W\right)\;,$
(49)

with

$\displaystyle A^{M}_{1}\left(W\right)$
$\displaystyle=+\left(\frac{g^{2}}{8}\right)\sum_{j}\left(V_{\rm
CKM}\right)^{*}_{j2}\left(V_{\rm CKM}\right)_{j3}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\times\left[\mathcal{I}\left(m_{b},m_{s},m_{u_{j}},m_{W}\right)+\mathcal{I}\left(m_{b},m_{s},-m_{u_{j}},m_{W}\right)\right]\;,$
(50) $\displaystyle A^{E}_{1}\left(W\right)$
$\displaystyle=-i\left(\frac{g^{2}}{8}\right)\sum_{j}\left(V_{\rm
CKM}\right)^{*}_{j2}\left(V_{\rm CKM}\right)_{j3}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\times\left[\mathcal{I}\left(m_{b},-m_{s},m_{u_{j}},m_{W}\right)+\mathcal{I}\left(m_{b},-m_{s},-m_{u_{j}},m_{W}\right)\right]\;,$
(51)

and

$\displaystyle A^{M}_{2}\left(W\right)$
$\displaystyle=\left(-Q_{u}\right)\left(\frac{g^{2}}{8}\right)\sum_{j}\left(V_{\rm
CKM}\right)^{*}_{j2}\left(V_{\rm CKM}\right)_{j3}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\times\left[\mathcal{J}\left(m_{b},m_{s},m_{u_{j}},m_{W}\right)+\mathcal{J}\left(m_{b},m_{s},-m_{u_{j}},m_{W}\right)\right]\;,$
(52) $\displaystyle A^{E}_{2}\left(W\right)$
$\displaystyle=\left(-i\right)\left(-Q_{u}\right)\left(\frac{g^{2}}{8}\right)\sum_{j}\left(V_{\rm
CKM}\right)^{*}_{j2}\left(V_{\rm CKM}\right)_{j3}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\times\left[\mathcal{J}\left(m_{b},-m_{s},m_{u_{j}},m_{W}\right)+\mathcal{J}\left(m_{b},-m_{s},-m_{u_{j}},m_{W}\right)\right]\;.$
(53)

For the dark Higgs $\mathcal{D}$ diagram in Fig. 2, we have

$\displaystyle A^{M}\left(\mathcal{D}\right)$ $\displaystyle=$
$\displaystyle(-Q_{d})\left[\sum_{j}\left(S^{\mathcal{D}}_{d}\right)^{*}_{j2}\left(S^{\mathcal{D}}_{d}\right)_{j3}\mathcal{K}(m_{b},m_{s},m_{d^{H}_{j}},m_{\mathcal{D}})\right.$
(54)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left.\sum_{j}\left(P^{\mathcal{D}}_{d}\right)^{*}_{j2}\left(P^{\mathcal{D}}_{d}\right)_{j3}\mathcal{K}(m_{b},m_{s},-m_{d^{H}_{j}},m_{\mathcal{D}})\right]\;,$
$\displaystyle A^{E}\left(\mathcal{D}\right)$ $\displaystyle=$ $\displaystyle
i(-Q_{d})\left[\sum_{j}\left(P^{\mathcal{D}}_{d}\right)^{*}_{j2}\left(S^{\mathcal{D}}_{d}\right)_{j3}\mathcal{K}(m_{b},-m_{s},m_{d^{H}_{j}},m_{\mathcal{D}})\right.$
(55)
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;+\left.\sum_{j}\left(S^{\mathcal{D}}_{d}\right)^{*}_{j2}\left(P^{\mathcal{D}}_{d}\right)_{j3}\mathcal{K}(m_{b},-m_{s},-m_{d^{H}_{j}},m_{\mathcal{D}})\right]\;.$

For the charged Higgs $\mathcal{H}^{\pm}$, from the two diagrams in Fig. 3, we
get

$A^{(M,E)}\left(\mathcal{H}\right)=A^{(M,E)}_{1}\left(\mathcal{H}\right)+A^{(M,E)}_{2}\left(\mathcal{H}\right)\;,$
(56)

with

$\displaystyle A^{M}_{1}\left(\mathcal{H}\right)$ $\displaystyle=$
$\displaystyle\sum_{j}\left(y^{\mathcal{H}}_{u}\right)^{*}_{j2}\left(y^{\mathcal{H}}_{u}\right)_{j3}$
(57)
$\displaystyle\;\;\;\;\;\;\;\times\left[\mathcal{L}(m_{b},m_{s},m_{u^{H}_{j}},m_{\mathcal{H}})+\mathcal{L}(m_{b},m_{s},-m_{u^{H}_{j}},m_{\mathcal{H}})\right]\;,$
$\displaystyle A^{E}_{1}\left(\mathcal{H}\right)$ $\displaystyle=$
$\displaystyle-i\sum_{j}\left(y^{\mathcal{H}}_{u}\right)^{*}_{j2}\left(y^{\mathcal{H}}_{u}\right)_{j3}$
(58)
$\displaystyle\;\;\;\;\;\;\;\times\left[\mathcal{L}(m_{b},-m_{s},m_{u^{H}_{j}},m_{\mathcal{H}})+\mathcal{L}(m_{b},-m_{s},-m_{u^{H}_{j}},m_{\mathcal{H}})\right]\,,$

and

$\displaystyle A^{M}_{2}\left(\mathcal{H}\right)$ $\displaystyle=$
$\displaystyle\left(-Q_{u}\right)\sum_{j}\left(y^{\mathcal{H}}_{u}\right)^{*}_{j2}\left(y^{\mathcal{H}}_{u}\right)_{j3}$
(59)
$\displaystyle\;\;\;\;\;\;\;\times\left[\mathcal{K}(m_{b},m_{s},m_{u^{H}_{j}},m_{\mathcal{H}})+\mathcal{K}(m_{b},m_{s},-m_{u^{H}_{j}},m_{\mathcal{H}})\right]\;,$
$\displaystyle A^{E}_{2}\left(\mathcal{H}\right)$ $\displaystyle=$
$\displaystyle(-i)\left(-Q_{u}\right)\sum_{j}\left(y^{\mathcal{H}}_{u}\right)^{*}_{j2}\left(y^{\mathcal{H}}_{u}\right)_{j3}$
(60)
$\displaystyle\;\;\;\;\;\;\;\times\left[\mathcal{K}(m_{b},-m_{s},m_{u^{H}_{j}},m_{\mathcal{H}})+\mathcal{K}(m_{b},-m_{s},-m_{u^{H}_{j}},m_{\mathcal{H}})\right]\,.$

Finally, for the contributions from the dark matter gauge boson $\mathcal{W}$
in the unitary gauge as depicted in Fig. 4, we obtain

$\displaystyle A^{M}\left(\mathcal{W}\right)$ $\displaystyle=$
$\displaystyle(-Q_{d})\left(\frac{g_{H}^{2}}{8}\right)\sum_{j}\left(V^{H}_{d}\right)_{2j}\left(V^{H}_{d}\right)^{*}_{3j}$
(61)
$\displaystyle\;\;\;\;\;\times\left[\mathcal{J}(m_{b},m_{s},m_{d^{H}_{j}},m_{\mathcal{W}})+\mathcal{J}(m_{b},m_{s},-m_{d^{H}_{j}},m_{\mathcal{W}})\right]\;,$
$\displaystyle A^{E}\left(\mathcal{W}\right)$ $\displaystyle=$ $\displaystyle
i(-Q_{d})\left(\frac{g_{H}^{2}}{8}\right)\sum_{j}\left(V^{H}_{d}\right)_{2j}\left(V^{H}_{d}\right)^{*}_{3j}$
(62)
$\displaystyle\;\;\;\;\;\times\left[\mathcal{J}(m_{b},-m_{s},m_{d^{H}_{j}},m_{\mathcal{W}})+\mathcal{J}(m_{b},-m_{s},-m_{d^{H}_{j}},m_{\mathcal{W}})\right]\,.$

In the above equations, $\mathcal{I}$, $\mathcal{J}$, $\mathcal{K}$ and
$\mathcal{L}$ are Feynman parametrization loop integrals that can be found in
the following Appendix B.

##  Appendix B Feynman Parametrization Loop Integrals

For convenience, we collect here the loop integrals $\mathcal{I}$,
$\mathcal{J}$, $\mathcal{K}$ and $\mathcal{L}$ which were derived previously
in Tran:2022cwh  (See also Lindner:2016bgg ). Integrals $\mathcal{I}$ and
$\mathcal{J}$ entered in the vector gauge boson exchange diagrams, like those
in Figs. 1 and 4, while $\mathcal{K}$ and $\mathcal{L}$ entered in the scalar
exchange diagrams, like those in Figs. 2 and 3. We have kept all the external
($m_{i}$ and $m_{j}$) and internal ($m_{k}$ and $m_{X}$) masses in these
integrals. We have checked that if the external masses are small compared with
the internal ones like in the $s\rightarrow d$ transition from the $W$
exchange diagrams, series expansions of the expressions of $\mathcal{I}$ and
$\mathcal{J}$ presented below can be used to reproduce the well-known SM
results Inami:1980fz  of heavy quark effects to leading order in $m_{s}$ and
$m_{d}$.

To avoid word cluttering, we denote $z\equiv 1-x-y$ in what follows.

###  B.1 Integral $\mathcal{I}$ and $\mathcal{J}$

$\displaystyle\mathcal{I}$ $\displaystyle\left(m_{i},m_{j},m_{k},m_{X}\right)$
$\displaystyle=\int_{0}^{1}{\rm d}x\int_{0}^{1-x}{\rm
d}y\Biggl{\\{}\frac{1}{-xzm_{i}^{2}-xym_{j}^{2}+xm_{k}^{2}+(1-x)m_{X}^{2}}\Biggr{.}$
$\displaystyle\;\;\;\;\times\Biggl{[}\left(\bigl{(}y+2z\left(1-x\right)\bigr{)}+\bigl{(}z+2y\left(1-x\right)\bigr{)}\frac{m_{j}}{m_{i}}-\,3\left(1-x\right)\frac{m_{k}}{m_{i}}\right)\Biggr{.}$
$\displaystyle\;\;\;\;\;\;\;\;\;+\Biggl{.}\frac{m_{i}^{2}}{m^{2}_{X}}x^{2}\left(z^{2}+y^{2}\frac{m^{3}_{j}}{m_{i}^{3}}+yz\frac{m_{j}}{m_{i}}\left(1+\frac{m_{j}}{m_{i}}\right)-\frac{m_{j}m_{k}}{m_{i}^{2}}\right)\Biggr{]}$
$\displaystyle\;\;\;\;+\frac{1}{m^{2}_{X}}\left(x(1-z)+y+\bigl{(}x\left(1-y\right)+z\bigr{)}\frac{m_{j}}{m_{i}}-\frac{m_{k}}{m_{i}}\right)$
$\displaystyle\;\;\;\;+\frac{1}{m^{2}_{X}}\left(2-x\left(3-4z\right)-3y-z+\bigl{(}2-x\left(3-4y\right)-y-3z\bigr{)}\frac{m_{j}}{m_{i}}\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\Biggl{.}\times\log\left(\frac{m^{2}_{X}}{-xzm_{i}^{2}-xym_{j}^{2}+xm_{k}^{2}+(1-x)m_{X}^{2}}\right)\Biggr{\\}}\;.$
(63)

We note that this integral $\mathcal{I}$ is for the diagram with two internal
charged vector bosons $X$ coupled to the external photon computed using the
unitary gauge. The third line of Eq. (B.1) comes from the product of the
transverse pieces of the two vector boson propagators, while all the remaining
terms are due to the product of the transverse and longitudinal pieces of
these two propagators. The product of longitudinal pieces do not give rise to
the contributions for the transition magnetic and electric dipole form
factors.

$\displaystyle\mathcal{J}$ $\displaystyle\left(m_{i},m_{j},m_{k},m_{X}\right)$
$\displaystyle=-\int_{0}^{1}{\rm d}x\int_{0}^{1-x}{\rm
d}y\Biggl{\\{}\frac{1}{-xzm_{i}^{2}-xym_{j}^{2}+(1-x)m_{k}^{2}+xm_{X}^{2}}\Biggr{.}$
$\displaystyle\;\;\;\;\times\Biggl{\lceil}2x\left((1-z)+(1-y)\frac{m_{j}}{m_{i}}-2\frac{m_{k}}{m_{i}}\right)\Biggr{.}$
$\displaystyle\;\;\;\;\;\;\;\;\;+\frac{m_{i}^{2}}{m^{2}_{X}}\Biggl{(}(1-x)\left(\frac{m_{j}}{m_{i}}-\frac{m_{k}}{m_{i}}\right)\left(z+y\frac{m_{j}}{m_{i}}\right)\left(1-\frac{m_{k}}{m_{i}}\right)\Biggr{.}$
$\displaystyle\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-z\left(\frac{m_{j}}{m_{i}}-\frac{m_{k}}{m_{i}}\right)\left((1-x(1-z))+xy\frac{m_{j}^{2}}{m_{i}^{2}}\right)$
$\displaystyle\Biggl{.}\Biggl{.}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;-\,y\left(1-\frac{m_{k}}{m_{i}}\right)\left(xz+(1-x(1-y))\frac{m_{j}^{2}}{m_{i}^{2}}\right)\Biggr{)}\Biggr{\rfloor}$
$\displaystyle\;\;\;\;+\frac{1}{m^{2}_{X}}\left(y+z\frac{m_{j}}{m_{i}}-\left(1-x\right)\frac{m_{k}}{m_{i}}\right)$
$\displaystyle\;\;\;\;+\frac{1}{m^{2}_{X}}\left((1-3y)+(1-3z)\frac{m_{j}}{m_{i}}+\left(1-3\,x\right)\frac{m_{k}}{m_{i}}\right)$
$\displaystyle\;\;\;\;\;\;\;\;\;\Biggl{.}\times\log\left(\frac{m^{2}_{X}}{-xzm_{i}^{2}-xym_{j}^{2}+(1-x)m_{k}^{2}+xm_{X}^{2}}\right)\Biggr{\\}}\;.$
(64)

We note that this integral $\mathcal{J}$ is for the diagram with one internal
charged or neutral gauge boson $X$ exchange while the external photon couples
to the internal charged fermion. The diagram is also computed using the
unitary gauge. The third line of Eq. (B.1) comes from the transverse piece of
the vector boson propagator, while the remaining terms come entirely from the
longitudinal piece of the propagator.

###  B.2 Integral $\mathcal{K}$ and $\mathcal{L}$

$\displaystyle\mathcal{K}\left(m_{i},m_{j},m_{k},m_{X}\right)$
$\displaystyle=\int_{0}^{1}{\rm d}x\int_{0}^{1-x}{\rm d}y$
$\displaystyle\;\;\;\;\times\left[\frac{x\left(y+z\frac{m_{j}}{m_{i}}\right)+(1-x)\frac{m_{k}}{m_{i}}}{-xym_{i}^{2}-xzm_{j}^{2}+(1-x)m_{k}^{2}+xm_{X}^{2}}\right]\;.$
(65)

$\displaystyle\mathcal{L}\left(m_{i},m_{j},m_{k},m_{X}\right)$
$\displaystyle=-\int_{0}^{1}{\rm d}x\int_{0}^{1-x}{\rm d}y$
$\displaystyle\;\;\;\;\times\left[\frac{x\left(y+z\frac{m_{j}}{m_{i}}+\frac{m_{k}}{m_{i}}\right)}{-xym_{i}^{2}-xzm_{j}^{2}+xm_{k}^{2}+(1-x)m_{X}^{2}}\right]\;.$
(66)

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