#  One-loop contributions to decays $e_{b}\to e_{a}\gamma$ and $(g-2)_{e_{a}}$
anomalies, and Ward identity

L.T. Hue<EMAIL_ADDRESS>H. N. Long<EMAIL_ADDRESS>Subatomic
Physics Research Group, Science and Technology Advanced Institute, Van Lang
University, Ho Chi Minh City, Vietnam  Faculty of Applied Technology, School
of Technology, Van Lang University, Ho Chi Minh City, Vietnam  V. H. Binh
Institute of Physics, Vietnam Academy of Science and Technology, 10 Dao Tan,
Ba Dinh, Hanoi, Vietnam  H. L. T. Mai  Faculty of Physics Science, Can Tho
Medical College, Nguyen Van Cu Street, Can Tho, Vietnam  T. Phong Nguyen
111corresponding author<EMAIL_ADDRESS>Department of Physics, Can Tho
University, 3/2 Street, Can Tho, Vietnam

###### Abstract

In this paper, we will present analytic formulas to express one-loop
contributions to lepton flavor violating decays $e_{b}\to e_{a}\gamma$, which
are also relevant to the anomalous dipole magnetic moments of charged leptons
$e_{a}$. These formulas were computed in the unitary gauge, using the well-
known Passarino-Veltman notations. We also show that our results are
consistent with those calculated previously in the ’t Hooft-Veltman gauge, or
in the limit of zero lepton masses. At the one-loop level, we show that the
appearance of fermion-scalar-vector type diagrams in the unitary gauge will
violate the Ward Identity relating to an external photon. As a result, the
validation of the Ward Identity guarantees that the photon always couples with
two identical particles in an arbitrary triple coupling vertex containing a
photon.

##  I  Introduction

The lepton sector is one of the most interesting objects for experiments to
search for new physics (NP) beyond the prediction of the standard model (SM).
For example, the evidence of neutrino oscillation confirms that the SM must be
extended. Recently, the experimental data of anomalous magnetic moments (AMM)
of charged leptons $(g-2)_{e_{a}}/2\equiv a_{e_{a}}$ has been updated, where
the deviation between SM prediction and the lasted experiment data for muon is
Muong-2:2021ojo

$\Delta a^{\mathrm{NP}}_{\mu}\equiv
a^{\mathrm{exp}}_{\mu}-a^{\mathrm{SM}}_{\mu}=\left(251\pm 59\right)\times
10^{-11},$ (1)

corresponding to the $4.2\sigma$ deviation from standard model (SM) prediction
Aoyama:2020ynm  combined from various contributions Davier:2010nc ;
Davier:2017zfy ; Keshavarzi:2018mgv ; Colangelo:2018mtw ; Hoferichter:2019mqg
; Davier:2019can ; Keshavarzi:2019abf ; Kurz:2014wya ; Melnikov:2003xd ;
Masjuan:2017tvw ; Colangelo:2017fiz ; Hoferichter:2018kwz ; Gerardin:2019vio ;
Bijnens:2019ghy ; Colangelo:2019uex ; Colangelo:2014qya ; Blum:2019ugy ;
Aoyama:2012wk ; Aoyama:2019ryr ; Czarnecki:2002nt ; Gnendiger:2013pva . For
the electron anomaly, the deviation between SM and experiment is $1.6\sigma$
discrepancy Morel:2020dww .

On the other hand, $\Delta a_{e,\mu}$ are strongly constrained by the
experimental data obtained from searching for the charged lepton flavor
violating (cLFV) decays $e_{b}\rightarrow e_{a}\gamma$ are MEG:2016leq ;
BaBar:2009hkt :

$\displaystyle\mathrm{Br}(\tau\rightarrow\mu\gamma)$ $\displaystyle<4.4\times
10^{-8},\;\mathrm{Br}(\tau\rightarrow e\gamma)<3.3\times
10^{-8},\;\mathrm{Br}(\mu\rightarrow e\gamma)<4.2\times 10^{-13}.$ (2)

This important property was discussed previously, for example see discussions
for a general estimation in Ref. Crivellin:2018qmi , and many particular
models beyond the standard model (BSM) Lindner:2016bgg ; Dorsner:2020aaz ;
Hue:2021xap ; Hue:2021xzl ; Hong:2022xjg ; Li:2022zap . General formulas
expressing simultaneously both one-loop contributions to AMM and cLFV
amplitudes were introduced in the limits of new heavy scalar and/or gauge
boson exchanges $m_{B}^{2}\gg m^{2}_{a}$ with $m_{a}$ being the mass of a
charged lepton $e_{a}=e,\mu,\tau$ Crivellin:2018qmi . Other calculations in
the unitary gauge were discussed Yu:2021suw ; Leveille:1977rc  for the one-
loop contributions to $a_{e_{a}}$ with $m_{a}\neq 0$, without the relations
with the cLFV amplitudes. The analytic one-loop formulas for cLFV amplitudes
calculated in the ’t Hooft Feynman (HF) gauge were also shown in Ref.
Lavoura:2003xp , using the notations of the Passarino-Veltman (PV) functions
Passarino:1978jh ; tHooft:1978jhc  with $m_{a}\neq m_{b}$. The approximate
formulas with $m_{a}=m_{b}=0$ were introduced and consistent with those given
in Ref. Crivellin:2018qmi , as shown particularly in Ref. Hue:2017lak  for
3-3-1 models. The general analytic formulas of these PV functions were
introduced for numerical investigations. They are consistent with the results
generated by LoopTools Hahn:1998yk , which can be transformed into other PV
notations implemented in the Fortran numerical package Collier Denner:2016kdg
, used to investigate cLFV decays in a two Higgs doublet model (2HDM)
Jurciukonis:2021izn . Many particular expressions to compute the AMM and/or
cLFV decay amplitudes predicted by different particular BSM were constructed
Lindner:2016bgg . The relations among them can be checked by using suitable
transformations, starting from the set of particular PV notations in this
work. On the other hand, in a discussion on analytic formulas for one-loop
contributions to AMM, a class of fermion-scalar-vector ($FSV$) diagrams
consisting of a photon coupling with two different physical particles, namely
one scalar and one gauge boson, were considered even in the unitary gauge
Yu:2021suw . It leads us to whether the Ward identity (WI) for the external
photon is still valid with the presence of this diagram type. We emphasize
that the general results for one-loop contributions to decays $e_{b}\to
e_{a}\gamma$ and AMM of leptons introduced in many previous works do not
include these $FSV$ diagrams. Moreover, they imply the existence of the triple
photon coupling with two distinguishable physical particles that has never
been mentioned previously. In particular, many works introducing general one-
loop contributions for AMM of charged leptons Leveille:1977rc ;
Lindner:2016bgg ; Crivellin:2018qmi , or decays relating with photon such as
cLFV decays $e_{b}\to e_{a}\gamma$ Lavoura:2003xp ; Lindner:2016bgg ;
Crivellin:2018qmi , loop-induced Higgs decays $h\to\gamma\gamma$ Gunion:1989we
; Bunk:2013uea , $h\to Z\gamma,f\bar{f}\gamma$ Bunk:2013uea ; Hue:2017cph ;
Phan:2021xwc ; VanOn:2021myp , quark decays $q\to q^{\prime}\gamma$, $\dots$.
Excluding the $FSV$ vertex type will reduce a huge number of related one- and
two-loop diagrams as well as confirm the validation of general one-loop
calculation introduced previously.

In this work, we will show precisely the important steps to derive the one-
loop contributions to both AMM and cLFV decays. The calculation is performed
by hand, which is consistent with another cross-checking using FORM package
Vermaseren:2000nd . The final formulas are expressed exactly in terms of the
PV functions defined by LoopTools. The results are then easy to change into
all the other available forms using suitable transformations. The conventions
of the PV-functions are very convenient to derive the exact formulas before
solving particular pure mathematical problems. We also determine contributions
arising from a new form of photon coupling with vector bosons such as
leptoquarks and confirm the consistency between our results and those
introduced in Ref. Bunk:2013uea ; Barbieri:2015yvd ; Biggio:2016wyy .

Our paper is organized as follows. Section I explains our aim of this work.
Section II introduces notations and important formulas to establish the
relations between AMM and cLFV amplitudes. Section III shows discussions to
confirm the consistency of our results and previous works, and the validation
of the WI for the relevant analytic formulas. Section IV summarizes main
features of our work. Finally, we provide many appendices showing precisely
many intermediate steps and notations to derive the final results mentioned in
this work, including the analytic formulas of the PV functions consistent with
LoopTools given in appendix A.

##  II  General amplitudes and notations

It is well-known that analytic formulas of one-loop contributions to the cLFV
amplitudes $e_{b}(p_{2})\rightarrow e_{a}(p_{1})\gamma(q)$ and AMM of SM
charged leptons $e_{a}$ can be presented in the same expressions, see for
example Ref. Crivellin:2018qmi  corresponding to the presence of new heavy
particles in BSM. Possible one-loop Feynman diagrams contributing to
$a_{e_{a}}$ and cLFV decay amplitudes $e_{b}\to e_{a}\gamma$ in BSM are shown
in Fig. 1, where $F$ is a fermion coupling with the SM charged lepton
$e_{a}=e,\mu,\tau$; and the boson $B=h,V$ is a scalar or gauge boson,
respectively.

Figure 1: Feynman diagrams for one-loop contribution to $a_{e_{a}}$ and cLFV
amplitudes $e_{b}\to e_{a}\gamma$ in the unitary gauge.

For a detailed calculation, precise conventions for external momenta and
propagators are presented in appendix C. We note here that Ref. Yu:2021suw
argues another type of $FSV$ one-loop diagrams giving new contributions to the
AMM. They will be discussed in detail in this work.

Firstly, we adopt the Lagrangian generating one-loop diagrams in Fig. 1,
namely Crivellin:2018qmi

$\displaystyle\mathcal{L}_{h}$
$\displaystyle=\overline{F}(g_{a,Fh}^{L}P_{L}+g_{a,Fh}^{R}P_{R})e_{a}h+\mathrm{h.c.},$
(3) $\displaystyle\mathcal{L}_{V}$
$\displaystyle=\overline{F}\gamma^{\mu}(g_{a,FV}^{L}P_{L}+g_{a,FV}^{R}P_{R})e_{a}V_{\mu}+\mathrm{h.c.},$
(4)

where the fermion $F$ and the boson $B=V_{\mu},h$ have electric charges
$Q_{F}$ and $Q_{B}$, and masses $m_{F}$ and $m_{B}$, respectively. These
Lagrangians (3) and (4) are consistent with those in Ref. Lavoura:2003xp .
Moreover, the photon couplings with all physical particles should be mentioned
clearly, as given in Ref. Lavoura:2003xp , i.e., we will adopt the Feynman
rules that the photon always couples with two identical physical particles, as
given in table 1,

Vertex | Coupling | Vertex | Couplings | Vertex | Couplings
---|---|---|---|---|---
$A^{\mu}(p_{0})V^{\nu}(p_{+})V^{*\lambda}(p_{-})$ | $-ieQ_{V}\Gamma_{\mu\nu\lambda}(p_{0},p_{+},p_{-})$ | $A^{\mu}h(p_{+})h^{*}(p_{-})$ | $ieQ_{h}(p_{+}-p_{-})_{\mu}$ | $A^{\mu}\overline{F}F$ | $ieQ_{F}\gamma_{\mu}$
Table 1: Feynman rules for cubic couplings of photon $A^{\mu}$, where
$p_{0,\pm}$ are incoming momenta into the relevant vertex.

where
$\Gamma_{\mu\nu\lambda}(p_{0},p_{+},p_{-})=g_{\mu\nu}(p_{0}-p_{+})_{\lambda}+g_{\nu\lambda}(p_{+}-p_{-})_{\mu}+g_{\lambda\mu}(p_{-}-p_{0})_{\nu}$
is the standard form. The more general form of
$\Gamma_{\mu\nu\lambda}(p_{0},p_{+},p_{-})$ introduced in Refs. Bunk:2013uea ;
Barbieri:2015yvd ; Biggio:2016wyy  will be discussed in detail later.

All couplings listed in Lagrangians (3), (4), and table 1 result in the
following form factors relevant with one-loop contributions:

$\displaystyle c_{RB}^{ab}=$
$\displaystyle\dfrac{e}{16\pi^{2}}g_{a,FB}^{L*}g_{b,FB}^{R}m_{F}\times\dfrac{f_{B}(x_{B})+Q_{F}g_{B}(x_{B})}{m_{B}^{2}}$
$\displaystyle+\dfrac{e}{16\pi^{2}}\left(m_{b}g_{a,FB}^{L*}g_{b,FB}^{L}+m_{a}g_{a,FB}^{R*}g_{b,FB}^{R}\right)\times\dfrac{\tilde{f}_{B}(x_{B})+Q_{F}\tilde{g}_{B}(x_{B})}{m_{B}^{2}},$
(5)

where $x_{B}\equiv m_{F}^{2}/m_{B}^{2}$. The four scalar functions $f_{B}(x)$,
$g_{B}(x)$, $\tilde{f}_{B}(x)$, and $\tilde{g}_{B}(x)$ are listed in Eq. (58)
of appendix A, as the approximate formulas in the limit $m_{a},m_{b}\ll
m_{B}$. The formula in Eq. (II) does not contain contributions from the $FSV$
diagrams mentioned in Ref. Yu:2021suw , because of the absence of photon
coupling $AVh$. The corresponding formulas of AMM and cLFV decay rates are:

$\displaystyle a_{e_{a}}$
$\displaystyle\equiv-\dfrac{2m_{a}}{e}\left(c^{aa}_{R}+c^{aa*}_{R}\right)=-\dfrac{4m_{a}}{e}\mathrm{Re}[c^{aa}_{R}],$
(6) $\displaystyle\mathrm{Br}(e_{b}\to e_{a}\gamma)$
$\displaystyle=\frac{m^{3}_{b}}{4\pi\Gamma_{b}}\left(\left|c^{ab}_{R}\right|^{2}+\left|c^{ba}_{R}\right|^{2}\right),$
(7)

where $m_{a}$, $m_{b}$, and $\Gamma_{b}$ are the masses and total decay width
of the leptons $e_{a}$, $e_{b}$, and

$\displaystyle c^{ab}_{R}$ $\displaystyle\equiv\sum_{B,F}c^{ab}_{RB}.$ (8)

The amplitude for a vertex $\bar{e}_{a}e_{a}A_{\mu}$ in Ref. Peskin:1995ev  is
consistent with the following form presenting both AMM and cLFV amplitudes
Escribano:1996wp ; Eidelman:2016aih

$i\mathcal{M}=-ie\overline{u_{a}}(p_{1})\left[\gamma^{\mu}F_{1}-\frac{\sigma^{\mu\nu}q_{\nu}}{2m_{a}}\left(iF_{2}+\gamma^{5}F_{3}\right)\right]u_{b}(p_{2})\varepsilon^{*}_{\mu},$
(9)

where
$\sigma^{\mu\nu}\equiv\frac{i}{2}\left[\gamma^{\mu}\gamma^{\nu}-\gamma^{\nu}\gamma^{\mu}\right]$;
$F_{1,2,3}$ are scalar form factors; $\varepsilon^{*}_{\mu}$ and $q_{\nu}$ is
the polarized vector of the external photon. The derivation of Eq. (9)
respecting the WI from the most general form was explained clearly in Ref.
Eidelman:2016aih . The form factors $F_{2,3}$ get contributions only from loop
corrections. They relate with the well-known experimental quantities called
the anomalous magnetic moment $a_{e_{a}}$ and electric dipole moment
$d_{e_{a}}$ for $b=a$, respectively. Specifically, we have $F_{1}=1$ for the
on-shell photon, and

$\quad a_{e_{a}}=F_{2};\quad d_{e_{a}}=-\frac{e}{2m_{a}}F_{3}.$ (10)

Regarding the LFV decay $e_{b}\rightarrow e_{a}\gamma$ the amplitude can also
be written in the same form Cheng:1984vwu ; Lavoura:2003xp , suggesting that
$F_{2}$ can be calculated based on the one-loop corrections to LFV decays. In
particular, the second term of the amplitude (9) can be expanded as follows
Hue:2017lak

$\displaystyle\mathcal{M}$
$\displaystyle=(2p_{1}.\varepsilon^{*})\overline{u_{a}}\left(C_{(ab)L}P_{L}+C_{(ab)R}P_{R}\right)u_{b}+\overline{u_{a}}\left[D_{(ab)L}\not{\varepsilon}^{*}P_{L}+D_{(ab)R}\not{\varepsilon}^{*}P_{R}\right]u_{b},$
(11)

where $m_{a}=m_{b}$ and we can prove that
$C_{(ab)L}P_{L}+C_{(ab)R}P_{R}=\frac{e}{2m_{a}}(F_{2}-i\gamma^{5}F_{3})$. The
WI for the external photon gives

$D_{(ab)L}=-(m_{b}C_{(ab)R}+m_{a}C_{(ab)L}),\;D_{(ab)R}=-(m_{b}C_{(ab)L}+m_{a}C_{(ab)R}).$
(12)

We note that although WI does not require the condition of on-shell photon
$q^{2}=0$ in general, it was also used to derive the two relations given in
Eq. (12), which simplify our calculation in the unitary gauge 222We thank the
referee for reminding us this point. The general case of $q^{2}=0$ is beyond
our scope, see Ref. Jurciukonis:2021izn  for a detailed discussion of this
case in the 2HDM framework. The hermiticity that $C_{(aa)R}=C^{*}_{(aa)L}$
Eidelman:2016aih  gives

$\displaystyle a_{e_{a}}$
$\displaystyle=\frac{m_{a}(C_{(aa)L}+C_{(aa)R})}{e}=\frac{2m_{a}\mathrm{Re}{[C_{(aa)L,R}]}}{e},$
$\displaystyle d_{e_{a}}$
$\displaystyle=i(C_{(aa)R}-C_{(aa)L})=\mathrm{Im}{[C_{(aa)L}]}=-\mathrm{Im}{[C_{(aa)R}]}.$
(13)

Hence, the following relations between two different notations must be
satisfied:

$c^{ab}_{R}=-\frac{1}{2}C_{(ab)R}\;\mathrm{and}\;c^{ba}_{R}=-\frac{1}{2}C_{(ab)L}.$
(14)

From the above discussion, we see that one-loop contributions to the
$a_{e_{a}}$ and $d_{e_{a}}$ can be written in terms of well-known PV
functions, see detailed discussions in Ref. Hue:2017lak  or general formula
introduced for calculations of the cLFV decay rates Lavoura:2003xp , with the
identification that $\sigma_{L,R}\equiv-C_{(ab)L,R}$. In the limit of $0\simeq
m_{a},m_{b}\ll m_{B}$, the numerical values of $a_{e_{a}}$ can be evaluated
using the numerical packages such as LoopTools Hahn:1998yk  or Collier
Denner:2016kdg . Although the exact analytic formulas of one-loop three-point
functions presented in Ref. Hue:2017lak  can not be applied to calculate
$a_{e_{a}}$, the limit of $m_{b}\to m_{a}$ can be used to solve this problem.
The analytic formulas of $a_{e_{a}}$ were introduced completely in Ref.
Yu:2021suw .

Because of the relations in Eq. (12), only $C_{(ab)L,R}$ is needed to
determine $a_{e_{a}}$ and Br$(e_{b}\to e_{a}\gamma)$. Because all two-point
diagrams give contributions to just $D_{(ab)L,R}$, $C_{(ab)L,R}$ are
calculated by considering only three-point diagrams. In this work, the
analytic formulas of $D_{(ab)L,R}$ will be determined directly from all
diagrams in Fig. 1 to check the validation of the WI in the presence of the
$FSV$.

The analytic formulas for one-loop contributions to the cLFV decay amplitudes
presented in this work are more general than the results introduced in Ref.
Hue:2017lak  for general 3-3-1 models. Many important steps in our
calculations were shown in appendix C. Using this unitary gauge, the
assumption for a particular form of the Goldstone boson couplings given in
Ref. Lavoura:2003xp  is unnecessary. In contrast, we use the same photon
couplings to other physical particles in an arbitrary BSM, as given in table
1. Namely, a tree-level photon coupling always contains two identical physical
particles. This implies that the contributions from the $FSV$ diagrams are not
included.

Using the notations of PV-functions defined in appendix A, the $Fhh$
contributions from diagram (1) in Fig. 1 are:

$\displaystyle C^{Fhh}_{(ab)L}=$
$\displaystyle\frac{-eQ_{h}}{16\pi^{2}}\left[m_{a}g_{a,Fh}^{L*}g_{b,Fh}^{L}X_{1}^{f}+m_{b}g_{a,Fh}^{R*}g_{b,Fh}^{R}X_{2}^{f}-m_{F}g_{a,Fh}^{R*}g_{b,Fh}^{L}X_{0}^{f}\right],$
$\displaystyle C^{Fhh}_{(ab)R}=$
$\displaystyle\frac{-eQ_{h}}{16\pi^{2}}\left[m_{a}g_{a,Fh}^{R*}g_{Fh}^{bR}X_{1}^{f}+m_{b}g_{a,Fh}^{L*}g_{b,Fh}^{L}X_{2}^{f}-m_{F}g_{a,Fh}^{L*}g_{b,Fh}^{R}X_{0}^{f}\right],$
(15)

where $X^{f}_{0},X^{f}_{1},\dots$ are linear combinations of the PV-functions
$C_{0,00,i,ij}(m_{F}^{2},m_{h}^{2},m_{h}^{2})$ defined precisely in appendix
A.

The diagram (2) in Fig. 1 gives $hFF$ contributions as follows:

$\displaystyle C^{hFF}_{(ab)L}=$
$\displaystyle\frac{-eQ_{F}}{16\pi^{2}}\left[m_{a}g_{a,Fh}^{L*}g_{b,Fh}^{bL}X_{1}^{h}+m_{b}g_{a,Fh}^{R*}g_{b,Fh}^{R}X_{2}^{h}+m_{F}g_{a,Fh}^{R*}g_{b,Fh}^{L}X_{3}^{h}\right],$
$\displaystyle C^{hFF}_{(ab)R}=$
$\displaystyle\frac{-eQ_{F}}{16\pi^{2}}\left[m_{a}g_{a,Fh}^{R*}g_{b,Fh}^{R}X_{1}^{h}+m_{b}g_{a,Fh}^{L*}g_{b,Fh}^{L}X_{2}^{h}+m_{F}g_{a,Fh}^{L^{*}}g_{b,Fh}^{R}X_{3}^{h}\right],$
(16)

where $X^{h}_{1,2,3}$ are linear combinations of
$C_{0,i,ij}(m_{h}^{2},m_{F}^{2},m_{F}^{2})$. The above results are completely
consistent with those introduced in Ref. Lavoura:2003xp , except an overall
sign and the signs before the PV-functions $\bar{c}_{1,2}$, arising from the
different definitions of the external momenta $p_{i}$ in the denominators of
the one-loop integrals. We also give the analytic formulas of
$D^{Fhh}_{(ab)L,R}$ and $D^{hFF}_{(ab)L,R}$, used to confirm the WI given in
Eq. (12) for the only-scalar contributions. The PV-functions derived from
diagram (2) defined as $X^{h}_{i}$ are different from $X^{f}_{i}$ defined for
three diagrams (1), (3), and (4). In contrast, the equal functions are denoted
as follows:

$B^{(i)}_{0}\equiv
B^{(i)f}_{0}=B^{(i)h}_{0}=B_{0}(p_{i}^{2},m_{h}^{2},m_{F}^{2}),\;X_{0}\equiv
X_{0}^{f}=X_{0}^{h},\;i=1,2.$

The form factors $D_{(ab)L,R}$ originated from scalar contributions are:

$\displaystyle D_{(ab)L}^{Fhh}=$
$\displaystyle\frac{-eQ_{H}}{16\pi^{2}}\left\\{g^{L*}_{a,Fh}g^{L}_{b,Fh}\times
2C^{f}_{00}\right\\}$
$\displaystyle+\frac{-eQ_{e}}{16\pi^{2}(m_{a}^{2}-m_{b}^{2})}\left\\{\left(m_{b}g^{L*}_{a,Fh}g^{R}_{b,Fh}+m_{a}g^{R*}_{a,Fh}g^{L}_{b,Fh}\right)m_{F}\left(B^{(1)}_{0}-B^{(2)}_{0}\right)\right.$
$\displaystyle-\left.g^{L*}_{a,Fh}g^{L}_{b,Fh}\left(m^{2}_{a}B^{(1)f}_{1}-m^{2}_{b}B^{(2)f}_{1}\right)-m_{a}m_{b}g^{R*}_{a,Fh}g^{R}_{b,Fh}\left(B^{(1)f}_{1}-B^{(2)f}_{1}\right)\right\\},$
$\displaystyle D_{(ab)R}^{Fhh}=$ $\displaystyle
D_{(ab)L}^{FHH}\left[g^{L}_{a,Fh}\leftrightarrow
g^{R}_{a,Fh},g^{L}_{b,Fh}\leftrightarrow g^{R}_{b,Fh}\right],$ $\displaystyle
D_{(ab)L}^{hFF}=$
$\displaystyle-\frac{eQ_{F}}{16\pi^{2}}\left\\{g^{L*}_{a,Fh}g^{L}_{b,Fh}\left[m_{F}^{2}C^{h}_{0}+(2-d)C^{h}_{00}-m_{a}^{2}X_{1}^{h}-m_{b}^{2}X_{2}^{h}\right]\right.$
$\displaystyle\left.+g^{R*}_{a,Fh}g^{R}_{b,Fh}m_{a}m_{b}X_{0}+\left[g^{R*}_{a,Fh}g^{L}_{b,Fh}m_{a}+g^{L*}_{a,Fh}g^{R}_{b,Fh}m_{b}\right]m_{F}C^{h}_{0}\right\\},$
$\displaystyle D_{(ab)R}^{hFF}=$ $\displaystyle
D_{(ab)L}^{hFF}\left[g^{L}_{a,Fh}\leftrightarrow
g^{R}_{a,Fh},g^{L}_{b,Fh}\leftrightarrow g^{R}_{b,Fh}\right],$ (17)

where $X^{h}_{1,2,3}$ are linear combinations of $C^{h}_{0,i,ij}\equiv
C_{0,i,ij}(m_{h}^{2},m_{F}^{2},m_{F}^{2})$, $C^{f}_{00}\equiv
C_{00}(m_{F}^{2},m_{h}^{2},m_{h}^{2})$, and $B^{(i)f}_{1}\equiv
B^{(i)}_{1}(m_{F}^{2},m_{h}^{2})$ given in Eq. (37).

It is noted that the $Fhh$ contributions are the sum of three diagrams (1),
(3), and (4), while the $hFF$ contributions are from only diagram (2). We
emphasize that the electric charge conservation $Q_{F}=Q_{h}+Q_{e}$ is one of
the necessary requirements to guarantee the WI given in Eq. (12), see a
detailed proof in appendix C. We can see this crudely from the necessary
condition that div$[D^{hFF}_{(ab)L}]+\mathrm{div}[D^{Fhh}_{(ab)L}]\sim
g_{a}^{L*}g^{L}_{b}(Q_{e}+Q_{h}-Q_{F})=0$ and
div$[D^{hFF}_{(ab)R}]+\mathrm{div}[D^{Fhh}_{(ab)R}]\sim
g_{a}^{R*}g^{R}_{b}(Q_{e}+Q_{h}-Q_{F})=0$. This conclusion supports completely
the only case of electric conservation among the remaining ones mentioned in
Ref. Lavoura:2003xp .

Regarding Lagrangian (4), which results in four diagrams in the second line of
Fig. 1, diagram (5) gives the following $FVV$ contributions:

$\displaystyle C_{(ab)L}^{FVV}=-\frac{eQ_{V}}{16\pi^{2}}$
$\displaystyle\left\\{g_{a,FV}^{R*}g_{b,FV}^{L}m_{F}\left[3X_{3}^{f}+\frac{1}{2m_{V}^{2}}\right]-g_{a,FV}^{L*}g_{b,FV}^{R}m_{F}\times\frac{m_{a}m_{b}}{m_{V}^{2}}X^{f}_{012}\right.$
$\displaystyle+g_{a,FV}^{L*}g_{b,FV}^{L}m_{a}\left[2(X^{f}_{1}-X^{f}_{3})+\frac{m_{F}^{2}X^{f}_{01}+m_{b}^{2}X^{f}_{2}}{m_{V}^{2}}\right]$
$\displaystyle\left.+g_{a,FV}^{R*}g_{b,FV}^{R}m_{b}\left[2(X^{f}_{2}-X^{f}_{3})+\frac{m_{F}^{2}X^{f}_{02}+m_{a}^{2}X^{f}_{1}}{m_{V}^{2}}\right]\right\\},$
(18)

where $X^{f}_{i}$ is the linear combinations of
$C_{0,ij}(m_{F}^{2},m_{V}^{2},m_{V}^{2})$, given in Eq. (40), and

$\displaystyle C_{(ab)R}^{FVV}=-\frac{eQ_{V}}{16\pi^{2}}$
$\displaystyle\left\\{g_{a,FV}^{L*}g_{b,FV}^{R}m_{F}\left[3X^{f}_{3}+\frac{1}{2m_{V}^{2}}\right]-g_{a,FV}^{R*}g_{b,FV}^{L}m_{F}\times\frac{m_{a}m_{b}}{m_{V}^{2}}X^{f}_{012}\right.$
$\displaystyle+g_{a,FV}^{R*}g_{b,FV}^{R}m_{a}\left[2(X^{f}_{1}-X^{f}_{3})+\frac{m_{F}^{2}X^{f}_{01}+m_{b}^{2}X^{f}_{2}}{m_{V}^{2}}\right]$
$\displaystyle\left.+g_{a,FV}^{L*}g_{b,FV}^{L}m_{b}\left[2(X^{f}_{2}-X^{f}_{3})+\frac{m_{F}^{2}X^{f}_{02}+m_{a}^{2}X^{f}_{1}}{m_{V}^{2}}\right]\right\\}.$
(19)

Diagram (6) gives $VFF$ contributions:

$\displaystyle C_{(ab)L}^{VFF}=-\frac{eQ_{F}}{16\pi^{2}}$
$\displaystyle\left\\{m_{a}g_{a,FV}^{L*}g_{b,FV}^{L}\left[\frac{}{}2X^{v}_{01}+\frac{m_{F}^{2}\left(X^{v}_{1}-X^{v}_{3}\right)+m_{b}^{2}X^{v}_{2}}{m_{V}^{2}}\right]\right.$
$\displaystyle\;+m_{b}g_{a,FV}^{R*}g_{b,FV}^{R}\left[\frac{}{}2X^{v}_{02}+\frac{m_{F}^{2}\left(X^{v}_{2}-X^{v}_{3}\right)+m_{a}^{2}X^{v}_{1}}{m_{V}^{2}}\right]$
$\displaystyle\;-g_{a,FV}^{R*}g_{b,FV}^{L}m_{F}\left[4X_{0}+\frac{m_{a}^{2}X^{v}_{1}+m_{b}^{2}X^{v}_{2}-m^{2}_{F}X^{v}_{3}}{m_{V}^{2}}\right]$
$\displaystyle\left.\;-g_{a,FV}^{L*}g_{b,FV}^{R}\frac{m_{a}m_{b}}{m_{V}^{2}}\times
m_{F}(X^{v}_{12}-X^{v}_{3})\right\\},$ (20)

where all $X_{i}^{v}$ are expressed in terms of PV functions
$C^{VFF}_{0,i,ij}=C_{0,i,ij}(m_{V}^{2},m_{F}^{2},m_{F}^{2})$, and

$\displaystyle C_{(ab)R}^{VFF}=-\frac{eQ_{F}}{16\pi^{2}}$
$\displaystyle\left\\{m_{a}g_{a,FV}^{R*}g_{b,FV}^{R}\left[2X^{v}_{01}+\frac{m_{F}^{2}\left(X^{v}_{1}-X^{v}_{3}\right)+m_{b}^{2}X^{v}_{2}}{m_{V}^{2}}\right]\right.$
$\displaystyle\;+m_{b}g_{a,FV}^{L*}g_{b,FV}^{L}\left[2X^{v}_{02}+\frac{m_{F}^{2}\left(X^{v}_{2}-X^{v}_{3}\right)+m_{a}^{2}X^{v}_{1}}{m_{V}^{2}}\right]$
$\displaystyle\;-g_{a,FV}^{L*}g_{b,FV}^{R}m_{F}\left[4X^{v}_{0}+\frac{m_{a}^{2}X^{v}_{1}+m_{b}^{2}X^{v}_{2}-m^{2}_{F}X^{v}_{3}}{m_{V}^{2}}\right]$
$\displaystyle\left.\;-g_{a,FV}^{R*}g_{b,FV}^{L}\frac{m_{b}m_{a}}{m_{V}^{2}}\times
m_{F}(X^{v}_{12}-X^{v}_{3})\right\\}.$ (21)

Finally, using the simple notations $g^{L,R}_{a}\equiv g^{L,R}_{a,FV}$, the
formulas of $D_{(ab)L}$ and $D_{(ab)R}$ are

$\displaystyle D^{(78)}_{(ab)L}=$ $\displaystyle
D^{(7)}_{(ab)L}+D^{(8)}_{(ab)L}$ $\displaystyle=$
$\displaystyle\frac{eQ_{e}}{16\pi^{2}(m_{a}^{2}-m_{b}^{2})}\left\\{\frac{}{}\left(g^{L*}_{a}g^{R}_{b}m_{b}+g^{R*}_{a}g^{L}_{b}m_{a}\right)3m_{F}\left[B^{(1)}_{0}-B^{(2)}_{0}\right]\right.$
$\displaystyle-\left.m_{b}\left(m_{a}g^{R*}_{a}g^{R}_{b}+m_{b}g^{L*}_{a}g^{L}_{b}\right)\left[\left(2+\frac{m_{F}^{2}+m_{b}^{2}}{m_{V}^{2}}\right)B^{(2)v}_{1}+\frac{A_{0}(m_{V}^{2})+2m_{F}^{2}B^{(1)}_{0}}{m_{V}^{2}}+1\right]\right.$
$\displaystyle+\left.m_{a}\left(m_{b}g^{R*}_{a}g^{R}_{b}+m_{a}g^{L*}_{a}g^{L}_{b}\right)\left[\left(2+\frac{m_{F}^{2}+m_{a}^{2}}{m_{V}^{2}}\right)B^{(1)v}_{1}+\frac{A_{0}(m_{V}^{2})+2m_{F}^{2}B^{(2)}_{0}}{m_{V}^{2}}+1\right]\right\\},$
(22) $\displaystyle D^{(78)}_{(ab)R}=$ $\displaystyle
D^{(78)}_{(ab)L}\left[g^{L}_{a}\leftrightarrow
g^{R}_{a},\;g^{L}_{b}\leftrightarrow g^{R}_{b}\right].$ $\displaystyle
D_{(ab)L}^{FVV}=$
$\displaystyle-\frac{eQ_{V}}{16\pi^{2}}\left\\{\frac{}{}g_{a}^{L*}g_{b}^{L}\left[\frac{}{}2(d-2)C^{f}_{00}+2(m_{a}^{2}+m_{b}^{2})X^{f}_{3}\right.\right.$
$\displaystyle\left.\left.-\frac{1}{m_{V}^{2}}\left(m_{F}^{2}(B^{(1)}_{0}+B^{(2)}_{0}-2C^{f}_{00})+A_{0}(m_{V}^{2})+m_{a}^{2}B^{(1)f}_{1}+m_{b}^{2}B^{(2)f}_{1}\right)\right]\right.$
$\displaystyle\;+g_{a}^{R*}g_{b}^{R}m_{a}m_{b}\left[4X^{f}_{3}+\frac{2C^{f}_{00}}{m_{V}^{2}}\right]+g_{a}^{R*}g_{b}^{L}\times
m_{a}m_{F}\left[3C^{f}_{0}-\frac{1}{2m_{V}^{2}}+\frac{m_{b}^{2}X^{f}_{012}}{m_{V}^{2}}\right]$
$\displaystyle\left.\;+g_{a}^{L*}g_{b}^{R}\times
m_{b}m_{F}\left[3C^{f}_{0}-\frac{1}{2m_{V}^{2}}+\frac{m_{a}^{2}X^{f}_{012}}{m_{V}^{2}}\right]\right\\},$
$\displaystyle D_{(ab)R}^{FVV}=$ $\displaystyle
C_{(ab)L}^{FVV}\left[g^{L}_{a}\leftrightarrow
g^{R}_{a},g^{L}_{b}\leftrightarrow g^{R}_{a}\right],$ (23)

where all $X_{i}^{f}$ are expressed in terms of PV functions
$C^{f}_{0,ij}\equiv C_{0,ij}(m_{F}^{2},m_{V}^{2},m_{V}^{2})$ and
$B^{(i)f}_{1}$ is given in Eq. (37).

The remaining formulas of $D_{(ab)L,R}$ from diagram (6) of Fig. 1 are

$\displaystyle D^{VFF}_{(ab)L}=$
$\displaystyle\frac{eQ_{F}}{16\pi^{2}}\left\\{g^{L*}_{a}g^{L}_{b}\left[\frac{}{}-2m_{F}^{2}C_{0}+(d-2)^{2}C_{00}^{v}+2m_{a}^{2}X^{v}_{01}+2m_{b}^{2}X^{v}_{02}\right.\right.$
$\displaystyle\left.\left.\qquad\qquad\qquad\quad-\frac{1}{m_{V}^{2}}\left[(2-d)m_{F}^{2}C^{v}_{00}+A_{0}(m_{V}^{2})+m_{F}^{2}\left(B^{(1)}_{0}+B^{(2)}_{0}\right)\right.\right.\right.$
$\displaystyle\left.\left.\left.\qquad\qquad\qquad\qquad\qquad-
m_{a}^{2}\left(B^{(1)v}_{0}+B^{(1)}_{1}\right)-m_{b}^{2}\left(B^{(2)v}_{0}+B^{(2)v}_{1}\right)+m_{a}^{2}m_{b}^{2}X_{0}\right.\right.\right.$
$\displaystyle\left.\left.\left.\qquad\qquad\qquad\qquad\qquad-
m_{F}^{2}\left((m_{a}^{2}+m_{b}^{2}-m_{F}^{2})C_{0}+m_{a}^{2}X^{v}_{1}+m_{b}^{2}X^{v}_{2}\right)\frac{}{}\right]\right]\right.$
$\displaystyle\left.\qquad\quad+g^{R*}_{a}g^{R}_{b}m_{a}m_{b}\left[2X_{0}-\frac{1}{m_{V}^{2}}\left((2-d)C^{v}_{00}+m_{F}^{2}X^{v}_{3}-m_{a}^{2}X^{v}_{1}-m_{b}^{2}X^{v}_{2}\right)\right]\right.$
$\displaystyle\left.\qquad\quad+\frac{g^{R*}_{a}g^{L}_{b}m_{a}m_{F}}{m_{V}^{2}}\left[-B^{(1)v}_{1}+(2-d)C_{00}-m_{a}^{2}X^{v}_{1}+m_{b}^{2}(X^{v}_{3}-X^{v}_{2})\right]\right.$
$\displaystyle\left.\qquad\quad+\frac{g^{L*}_{a}g^{R}_{b}m_{b}m_{F}}{m_{V}^{2}}\left[-B^{(2)v}_{1}+(2-d)C^{v}_{00}-m_{b}^{2}X^{v}_{2}+m_{a}^{2}(X^{v}_{3}-X^{v}_{1})\right]\right\\},$
$\displaystyle D^{VFF}_{(ab)R}=$ $\displaystyle
D^{VFF}_{(ab)L}\left[g_{a}^{L}\leftrightarrow
g_{a}^{R},\;g_{b}^{L}\leftrightarrow g_{b}^{R}\right],$ (24)

where all $X_{i}^{v}$ are expressed in terms of PV functions
$C^{v}_{0,ij}\equiv C_{0,ij}(m_{V}^{2},m_{F}^{2},m_{F}^{2})$ and
$B^{(i)v}_{1}$ is given in Eq. (37).

We note that all results presented here are crosschecked by FORM package
Vermaseren:2000nd , using intermediate steps given in appendix C. There is a
property that $C^{X}_{(ab)R}=C^{X}_{(ab)L}\left[g_{a}^{L}\leftrightarrow
g_{a}^{R},\;g_{b}^{L}\leftrightarrow g_{b}^{R}\right]$ for all
$X=Fhh,hFF,FVV,VFF$. The above results of one-loop contribution to
$C_{(ab)L,R}$ are totally consistent with those introduced in Ref.
Lavoura:2003xp , after some transformations of notations presented in appendix
B. In the limit of $m_{h}^{2},m_{V}^{2}\gg m^{2}_{a},m^{2}_{b}$, i.e.,
$m^{2}_{a}/m_{B}^{2},m^{2}_{b}/m_{h}^{2}\simeq 0$ with $B=h,V$, we get
consistent results with those given in Ref. Crivellin:2018qmi ;
Freitas:2014pua ; Stockinger:2006zn . To derive the above results for gauge
boson exchanges, we start with many important features different from those
mentioned in Ref. Lavoura:2003xp , namely: i) we do not use the typical form
of couplings relating to Goldstone bosons going along with the presence of new
gauge bosons, ii) we have to use the massless property of the on-shell photon
$q^{2}=0$, iii) to confirm the WI for all diagrams given in Fig. 1, we need
the charge conservation law corresponding to the Lagrangian (1):
$Q_{F}=Q_{V}+Q_{e}$. Therefore, our calculation is another independent
approach to confirm the result given in Ref. Lavoura:2003xp . The details of
the calculation to confirm the WI for all one-loop contributions are given in
appendix C. We remind that our results are derived from the photon couplings
listed in the table 1, and do not contain the contributions from the FSV
diagrams. In the following, we pay attention to the possibility of adding the
FSV diagrams or the new forms of the photon couplings.

##  III  Discussion on WI and previous results

###  III.1 WI to constrain the form of photon couplings

Now we focus on the feature that the WI of the on-shell photon will constrain
strongly the forms of the cubic photon couplings with two physical particles
in a renormalized Lagrangian. Now we consider the existence of the photon
coupling types at tree level:

$\displaystyle\mathcal{L}^{\gamma XX}=$ $\displaystyle
eQ_{F}A^{\mu}\left[\overline{F_{1}}\gamma^{\mu}F_{2}+\mathrm{h.c.}\right]+ieQ_{h}A^{\mu}\left[\left(h_{1}^{*}\partial_{\mu}h_{2}-h_{2}\partial_{\mu}h_{1}^{*}\right)+\mathrm{h.c.}\right]$
$\displaystyle-\left[eQ_{V}A^{\mu}V^{\nu}_{1}V^{\lambda*}_{2}\Gamma_{\mu\nu\lambda}(p_{0},p_{+}p_{-})+\mathrm{h.c.}\right]+\left[g_{\gamma
hV}g_{\mu\nu}h^{-Q}A^{\mu}V^{Q\nu}+\mathrm{h.c.}\right],$ (25)

where all couplings are more general than those well-known as the standard
forms given in Table 1. In addition, the last term corresponds to the photon
coupling to a scalar $h\equiv S$ and a gauge boson $V$ mentioned in Eq. (110).
The above Lagrangian results in the following decays from the heavy particle
to the lighter one: i) $F_{2}\to F_{1}\gamma$, ii) $h_{2}\to h_{1}\gamma$,
iii) $V_{2}\to V_{1}\gamma$, and iv) $V\to h\gamma$. The WI for these decay
amplitudes at tree level is $\mathcal{M}^{\mu}(X_{1}\to
X_{2}\gamma)p_{0\mu}=0$ with $p_{0\mu}$ being the external photon momentum. It
can be derived that:

  * •

Using the same convention of external momenta given in Fig. 1, we have
$\mathcal{M}^{\mu}(F_{2}\to
F_{1}\gamma)q_{\mu}\sim(m_{F_{2}}-m_{F_{1}})\overline{u}_{F_{2}}(p_{2})u_{F_{1}}(p_{1})=0$,
where $p_{0}\equiv-q$. Therefore, $m_{F_{2}}=m_{F_{1}}$. This case is
automatically satisfied for the tree-level AMM amplitude.

  * •

$\mathcal{M}^{\mu}(h_{2}\to
h_{1}\gamma)p_{0\mu}\sim(p_{2}-p_{1}).(p_{2}+p_{1})=(m^{2}_{h_{2}}-m^{2}_{h_{1}})=0$,
where all on-shell momenta are incoming the vertex $A^{\mu}h^{*}_{1}h_{2}$,
implying that $p_{0}=-(p_{1}+p_{2})$ and $p_{1,2}^{2}=m^{2}_{h_{1,2}}$. The
consequence is $m_{h_{1}}=m_{h_{2}}$.

  * •

$\mathcal{M}^{\mu}(V\to h\gamma)p_{0\mu}\sim\varepsilon_{v}.p_{0}=0$, where
$\varepsilon_{v}$ and $p_{0}$ are the polarization of gauge boson $V$ and the
external momentum of the photon $A_{\mu}$. Hence, the presence of a $AhV$
vertex does not automatically satisfy the WI. One-loop contributions for all
diagrams arising from this vertex must be checked for the validation of WI. In
Ref. Yu:2021suw , the presence of these vertices was mentioned in a Higgs
triplet model (HTM). A detailed calculation in appendix E shows an opposite
conclusion that this vertex vanishes at tree level 333To the best of our
knowledge, we have not seen any UV models beyond the SM that have violated
$U(1)_{em}$ couplings of the form $S$-$V$-$\gamma$, which is the necessary
source for generating $FSV$-type diagrams..

  * •

$\mathcal{M}_{\mu}(V_{1}\to
V_{2}\gamma)p^{\mu}_{0}\sim\varepsilon^{\nu}_{1}\varepsilon^{\lambda*}_{2}p^{\mu}_{0}\Gamma_{\mu\nu\lambda}(p_{0},p_{1},p_{2})=0$,
where $\varepsilon_{1,2}$, and $p_{1,2,0}$ are the polarization of the gauge
boson $V_{1,2}$ and the external momentum of the gauge bosons $V_{1,2}$ and
photon $A_{\mu}$, respectively. We will use the following properties of the
external gauge bosons $V_{i}(i=1,2)$ and photon: $\varepsilon_{i}.p_{i}=0$,
$p_{0}^{2}=0$, $p_{i}^{2}=m^{2}_{V_{i}}$, and the momentum conservation
$p_{0}+p_{1}+p_{2}=0$ following notations in table 1. After some intermediate
calculating steps, we have:

$\displaystyle\mathcal{M}_{\mu}(V_{1}\to V_{2}\gamma)p^{\mu}_{0}\sim$
$\displaystyle(p_{0}.\varepsilon_{1})\left[(p_{0}-p_{1}).\varepsilon^{*}_{2}\right]+(\varepsilon_{1}.\varepsilon^{*}_{2})\left[(p_{1}-p_{2}).p_{0}\right]+(p_{0}.\varepsilon^{*}_{2})\left[(p_{2}-p_{0}).\varepsilon_{1}\right]$
$\displaystyle=$
$\displaystyle(\varepsilon_{1}.\varepsilon^{*}_{2})\left[m^{2}_{V_{2}}-m^{2}_{V_{1}}\right]=0.$
(26)

Hence, $m_{V_{1}}=m_{V_{2}}$ is necessary. From this, we consider the more
general photon coupling with a gauge boson Biggio:2016wyy  describing the
couplings of a leptoquark field Barbieri:2015yvd

$\displaystyle\Gamma^{\prime}_{\mu\nu\lambda}(p_{0},p_{1},p_{2})=$
$\displaystyle
g_{\mu\nu}(k_{v}p_{0}-p_{1})_{\lambda}+g_{\nu\lambda}(p_{1}-p_{2})_{\mu}+g_{\lambda\mu}(p_{2}-k_{v}p_{0})_{\nu}$
$\displaystyle=$
$\displaystyle\Gamma_{\mu\nu\lambda}(p_{0},p_{1},p_{2})+\delta
k_{v}\left(g_{\mu\nu}p_{0\lambda}-g_{\lambda\mu}p_{0\nu}\right),$ (27)

with $\delta k_{v}=k_{v}-1$ showing the deviation from the standard vertex
listed in table 1. This may change the one-loop contributions of the diagram
(5) in Fig. 1, hence change the formulas of $C^{FVV}_{(ab)L,R}$ given in Eqs.
(II) and (II), respectively. One can prove immediately that the vertex
deviation

$\displaystyle\delta\Gamma_{\mu\nu\lambda}(p_{0},p_{1},p_{2})\equiv\Gamma^{\prime}_{\mu\nu\lambda}(p_{0},p_{1},p_{2})-\Gamma_{\mu\nu\lambda}(p_{0},p_{1},p_{2})=\delta
k_{v}\left(g_{\mu\nu}p_{0\lambda}-g_{\lambda\mu}p_{0\nu}\right)$ (28)

guarantees the WI. The new one-loop contributions arising from $\delta\Gamma$
are also satisfied the WI, see analytic formulas given in Eq. (C.2).

Now we start from the point that all results of one loop contributions given
from Eq. (II) to Eq. (II) based on the standard forms of photon couplings
given in table 1, where a photon always couples with two identical physical
fields. On the other hand, a recent work Yu:2021suw  assumed the existence of
a new photon coupling kind $ASV$, which may appear in some BSM, in which the
photon couples with one gauge boson $V$ and one scalar $S$. The appearance of
a boson $V$ or $S$ will generate by itself the one-loop contributions that
always guarantee the WI by the respective set of four diagrams given in Fig.
1. Hence, the two FSV diagrams must give contributions satisfying the WI
themselves, namely

$\displaystyle
D^{FSV}_{(ab)_{L}}+m_{a}C^{FSV}_{(ab)_{L}}+m_{b}C^{FSV}_{(ab)_{R}}=D^{FSV}_{(ab)_{R}}+m_{a}C^{FSV}_{(ab)_{R}}+m_{b}C^{FSV}_{(ab)_{L}}=0.$
(29)

As a result, the divergent parts of $h\equiv S$ given in appendix D for both
$L$ and $R$ parts give:

$\displaystyle 0=$ $\displaystyle g_{\gamma
hV}\left[2g^{L*}_{ah}g^{L}_{bV}m_{F}-g^{L*}_{ah}g^{R}_{bV}m_{b}-g^{R*}_{ah}g^{L}_{bV}m_{a}\right]$
$\displaystyle=$ $\displaystyle g_{\gamma
hV}\left[2g^{R*}_{ah}g^{R}_{bV}m_{F}-g^{R*}_{ah}g^{L}_{bV}m_{b}-g^{L*}_{ah}g^{R}_{bV}m_{a}\right].$
(30)

Considering the case of $g_{\gamma hV}\neq 0$. Then, all quantities
$g^{L}_{ah},g^{R}_{ah}$, $g^{L}_{bV}$, and $g^{R}_{bV}$ are zeros if at least
one of them is zero. More strictly, we require that the two Eqs. (29) must be
held for both divergent and finite parts arising from $D_{(ab)L,R}$ and
$C_{(ab)L,R}$ given in appendix D. Consequently, $g_{\gamma hV}=0$, i.e., the
$FSV$ diagram type does not satisfy the WI.

Regarding the vertex deviation of the $AVV$ couplings defined in Eq. (28), the
new one-loop contributions relating to $C^{FVV}_{(ab)L,R}$ and
$D^{FVV}_{(ab)L,R}$ are shown in Eq. (C.2) of appendix C. Our results are
consistent with previous works Barbieri:2015yvd ; Biggio:2016wyy . Although
they satisfy the WI, they contain divergences. For example, the divergent part
of $\delta C^{FVV}_{L}$ is

$\displaystyle\mathrm{div}\left[-\delta C^{FVV}_{L}\right]=$
$\displaystyle\frac{\delta
k_{v}eQ_{V}}{32\pi^{2}m_{V}^{2}}\left[g^{L*}_{a}g^{L}_{b}m_{a}+g^{R*}_{a}g^{R}_{b}m_{b}-2g^{R*}_{a}g^{L}_{b}m_{F}\right].$
(31)

Hence, $\delta k_{v}=0$ is equivalent to the renormalizable condition of the
theory, see a more detailed explanation in Ref. Biggio:2016wyy . This confirms
that the $AVV$ coupling listed in table 1 is still valid for a general UV-
complete model. Consequently, $\delta C^{FVV}_{L}=0$, implying that the
results of $C^{FVV}_{(ab)L,R}$ given in Eqs. (II) and (II) are unchanged for
many renormalizable theories.

###  III.2 Discussions on previous results

It is easy to derive that $C_{(ab)L,R}=\sigma_{L,R}$ corresponding to the
notations given in Ref. Lavoura:2003xp , see a detailed explanation in
appendix B. This confirms a perfect consistency of the two results obtained
from different original assumptions that we have indicated above. In addition,
these results are also consistent with those given in Ref. Crivellin:2018qmi
in the limit of heavy boson masses in the loops, which are very useful for
studying the correlations of AMM and cLFV decays.

In some BSM, SM light quark may play the role of the light fermions $u,d\equiv
F$ in the Yukawa couplings Dorsner:2020aaz , hence the condition $m_{F}^{2}\gg
m^{2}_{a},m^{2}_{b}$ is not held. But numerical illustrations Hue:2017lak  to
investigate cLFV decays $e_{b}\to e_{a}\gamma$ with very light neutrinos show
that the case of $m^{2}_{F}\ll m^{2}_{a}$ is also valid for approximation
formulas with $m^{2}_{a}=m^{2}_{b}=0$, provided $m^{2}_{a},m^{2}_{b}\ll
m^{2}_{h},m^{2}_{V}$. An analytic approximation to explain this result was
given in, for example, Ref. Hue:2015fbb .

For analytic formulas of cLFV and $a_{e_{a}}$ introduced in Ref.
Lindner:2016bgg , They can be changed into the form of PV-functions consistent
with our results. An exceptional case mentioned is the coupling of a doubly
charged boson with two identical leptons. For example, the Lagrangian
containing couplings of a doubly charged Higgs boson is Lindner:2016bgg :

$\displaystyle\mathcal{L}_{\mathrm{int}}=g^{ij}_{s3}\phi^{++}\overline{\ell^{C}_{i}}\ell_{j}+g^{ij}_{p3}\phi^{++}\overline{\ell^{C}_{i}}\gamma^{5}\ell_{j}+\mathrm{h.c.},$
(32)

where we can identify that $g^{R}_{a,Fh}=g^{ij}_{s3}+g^{ij}_{p3}$ and
$g^{R}_{a,Fh}=g^{ij}_{s3}-g^{ij}_{p3}$. But the Feynman rules for the vertex
$\overline{\ell^{C}_{i}}\ell_{j}\phi^{++}$ containing two identical leptons
give an extra factor 2, implying that $C_{(ab)L,R}$ given in Eqs. (II) and
(II) must be added a factor 4. Instead of many particular formulas to
calculate one-loop contributions relating to different charged particles, the
one-loop results for $(g-2)_{e_{a}}$ and $e_{b}\to e_{a}\gamma$ decays can be
generalized for $a_{e_{a}}$ with an arbitrary electric charge $Q_{F}$ of a new
fermion and the boson with $Q_{B}=Q_{F}-Q_{e}$ with $B=h,V$. Namely, the
$a_{e_{a}}$ formulas are

$\displaystyle a_{e_{a}}(h)=$
$\displaystyle\frac{Q_{h}m_{a}}{16\pi^{2}}\int_{0}^{1}dx\times\frac{x(x-1)\left[2\mathrm{Re}[g^{RL}]m_{F}+(g^{LL}+g^{RR})m_{a}x\right]}{(1-x)m_{F}^{2}+x\left[m_{h}^{2}+m_{a}^{2}(x-1)\right]}$
$\displaystyle+\frac{Q_{F}m_{a}}{16\pi^{2}}\int_{0}^{1}dx\times\frac{x^{2}\left[-2g^{RL}[g^{RL}]m_{F}+(g^{LL}+g^{RR})m_{a}(x-1)\right]}{(1-x)m_{h}^{2}+x\left[m_{F}^{2}+m_{a}^{2}(x-1)\right]},$
(33) $\displaystyle a_{e_{a}}(V)=$
$\displaystyle-\frac{Q_{V}m_{a}}{16\pi^{2}m_{V}^{2}}\int_{0}^{1}dx\times\left[\frac{\mathrm{Re}[g^{RL}]m_{F}\left[m_{F}^{2}(x-1)+m_{V}^{2}x(6x-1)+m_{a}^{2}x(3-5x+2x^{2})\right]}{(1-x)m_{F}^{2}+x\left[m_{V}^{2}+m_{a}^{2}(x-1)\right]}\right.$
$\displaystyle\left.-\frac{m_{a}(g^{LL}+g^{RR})\left[m_{F}^{2}(2-3x+x^{2})+m_{V}^{2}2x(x+1)+m_{a}^{2}x(x-1)\right]}{(1-x)m_{F}^{2}+x\left[m_{V}^{2}+m_{a}^{2}(x-1)\right]}\right]$
$\displaystyle+\frac{Q_{F}m_{a}}{16\pi^{2}m_{V}^{2}}\int_{0}^{1}dx\left[\frac{2g^{RL}[g^{RL}]m_{F}x\left[m_{F}^{2}x-4m_{V}^{2}(1-x)+m_{a}^{2}x(2x-1)\right]}{(1-x)m_{V}^{2}+x\left[m_{F}^{2}+m_{a}^{2}(x-1)\right]}\right.$
$\displaystyle\left.+\frac{(g^{LL}+g^{RR})m_{a}x\left[m_{F}^{2}x(1+x)+2m_{V}^{2}(2-3x+x^{2})+m_{a}^{2}x(x-1)\right]}{(1-x)m_{V}^{2}+x\left[m_{F}^{2}+m_{a}^{2}(x-1)\right]}\right],$
(34)

where $g^{RL}=g^{R*}_{a,FB}g^{L}_{a,FB}$, $g^{LL}=g^{L*}_{a,FB}g^{L}_{a,FB}$,
and $g^{RR}=g^{R*}_{a,FB}g^{R}_{a,FB}$ with $B=h,V$. The coupling
identifications are $g^{R}_{a,Fh}=g^{aa}_{sk}+g^{aa}_{pk}$ and
$g^{R}_{a,Fh}=g^{aa}_{sk}-g^{aa}_{pk}$ for $k=1,2,3$ relating to neutral,
singly, and doubly charged Higgs bosons. Similarly to the gauge bosons,
$g^{R}_{a,FV}=g^{aa}_{vk}+g^{aa}_{ak}$ and
$g^{R}_{a,FV}=g^{aa}_{vk}-g^{aa}_{ak}$ for $Q_{V}=1,0,-1,2$ corresponding
$k=1,2,3,4$. The two formulas (33) and (34) are derived by inserting the PV
functions given in appendix A in the limit $p_{1}^{2}=p_{2}^{2}=m_{a}^{2}$
into $C_{(ab)L,R}$. We have checked that our results are consistent with all
$HFF$, $FHH$, and $VFF$ contributions relating to the diagrams (1), (2), and
(6) in Fig. 1, respectively. For the one-loop $FVV$ contributions arising from
the diagram (5), there is a difference between our result and that in Ref.
Lindner:2016bgg , namely

$\delta(a_{e_{a}})(FVV)=\frac{Q_{V}m_{a}m_{F}}{16\pi^{2}m_{V}^{2}}(|g^{aa}_{vk}|^{2}-|g^{aa}_{ak}|^{2})\int_{0}^{1}dx(2x+1)=\frac{Q_{V}m_{a}m_{F}}{8\pi^{2}m_{V}^{2}}(|g^{aa}_{vk}|^{2}-|g^{aa}_{ak}|^{2}).$

It shows that the two results are consistent if $g^{aa}_{vk}=\pm
g^{aa}_{ak}$,i.e., $g^{L}_{a,FB}g^{R}_{a,FB}=0$, which appears in many BSM
such as the SM, 3-3-1 models,… We also see that the $FVV$ contribution to
$a_{e_{a}}$ of the doubly gauge boson given in Ref. Lindner:2016bgg  has an
opposite sign with our result.

We note that our results are also valid as the exact solutions for studying
the AMM and $e_{b}\to e_{a}\gamma$ decay in BSM consisting of very light
bosons $m_{B}\ll m^{2}_{a},m^{2}_{b}$ such as an axion-like particle (ALP)
Bauer:2019gfk ; Cornella:2019uxs , or a new scalar singlet Liu:2018xkx .

##  IV  Conclusion

Using the unitary gauge, we confirm the exact results of analytic formulas in
terms of PV functions for one-loop contributions to the cLFV decay rates
$e_{b}\to e_{a}\gamma$ given in Ref. Lavoura:2003xp , which are also
applicable to compute the AMM of charged leptons. These results are consistent
with those given in Ref. Crivellin:2018qmi  in the limit of heavy bosons
$m_{B}\gg m_{a},m_{b}$. The general expressions in terms of PV-functions are
very convenient to change into available forms. Our calculations here have
many new features as follows. Our calculation is independent of the Goldstone
boson couplings of the new gauge bosons. The Ward Identity of the external
photon allows only the couplings of a photon with two identical physical
particles, as given in table 1. At tree-level, the $ASV$ couplings do not
satisfy the WI if $\varepsilon_{v}.p_{0}\neq 0$, where $\varepsilon_{v}$ and
$p_{0}$ are the polarization of gauge boson $V$ and the external momentum of
the photon, respectively. The one-loop $FSV$ contributions arising from this
vertex type to cLFV amplitudes and AMM do violate the WI. Therefore, the
results given in Ref. Lavoura:2003xp ; Crivellin:2018qmi  are valid in all
renormalizable BSM respecting the WI. They are still applied for other similar
decays of quarks $q\to q^{\prime}\gamma$. The photon-scalar-vector $ASV$
vertex does not appear in BSM satisfying the WI. Our conclusion is very useful
for constructing loop calculations relating to photon couplings, where only
the vertex types listed in Table 1 are valid.

## Acknowledgments

The authors thank the referee for suggesting an open question about the
existence of the $S$-$V$-$\gamma$ couplings in the UV models, which we will
solve more generally in our future work. This research is funded by the
Vietnam National Foundation for Science and Technology Development (NAFOSTED)
under the grant number 103.01-2019.387.

##  Appendix A  PV functions for one loop contributions defined by LoopTools

###  A.1 General notations

The PV-functions used here were listed in Ref. Hue:2017lak , namely

$\displaystyle
A_{0}(m^{2})=\frac{(2\pi\mu)^{4-d}}{i\pi^{2}}\int\frac{d^{d}k}{k^{2}-m^{2}+i\delta},$
$\displaystyle
B_{\\{0,\mu\\}}(p^{2}_{i},M^{2}_{1},M^{2}_{2})=\frac{(2\pi\mu)^{4-d}}{i\pi^{2}}\int\frac{d^{d}k\times\\{1,k_{\mu},k_{\mu}k_{\nu}\\}}{D_{0}D_{i}},\;i=1,2,$
$\displaystyle
C_{\\{0,\mu,\mu\nu\\}}=\frac{(2\pi\mu)^{4-d}}{i\pi^{2}}\int\frac{d^{d}k\\{1,k_{\mu},k_{\mu}k_{\nu}\\}}{D_{0}D_{1}D_{2}},$
$\displaystyle
B_{\mu}(p^{2}_{i},M^{2}_{1},M^{2}_{2})=\left(-p_{i\mu}\right)B^{(i)}_{1},$
$\displaystyle
C_{\mu}=\left(-p_{1\mu}\right)C_{1}+\left(-p_{2\mu}\right)C_{2},$
$\displaystyle
C_{\mu\nu}=g_{\mu\nu}C_{00}+p_{1\mu}p_{1\nu}C_{11}+p_{2\mu}p_{2\nu}C_{22}+(p_{1\mu}p_{2\nu}+p_{2\mu}p_{1\nu})C_{12},$
(35)

where $D_{0}\equiv k^{2}-M_{1}^{2}+i\delta$,
$D_{i}\equiv(k-p_{i})^{2}-M_{2}^{2}+i\delta$,
$C_{0,\mu,\mu\nu}=C_{0,\mu,\mu\nu}(p_{1}^{2},0,p_{2}^{2};M_{1}^{2},M_{2}^{2},M_{2}^{2})$,
$\mu$ is an arbitrary mass parameter introduced via dimensional regularization
tHooft:1972tcz . In this work, we discuss only the case of external photon
$q^{2}=(p_{2}-p_{1})^{2}=0$. The scalar functions
$A_{0},B_{0},B^{(i)}_{1},C_{0},C_{00},C_{i}$, and $C_{ij}$ ($i,j=1,2$) are
well-known PV functions, which are consistent with those defined by LoopTools
Hahn:1998yk . The well-known relations are:

$\displaystyle B^{(i)}_{0}\equiv
B^{(i)}_{0}(p_{i}^{2};M_{1}^{2},M_{2}^{2})=B^{(i)}_{0}(p_{i}^{2};M_{2}^{2},M_{1}^{2}),$
$\displaystyle B^{(i)}_{1}\equiv
B^{(i)}_{1}(p_{i}^{2};M_{1}^{2},M_{2}^{2})=-\frac{1}{2p_{i}^{2}}\left[A_{0}(M_{2}^{2})-A_{0}(M_{1}^{2})+f_{i}B^{(i)}_{0}\right],$
(36)

where $f_{i}=p_{i}^{2}+M_{2}^{2}-M_{1}^{2}$. Depending on the particle
exchanges in Feynman diagrams, the $B^{(i)}_{1}$-function given in Eq. (A.1)
is denoted more precisely as follows:

$\displaystyle B^{(i)f}_{1}\equiv
B^{(i)}_{1}(p_{i}^{2};m_{F}^{2},m_{B}^{2}),\;B^{(i)v}_{1}\equiv
B^{(i)}_{1}(p_{i}^{2};m_{V}^{2},m_{F}^{2}),\;B^{(i)h}_{1}\equiv
B^{(i)}_{1}(p_{i}^{2};m_{h}^{2},m_{F}^{2}).$ (37)

The scalar functions $A_{0}$, $B_{0}$, and $C_{0}$ can be calculated using the
techniques of tHooft:1978jhc . Other PV functions needed in this work are

$\displaystyle
B_{0,\mu,\mu\nu}(M_{2})=\frac{(2\pi\mu)^{4-d}}{i\pi^{2}}\int\frac{d^{d}k~{}\\{1,\;k_{\mu},\;k_{\mu}k_{\nu}\\}}{D_{1}D_{2}}.$
(38)

For simplicity, we define the following notations appearing in many important
formulas:

$\displaystyle X_{0}$ $\displaystyle\equiv C_{0}+C_{1}+C_{2},$ $\displaystyle
X_{1}$ $\displaystyle\equiv C_{11}+C_{12}+C_{1},$ $\displaystyle X_{2}$
$\displaystyle\equiv C_{12}+C_{22}+C_{2},$ $\displaystyle X_{3}$
$\displaystyle\equiv C_{1}+C_{2}=X_{0}-C_{0},$ $\displaystyle X_{012}$
$\displaystyle\equiv X_{0}+X_{1}+X_{2},\;X_{ij}=X_{i}+X_{j}.$ (39)

Depending on the form of the PV-functions, we have

$\displaystyle X^{f}_{i}$
$\displaystyle=X_{i}(m_{F}^{2},m_{B}^{2},m_{B}^{2}),\;X^{h}_{i}\sim
X_{i}(m_{B}^{2},m_{F}^{2},m_{F}^{2}),\;X^{v}_{i}\sim
X_{i}(m_{V}^{2},m_{F}^{2},m_{F}^{2})$ (40)

corresponding to the diagram types of $FBB$, $HFF$, and $VFF$ with $B=h,V$.

###  A.2  $p_{1}^{2}\neq p_{2}^{2}\neq 0$ and $p_{1}^{2},p_{2}^{2}\neq 0$.

From the definitions of PV-functions given in Eq. (35), it can be proved that:

$\displaystyle B^{(0)}_{0}\equiv B_{0}^{(0)}(M_{2})\equiv
B_{0}(0;M_{2},M_{2})=C_{UV}-\ln(M_{2}^{2})+\mathcal{O}(\epsilon),$
$\displaystyle A_{0}(M)=M^{2}\left[B^{(0)}_{0}(M)+1\right],$ (41)
$\displaystyle
B^{\mu}(M_{2})=\frac{1}{2}B^{(0)}_{0}(p^{\mu}_{1}+p^{\mu}_{2}),$ (42)
$\displaystyle
B^{\mu\nu}(M_{2})=\frac{g^{\mu\nu}}{2}M^{2}_{2}\left[B^{(0)}_{0}+1\right]+\frac{1}{6}B^{(0)}_{0}\left(2p_{1}^{\mu}p_{1}^{\nu}+p_{1}^{\mu}p_{2}^{\nu}+p_{2}^{\mu}p_{1}^{\nu}+2p_{2}^{\mu}p_{2}^{\nu}\right),$
$\displaystyle
C_{00}=\frac{1}{4}\left[2M_{2}^{2}C_{0}+\frac{\left(M_{2}^{2}-M_{1}^{2}+m_{a}^{2}\right)B^{(1)}_{0}-\left(M_{2}^{2}-M_{1}^{2}+m_{b}^{2}\right)B^{(2)}_{0}}{m_{a}^{2}-m_{b}^{2}}+1\right],$
(43)

where $C_{UV}$ is defined as the divergent part of the PV functions when $D\to
4$, $C_{UV}=1/\epsilon-\gamma_{E}+\log(4\pi\mu^{2})$ with $\gamma_{E}$ being
Euler’s constant and $D=4-2\epsilon$. It is well-known that the PV-functions
having non-zero divergent parts are:

$\displaystyle\mathrm{div}\left[B^{(0)}_{0}\right]$
$\displaystyle=\mathrm{div}\left[B^{(1)}_{0}\right]=\mathrm{div}\left[B^{(2)}_{0}\right]=-2\mathrm{div}\left[B^{(1)}_{1}\right]=-2\mathrm{div}\left[B^{(2)}_{1}\right]=4\mathrm{div}\left[C_{00}\right]=C_{UV},$
$\displaystyle\mathrm{div}\left[A_{0}(M)\right]$ $\displaystyle=M^{2}C_{UV}.$
(44)

As mentioned in Ref. Hue:2017lak , we can derive all formulas of $C_{i}$, and
$C_{ij}$ as functions of $A_{0}$, $B^{(i)}_{0}$, and $C_{0}$ consistent with
Ref. Hue:2017lak , using the following relations:

$\displaystyle
2m^{2}_{a}C_{1}+(m_{a}^{2}+m_{b}^{2})C_{2}=-f_{a}C_{0}-B^{(0)}_{0}+B^{(2)}_{0},$
$\displaystyle(m_{a}^{2}+m_{b}^{2})C_{1}+2m^{2}_{b}C_{2}=-f_{b}C_{0}-B^{(0)}_{0}+B^{(1)}_{0},$
$\displaystyle
2C_{00}+2m^{2}_{a}C_{11}+(m_{a}^{2}+m_{b}^{2})C_{12}=\frac{1}{2}B^{(0)}_{0}-f_{a}C_{1},$
$\displaystyle
2m^{2}_{a}C_{12}+(m_{a}^{2}+m_{b}^{2})C_{22}=\frac{1}{2}B^{(0)}_{0}+B^{(2)}_{1}-f_{a}C_{2}$
$\displaystyle
2C_{00}+(m_{a}^{2}+m_{b}^{2})C_{12}+2m^{2}_{b}C_{22}=\frac{1}{2}B^{(0)}_{0}-f_{b}C_{2},$
$\displaystyle(m_{a}^{2}+m_{b}^{2})C_{11}+2m^{2}_{b}C_{12}=\frac{1}{2}B^{(0)}_{0}+B^{(1)}_{1}-f_{b}C_{1},$
$\displaystyle
4C_{00}-\frac{1}{2}+m_{a}^{2}C_{11}+(m_{a}^{2}+m_{b}^{2})C_{12}+m_{b}^{2}C_{22}=B^{(0)}_{0}+M_{1}^{2}C_{0},$
(45)

where $f_{a,b}=M_{2}^{2}-M_{1}^{2}+m_{a,b}^{2}$, and $C_{12}=C_{21}$ are used.
In this work, we need just combinations of these PV-functions for our
immediate steps. In particular, we can prove that:

$\displaystyle X_{0}$
$\displaystyle=-\frac{B^{(1)}_{0}-B^{(2)}_{0}}{m_{a}^{2}-m_{b}^{2}},$
$\displaystyle X_{12}$
$\displaystyle=-\frac{B^{(1)}_{1}-B^{(2)}_{1}}{m_{a}^{2}-m_{b}^{2}}$
$\displaystyle=\frac{A_{0}(M_{1}^{2})-A_{0}(M_{2}^{2})}{2m_{a}^{2}m_{b}^{2}}+\frac{(M_{1}^{2}-M_{2}^{2})}{2(m_{a}^{2}-m_{b}^{2})}\left(\frac{B^{(1)}_{0}}{m_{a}^{2}}-\frac{B^{(2)}_{0}}{m_{b}^{2}}\right)-\frac{1}{2}X_{0},$
$\displaystyle m_{a}^{2}B^{(1)}_{1}-m_{b}^{2}B^{(2)}_{1}$
$\displaystyle=-\frac{1}{2}\left[\left(m_{a}^{2}+M_{2}^{2}-M_{1}^{2}\right)B^{(1)}_{0}-\left(m_{b}^{2}+M_{2}^{2}-M_{1}^{2}\right)B^{(2)}_{0}\right],$
$\displaystyle\mathbf{b}_{1}$
$\displaystyle\equiv\frac{m_{a}^{2}B^{(1)}_{1}-m_{b}^{2}B^{(2)}_{1}}{(m_{a}^{2}-m_{b}^{2})}=-(2C_{00}+m_{a}^{2}X_{1}+m_{b}^{2}X_{2}),$
$\displaystyle(2-d)C_{00}+M_{2}^{2}C_{0}$
$\displaystyle=-2C_{00}+\frac{1}{2}+M_{2}^{2}C_{0}$
$\displaystyle=-\frac{\left(m_{a}^{2}+M_{1}^{2}-M_{2}^{2}\right)B^{(1)}_{0}-\left(m_{b}^{2}+M_{1}^{2}-M_{2}^{2}\right)B^{(2)}_{0}}{2(m_{a}^{2}-m_{b}^{2})}$
$\displaystyle=\mathbf{b}_{1}+(M_{2}^{2}-M_{1}^{2})X_{0},$ (46)

where $A_{0}(M_{2}^{2})=M_{2}^{2}(B^{(0)}_{0}+1)$ and
$A_{0}(M_{1}^{2})=M_{1}^{2}(B^{(0)}_{0}+1+\ln(M_{2}^{2}/M_{1}^{2}))$.

It was proved previously, for example Hue:2017lak , that

$\displaystyle
B_{0}(p^{2};M^{2}_{1},M^{2}_{2})=B_{0}(p^{2};M^{2}_{2},M^{2}_{1})=C_{UV}-\ln(M^{2}_{2})+2-\sum_{\sigma=\pm}(1-\frac{1}{x_{\sigma}})\ln\left(1-x_{\sigma}\right),$
$\displaystyle
C_{0}(m_{a}^{2},0,m_{b}^{2};M^{2}_{1},M^{2}_{2},M^{2}_{2})=-\frac{1}{m_{a}^{2}-m_{b}^{2}}\sum_{\sigma=\pm}\left[\mathrm{Li}_{2}(y_{a\sigma})-\mathrm{Li}_{2}(y_{b\sigma})\right],$
(47)

where $p=p_{a},p_{b}$; and

$\displaystyle x_{\pm}$
$\displaystyle=\frac{1}{2M_{2}^{2}}\left[(M_{2}^{2}-M_{1}^{2}+p^{2})\pm\sqrt{(M_{2}^{2}-M_{1}^{2}+p^{2})^{2}-4M_{2}^{2}p^{2}}\right],$
$\displaystyle y_{a\pm}$
$\displaystyle=\frac{1}{2M_{2}^{2}}\left[(M_{2}^{2}-M_{1}^{2}+m^{2}_{a})\pm\Lambda\right],$
$\displaystyle y_{b\pm}$ $\displaystyle=x_{a\pm}\left[b\to a\right]$ (48)

with
$\Lambda=(M_{1}^{4}+M_{2}^{4}+m_{a}^{4}-2M_{1}^{2}M_{2}^{2}-2M_{1}^{2}m_{a}^{2}-2M_{2}^{2}m_{a}^{2})^{1/2}$.
The above formula of $C_{0}$ is also consistent with that introduced in loop-
induced decay amplitude of $h\to Z\gamma$ Djouadi:1996yq .

###  A.3 $m_{a}^{2}=p_{a}^{2}=p_{b}^{2}\neq 0$

Formulas for AMM in Ref. Yu:2021suw  require that analytic formulas of PV
functions with $m_{b}=m_{a}$. It seems that the results of PV-functions listed
in Ref. Hue:2017lak  are not valid. But the limit $m_{b}=m_{a}$ can be derived
mathematically. For example, the result of $C_{0}$ given in Eq. (A.2) leads to
a consequence that

$\displaystyle C_{0}(m_{a}^{2},0,m_{a}^{2};M^{2}_{1},M^{2}_{2},M^{2}_{2})$
$\displaystyle=\lim_{m_{b}\to
m_{a}}C_{0}(m_{a}^{2},0,m_{b}^{2};M^{2}_{1},M^{2}_{2},M^{2}_{2})$
$\displaystyle=-\frac{\partial}{\partial(m_{a}^{2})}\sum_{\sigma=\pm}\mathrm{Li}_{2}(y_{a\sigma})=\sum_{\sigma=\pm}\frac{y^{\prime}_{a\sigma}\ln(1-y_{a\sigma})}{y_{a\sigma}}$
$\displaystyle=\sum_{\sigma=\pm}\frac{\ln(1-y_{a\sigma})}{2M_{2}^{2}y_{a\sigma}}\times\left[1-\sigma\times\frac{M_{1}^{2}+M_{2}^{2}-m_{a}^{2}}{\Lambda}\right],$
(49)

where $f^{\prime}\equiv\partial f/(\partial m_{a}^{2})$ denotes a well-known
derivative notation. In addition, $B^{(1)}_{0}=B^{(2)}_{0}$ and
$B^{(1)}_{1}=B^{(2)}_{1}$ are automatically satisfied. Many formulas
containing $(m_{a}^{2}-m_{b}^{2})$ in the denominators corresponding a
derivative in the limit $m_{a}\to m_{b}$:

$\displaystyle X_{0}$
$\displaystyle=-B^{(1)\prime}_{0}=\sum_{\sigma=\pm}\frac{y^{\prime}_{a\sigma}\left[y_{a\sigma}+\ln(1-y_{a\sigma})\right]}{y_{a\sigma}^{2}},$
$\displaystyle X_{12}$ $\displaystyle=-B^{(1)\prime}_{1}.\dots,$ (50)

In this way, we can confirm all results introduced in Ref. Yu:2021suw . There
is another way to calculate form factors, using the Feynman trick:

$\displaystyle\frac{1}{D_{0}D_{1}D_{2}}$
$\displaystyle=\Gamma(3)\int^{1}_{0}\frac{dx~{}dy~{}dz~{}\delta(1-x-y-z)}{D^{3}},$
(51)

where

$\displaystyle D$
$\displaystyle=\left[k-(yp_{1}+zp_{2})\right]^{2}-M^{2}+i\delta,$
$\displaystyle M^{2}$
$\displaystyle=y(y+z-1)p_{1}^{2}+z(y+z-1)p_{2}^{2}+xM_{1}^{2}+(1-x)M_{2}^{2}.$
(52)

With
$M_{0}^{2}=(p_{2}^{2}-p_{1}^{2})xy-x(1-x)p_{2}^{2}+xM_{1}^{2}+(1-x)M_{2}^{2}$,
the PV functions are:

$\displaystyle C_{\\{0,1,2,11,22,12\\}}$
$\displaystyle=-\int_{0}^{1}dx\int_{0}^{1-x}\frac{dy\left\\{1,-y,-(1-x-y),y^{2},(1-x-y)y,(1-x-y)^{2}\right\\}}{M_{0}^{2}},$
$\displaystyle X_{0,1,2,3}$
$\displaystyle=-\int_{0}^{1}dx\int_{0}^{1-x}\frac{dy\times\\{x,-xy,-x(1-x-y),-(1-x)\\}}{M_{0}^{2}}.$
(53)

The expressions of $X_{i}$ in Eq. (A.3) are very convenient for the case of
$(g-2)$ anomaly, where $p_{1}^{2}=p_{2}^{2}=m_{a}^{2}$ results in
$M_{0}^{2}=-x(1-x)m_{a}^{2}+xM_{1}^{2}+(1-x)M_{2}^{2}$, which is independent
with $y$. Consequently, the

$\displaystyle X_{0,1,2,3}$
$\displaystyle=-\int_{0}^{1}dx\frac{\\{x(1-x),-x(1-x)^{2}/2,-x(1-x)^{2}/2,-(1-x)^{2}\\}}{M_{0}^{2}}$
$\displaystyle=-\int_{0}^{1}dx\frac{\\{x(1-x),-(1-x)x^{2}/2,-(1-)x^{2}/2,-x^{2}\\}}{M_{0}^{2}},$
(54)

Formulas of Eq. (A.3) are enough to check the consistency between our results
with those of $(g-2)$ anomalies and cLFV amplitudes mentioned in ref.
Lindner:2016bgg . Using the second line of Eq. (A.3), we can write the general
formulas of $a_{\mu}$ as shown in Eqs. (33) and (34).

Indeed, all integrals in Eqs. (33) and (34) can be solved analytically.
Starting from the general formulas of $M_{0}^{2}$ as functions of $x$:
$M_{0}^{2}(x)=m_{a}^{2}(x-x_{+})(x-x_{-})$ corresponding to the two solutions
$x_{\pm}$. All numerators in Eqs. (33) and (34) are always written in the
following forms:

$\displaystyle
ax^{2}+bx^{2}+c=a_{1}M_{0}^{2}+b_{1}\frac{dM_{0}^{2}}{dx}+c_{1}.$ (55)

The consequence is

$\displaystyle\int_{0}^{1}dx\times\frac{ax^{2}+bx^{2}+c}{M_{0}^{2}}=a_{1}+b_{1}\ln\frac{M_{1}^{2}}{M_{2}^{2}}+\frac{c_{1}}{\sqrt{\Lambda}}\ln\left[\frac{(1-x_{-})x_{+}}{(1-x_{+})x_{-}}\right].$
(56)

The result in this way must be consistent with those discussed in Ref.
Yu:2021suw , hence we do not show precisely here.

###  A.4 $p_{a}^{2}=p_{b}^{2}=0$

Results for the case of $p^{2}_{a}=p^{2}_{b}=0$ were provided in Ref.
Lavoura:2003xp , namely

$\displaystyle C_{0}$
$\displaystyle=a=\frac{M_{1}^{2}-M_{2}^{2}+M_{1}^{2}\ln\left[\frac{M_{2}^{2}}{M_{1}^{2}}\right]}{(M_{1}^{2}-M_{2}^{2})^{2}},$
$\displaystyle C_{1}$
$\displaystyle=C_{2}=c=-\frac{3M_{1}^{4}-4M_{1}^{2}M_{2}^{2}+M_{2}^{4}+2M_{1}^{4}\ln\left[\frac{M_{2}^{2}}{M_{1}^{2}}\right]}{4(M_{1}^{2}-M_{2}^{2})^{3}},$
$\displaystyle C_{11}$
$\displaystyle=C_{22}=2C_{12}=d=\frac{11M_{1}^{6}-18M_{1}^{4}M_{2}^{2}+9M_{1}^{2}M_{2}^{4}-2M_{2}^{6}+6M_{1}^{6}\ln\left[\frac{M_{2}^{2}}{M_{1}^{2}}\right]}{18(M_{1}^{2}-M_{2}^{2})^{4}}.$
(57)

This approximate formulas of PV functions give results consistent with those
given in Ref. Crivellin:2018qmi , namely

$\displaystyle f_{h}(x)=2\tilde{g}_{h}(x)=\dfrac{x^{2}-1-2x\log
x}{4(x-1)^{3}},$ $\displaystyle g_{h}(x)=\dfrac{x-1-\log x}{2(x-1)^{2}},$
$\displaystyle\tilde{f}_{h}(x)=\dfrac{2x^{3}+3x^{2}-6x+1-6x^{2}\log
x}{24(x-1)^{4}},$ (58) $\displaystyle
f_{V}(x)=\dfrac{x^{3}-12x^{2}+15x-4+6x^{2}\log x}{4(x-1)^{3}},$ $\displaystyle
g_{V}(x)=\dfrac{x^{2}-5x+4+3x\log x}{2(x-1)^{2}},$
$\displaystyle\tilde{f}_{V}(x)=\dfrac{-4x^{4}+49x^{3}-78x^{2}+43x-10-18x^{3}\log
x}{24(x-1)^{4}},$
$\displaystyle\tilde{g}_{V}(x)=\dfrac{-3(x^{3}-6x^{2}+7x-2+2x^{2}\log
x)}{8(x-1)^{3}},$

where $x\equiv m_{F}^{2}/m_{B}^{2}$. The diagrams $FBB$ and $BFF$ correspond
to different identifications that $\\{M_{1},M_{2}\\}=\\{m_{F},m_{B}\\}$ or and
$\\{M_{1},M_{2}\\}=\\{m_{B},m_{F}\\}$.

##  Appendix B  Notations in Ref. Lavoura:2003xp

Here we give a brief review of the approach of Ref. Lavoura:2003xp . Apart
from the general couplings of physical Higgs and gauge bosons given in Eqs.
(3) and (4), the photon couplings were assumed to be the standard forms given
in table 1. Furthermore, the couplings of the Goldstone boson $G_{V}$
corresponding to $V$ are assumed to be the following forms:

$\displaystyle\mathcal{L}_{G_{V}}=$
$\displaystyle\left\\{G_{V}\frac{i}{m_{V}}\overline{F}\left[\left(g^{R}_{a,FV}m_{a}-g^{L}_{a,FV}m_{F}\right)P_{L}+\left(g^{L}_{a,FV}m_{a}-g^{R}_{a,FV}m_{F}\right)P_{R}\right]e_{a}+\mathrm{h.c.}\right\\}$
$\displaystyle+eQ_{V}m_{V}A_{\mu}V^{*\mu}G_{V}-eQ_{V}m_{V}A_{\mu}V^{\mu}G^{*}_{V}.$
(59)

The above assumptions of the $G_{V}$ couplings are necessary for the
calculation of one-loop gauge contributions that were done in the ’t Hoof
Feynman gauge. These final results introduced in Ref. Lavoura:2003xp  were the
sum of all diagrams consisting of gauge and Goldstone boson exchanges.
Corresponding to the two one-loop diagram classes $FVV$ and $VFF$, we have the
following equivalence between two classes of notations

$\displaystyle\left\\{a,c_{1},c_{2},d_{1},d_{2},f,g\right\\}$
$\displaystyle\equiv\left\\{C_{0},C_{2},C_{1},C_{22},C_{11},C_{12},C_{00}\right\\}^{B},$
$\displaystyle\left\\{\bar{a},-\bar{c}_{1},-\bar{c}_{2},\bar{d}_{1},\bar{d}_{2},\bar{f},\bar{g}\right\\}$
$\displaystyle\equiv\left\\{C_{0},C_{2},C_{1},C_{11},C_{22},C_{12},C_{00}\right\\}^{f},$

where $B=h,V$ are gauge bosons in the loop. In addition, the different
notations in the definitions of the one-loop integrals given in Eq. (35), we
have $\\{m_{1},m_{2}\\}\equiv\\{m_{b},m_{a}\\}$ while
$\\{p_{1},p_{2}\\}\equiv\\{-p_{2},-p_{1}\\}$ and
$\\{p_{1},p_{2}\\}\equiv\\{p_{2},p_{1}\\}$ for the diagrams $VFF$ and $FVV$
respectively. The couplings in the Yukawa Lagrangian of physical bosons are
$L_{1}\equiv g_{b}^{L}$, $R_{1}\equiv g_{b}^{R}$, $L_{2}\equiv g_{a}^{L}$, and
$R_{2}\equiv g^{a}_{R}$, which result in the following equivalences:
$\lambda\equiv g^{L*}_{a}g^{L}_{b}=g^{LL}$, $\rho\equiv
g^{R*}_{a}g^{R}_{b}=g^{RR}$, $\zeta\equiv g^{L*}_{a}g^{R}_{b}=g^{LR}$, and
$v\equiv g^{R*}_{a}g^{L}_{b}=g^{RL}$. As a result, we can identify that:

$\displaystyle k_{1}$
$\displaystyle=m_{b}X_{2}^{B},\;k_{2}=m_{a}X_{1}^{B},\;k_{3}=m_{F}(c_{1}+c_{2})=m_{F}X_{3}^{B},$
$\displaystyle\bar{k}_{1}$
$\displaystyle=m_{b}X_{2}^{f},\;\bar{k}_{2}=m_{b}X_{1}^{f},\;k_{3}=-m_{F}X_{3}^{f}.$
(60)

For a gauge boson $B_{\mu}$, the one-loop form factors relate to the following
notations:

$\displaystyle y_{1}$
$\displaystyle=m_{b}\left[2X_{02}^{f}+\frac{m_{F}^{2}(X^{f}_{2}-X^{f}_{3})+m_{a}^{2}X^{f}_{1}}{m_{B}^{2}}\right],\;y_{2}=m_{a}\left[2X_{01}^{f}+\frac{m^{2}_{F}(X_{1}^{f}-X_{3}^{f})+m_{b}^{2}X_{2}}{m^{2}_{B}}\right],$
$\displaystyle y_{3}$
$\displaystyle=m_{F}\left[-4X_{0}^{f}+\frac{m^{2}_{F}X_{3}^{f}+m^{2}_{a}X_{1}^{f}+m^{2}_{b}X^{f}_{2}}{m^{2}_{B}}\right],\;y_{4}=-\frac{m_{a}m_{b}m_{F}(X_{12}^{f}-X_{3}^{f})}{m^{2}_{B}},$
(61)

and

$\displaystyle\bar{y}_{1}$
$\displaystyle=m_{b}\left[2(X_{2}^{f}-X_{3}^{f})+\frac{m^{2}_{F}X_{02}^{f}+m_{a}^{2}X_{1}^{f}}{m^{2}_{B}}\right],\;\bar{y}_{2}=m_{a}\left[2(X_{1}^{f}-X_{3}^{f})+\frac{m^{2}_{F}X_{01}^{f}+m_{b}^{2}X_{2}^{f}}{m^{2}_{B}}\right],$
$\displaystyle\bar{y}_{3}$
$\displaystyle=m_{F}\left[4X_{3}^{f}+\frac{-m^{2}_{F}X_{0}-m_{a}^{2}X_{1}-m_{b}^{2}X_{2}}{m^{2}_{B}}\right],\;\bar{y}_{4}=-\frac{m_{a}m_{b}m_{F}}{m^{2}_{B}}X_{012}.$
(62)

##  Appendix C  Important steps to derive $C_{(ab)L,R}$ and $D_{(ab)L,R}$ by
hand

The notations for calculating the amplitude corresponding to all diagrams of
both Higgs and gauge boson exchanges in Fig. 1 are shown in Fig. 2.

Figure 2: Momneta notations to derive the one-loop contributions

All diagrams in the same class will have the same conventions of external
momenta and propagators. There are three classes of diagrams: i) The first
class consists of four diagrams (1), (2), (5), and (6) in Fig. 1, and the two
diagrams (1) and (2) in Fig. 2; ii) the second class consists of three
diagrams: (3) and (7) in Fig. 1, and (3) in Fig. 2; iii) the last class
consists of the remaining diagrams in the two Figs. 1 and 2. Although all the
internal momenta have opposite signs with those denoted following LoopTools,
the PV-functions are defined with the same values. The relations relevant to
momenta are:

$\displaystyle k_{i}$
$\displaystyle=k-p_{i},\;p_{1}^{2}=m^{2}_{a},\;p_{2}=q+p_{1},\;p_{2}^{2}=m^{2}_{b},\;q^{2}=0,$
$\displaystyle q.\varepsilon^{*}$
$\displaystyle=0,\;p_{1}.\varepsilon^{*}=p_{2}.\varepsilon^{*},\;$ (63)

Only four diagrams (1), (2), (5), and (6) in Fig. 1 give non-zero
contributions to $C_{(ab)L,R}$, hence we firstly derive $C_{(ab)L,R}$ as the
factors of $(2p_{1}.\varepsilon^{*})$ in the amplitudes arising from these
diagrams. For convenience in detailed calculations, we use simple notations
for all the coupling factors $g^{aL,R}_{FB}\to g^{L,R}_{a}$. For integrals
containing divergences, we use the regular dimensional regularization defined
by the following replacement:

$\int\frac{d^{4}k}{(2\pi)^{4}}\to\frac{i}{16\pi^{2}}\times\frac{(2\pi\mu)^{4-d}}{i\pi^{2}}\int
d^{d}k\equiv\int Dk.$

The final results now are written in terms of the PV functions. In many
intermediate steps, we use many results for products of gamma matrices in the
dimension $d$ Peskin:1995ev , namely

$\displaystyle\gamma^{\mu}\gamma_{\mu}=d,$
$\displaystyle\gamma^{\mu}\gamma^{\nu}\gamma_{\mu}=(2-d)\gamma^{\nu}\to\gamma^{\mu}\not{p}\gamma^{\mu}=(2-d)\not{p},$
$\displaystyle\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma_{\mu}=4g^{\nu\rho}+(d-4)\gamma^{\nu}\gamma^{\rho}\to\gamma^{\mu}\not{p}_{1}\not{p}_{2}\gamma^{\mu}=4p_{1}.p_{2}+(d-4)\not{p}_{1}\not{p}_{2},$
$\displaystyle\gamma^{\mu}\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\gamma_{\mu}=-2\gamma^{\sigma}\gamma^{\rho}\gamma^{\nu}-(d-4)\gamma^{\nu}\gamma^{\rho}\gamma^{\sigma}\to\gamma^{\mu}\not{p}_{1}\not{p}_{2}\not{p}_{3}\gamma_{\mu}=-2\not{p}_{3}\not{p}_{2}\not{p}_{1}-(d-4)\not{p}_{1}\not{p}_{2}\not{p}_{3},\dots$

###  C.1 Scalar contributions

We list here 8 formulas of amplitudes corresponding to 8 particular diagrams
shown in Fig. 1. Namely, for three diagrams (1), (3), and (4) we have

$\displaystyle i\mathcal{M}_{1}=$ $\displaystyle-eQ_{H}\int
Dk\times\overline{u_{a}}[{g_{a}^{R}}^{*}P_{L}+{g_{a}^{L}}^{*}P_{R}]\frac{(m_{F}+\not{k})}{D_{0}D_{1}D_{2}}[g_{b}^{L}P_{L}+g_{b}^{R}P_{R}]u_{b}\times(2k_{1}.\varepsilon^{*}),$
(64) $\displaystyle i\mathcal{M}_{3}=$
$\displaystyle\frac{-eQ_{e}}{m^{2}_{a}-m^{2}_{b}}\int
Dk\times\overline{u_{a}}[{g_{a}^{R}}^{*}P_{L}+{g_{a}^{L}}^{*}P_{R}]\frac{(m_{F}+\not{k})}{D_{0}D_{1}}[g_{b}^{L}P_{L}+g_{b}^{R}P_{R}](m_{b}+\not{p}_{1})\not{\varepsilon}^{*}u_{b},$
(65) $\displaystyle i\mathcal{M}_{4}=$
$\displaystyle\frac{eQ_{e}}{m^{2}_{a}-m^{2}_{b}}\int
Dk\times\overline{u_{a}}\not{\varepsilon}^{*}(m_{a}+\not{p}_{2})[{g_{a}^{R}}^{*}P_{L}+{g_{a}^{L}}^{*}P_{R}]\frac{(m_{F}+\not{k})}{D_{0}D_{2}}[g_{b}^{L}P_{L}+g_{b}^{R}P_{R}]u_{b},$
(66)

where $D_{0}=k^{2}-m_{F}^{2}$ and $D_{i}=k_{i}^{2}-m_{h}^{2}$. The amplitude
for the diagram (2) is:

$\displaystyle i\mathcal{M}_{2}$ $\displaystyle=-eQ_{F}\int
Dk\times\overline{u_{a}}[{g_{a}^{R}}^{*}P_{L}+{g_{a}^{L}}^{*}P_{R}]\frac{(m_{F}-\not{k}_{1})\not{\varepsilon}^{*}(m_{F}-\not{k}_{2})}{D_{0}D_{1}D_{2}}[g_{b}^{L}P_{L}+g_{b}^{R}P_{R}]u_{b},$
(67)

where $D_{0}=k^{2}-m_{h}^{2}$ and $D_{i}=k_{i}^{2}-m_{F}^{2}$.

In the next calculation, we use the following simple notations:

$\displaystyle g^{LL}$ $\displaystyle\equiv g_{a}^{L*}g_{b}^{L},\;g^{RR}\equiv
g_{a}^{R*}g_{b}^{R},\;g^{RL}\equiv g_{a}^{R*}g_{b}^{L},\;g^{LR}\equiv
g_{a}^{L*}g_{b}^{R},$ $\displaystyle A_{1}$
$\displaystyle=g_{a}^{L*}g_{b}^{R}P_{R}+g_{a}^{R*}g_{b}^{L}P_{L},\;A_{2}=g_{a}^{L*}g_{b}^{L}P_{L}+g_{a}^{R*}g_{b}^{R}P_{R},$
(68)

where $g_{a}^{L,R}\equiv g_{a,Fh}^{L,R}$ and $g_{b}^{L,R}\equiv
g_{b,Fh}^{L,R}$ without any confusions with the gauge boson couplings
$g_{a,FV}^{L,R}$. It is not hard to write all amplitudes in terms of PV-
functions as follows:

$\displaystyle\mathcal{M}_{1}=$
$\displaystyle\frac{-eQ_{H}}{16\pi^{2}}\overline{u_{a}}\left\\{-2p_{1}.\varepsilon^{*}\left[A_{1}\right]m_{F}X_{0}+\left[2C^{f}_{00}\not{\varepsilon}^{*}+\left(X_{1}^{f}\not{p}_{1}+X_{2}^{f}\not{p}_{2}\right)(2p_{1}.\varepsilon^{*})\right][A_{2}]\right\\}u_{b},$
(69) $\displaystyle\mathcal{M}_{3}=$
$\displaystyle\frac{-eQ_{e}}{16\pi^{2}}\times\frac{\overline{u_{a}}[{g_{a}^{R}}^{*}P_{L}+{g_{a}^{L}}^{*}P_{R}](m_{F}B^{(1)}_{0}-B^{(1)f}_{1}\not{p}_{1})[g_{b}^{L}P_{L}+g_{b}^{R}P_{R}](m_{b}+\not{p}_{1})\not{\varepsilon}^{*}u_{b}}{(m^{2}_{a}-m^{2}_{b})},$
(70) $\displaystyle\mathcal{M}_{4}=$
$\displaystyle\frac{eQ_{e}}{16\pi^{2}}\times\frac{\overline{u_{a}}\not{\varepsilon}^{*}(m_{a}+\not{p}_{2})[{g_{a}^{R}}^{*}P_{L}+{g_{a}^{L}}^{*}P_{R}](m_{F}B^{(2)}_{0}-B^{(2)f}_{1}\not{p}_{2})[g_{b}^{L}P_{L}+g_{b}^{R}P_{R}]u_{b}}{(m^{2}_{a}-m^{2}_{b})},$
(71)

and

$\displaystyle\mathcal{M}_{2}=$ $\displaystyle-eQ_{F}\int
Dk\times\overline{u_{a}}\left\\{m_{F}^{2}\not{\varepsilon}^{*}+\not{k}_{1}\not{\varepsilon}^{*}\not{k}_{2}\right\\}[A_{2}]u_{b}$
$\displaystyle-eQ_{F}(-1)m_{F}\int
Dk\times\overline{u_{a}}\left\\{2k.\varepsilon^{*}-\not{p}_{1}\not{\varepsilon}^{*}-\not{\varepsilon}^{*}\not{p}_{2}\right\\}[A_{1}]u_{b}$
$\displaystyle=$
$\displaystyle\frac{-eQ_{F}}{16\pi^{2}}\overline{u_{a}}\left\\{\left[m_{F}^{2}C_{0}+(2-d)C_{00}\right]\not{\varepsilon}^{*}+(C_{11}+C_{1})\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{1}+(C_{22}+C_{2})\not{p}_{2}\not{\varepsilon}^{*}\not{p}_{2}\right.$
$\displaystyle\left.\qquad\qquad+(X_{0}+C_{12})\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}+C_{12}\not{p}_{2}\not{\varepsilon}^{*}\not{p}_{1}\right\\}\times[A_{2}]u_{b}$
$\displaystyle-\frac{eQ_{F}m_{F}}{16\pi^{2}}\overline{u_{a}}\left\\{(2p_{1}.\varepsilon^{*})(C_{1}+C_{2})+\left(\not{p}_{1}\not{\varepsilon}^{*}+\not{\varepsilon}^{*}\not{p}_{2}\right)C_{0}\right\\}[A_{1}]u_{b}.$
(72)

The validation of the WI given in Eq. (12) implies whether $f^{WI}_{L}=0$ is
correct with:

$\displaystyle f^{WI}_{L}\equiv$ $\displaystyle
D_{(ab)L}+m_{a}C_{(ab)L}+m_{b}C_{(ab)R}$ $\displaystyle=$ $\displaystyle
g^{LL}\left[\frac{Q_{e}\left(m_{a}^{2}B^{(1)}_{1}-m_{b}^{2}B^{(2)}_{1}\right)^{f}}{m_{a}^{2}-m_{b}^{2}}-\left(\frac{1}{2}-2C_{00}^{h}+m_{F}^{2}C_{0}^{h}\right)Q_{f}\right.$
$\displaystyle\left.\qquad\quad-
Q_{h}\left(m_{a}^{2}X_{1}+m_{b}^{2}X_{2}+2C_{00}\right)^{f}\frac{}{}\right]$
$\displaystyle+g^{RR}m_{a}m_{b}\left(\frac{Q_{e}\left(B^{(1)}_{1}-B^{(2)}_{1}\right)^{f}}{m_{a}^{2}-m_{b}^{2}}-Q_{f}X_{012}^{h}-Q_{h}X_{12}^{f}\right).$
(73)

We have used many formulas listed in Eqs. (A.1) and (A.2) to show that

$\displaystyle 0$ $\displaystyle=X_{12}^{f}+X^{h}_{12}+X_{0}\to
X^{h}_{012}=-X^{f}_{12},$ $\displaystyle\mathbf{b}_{1}^{f}$
$\displaystyle=-\left(m_{a}^{2}X_{1}+m_{b}^{2}X_{2}+2C_{00}\right)^{f}=\frac{1}{2}-2C_{00}^{h}+m_{F}^{2}C_{0}^{h}.$
(74)

Finally, the electric charge conservation $Q_{F}=Q_{e}+Q_{h}$ must be
satisfied so that Eq. (C.1) resulting in $f^{WI}_{L}=0$. On the other word,
the WI is valid for only one-loop Higgs contributions arising from the set of
four diagrams (1)-(4) in Fig. 1.

###  C.2 Vector contributions

To calculate the one-loop contributions from gauge boson exchanges
corresponding to Lagrangian (4), we denote $g_{a}^{L,R}\equiv g_{a,FV}^{L,R}$
and $g_{b}^{L,R}\equiv g_{b,FV}^{L,R}$ then use the notations given in Eq.
(C.1). The amplitudes relevant with gauge boson exchanges are:

$\displaystyle i\mathcal{M}_{5}=$ $\displaystyle\int
Dk\times\overline{u_{a}}i\gamma_{\alpha}[g_{a}^{L^{*}}P_{L}+g_{a}^{R^{*}}P_{R}]\frac{i(m_{F}+\not{k})}{D_{0}}i\gamma_{\beta}[g_{b}^{L}P_{L}+{g_{b}^{R}}P_{R}]u_{b}$
$\displaystyle\times\frac{-i}{D_{1}}\left(g^{\alpha\alpha^{\prime}}-\frac{k_{1}^{\alpha}k_{1}^{\alpha^{\prime}}}{m^{2}_{V}}\right)\left[-ieQ_{V}\Gamma_{\mu\alpha^{\prime}\beta^{\prime}}\left(-q,k_{1},-k_{2}\right)\varepsilon^{*\mu}\right]\frac{-i}{D_{2}}\left(g^{\beta\beta^{\prime}}-\frac{k_{2}^{\beta}k_{2}^{\beta^{\prime}}}{m^{2}_{V}}\right)$
$\displaystyle=$ $\displaystyle
eQ_{V}\int\frac{d^{4}k}{(2\pi)^{4}}\overline{u_{a}}\gamma_{\alpha}[g_{a}^{L^{*}}P_{L}+g_{a}^{R^{*}}P_{R}]\frac{(m_{F}+\not{k})}{D_{0}D_{1}D_{2}}\gamma_{\beta}[{g_{b}^{L}}P_{L}+{g_{b}^{R}}P_{R}]u_{b}$
$\displaystyle\times\left[\Gamma_{\mu\alpha^{\prime}\beta^{\prime}}\left(-q,k_{1},-k_{2}\right)\varepsilon^{*\mu}\right]\left(g^{\alpha\alpha^{\prime}}-\frac{k_{1}^{\alpha}k_{1}^{\alpha^{\prime}}}{m^{2}_{V}}\right)\left(g^{\beta\beta^{\prime}}-\frac{k_{2}^{\beta}k_{2}^{\beta^{\prime}}}{m^{2}_{V}}\right),$
(75) $\displaystyle i\mathcal{M}_{7}=$
$\displaystyle\frac{eQ_{e}}{m_{a}^{2}-m_{b}^{2}}\int
Dk\times\frac{1}{D_{0}D_{1}}\times\left(g^{\alpha\beta}-\frac{k^{\alpha}_{1}k^{\beta}_{1}}{m_{V}^{2}}\right)$
$\displaystyle\times\overline{u_{a}}\gamma_{\alpha}[g_{a}^{L^{*}}P_{L}+g_{a}^{R^{*}}P_{R}](m_{F}+\not{k})\gamma_{\beta}[{g_{b}^{L}}P_{L}+{g_{b}^{R}}P_{R}]\left(m_{b}+\not{p}_{1}\right)\not{\varepsilon}^{*}u_{b},$
(76) $\displaystyle i\mathcal{M}_{8}=$
$\displaystyle-\frac{eQ_{e}}{m_{a}^{2}-m_{b}^{2}}\int
Dk\times\frac{1}{D_{0}D_{2}}\times\left(g^{\alpha\beta}-\frac{k^{\alpha}_{2}k^{\beta}_{2}}{m_{V}^{2}}\right)$
$\displaystyle\times\overline{u_{a}}\not{\varepsilon}^{*}\left(m_{a}+\not{p}_{2}\right)\gamma_{\alpha}[g_{a}^{L^{*}}P_{L}+g_{a}^{R^{*}}P_{R}](m_{F}+\not{k})\gamma_{\beta}[{g_{b}^{L}}P_{L}+{g_{b}^{R}}P_{R}]u_{b},$
(77)

where $D_{0}=k^{2}-m_{F}^{2}$, $D_{i}=k_{i}^{2}-m_{V}^{2}$, and

$\displaystyle\Gamma_{\mu\alpha^{\prime}\beta^{\prime}}\left(-q,k_{1},-k_{2}\right)=g_{\alpha^{\prime}\beta^{\prime}}\left(k_{1}+k_{2}\right)_{\mu}+g_{\beta^{\prime}\mu}\left(-k_{2}+q\right)_{\alpha^{\prime}}+g_{\mu\alpha^{\prime}}\left(-q-k_{1}\right)_{\beta^{\prime}}.$
(78)

The amplitude for the diagram (6) is:

$\displaystyle i\mathcal{M}_{6}=$ $\displaystyle eQ_{F}\int
Dk\times\frac{1}{D_{0}D_{1}D_{2}}\times\left(g^{\alpha\beta}-\frac{k^{\alpha}k^{\beta}}{m^{2}_{V}}\right)$
$\displaystyle\times\overline{u_{a}}\gamma_{\alpha}[g_{a}^{L^{*}}P_{L}+g_{a}^{R^{*}}P_{R}](m_{F}-\not{k}_{1})\not{\varepsilon}^{*}(m_{F}-\not{k}_{2})\gamma_{\beta}[{g_{b}^{L}}P_{L}+{g_{b}^{R}}P_{R}]u_{b},$
(79)

where $D_{0}=k^{2}-m_{V}^{2}$ and $D_{i}=k_{i}^{2}-m_{F}^{2}$.

Considering diagram (7), we have:

$\displaystyle i\mathcal{M}_{7}=$
$\displaystyle\frac{eQ_{e}}{m_{a}^{2}-m_{b}^{2}}\int
Dk\times\frac{\overline{u_{a}}\left[\gamma_{\alpha}\gamma_{\beta}m_{F}\left[A_{1}\right]+\gamma_{\alpha}\not{k}\gamma_{\beta}\left[A_{2}\right]\right]\left(m_{b}+\not{p}_{1}\right)\not{\varepsilon}^{*}u_{b}}{D_{0}D_{1}}\times\left(g^{\alpha\beta}-\frac{k^{\alpha}_{1}k^{\beta}_{1}}{m_{V}^{2}}\right)$
$\displaystyle=$ $\displaystyle\frac{eQ_{e}}{m_{a}^{2}-m_{b}^{2}}\int
Dk\times\frac{1}{D_{0}D_{1}}$
$\displaystyle\times\overline{u_{a}}\left[m_{F}\left(d-\frac{k_{1}^{2}}{m_{V}^{2}}\right)\left[A_{1}\right]+\left((2-d)\not{k}-\frac{\not{k}_{1}\not{k}\not{k}_{1}}{m_{V}^{2}}\right)\left[A_{2}\right]\right]\left(m_{b}+\not{p}_{1}\right)\not{\varepsilon}^{*}u_{b}$
$\displaystyle=$
$\displaystyle\frac{ieQ_{e}}{16\pi^{2}(m_{a}^{2}-m_{b}^{2})}\overline{u_{a}}\left\\{m_{F}\left[A_{1}\right]\left[(d-1)B^{(1)}_{0}-\frac{A_{0}(m_{F}^{2})}{m_{V}^{2}}\right]\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad+m_{a}\left[\left(-(2-d)+\frac{m_{F}^{2}+m_{a}^{2}}{m_{V}^{2}}\right)B^{(1)}_{1}+\frac{A_{0}(m_{V}^{2})+2m_{F}^{2}B^{(1)}_{0}}{m_{V}^{2}}\right]\left[A_{2}\right]\right\\}$
$\displaystyle\times\left(m_{b}+\not{p}_{1}\right)\not{\varepsilon}^{*}u_{b},$
(80)

where we have used the following results

$\displaystyle\not{k}_{1}\not{k}\not{k}_{1}$
$\displaystyle=\left(D_{0}+m_{F}^{2}\right)\not{k}-2\left(D_{0}+m_{F}^{2}\right)\not{p}_{1}+\not{p}_{1}\not{k}\not{p}_{1},$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\times\frac{k_{\mu}}{D_{1}}$
$\displaystyle=A_{0}(m_{V}^{2})p_{1\mu}.$

Then the one-loop contribution form factors from diagram (7) are:

$\displaystyle D_{(ab)L,7}$
$\displaystyle=\frac{eQ_{e}}{16\pi^{2}(m_{a}^{2}-m_{b}^{2})}\left\\{\left(g^{RL}m_{a}+g^{LR}m_{b}\right)m_{F}\left[(d-1)B^{(1)}_{0}-\frac{A_{0}(m_{F}^{2})}{m_{V}^{2}}\right]\right.$
$\displaystyle+\left.m_{a}\left(m_{a}g^{LL}+m_{b}g^{RR}\right)\left[\left(-(2-d)+\frac{m_{F}^{2}+m_{a}^{2}}{m_{V}^{2}}\right)B^{(1)}_{1}+\frac{A_{0}(m_{V}^{2})+2m_{F}^{2}B^{(1)}_{0}}{m_{V}^{2}}\right]\right\\},$
$\displaystyle D_{(ab)R,7}=$ $\displaystyle
D_{(ab)L,7}\left[g^{L}_{a}\leftrightarrow g^{R}_{a},g^{L}_{b}\leftrightarrow
g^{R}_{b}\right].$ (81)

The same calculation for diagram (8) gives the following one-loop contribution
form factor:

$\displaystyle D_{(ab)L,8}$
$\displaystyle=-\frac{eQ_{e}}{16\pi^{2}(m_{a}^{2}-m_{b}^{2})}\left\\{\left(g^{RL}m_{a}+g^{LR}m_{b}\right)m_{F}\left[(d-1)B^{(2)}_{0}-\frac{A_{0}(m_{F}^{2})}{m_{V}^{2}}\right]\right.$
$\displaystyle+\left.m_{b}\left(m_{a}g^{RR}+m_{b}g^{LL}\right)\left[\left(-(2-d)+\frac{m_{F}^{2}+m_{b}^{2}}{m_{V}^{2}}\right)B^{(2)}_{1}+\frac{A_{0}(m_{V}^{2})+2m_{F}^{2}B^{(2)}_{0}}{m_{V}^{2}}\right]\right\\},$
$\displaystyle D_{(ab)R,8}$
$\displaystyle=D_{(ab)L,8}\left[g^{L}_{a}\leftrightarrow
g^{R}_{a},g^{L}_{b}\leftrightarrow g^{R}_{b}\right].$ (82)

Using $d=4-2\epsilon$ and the divergent parts of PV-functions given in Eq.
(A.2), we get the formulas of $D_{(ab)L,78}$ given in Eq. (22).

Diagram (5)

From the equalities $q^{2}=0$, $q.\varepsilon^{*}=0$, and $k_{1}=q+k_{2}$, it
is easy to prove that

$\displaystyle\left[\Gamma_{\mu\alpha^{\prime}\beta^{\prime}}\left(-q,k_{1},-k_{2}\right)\varepsilon^{*\mu}\right]k_{1}^{\alpha}k_{1}^{\alpha^{\prime}}k_{2}^{\beta}k_{2}^{\beta^{\prime}}$
$\displaystyle=k_{1}^{\alpha}k_{2}^{\beta}\left\\{(k_{1}.k_{2})\left[(k_{1}+k_{2}).\varepsilon^{*}\right]+(k_{2}.\varepsilon^{*})\left[k_{1}.(-k_{2}+q)\right]+(k_{1}.\varepsilon^{*})\left[k_{2}.(-q-k_{1})\right]\right\\}$
$\displaystyle\sim(k_{1}.k_{2})\left[2k_{1}.\varepsilon^{*}\right]+(k_{1}.\varepsilon^{*})\left[q^{2}-k_{2}^{2}\right]+(k_{1}.\varepsilon^{*})\left[q^{2}-k^{2}_{1}\right]=0.$
(83)

As a result, the amplitude (75) is written as follows:

$\displaystyle i\mathcal{M}_{5}$ $\displaystyle=eQ_{V}\int
Dk\frac{\overline{u_{a}}\gamma_{\alpha}\left[A\right]\gamma_{\beta}u_{b}}{D_{0}D_{1}D_{2}}\left[\Gamma_{\mu\alpha^{\prime}\beta^{\prime}}\left(-q,k_{1},-k_{2}\right)\varepsilon^{*\mu}\right]\left(g^{\alpha\alpha^{\prime}}g^{\beta\beta^{\prime}}-\frac{g^{\beta\beta^{\prime}}k_{1}^{\alpha}k_{1}^{\alpha^{\prime}}+g^{\alpha\alpha^{\prime}}k_{2}^{\beta}k_{2}^{\beta^{\prime}}}{m^{2}_{V}}\right),$
(84)

where

$A=m_{F}\left(g_{a}^{L*}g_{b}^{R}P_{L}+g_{a}^{R*}g_{b}^{L}P_{R}\right)+\left[A_{2}\right]\not{k}.$
(85)

The first term in the integrand is

$\displaystyle(1)=$
$\displaystyle\overline{u_{a}}\left\\{4(k_{1}+k_{2}).\varepsilon^{*}+(-\not{k}+2\not{p}_{2}-\not{p}_{1})\not{\varepsilon}^{*}+\not{\varepsilon}^{*}(-\not{k}+2\not{p}_{1}-\not{p}_{2})\right\\}\times
m_{F}\left[A_{1}\right]u_{b}$
$\displaystyle+\overline{u_{a}}\left\\{(2-d)(2k_{1}.\varepsilon^{*})\not{k}+(-\not{k}+2\not{p}_{2}-\not{p}_{1})\not{k}\not{\varepsilon}^{*}+\not{\varepsilon}^{*}\not{k}(-\not{k}+2\not{p}_{1}-\not{p}_{2})\right\\}\times\left[A_{2}\right]u_{b}$
$\displaystyle=$
$\displaystyle\overline{u_{a}}\left\\{6k.\varepsilon^{*}-3(\not{p}_{1}\not{\varepsilon}^{*}+\not{\varepsilon}^{*}\not{p}_{2})\right\\}m_{F}\left[A_{1}\right]u_{b}$
$\displaystyle+\overline{u_{a}}\left\\{(2-d)(2k_{1}.\varepsilon^{*})\not{k}+(2\not{p}_{2}-\not{p}_{1})\not{k}\not{\varepsilon}^{*}+\not{\varepsilon}^{*}\not{k}(2\not{p}_{1}-\not{p}_{2})-2k^{2}\not{\varepsilon}^{*}\right\\}\left[A_{2}\right]u_{b}.$
(86)

After integrating out, the formula is

$\displaystyle(1)=$
$\displaystyle\overline{u_{a}}\left\\{(2p_{1}.\varepsilon^{*})\times(-3m_{F})X_{3}-3m_{F}C_{0}(\not{p}_{1}\not{\varepsilon}^{*}+\not{\varepsilon}^{*}\not{p}_{2})\right\\}\left[A_{1}\right]u_{b}$
$\displaystyle+\overline{u_{a}}\left\\{(2-d)2\varepsilon^{\alpha*}(C_{\alpha\beta}-C_{\beta}p_{1\alpha})\gamma^{\beta}+C_{\alpha}\left[(2\not{p}_{2}-\not{p}_{1})\gamma^{\alpha}\not{\varepsilon}^{*}+\not{\varepsilon}^{*}\gamma^{\alpha}(2\not{p}_{1}-\not{p}_{2})\right]\right.$
$\displaystyle\left.-2\left(B^{(0)}_{0}+m_{F}^{2}C_{0}\right)\not{\varepsilon}^{*}\right\\}\times\left[A_{2}\right]u_{b}$
$\displaystyle=$
$\displaystyle\overline{u_{a}}(-3m_{F})\times\left\\{(2p_{1}.\varepsilon^{*})X_{3}+C_{0}(\not{p}_{1}\not{\varepsilon}^{*}+\not{\varepsilon}^{*}\not{p}_{2})\right\\}\left[A_{1}\right]u_{b}$
$\displaystyle+\overline{u_{a}}\left\\{\not{\varepsilon}^{*}\left[2(2-d)C_{00}-2(B^{(0)}_{0}+m_{F}^{2}C_{0})-(3m_{a}^{2}+2m_{b}^{2})C_{1}-(2m_{a}^{2}+3m_{b}^{2})C_{2}\right]\right.$
$\displaystyle\left.+\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}(-3X_{3})\right\\}\left[A_{2}\right]u_{b}$
$\displaystyle+\overline{u_{a}}(2p_{1}.\varepsilon^{*})\left\\{\left[-2(C_{11}+C_{12})+C_{2}\right]\not{p}_{1}+\left[-2(C_{12}+C_{22})+C_{1}\right]\not{p}_{2}\right\\}\left[A_{2}\right]u_{b}.$
(87)

The second term in the integrand is

$\displaystyle\left(-\frac{1}{m_{V}^{2}}\right)^{-1}\times(2)$
$\displaystyle=$
$\displaystyle\Gamma_{\mu\alpha^{\prime}\beta^{\prime}}\left(-q,k_{1},-k_{2}\right)\varepsilon^{*\mu}\left(g^{\beta\beta^{\prime}}k_{1}^{\alpha}k_{1}^{\alpha^{\prime}}+g^{\alpha\alpha^{\prime}}k_{2}^{\beta}k_{2}^{\beta^{\prime}}\right)\times\overline{u_{a}}\gamma_{\alpha}\left[A\right]\gamma_{\beta}u_{b}$
$\displaystyle=$
$\displaystyle\;\overline{u_{a}}\not{k}_{1}\left[A\right]\left[(k_{1}.\varepsilon^{*})\not{k}_{2}-\not{\varepsilon}^{*}k_{2}^{2}\right]u_{b}+\overline{u_{a}}\left[(k_{1}.\varepsilon^{*})\not{k}_{1}-\not{\varepsilon}^{*}k_{1}^{2}\right]\left[A\right]\not{k}_{2}u_{b}$
$\displaystyle=$
$\displaystyle\;\overline{u_{a}}m_{F}\left[A_{1}\right]\left[2(k_{1}.\varepsilon^{*})\not{k}_{1}\not{k}_{2}-k_{2}^{2}\not{k}_{1}\not{\varepsilon}^{*}-k_{1}^{2}\not{\varepsilon}^{*}\not{k}_{2}\right]u_{b}$
$\displaystyle+\overline{u_{a}}\left[2(k_{1}.\varepsilon^{*})\not{k}_{1}\not{k}\not{k}_{2}-k_{2}^{2}\not{k}_{1}\not{k}\not{\varepsilon}^{*}-k_{1}^{2}\not{\varepsilon}^{*}\not{k}\not{k}_{2}\right]\left[A_{2}\right]u_{b}$
$\displaystyle=$
$\displaystyle\;\overline{u_{a}}m_{F}\left[A_{1}\right]\left[\not{k}_{1}\not{\varepsilon}^{*}\not{q}\not{k}_{2}\right]u_{b}+\overline{u_{a}}\left[2(k_{1}.\varepsilon^{*})\not{k}_{1}\not{k}\not{k}_{2}-k_{2}^{2}\not{k}_{1}\not{k}\not{\varepsilon}^{*}-k_{1}^{2}\not{\varepsilon}^{*}\not{k}\not{k}_{2}\right]\left[A_{2}\right]u_{b}.$
(88)

The first term in Eq. (C.2) gives

$\displaystyle\not{k}_{1}\not{\varepsilon}^{*}\not{q}\not{k}_{2}$
$\displaystyle=\left(\not{k}-\not{p}_{1}\right)\not{\varepsilon}^{*}\not{q}\left(\not{k}-\not{p}_{2}\right)=\not{k}\not{\varepsilon}^{*}\not{q}\not{k}-\not{p}_{1}\not{\varepsilon}^{*}\not{q}\not{k}-\not{k}\not{\varepsilon}^{*}\not{q}\not{p}_{2}+\not{p}_{1}\not{\varepsilon}^{*}\not{q}\not{p}_{2}$
$\displaystyle=C_{\alpha\beta}\gamma^{\alpha}\not{\varepsilon}^{*}\not{q}\gamma^{\beta}-C_{\alpha}\not{p}_{1}\not{\varepsilon}^{*}\not{q}\gamma^{\alpha}-C_{\alpha}\gamma^{\alpha}\not{\varepsilon}^{*}\not{q}\not{p}_{2}+C_{0}\not{p}_{1}\not{\varepsilon}^{*}\not{q}\not{p}_{2}$
$\displaystyle+\left(C_{1}\not{p}_{1}+C_{2}\not{p}_{2}\right)\not{\varepsilon}^{*}\not{q}\not{p}_{2}+\not{p}_{1}\not{\varepsilon}^{*}\not{q}\left(C_{1}\not{p}_{1}+C_{2}\not{p}_{2}\right)+C_{0}\not{p}_{1}\not{\varepsilon}^{*}\not{q}\not{p}_{2}$
$\displaystyle=C_{00}\left[\varepsilon^{*}.q-(4-d)\not{\varepsilon}^{*}\not{q}\right]+\left(C_{12}\not{p}_{2}+C_{11}\not{p}_{1}+C_{1}\not{p}_{1}\right)\not{\varepsilon}^{*}\not{q}\not{p}_{1}$
$\displaystyle+\left(C_{12}\not{p}_{1}+C_{22}\not{p}_{2}+C_{1}\not{p}_{1}+C_{2}\not{p}_{2}+C_{2}\not{p}_{1}+C_{0}\not{p}_{1}\right)\not{\varepsilon}^{*}\not{q}\not{p}_{2}.$
(89)

Because the divergent part $C_{00}=\Delta_{\epsilon}/4=1/(4\epsilon)$, which
$d=4-2\epsilon$, hence $C_{00}(4-d)=1/2$. The result is:

$\displaystyle\not{k}_{1}\not{\varepsilon}^{*}\not{q}\not{k}_{2}=$
$\displaystyle-\frac{1}{2}\not{\varepsilon}\not{q}+\left[C_{12}\left(\not{p}_{1}+\not{q}\right)+\left(C_{11}+C_{1}\right)\not{p}_{1}\right]\not{\varepsilon}^{*}\not{q}\left(\not{p}_{2}-\not{q}\right)$
$\displaystyle+\left[\left(C_{12}+X_{0}\right)\not{p}_{1}+\left(C_{22}+C_{2}\right)\left(\not{p}_{1}+\not{q}\right)\right]\not{\varepsilon}^{*}\not{q}\not{p}_{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\not{\varepsilon}^{*}\not{q}+X_{012}\not{p}_{1}\not{\varepsilon}^{*}\not{q}\not{p}_{2},$
(90)

where we have used $\varepsilon^{*}.q=q^{2}=0$ and
$\not{q}\not{\varepsilon}^{*}\not{q}=2\varepsilon^{*}.q\not{q}-q^{2}\not{\varepsilon}^{*}=0$.
The final result is

$\displaystyle\overline{u_{a}}m_{F}\left[A_{1}\right]\left[\not{k}_{1}\not{\varepsilon}^{*}\not{q}\not{k}_{2}\right]u_{b}=$
$\displaystyle\overline{u_{a}}m_{F}\left\\{\not{p}_{1}\not{\varepsilon}^{*}\left[-\frac{1}{2}+m_{b}^{2}X_{012}\right]+\not{\varepsilon}^{*}\not{p}_{2}\left[-\frac{1}{2}+m_{a}^{2}X_{012}\right]\right.$
$\displaystyle\left.\qquad\quad+(2p_{1}.\varepsilon^{*})\left[\frac{1}{2}-X_{012}\not{p}_{1}\not{p}_{2}\right]\right\\}\left[A_{1}\right]u_{b}.$
(91)

Consider the last two terms in the last line of the formula (C.2)

$\displaystyle-
k_{2}^{2}\not{k}_{1}\not{k}\not{\varepsilon}^{*}-k_{1}^{2}\not{\varepsilon}^{*}\not{k}\not{k}_{2}$
$\displaystyle=$ $\displaystyle-
k_{2}^{2}\left(k^{2}-\not{p}_{1}\not{k}\right)\not{\varepsilon}^{*}-\not{\varepsilon}^{*}\left(k^{2}-\not{k}\not{p}_{2}\right)k_{1}^{2}$
$\displaystyle=$
$\displaystyle-k^{2}\left(k_{1}^{2}+k^{2}_{2}\right)\not{\varepsilon}^{*}+\left(D_{2}+m_{V}^{2}\right)\not{p}_{1}\not{k}\not{\varepsilon}^{*}+\left(D_{1}+m_{V}^{2}\right)\not{\varepsilon}^{*}\not{k}\not{p}_{2}$
$\displaystyle\to$
$\displaystyle-\not{\varepsilon}^{*}\frac{(D_{0}+m_{F}^{2})(D_{1}+D_{2}+2m_{V}^{2})}{D_{0}D_{1}D_{2}}+\frac{\not{p}_{1}\not{k}\not{\varepsilon}^{*}}{D_{0}D_{1}}+\frac{\not{\varepsilon}^{*}\not{k}\not{p}_{2}}{D_{0}D_{2}}+m_{V}^{2}\left(\frac{\not{p}_{1}\not{k}\not{\varepsilon}^{*}}{D_{0}D_{1}D_{2}}+\frac{\not{\varepsilon}^{*}\not{k}\not{p}_{2}}{D_{0}D_{1}D_{2}}\right)$
$\displaystyle=$
$\displaystyle-\not{\varepsilon}^{*}\left[2m_{V}^{2}(B^{(0)}_{0}+1)+2m_{V}^{2}B^{(0)}_{0}+m_{F}^{2}(B^{(1)}_{0}+B^{(2)}_{0}+2m_{V}^{2}C_{0})\right]$
$\displaystyle-\not{p}_{1}\left[B^{(1)}_{1}\not{p}_{1}+m_{V}^{2}\left(C_{1}\not{p}_{1}+C_{2}\not{p}_{2}\right)\right]\not{\varepsilon}^{*}-\not{\varepsilon}^{*}\left[B^{(2)}_{1}\not{p}_{2}+m_{V}^{2}\left(C_{1}\not{p}_{1}+C_{2}\not{p}_{2}\right)\right]\not{p}_{2}$
$\displaystyle=$
$\displaystyle-\not{\varepsilon}^{*}\left[m_{V}^{2}(4B^{(0)}_{0}+2+2m_{F}^{2}C_{0}+m_{a}^{2}C_{1}+m_{b}^{2}C_{2})+m_{a}^{2}B^{(1)}_{1}+m_{b}^{2}B^{(2)}_{1}+m_{F}^{2}(B^{(1)}_{0}+B^{(2)}_{0})\right]$
$\displaystyle-
m_{V}^{2}\left(C_{2}\not{p}_{1}\not{p}_{2}\not{\varepsilon}^{*}+C_{1}\not{\varepsilon}^{*}\not{p}_{1}\not{p}_{2}\right)$
$\displaystyle=$
$\displaystyle-\not{\varepsilon}^{*}\left[m_{V}^{2}(4B^{(0)}_{0}+2+2m_{F}^{2}C_{0}+m_{a}^{2}C_{1}+m_{b}^{2}C_{2})+m_{a}^{2}B^{(1)}_{1}+m_{b}^{2}B^{(2)}_{1}+m_{F}^{2}(B^{(1)}_{0}+B^{(2)}_{0})\right]$
$\displaystyle+m_{V}^{2}X_{3}\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}+(2p_{1}.\varepsilon^{*})(-m_{V}^{2})\left[C_{2}\not{p}_{1}+C_{1}\not{p}_{2}\right].$
(92)

Lastly, consider the first term in the last line of the formula (C.2):

$\displaystyle
2(k_{1}.\varepsilon^{*})\not{k}_{1}\not{k}\not{k}_{2}=\left(k.\varepsilon^{*}-2p_{1}.\varepsilon^{*}\right)\times\left(\not{k}-\not{p}_{1}\right)\not{k}\left(\not{k}-\not{p}_{2}\right)$
$\displaystyle=$
$\displaystyle\left(-2p_{1}.\varepsilon^{*}+2k.\varepsilon^{*}\right)\times\left(k^{2}\not{k}-k^{2}\not{p}_{1}-k^{2}\not{p}_{2}+\not{p}_{1}\not{k}\not{p}_{2}\right)$
$\displaystyle\to$
$\displaystyle\left(-2p_{1}.\varepsilon^{*}+2k.\varepsilon^{*}\right)\times\left[\left(\frac{1}{D_{1}D_{2}}+\frac{m_{F}^{2}}{D_{0}D_{1}D_{2}}\right)\left(\not{k}-\not{p}_{1}-\not{p}_{2}\right)+\frac{\not{p}_{1}\not{k}\not{p}_{2}}{D_{0}D_{1}D_{2}}\right]$
$\displaystyle=$
$\displaystyle\left(-2p_{1}.\varepsilon^{*}\right)\times\left\\{\left[-\frac{1}{2}B^{(0)}_{0}-m_{F}^{2}C_{0}\right](\not{p}_{1}+\not{p}_{2})-m_{F}^{2}(C_{1}\not{p}_{1}+C_{2}\not{p}_{2})-\not{p}_{1}(C_{1}\not{p}_{1}+C_{2}\not{p}_{2})\not{p}_{2}\right\\}$
$\displaystyle+\left(2\varepsilon^{*}_{\mu}\right)\left[\left(\frac{1}{D_{1}D_{2}}+\frac{m_{F}^{2}}{D_{0}D_{1}D_{2}}\right)\not{k}k^{\mu}-\left(\frac{1}{D_{1}D_{2}}+\frac{m_{F}^{2}}{D_{0}D_{1}D_{2}}\right)(\not{p}_{1}+\not{p}_{2})k^{\mu}+\frac{\not{p}_{1}\not{k}\not{p}_{2}k^{\mu}}{D_{0}D_{1}D_{2}}\right]$
$\displaystyle=$
$\displaystyle\left(2p_{1}.\varepsilon^{*}\right)\left\\{\left[\frac{B^{(0)}_{0}}{2}+m_{F}^{2}C_{0}\right]\left(\not{p}_{1}+\not{p}_{2}\right)+\left(m_{F}^{2}C_{1}+m_{b}^{2}C_{2}\right)\not{p}_{1}+\left(m_{F}^{2}C_{2}+m_{a}^{2}C_{1}\right)\not{p}_{2}\right\\}$
$\displaystyle+\left(2\varepsilon^{*}_{\mu}\right)\times\left\\{\left(B^{\mu\nu}+m_{F}^{2}C^{\mu\nu}\right)\gamma_{\nu}-\left(B^{\mu}+m_{F}^{2}C^{\mu}\right)\left(\not{p}_{1}+\not{p}_{2}\right)+C^{\mu\nu}\not{p}_{1}\gamma_{\nu}\not{p}_{2}\right\\},$
(93)

where $B^{\mu}=B^{\mu}(0,m_{V}^{2},m_{V}^{2})$ and
$B^{\mu\nu}=B^{\mu\nu}(0,m_{V}^{2},m_{V}^{2})$. The last line in Eq. (C.2) is
expressed in terms of the PV functions as follows

$\displaystyle\left(2\varepsilon^{*}_{\mu}\right)\left\\{\gamma_{\nu}\left[\frac{g^{\mu\nu}}{2}\left(B^{(0)}_{0}+1\right)+\frac{1}{6}B^{(0)}_{0}\left(2p_{1}^{\mu}p_{1}^{\nu}+p_{1}^{\mu}p_{2}^{\nu}+p_{2}^{\mu}p_{1}^{\nu}+2p_{2}^{\mu}p_{2}^{\nu}\right)\right]\right.$
$\displaystyle\qquad\qquad\quad+m_{F}^{2}\gamma_{\nu}\left[C_{00}g^{\mu\nu}+C_{11}p_{1}^{\mu}p_{1}^{\nu}+C_{12}p_{1}^{\mu}p_{2}^{\nu}+C_{12}p_{2}^{\mu}p_{1}^{\nu}+C_{22}p_{2}^{\mu}p_{2}^{\nu}\right]$
$\displaystyle\qquad\qquad\quad-\left[\frac{1}{2}B^{(0)}_{0}(p_{1}+p_{2})^{\mu}-m_{F}^{2}\left(C_{1}p_{1}^{\mu}+C_{2}p_{2}^{\mu}\right)\right]\left(\not{p}_{1}+\not{p}_{2}\right)$
$\displaystyle\left.\qquad\qquad\quad+\left[C_{00}g^{\mu\nu}+C_{11}p_{1}^{\mu}p_{1}^{\nu}+C_{12}p_{1}^{\mu}p_{2}^{\nu}+C_{12}p_{2}^{\mu}p_{1}^{\nu}+C_{22}p_{2}^{\mu}p_{2}^{\nu}\right]\not{p}_{1}\gamma_{\nu}\not{p}_{2}\right\\}$
$\displaystyle=$ $\displaystyle
m_{V}^{2}\not{\varepsilon}^{*}\left(B^{(0)}_{0}+1\right)+(p_{1}.\varepsilon^{*})B^{(0)}_{0}\left(\not{p}_{1}+\not{p}_{2}\right)$
$\displaystyle+m_{F}^{2}\left[2C_{00}\not{\varepsilon}^{*}+\left(2p_{1}.\varepsilon^{*}\right)\left(C_{11}\not{p}_{1}+C_{12}\not{p}_{2}+C_{12}\not{p}_{1}+C_{22}\not{p}_{2}\right)\right]$
$\displaystyle-\left[B^{(0)}_{0}\left(2p_{1}.\varepsilon^{*}\right)-m_{F}^{2}\left(2p_{1}.\varepsilon^{*}\right)\left(C_{1}+C_{2}\right)\right]\left(\not{p}_{1}+\not{p}_{2}\right)$
$\displaystyle+\not{p}_{1}\left[2\not{\varepsilon}^{*}C_{00}+\left(2p_{1}.\varepsilon^{*}\right)\left(C_{11}\not{p}_{1}+C_{12}\not{p}_{1}+C_{12}\not{p}_{2}+C_{22}\not{p}_{2}\right)\right]\not{p}_{2}.$
(94)

Hence the final result of Eq. (C.2) is

$\displaystyle 2(k_{1}.\varepsilon^{*})\not{k}_{1}\not{k}\not{k}_{2}=$
$\displaystyle\not{\varepsilon}^{*}\left[m_{V}^{2}(B^{(0)}_{0}+1)+2m_{F}^{2}C_{00}\right]+\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}\left(2C_{00}\right)$
$\displaystyle+(2p_{1}.\varepsilon^{*})\left[m_{F}^{2}X_{01}+m_{b}^{2}X_{2}\right]\not{p}_{1}+(2p_{1}.\varepsilon^{*})\left[m_{F}^{2}X_{01}+m_{a}^{2}X_{1}\right]\not{p}_{2}.$
(95)

The sum of three terms given in Eqs. (C.2), (C.2), and (C.2) gives
$C_{(ab)L,R}$ corresponding to the diagrams (5) given in Eqs. (II) and (II).
The formulas of $D_{(ab)L,5}$ and $D_{(ab)R,5}$ are given in Eq. (23).

Regarding to the case of photon couplings in Eq. (• ‣ III.1), the equality
given in Eq. (C.2) is still valid because the new part
$\Delta\Gamma_{\mu\alpha^{\prime}\beta^{\prime}}=\Gamma_{\mu\alpha^{\prime}\beta^{\prime}}-\Gamma^{\prime}_{\mu\alpha^{\prime}\beta^{\prime}}=\delta
k_{v}\left(g_{\mu\alpha^{\prime}}q_{\beta^{\prime}}-g_{\beta^{\prime}\mu}q_{\alpha^{\prime}}\right)$
satisfies
$\left(g_{\mu\alpha^{\prime}}q_{\beta^{\prime}}-g_{\beta^{\prime}\mu}q_{\alpha^{\prime}}\right)\varepsilon^{*\mu}k_{1}^{\alpha^{\prime}}k^{\beta^{\prime}}_{2}=q^{2}(\varepsilon^{*}.k_{2})-(q.k_{2})(\varepsilon^{*}.q)=0$.
The other relevant part of $\mathcal{M}_{5}$ is:

$\displaystyle-\gamma_{\alpha}\left[A\right]\gamma_{\beta}\times\Delta\Gamma_{\mu\alpha^{\prime}\beta^{\prime}}\varepsilon^{\mu*}\left(g^{\alpha\alpha^{\prime}}g^{\beta\beta^{\prime}}-\frac{g^{\beta\beta^{\prime}}k_{1}^{\alpha}k_{1}^{\alpha^{\prime}}+g^{\alpha\alpha^{\prime}}k_{2}^{\beta}k_{2}^{\beta^{\prime}}}{m^{2}_{V}}\right)$
$\displaystyle=$
$\displaystyle\left(\not{q}\not{\varepsilon}^{*}-\not{\varepsilon}^{*}\not{q}\right)m_{F}A_{1}+\left(\not{q}\not{k}\not{\varepsilon}^{*}-\not{\varepsilon}^{*}\not{k}\not{q}\right)A_{2}$
$\displaystyle-\frac{1}{m_{V}^{2}}\left\\{\left[(k_{1}.q)(\not{k}_{1}\not{\varepsilon}^{*}-\not{\varepsilon}^{*}\not{k}_{2})+(k_{1}.\varepsilon^{*})(\not{q}\not{k}_{2}-\not{k}_{1}\not{q})\right]m_{F}A_{1}\right.$
$\displaystyle\left.\quad\quad\quad+\left[(k_{1}.q)(-\not{p}_{1}\not{k}\not{\varepsilon}^{*}+\not{\varepsilon}^{*}\not{k}\not{p}_{2})+(k_{1}.\varepsilon^{*})(\not{p}_{1}\not{k}\not{q}-\not{q}\not{k}\not{p}_{2})\right]A_{2}\right\\}.$
(96)

The final result of new contributions to $i\delta\mathcal{M}_{5}$ is:

$\displaystyle i\delta\mathcal{M}_{5}=$
$\displaystyle\int\frac{d^{4}k}{(2\pi)^{4}}\frac{\overline{u_{a}}\gamma_{\alpha}\left[A\right]\gamma_{\beta}u_{b}}{D_{0}D_{1}D_{2}}\times\Delta\Gamma_{\mu\alpha^{\prime}\beta^{\prime}}\left(g^{\alpha\alpha^{\prime}}g^{\beta\beta^{\prime}}-\frac{g^{\beta\beta^{\prime}}k_{1}^{\alpha}k_{1}^{\alpha^{\prime}}+g^{\alpha\alpha^{\prime}}k_{2}^{\beta}k_{2}^{\beta^{\prime}}}{m^{2}_{V}}\right)$
$\displaystyle=$
$\displaystyle-\frac{ieQ_{V}\delta_{v}}{16\pi^{2}}\left\\{\frac{}{}\overline{u_{a}}\left[\left(4p_{1}.\varepsilon^{*}-2\not{p}_{1}\not{\varepsilon}^{*}-2\not{\varepsilon}^{*}\not{p}_{2}\right)C_{0}m_{F}A_{1}\right]u_{b}\right.$
$\displaystyle\left.+\overline{u_{a}}\left[(2p_{1}.\varepsilon^{*})(\not{p}_{1}+\not{p}_{2})-(m_{a}^{2}+m_{b}^{2})\not{\varepsilon}^{*}-2\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}\right]X_{3}A_{2}u_{b}\right.$
$\displaystyle\left.-\frac{1}{m_{V}^{2}}\overline{u_{a}}\left[C_{00}\left(\not{q}\not{\varepsilon}^{*}-\not{\varepsilon}^{*}\not{q}\right)+(C_{11}+C_{12})\left[-2(p_{1}.q)\not{\varepsilon}^{*}\not{p}_{1}+2(p_{1}.\varepsilon^{*})\not{q}\not{p}_{1}\right]\right.\right.$
$\displaystyle\left.\left.\qquad\qquad+(C_{22}+C_{12})\left[-2(p_{2}.q)\not{\varepsilon}^{*}\not{p}_{2}+2(p_{2}.\varepsilon^{*})\not{q}\not{p}_{2}\right]\right.\right.$
$\displaystyle\left.\left.\qquad\qquad+(p_{1}.q)\left[-X_{0}\left(-\not{p}_{1}\not{\varepsilon}^{*}+\not{\varepsilon}^{*}\not{p}_{2}\right)+(2p_{1}.\varepsilon^{*})(C_{2}-C_{1})+2\left(C_{1}\not{p}_{1}\not{\varepsilon}^{*}-C_{2}\not{\varepsilon}^{*}\not{p}_{2}\right)\right]\right.\right.$
$\displaystyle\left.\left.\qquad\qquad-(p_{1}.\varepsilon^{*})\left(2\not{p}_{1}\not{p}_{2}-m_{a}^{2}-m_{b}^{2}\right)(2X_{3}+C_{0})\right]m_{F}A_{1}u_{b}\right.$
$\displaystyle\left.-\frac{1}{m_{V}^{2}}\overline{u_{a}}\left[\frac{}{}C_{00}\left(2(m_{a}^{2}+m_{b}^{2})\not{\varepsilon}^{*}+4\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}-2(\not{p}_{1}+\not{p}_{2})\times(2p_{1}.\varepsilon^{*})\right)\right.\right.$
$\displaystyle\left.\left.\qquad\qquad+\frac{m_{b}^{2}-m_{a}^{2}}{2}\times
X_{1}\left(-m_{a}^{2}\not{\varepsilon}^{*}-\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}+(2p_{1}.\varepsilon^{*})\not{p}_{1}\right)\right.\right.$
$\displaystyle\left.\left.\qquad\qquad+\frac{m_{b}^{2}-m_{a}^{2}}{2}\times
X_{2}\left(m_{b}^{2}\not{\varepsilon}^{*}+\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}-(2p_{1}.\varepsilon^{*})\not{p}_{2}\right)\right]A_{2}u_{b}\right\\}.$
(97)

Ignoring the factor $\frac{eQ_{V}\delta k_{v}}{16\pi^{2}}$, the form factors
are:

$\displaystyle-\delta C^{FVV}_{(ab)L}=$ $\displaystyle
g^{LL}m_{a}\left[X_{3}+\frac{4C_{00}-(m_{b}^{2}-m_{a}^{2})X_{1}}{2m_{V}^{2}}\right]+g^{RR}m_{b}\left[X_{3}+\frac{4C_{00}+(m_{b}^{2}-m_{a}^{2})X_{2}}{2m_{V}^{2}}\right]$
$\displaystyle+g^{RL}m_{F}\left[2C_{0}-\frac{8C_{00}+m_{a}^{2}(2X_{1}+X_{0})+m_{b}^{2}(2X_{2}+X_{0})}{2m_{V}^{2}}\right]+g^{LR}\frac{m_{F}m_{a}m_{b}X_{012}}{m_{V}^{2}},$
$\displaystyle-\delta C^{FVV}_{(ab)R}=$ $\displaystyle\delta
C_{L,5}\left[g_{a}^{L}\leftrightarrow g_{a}^{R},g_{b}^{L}\leftrightarrow
g_{b}^{R}\right],$ $\displaystyle-\delta D^{FVV}_{(ab)L}=$ $\displaystyle
g^{LL}\left[-(m_{a}^{2}+m_{b}^{2})X_{3}-\frac{4(m_{a}^{2}+m_{b}^{2})C_{00}+(m_{b}^{2}-m_{a}^{2})(-m_{a}^{2}X_{1}+m_{b}^{2}X_{2})}{2m_{V}^{2}}\right]$
$\displaystyle+g^{RR}m_{a}m_{b}\left[-2X_{3}-\frac{8C_{00}+(m_{b}^{2}-m_{a}^{2})(-X_{1}+X_{2})}{2m_{V}^{2}}\right]$
$\displaystyle+g^{RL}m_{a}m_{F}\left[-2C_{0}-\frac{-8C_{00}+(m_{b}^{2}-m_{a}^{2})(2X_{1}+X_{0})}{2m_{V}^{2}}\right]$
$\displaystyle+g^{LR}m_{b}m_{F}\left[-2C_{0}+\frac{8C_{00}+(m_{b}^{2}-m_{a}^{2})(2X_{2}+X_{0})}{2m_{V}^{2}}\right],$
$\displaystyle\delta D^{FVV}_{(ab)R}=$ $\displaystyle\delta
D^{FVV}_{(ab)L}\left[g_{a}^{L}\leftrightarrow
g_{a}^{R},g_{b}^{L}\leftrightarrow g_{b}^{R}\right].$ (98)

All results given in Eq. (C.2) were cross checked using FORM package
Vermaseren:2000nd . All formulas in Eq. (C.2) satisfy automatically the WI,
namely $\delta D^{FVV}_{(ab)L}+m_{a}\delta C^{FVV}_{(ab)L}+m_{b}\delta
C^{FVV}_{(ab)R}=0$.

Diagram (6)

After using the property of chiral operators $P_{L,R}$, the amplitude (C.2) is
written as

$\displaystyle i\mathcal{M}_{6}=$ $\displaystyle
eQ_{F}\int\frac{d^{4}k}{(2\pi)^{4}}\frac{1}{D_{0}D_{1}D_{2}}\times\overline{u_{a}}\left[\left(m_{F}^{2}\gamma_{\alpha}\not{\varepsilon}^{*}\gamma_{\beta}+\gamma_{\alpha}\not{k}_{1}\not{\varepsilon}^{*}\not{k}_{2}\gamma_{\beta}\right)\left[A_{2}\right]\right.$
$\displaystyle\left.\qquad\;-m_{F}\left[A_{1}\right]\left(g_{a}^{L^{*}}g_{b}^{R}P_{R}+g_{a}^{R^{*}}g_{b}^{L}P_{L}\right)\left(\gamma_{\alpha}\not{k}_{1}\not{\varepsilon}^{*}\gamma_{\beta}+\gamma_{\alpha}\not{\varepsilon}^{*}\not{k}_{2}\gamma_{\beta}\right)\right]u_{b}.$
(99)

The numerator is divided into the two parts $N_{1}\sim g^{\alpha\beta}$ and
$N_{2}\sim-k^{\alpha}k^{\beta}/m_{V}^{2}$. After extracting $g^{\alpha\beta}$,
the first part is

$\displaystyle N_{1}=$
$\displaystyle\overline{u_{a}}\left\\{\left[(2-d)m_{F}^{2}\not{\varepsilon}^{*}-2\not{k}_{2}\not{\varepsilon}^{*}\not{k}_{1}+(4-d)\not{k}_{1}\not{\varepsilon}^{*}\not{k}_{2}\right]\left[A_{2}\right]\right.$
$\displaystyle\left.\quad\;-m_{F}\left[A_{1}\right]\left[4\varepsilon^{*}.(k_{1}+k_{2})-(4-d)\left(\not{k}_{1}\not{\varepsilon}^{*}+\not{\varepsilon}^{*}\not{k}_{2}\right)\right]\right\\}u_{b}.$
(100)

Ignoring the overall factor $eQ_{F}/(16\pi^{2})$, the formula in terms of
tensor notations is

$\displaystyle N_{1}=$
$\displaystyle\overline{u_{a}}\not{\varepsilon}^{*}\left[A_{2}\right]u_{b}\left[-2m_{F}^{2}C_{0}+(d-4)(d-2)C_{00}\right]+\left(2p_{1}.\varepsilon^{*}\right)\overline{u_{a}}\left[A_{1}\right]\left(4m_{F}X_{0}\right)u_{b}$
$\displaystyle+\overline{u_{a}}\left[(2-d)C_{\alpha\beta}\gamma^{\alpha}\not{\varepsilon}^{*}\gamma^{\beta}+2C_{\alpha}\left(\not{p}_{2}\not{\varepsilon}^{*}\gamma^{\alpha}+\gamma^{\alpha}\not{\varepsilon}^{*}\not{p}_{1}\right)-2C_{0}\not{p}_{2}\not{\varepsilon}^{*}\not{p}_{1}\right]\times\left[A_{2}\right]u_{b}.$
(101)

After expanding the tensors in terms of scalar PV-functions, the final result
is

$\displaystyle N_{1}=$
$\displaystyle\overline{u_{a}}\not{\varepsilon}^{*}\left[A_{2}\right]u_{b}\left[-2m_{F}^{2}C_{0}+(d-2)^{2}C_{00}+2m_{a}^{2}X_{01}+m_{b}^{2}X_{02}\right]+\overline{u_{a}}\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}\left[A_{2}\right]u_{b}\times(2X_{0})$
$\displaystyle+\left(2\varepsilon^{*}.p_{1}\right)\overline{u_{a}}(-2)\left[X_{01}\not{p}_{1}+X_{02}\not{p}_{2}\right]\left[A_{2}\right]u_{b}+\left(2\varepsilon^{*}.p_{1}\right)\overline{u_{a}}\left\\{4m_{F}X_{0}\left[A_{1}\right]\right\\}u_{b}.$
(102)

Considering the second term proportional to $k^{\alpha}k^{\beta}$, we have

$\displaystyle-m_{V}^{2}N_{2}$
$\displaystyle=\overline{u_{a}}\left(m_{F}^{2}\not{k}\not{\varepsilon}^{*}\not{k}+\not{k}\not{k}_{1}\not{\varepsilon}^{*}\not{k}_{2}\not{k}\right)\left[A_{2}\right]u_{b}-m_{F}\overline{u_{a}}\left(\not{k}\not{k}_{1}\not{\varepsilon}^{*}\not{k}+\not{k}\not{\varepsilon}^{*}\not{k}_{2}\not{k}\right)\left[A_{1}\right]u_{b}.$
(103)

The two relations
$\not{k}\not{k}_{1}=D_{1}+m_{F}^{2}-m_{a}^{2}+\not{p}_{1}\not{k}$ and
$\not{k}_{2}\not{k}=D_{2}+m_{F}^{2}-m_{b}^{2}+\not{k}\not{p}_{2}$ give

$\displaystyle N_{2}\sim$
$\displaystyle\overline{u_{a}}\left(m_{F}^{2}\not{k}\not{\varepsilon}^{*}\not{k}\right)\left[A_{2}\right]u_{b}+\overline{u_{a}}\left(D_{1}+m_{F}^{2}-m_{1}^{2}+\not{p}_{1}\not{k}\right)\not{\varepsilon}^{*}\left(D_{2}+m_{F}^{2}-m_{2}^{2}+\not{k}\not{p}_{2}\right)\left[A_{2}\right]u_{b}$
$\displaystyle-
m_{F}\overline{u_{a}}\left[\left(D_{1}+m_{F}^{2}-m_{a}^{2}+\not{p}_{1}\not{k}\right)\not{\varepsilon}^{*}\not{k}+\not{k}\not{\varepsilon}^{*}\left(D_{2}+m_{F}^{2}-m_{b}^{2}+\not{k}\not{p}_{2}\right)\right]\left[A_{1}\right]u_{b}$
$\displaystyle\equiv$
$\displaystyle\overline{u_{a}}\left[\left(L_{1}+L_{2}\right)\left[A_{2}\right]-m_{F}\left[A_{1}\right]L_{3}\right]u_{b},$
(104)

where

$\displaystyle L_{1}=$ $\displaystyle
m_{F}^{2}\left\\{\not{\varepsilon}^{*}\left[(2-d)C_{00}-m_{a}^{2}\left(C_{11}+C_{12}\right)-m_{b}^{2}\left(C_{12}+C_{22}\right)\right]\right.$
$\displaystyle\left.\quad\;+(2p_{1}.\varepsilon^{*})\left[\left(C_{11}+C_{12}\right)\not{p}_{1}+\left(C_{12}+C_{22}\right)\not{p}_{2}\right]\right\\},$
$\displaystyle L_{2}=$
$\displaystyle\frac{1}{D_{0}D_{1}D_{2}}\left(D_{1}+m_{F}^{2}-m_{a}^{2}+\not{p}_{1}\not{k}\right)\not{\varepsilon}^{*}\left(D_{2}+m_{F}^{2}-m_{b}^{2}+\not{k}\not{p}_{2}\right)$
$\displaystyle=$
$\displaystyle\not{\varepsilon}^{*}\left[\frac{1}{D_{0}}+\frac{m_{F}^{2}-m_{b}^{2}+\not{k}\not{p}_{2}}{D_{0}D_{2}}+\frac{m_{F}^{2}-m_{a}^{2}}{D_{0}D_{1}}+\frac{\left(m_{F}^{2}-m_{a}^{2}\right)\left(m_{F}^{2}-m_{b}^{2}\right)}{D_{0}D_{1}D_{2}}\right]+\frac{\not{p}_{1}\not{k}\not{\varepsilon}^{*}}{D_{0}D_{1}}$
$\displaystyle+\frac{\not{p}_{1}\not{k}\not{\varepsilon}^{*}\not{k}\not{p}_{2}}{D_{0}D_{1}D_{2}}+\frac{\not{p}_{1}\not{k}\not{\varepsilon}^{*}\left(m_{F}^{2}-m_{b}^{2}\right)}{D_{0}D_{1}D_{2}}+\frac{\left(m_{F}^{2}-m_{a}^{2}\right)\not{\varepsilon}^{*}\not{k}\not{p}_{2}}{D_{0}D_{1}D_{2}}$
$\displaystyle=$
$\displaystyle\not{\varepsilon}^{*}\left[A_{0}(m_{V}^{2})+\left(m_{F}^{2}-m_{b}^{2}\right)B^{(2)}_{0}-m_{b}^{2}B^{(2)}_{1}+\left(m_{F}^{2}-m_{a}^{2}\right)B^{(1)}_{0}-m_{a}^{2}B^{(1)}_{1}\right.$
$\displaystyle\left.+\left(m_{F}^{2}-m_{a}^{2}\right)\left(m_{F}^{2}-m_{b}^{2}\right)C_{0}\right]$
$\displaystyle+C_{\alpha\beta}(\not{p}_{1}\gamma^{\alpha}\not{\varepsilon}^{*}\gamma^{\beta}\not{p}_{2})+C_{\alpha}(\not{p}_{1}\gamma^{\alpha}\not{\varepsilon}^{*})\left(m_{F}^{2}-m_{b}^{2}\right)+C_{\alpha}(\not{\varepsilon}^{*}\gamma^{\alpha}\not{p}_{2})\left(m_{F}^{2}-m_{a}^{2}\right),$
$\displaystyle L_{3}=$
$\displaystyle\frac{\not{\varepsilon}^{*}\not{k}}{D_{0}D_{2}}+\frac{\not{k}\not{\varepsilon}^{*}}{D_{0}D_{1}}+\frac{m_{F}^{2}(2k.\varepsilon^{*})-m_{a}^{2}\not{\varepsilon}^{*}\not{k}-m_{b}^{2}\not{k}\not{\varepsilon}^{*}}{D_{0}D_{1}D_{2}}+\frac{\not{p}_{1}\not{k}\not{\varepsilon}^{*}\not{k}+\not{k}\not{\varepsilon}^{*}\not{k}\not{p}_{2}}{D_{0}D_{1}D_{2}}$
$\displaystyle=$
$\displaystyle-B^{(2)}_{1}\not{\varepsilon}^{*}\not{p}_{2}-B^{(1)}_{1}\not{p}_{1}\not{\varepsilon}^{*}-(2p_{1}.\varepsilon^{*})(C_{1}+C_{2})m_{F}^{2}$
$\displaystyle-
C_{\alpha}\left(m_{a}^{2}\not{\varepsilon}^{*}\gamma^{\alpha}+m_{b}^{2}\gamma^{\alpha}\not{\varepsilon}^{*}\right)+C_{\alpha\beta}\left(\not{p}_{1}\gamma^{\alpha}\not{\varepsilon}^{*}\gamma^{\beta}+\gamma^{\alpha}\not{\varepsilon}^{*}\gamma^{\beta}\not{p}_{2}\right).$

It can be proved that:

$\displaystyle
C_{\alpha\beta}(\not{p}_{1}\gamma^{\alpha}\not{\varepsilon}^{*}\gamma^{\beta}\not{p}_{2})=\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}\left[(2-d)C_{00}-m_{a}^{2}(C_{11}+C_{12})-m_{b}^{2}(C_{22}+C_{12})\right]$
$\displaystyle\qquad\qquad\qquad\qquad+(2p_{1}.\varepsilon^{*})\left[m_{b}^{2}(C_{22}+C_{12})\not{p}_{1}+m_{a}^{2}(C_{11}+C_{12})\not{p}_{2}\right],$
$\displaystyle
C_{\alpha}(\not{p}_{1}\gamma^{\alpha}\not{\varepsilon}^{*})=-m_{a}^{2}C_{1}\not{\varepsilon}^{*}-C_{2}\left[(2p_{1}.\varepsilon^{*})\not{p}_{1}-\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}\right],$
$\displaystyle
C_{\alpha}(\not{\varepsilon}^{*}\gamma^{\alpha}\not{p}_{2})=-m_{b}^{2}C_{2}\not{\varepsilon}^{*}-C_{1}\left[(2p_{1}.\varepsilon^{*})\not{p}_{2}-\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}\right],$
$\displaystyle
C_{\alpha}\left(m_{a}^{2}\not{\varepsilon}^{*}\gamma^{\alpha}+m_{b}^{2}\gamma^{\alpha}\not{\varepsilon}^{*}\right)$
$\displaystyle=$
$\displaystyle(2p_{1}.\varepsilon^{*})\left[-m_{a}^{2}C_{1}-m_{b}^{2}C_{2}\right]+\not{p}_{1}\not{\varepsilon}^{*}(m_{a}^{2}-m_{b}^{2})C_{1}+\not{\varepsilon}^{*}\not{p}_{2}(m_{b}^{2}-m_{a}^{2})C_{2},$
$\displaystyle
C_{\alpha\beta}\left(\not{p}_{1}\gamma^{\alpha}\not{\varepsilon}^{*}\gamma^{\beta}+\gamma^{\alpha}\not{\varepsilon}^{*}\gamma^{\beta}\not{p}_{2}\right)$
$\displaystyle=$
$\displaystyle(2p_{1}.\varepsilon^{*})\left[m_{a}^{2}(C_{11}+C_{12})+m_{b}^{2}(C_{22}+C_{12})+\not{p}_{1}\not{p}_{2}(C_{11}+2C_{12}+C_{22})\right]$
$\displaystyle+(\not{p}_{1}\not{\varepsilon}^{*}+\not{\varepsilon}^{*}\not{p}_{2})\left[(2-d)C_{00}-m_{a}^{2}(C_{11}+C_{12})-m_{b}^{2}(C_{22}+C_{12})\right].$
(105)

Final results are:

$\displaystyle L_{1}=$ $\displaystyle
m_{F}^{2}\left\\{\frac{}{}\not{\varepsilon}^{*}\left[(2-d)C_{00}-m_{a}^{2}\left(C_{11}+C_{12}\right)-m_{b}^{2}\left(C_{12}+C_{22}\right)\right]\right.$
$\displaystyle\left.+(2p_{1}.\varepsilon^{*})\left[\left(C_{11}+C_{12}\right)\not{p}_{1}+\left(C_{12}+C_{22}\right)\not{p}_{2}\right]\right\\},$
$\displaystyle L_{2}=$
$\displaystyle\not{\varepsilon}^{*}\left\\{m_{V}^{2}(B^{(0)}_{0}+1)+m_{F}^{2}(B^{(1)}_{0}+B^{(2)}_{0})-m_{a}^{2}(B^{(1)}_{0}+B^{(1)}_{1})-m_{b}^{2}(B^{(2)}_{0}+B^{(2)}_{1})\right.$
$\displaystyle\left.\quad+m_{F}^{4}C_{0}-m_{F}^{2}\left[(m_{a}^{2}+m_{b}^{2})C_{0}+m_{a}^{2}C_{1}+m_{b}^{2}C_{2}\right]+m_{a}^{2}m_{b}^{2}X_{0}\right\\}$
$\displaystyle+\not{p}_{1}\not{\varepsilon}^{*}\not{p}_{2}\left[(2-d)C_{00}+m_{F}^{2}X_{3}-m_{a}^{2}X_{1}-m_{b}^{2}X_{2}\right]$
$\displaystyle+(2p_{1}.\varepsilon^{*})\left[\left(m_{b}^{2}X_{2}-m^{2}_{F}C_{2}\right)\not{p}_{1}+\left(m_{a}^{2}X_{1}-m^{2}_{F}C_{1}\right)\not{p}_{2}\right],$
$\displaystyle L_{3}=$
$\displaystyle\not{p}_{1}\not{\varepsilon}^{*}\left[-B^{(1)}_{1}+(2-d)C_{00}-m_{a}^{2}X_{1}+m_{b}^{2}(X_{3}-X_{2})\right]$
$\displaystyle+\not{\varepsilon}^{*}\not{p}_{2}\left[-B^{(2)}_{1}+(2-d)C_{00}-m_{b}^{2}X_{2}+m_{a}^{2}(X_{3}-X_{1})\right]$
$\displaystyle+(2p_{1}.\varepsilon^{*})\left[m_{a}^{2}X_{1}+m_{b}^{2}X_{2}-m_{F}^{2}X_{3}+\not{p}_{1}\not{p}_{2}\left(X_{1}+X_{2}-X_{3}\right)\right].$
(106)

The above calculation is enough to derive relevant contributions to
$C^{VFF}_{L,R}$ given in Eqs. (II) and (II), and $D^{VFF}_{L,R}$ given in
(II).

Ward identity for the only gauge boson exchanges

Before coming to discuss the WI, we use the relations given in Eq. (A.2) to
write all the one-loop factors (22), (23), and (II) from gauge boson exchanges
in the following simple forms, ignoring the overall factor $e/(16\pi^{2})$:

$\displaystyle D^{FV}_{(ab)L,78}=$ $\displaystyle
Q_{e}\left(g^{RL}m_{a}+g^{LR}m_{b}\right)(-3m_{F}X_{0})+Q_{e}g^{RR}m_{a}m_{b}\left[-2X^{f}_{12}+\frac{\mathbf{b}^{f}_{1}-m_{F}^{2}(2X_{0}+X^{f}_{12})}{m_{V}^{2}}\right]$
$\displaystyle+Q_{e}g^{LL}\left\\{\left(2+\frac{m_{F}^{2}+m_{a}^{2}+m_{b}^{2}}{m_{V}^{2}}\right)\mathbf{b}^{f}_{1}+1+\frac{A_{0}(m_{V}^{2})+m_{a}^{2}m_{b}^{2}X^{f}_{12}}{m_{V}^{2}}\right.$
$\displaystyle\qquad\qquad\quad\left.+\frac{2m_{F}^{2}(m_{a}^{2}B^{(1)}_{0}-m_{b}^{2}B^{(2)}_{0})}{(m_{a}^{2}-m_{b}^{2})m_{V}^{2}}\right\\}.$
(107)

The WI for the $FVV$ and $FVV$ diagrams are $f^{WI}_{FVV}\equiv
D^{FVV}_{(ab)L}+m_{a}C^{FVV}_{(ab)L}+m_{b}C^{FVV}_{(ab)R}$ and
$f^{WI}_{VFF}\equiv
D^{VFF}_{(ab)L}+m_{a}C^{VFF}_{(ab)L}+m_{b}C^{VFF}_{(ab)R}$, respectively. The
relations given in Eq. (A.2) give:

$\displaystyle
2C^{f}_{00}+m_{a}^{2}X_{1}^{f}+m_{b}^{2}X_{2}^{f}=-\mathbf{b}^{f}_{1},$
$\displaystyle X^{v}_{012}=-X^{f}_{12},$ $\displaystyle
m_{F}^{2}(X^{v}_{12}-X^{v}_{3})+m_{a}^{2}X_{1}^{v}+m_{b}^{2}X_{2}^{v}+\mathbf{b}_{1}^{v}+1/2=m_{F}^{2}(X^{v}_{12}-X^{v}_{3})-m_{F}^{2}C_{0}^{v}$
$\displaystyle=m_{F}^{2}(X^{v}_{012}-2X_{0})=-m_{F}^{2}(X^{f}_{12}+2X_{0}),$
$\displaystyle
1/2+m_{a}^{2}X^{v}_{1}+m_{b}^{2}X^{v}_{2}-m_{F}^{2}X_{3}^{v}=(m_{F}^{2}-m_{V}^{2})X_{0}-m_{F}^{2}C_{0}^{v}-m_{F}^{2}X_{3}^{v}=-m_{V}^{2}X_{0}.$
(108)

Combining the above formulas and results of $C_{i,ij}$ functions listed in
Ref. Hue:2017lak , the WI of all diagrams with boson exchanges is derived as
follows

$\displaystyle f^{WI}_{V}=$ $\displaystyle
D_{(ab)L,78}+f^{WI}_{FVV}+f^{WI}_{VFF}$ $\displaystyle\sim$
$\displaystyle\left(Q_{e}+Q_{V}-Q_{F}\right)$
$\displaystyle\times\left\\{g^{LL}\left[\frac{3-B^{(0)}_{0}(m_{V}^{2})}{2}+\frac{A_{0}(m_{F}^{2})}{2m_{V}^{2}}-\frac{(m_{a}^{2}+m_{F}^{2}-2m_{V}^{2})m_{V}^{2}+(m_{a}^{2}-m_{F}^{2})^{2}}{2(m_{a}^{2}-m_{b}^{2})m_{V}^{2}}\times
B^{(1)}_{0}\right.\right.$
$\displaystyle\left.\left.\qquad\qquad+\frac{(m_{b}^{2}+m_{F}^{2}-2m_{V}^{2})m_{V}^{2}+(m_{b}^{2}-m_{F}^{2})^{2}}{2(m_{a}^{2}-m_{b}^{2})m_{V}^{2}}\times
B^{(2)}_{0}\right]\right.$
$\displaystyle\left.+g^{RR}\left[\frac{(m_{V}^{2}B^{(0)}_{0}-A_{0}(m_{F}^{2}))\times(m_{F}^{2}+2m_{V}^{2})-(m_{F}^{2}-4m_{V}^{2})m_{V}^{2}}{2m_{V}^{2}}\right.\right.$
$\displaystyle\left.\left.\qquad\quad-\frac{m_{b}^{2}\left[(m_{a}^{2}+m_{F}^{2}-2m_{V}^{2})m_{V}^{2}+(m_{a}^{2}-m_{F}^{2})^{2}\right]}{2(m_{a}^{2}-m_{b}^{2})m_{V}^{2}}\times
B^{(1)}_{0}\right.\right.$
$\displaystyle\left.\left.\qquad\quad+\frac{m_{a}^{2}\left[(m_{b}^{2}+m_{F}^{2}-2m_{V}^{2})m_{V}^{2}+(m_{b}^{2}-m_{F}^{2})^{2}\right]}{2(m_{a}^{2}-m_{b}^{2})m_{V}^{2}}\times
B^{(2)}_{0}\right]\right.$
$\displaystyle\left.+\left(g^{RL}m_{a}+g^{LR}m_{b}\right)(3m_{F}X_{0})\frac{}{}\right\\}.$
(109)

The final result is $f^{WI}_{V}\sim Q_{F}-(Q_{e}+Q_{V})=0$. In conclusion, the
contributions from the four diagrams with only gauge boson exchanges satisfy
the WI when the electric charge conservation is valid.

##  Appendix D  Ward Identity for the diagrams of FSV-type in the unitary
gauge

This type of diagrams were mentioned firstly in Ref. Yu:2021suw  for the
general case of their contributions to BSM. The $\gamma-S-V$ vertices come the
kinetic terms of the scalars:

$\displaystyle
L^{D}(S)=\left(\partial_{\mu}S-iP_{\mu}S\right)^{\dagger}\left(\partial^{\mu}S-iP^{\mu}S\right)=\left[g_{\gamma
SV}g_{\mu\nu}S^{-Q}A^{\mu}V^{Q\nu}+\mathrm{h.c.}\right]+\dots,$ (110)

where $P_{\mu}$ containing the photon $A_{\mu}$ and $V_{\mu}$ is the covariant
part of the covariant derivative of the Higgs multiplets. The Feynman diagrams
in the general gauge $R_{\xi}$ are shown in Fig. 3.

Figure 3: One-loop three-point FSV diagrams in the gauge $R_{\xi}$

Here only two diagrams (1) and (2) give non-zeros contributions in the unitary
gauge, which correspond to the two diagrams (b) and (a) in Fig. 5 introduced
in Ref. Yu:2021suw . In this gauge, the contributions of these two diagrams
are:

$\displaystyle i\mathcal{M}_{9}=$ $\displaystyle g_{\gamma
SV}\int\frac{d^{4}k}{(2\pi)^{4}}\times\frac{\overline{u_{a}}[g_{a}^{L^{*}}P_{R}+g_{a}^{R^{*}}P_{L}](m_{F}+\not{k})\gamma_{\alpha}[{g_{b}^{L}}P_{L}+{g_{b}^{R}}P_{R}]u_{b}}{D_{0}D_{1}D_{2}}\left(\varepsilon^{*\alpha}-\frac{(\varepsilon^{*}.k_{2})k_{2}^{\alpha}}{m^{2}_{V}}\right)$
$\displaystyle=$ $\displaystyle g_{\gamma
SV}\int\frac{d^{4}k}{(2\pi)^{4}}\times\frac{1}{D_{0}D_{1}D_{2}}$
$\displaystyle\times\overline{u_{a}}\left\\{\not{\varepsilon}^{*}\left[m_{F}\left[A_{2}\right]+\not{k}\left[A_{1}\right]\right]-\frac{\not{k}_{2}(k_{2}.\varepsilon^{*})}{m_{V}^{2}}m_{F}\left[A_{2}\right]-\left[A_{1}\right]\frac{\left(D_{0}+m_{F}^{2}-\not{k}\not{p}_{2}\right)(k_{2}.\varepsilon^{*})}{m_{V}^{2}}\right\\}u_{b}$
$\displaystyle=$ $\displaystyle\frac{ig_{\gamma
SV}}{16\pi^{2}}\overline{u_{a}}\left\\{\not{\varepsilon}^{*}\left[C_{0}m_{F}\left[A_{2}\right]-(C_{1}\not{p}_{1}+C_{2}\not{p}_{2})\left[A_{1}\right]\right]\right.$
$\displaystyle\left.-\frac{m_{F}}{m_{V}^{2}}\left[(\gamma^{\mu}\varepsilon^{*\nu})C_{\mu\nu}-(C_{\mu}\gamma^{\mu})(p_{2}.\varepsilon^{*})+\not{p}_{2}X_{0}(p_{1}.\varepsilon^{*})\right]\left[A_{2}\right]\right.$
$\displaystyle\left.+\frac{1}{m_{V}^{2}}\left[A_{1}\right]\left[C_{00}\not{\varepsilon}^{*}\not{p}_{2}+(p_{1}.\varepsilon^{*})\left(m_{F}^{2}X_{0}+X_{1}\not{p}_{1}\not{p}_{2}+m_{b}^{2}X_{2}\right)\right]\right\\}u_{b},$
(111)

where we have used $k_{2}.\varepsilon^{*}/(D_{1}D_{2})\to 0$. The formulas of
$D_{L,R}$ and $C_{L,R}$ are:

$\displaystyle eD^{Fhv}_{(ab)L,9}\times\left(\frac{g_{\gamma
SV}}{16\pi^{2}}\right)^{-1}=$ $\displaystyle
g^{LL}m_{F}\left(C_{0}-\frac{C_{00}}{m_{V}^{2}}\right)-g^{RL}m_{a}C_{1}+g^{LR}m_{b}\left(C_{2}+\frac{C_{00}}{m_{V}^{2}}\right),$
$\displaystyle D^{Fhv}_{(ab)R,9}=$ $\displaystyle
D^{Fhv}_{(ab)L,9}\left[g^{L}_{a}\leftrightarrow
g^{R}_{a},g^{L}_{b}\leftrightarrow g^{R}_{b}\right],$ $\displaystyle
eC^{Fvh}_{(ab)L,9}\times\left(\frac{g_{\gamma SV}}{16\pi^{2}}\right)^{-1}=$
$\displaystyle-g^{RL}C_{2}-\frac{m_{F}}{2m_{V}^{2}}\left[g^{LL}m_{a}X_{1}+g^{RR}m_{b}X_{02}\right]$
$\displaystyle+\frac{1}{2m_{V}^{2}}\left[g^{RL}(m_{F}^{2}X_{0}+m_{b}^{2}X_{2})+g^{LR}m_{a}m_{b}X_{1}\right],$
$\displaystyle C^{Fhv}_{(ab)R,9}=$ $\displaystyle
C_{(ab)L}\left[g^{L}_{a}\leftrightarrow g^{R}_{a},g^{L}_{b}\leftrightarrow
g^{R}_{b}\right],$ (112)

where $X_{i}^{Fhv}\equiv
X_{i}(m_{a}^{2},0,m_{b}^{2};m_{F}^{2},m_{h}^{2},m_{V}^{2})$. Similarly, the
results for diagram (10) are:

$\displaystyle eD^{Fvh}_{(ab)L,10}\times\left(\frac{g_{\gamma
SV}}{16\pi^{2}}\right)^{-1}=$ $\displaystyle
g^{LL}m_{F}\left(C_{0}-\frac{C_{00}}{m_{V}^{2}}\right)-g^{LR}m_{b}C_{2}+g^{RL}m_{a}\left(C_{1}+\frac{C_{00}}{m_{V}^{2}}\right),$
$\displaystyle D^{Fvh}_{(ab)R,10}=$ $\displaystyle
D^{Fvh}_{(ab)L,10}\left[g^{L}_{a}\leftrightarrow
g^{R}_{a},g^{L}_{b}\leftrightarrow g^{R}_{b}\right],$ $\displaystyle
eC^{Fvh}_{(ab)L,10}\times\left(\frac{g_{\gamma SV}}{16\pi^{2}}\right)^{-1}=$
$\displaystyle-g^{R*}_{a}g^{L}_{b}C_{1}-\frac{m_{F}}{2m_{V}^{2}}\left[g^{R*}_{a}g^{R}_{b}m_{b}X_{2}+g^{L*}_{a}g^{L}_{b}m_{a}X_{01}\right]$
$\displaystyle+\frac{1}{2m_{V}^{2}}\left[g^{R*}_{a}g^{L}_{b}(m_{F}^{2}X_{0}+m_{a}^{2}X_{1})+g^{L*}_{a}g^{R}_{b}m_{a}m_{b}X_{2}\right],$
$\displaystyle eC^{Fvh}_{(ab)R,10}\times\left(\frac{g_{\gamma
SV}}{16\pi^{2}}\right)^{-1}=$ $\displaystyle
C^{Fvh}_{(ab)L}\left[g^{L}_{a}\leftrightarrow
g^{R}_{a},g^{L}_{b}\leftrightarrow g^{R}_{b}\right],$ (113)

where $X^{Fvh}_{i}\equiv
X_{i}(m_{a}^{2},0,m_{b}^{2};m_{F}^{2},m_{V}^{2},m_{h}^{2})$. The above
formulas are consistent with calculation using FORM. The corresponding
formulas of WI are

$\displaystyle\frac{f_{WI}^{Fhv}}{k_{\gamma SV}}$
$\displaystyle=g^{LL}m_{F}\left[2(m_{V}^{2}C_{0}-C_{00})+m_{a}m_{b}X_{012}\right]^{fhv}-g^{RR}m_{F}\left[m_{a}^{2}X_{1}+m_{b}^{2}X_{02}\right]^{fhv}$
$\displaystyle+g^{RL}\left[-2m_{V}^{2}(m_{a}C_{1}+m_{b}C_{2})+m_{b}\left(m_{F}^{2}X_{0}+m_{a}^{2}X_{1}+m_{b}^{2}X_{2}\right)\right]^{fhv}$
$\displaystyle+g^{LR}\left[2m_{V}^{2}(m_{b}C_{2}-m_{a}C_{1})+2m_{b}C_{00}+m_{a}\left(m_{f}^{2}X_{0}+m_{b}^{2}X_{12}\right)\right]^{fhv},$
$\displaystyle\frac{f_{WI}^{Fvh}}{k_{\gamma SV}}$
$\displaystyle=g^{LL}m_{F}\left[2(m_{V}^{2}C_{0}-C_{00})-m_{a}m_{b}X_{012}\right]-g^{RR}m_{F}\left[m_{a}^{2}X_{01}+m_{b}^{2}X_{2}\right]^{fvh}$
$\displaystyle+g^{RL}\left[2m_{V}^{2}(m_{a}C_{1}-m_{b}C_{2})+2m_{a}C_{00}+m_{b}\left(m_{f}^{2}X_{0}+m_{a}^{2}X_{12}\right)\right]^{fvh}$
$\displaystyle+g^{LR}\left[-2m_{V}^{2}(m_{a}C_{1}+m_{b}C_{2})+m_{a}\left(m_{f}^{2}X_{0}+m_{a}^{2}X_{1}+m_{b}^{2}X_{2}\right)\right]^{fvh},$
(114)

where $k_{\gamma SV}=g_{\gamma SV}/(32\pi^{2}m_{V}^{2})$. The WI is valid if
only $f_{WI}^{Fhv}+f_{WI}^{Fvh}=0$. We can see crudely that all $C_{(ab)L,9}$,
$C_{(ab)R,9}$, $C_{(ab)L,10}$, and $C_{(ab)R,10}$ are convergent. In contrast,
all $D_{(ab)L,9}$, $D_{(ab)R,9}$, $D_{(ab)L,10}$, and $D_{(ab)R,10}$ contain
divergent terms. Therefore, the necessary condition to guarantee the
validation of the WI given in Eq. (12) is that all of these divergent terms
must vanish. Strictly, the WI is valid if only $g_{\gamma SV}$=0 or
$g^{L}_{a}=g^{R}_{a}=0$. Because at least one of $g^{L}_{a}$ or $g^{R}_{a}$
must be non-zero, the condition $g_{\gamma SV}$=0 is the only valid choice,
i.e., the vertex-type $\gamma$-$S$-$V$ does not appear in the all BSM
guaranteeing the WI for the external photon. This conclusion is also true for
the case $a=b$, corresponding to the one-loop contribution to the AMM of the
leptons.

Finally, using the assumption of the Lagrangian for couplings of the Goldstone
boson given in Eq. (B), we can determine the one-loop contributions of the FSV
diagrams mentioned above, using the general gauge $R_{\xi}$. The propagator of
the gauge boson $V$ can be written in terms of two separated parts:

$\displaystyle\Delta^{(\xi)\mu\nu}_{V}(k^{2})$
$\displaystyle\equiv\Delta^{(u)\mu\nu}_{V}(k^{2})+\Delta^{(T)\mu\nu}_{\xi,V}(k^{2}),$
$\displaystyle\Delta^{(u)\mu\nu}_{V}(k^{2})$
$\displaystyle=\frac{-i}{k^{2}-m^{2}_{V}}\left(g^{\mu\nu}-\frac{k^{\mu}k^{\nu}}{m^{2}_{V}}\right),$
$\displaystyle\Delta^{(T)\mu\nu}_{\xi,V}(k^{2})$
$\displaystyle=\frac{-i}{m^{2}_{V}}\times\frac{k^{\mu}k^{\nu}}{k^{2}-\xi\,m^{2}_{V}}=\frac{-k^{\mu}k^{\nu}}{m_{V}^{2}}\times
i\Delta^{0\xi}_{G_{V}},$ (115)

where $\Delta^{(u)\mu\nu}_{V}(k^{2})$ is the propagator in the unitary gauge,
and $\Delta^{0\xi}_{G_{V}}$ relates to the propagator of $G_{V}$ as follows:

$\displaystyle\Delta^{\xi}_{G_{V}}$
$\displaystyle=i\Delta^{0\xi}_{G_{V}}=\frac{i}{k^{2}-\xi
m_{V}^{2}}=\left[\begin{array}[]{cc}0&\xi\to\infty:\;\mathrm{Unitary}(u),\\\
\frac{i}{k^{2}-m_{V}^{2}},&\xi=1:\;\mathrm{{}^{\prime}t\;Hooft-
Feynman}(HF)\end{array}\right..$ (118)

For two diagrams (3) and (4) in Fig. 3, the Feynman rules for the couplings
$\gamma-S-G_{V}$ are the same form as those given in Lagrangian (III.1),
namely $S\equiv h_{1}$ and $G_{V}\equiv h_{2}$. The reason is that all mass
eigenstates of the scalar with the same electric charges come from the same
squared mass matrix. Therefore, $\mathcal{L}^{\gamma
hG_{V}}=ieQ_{H}A^{\mu}\left[\left(h^{*}\partial_{\mu}G_{V}-G_{V}\partial_{\mu}h^{*}\right)+\mathrm{h.c.}\right]$.
Formulas corresponding to diagrams (1) and (3) of Fig. 3 in the general gauge
$R_{\xi}$ are

$\displaystyle i\mathcal{M}^{(\xi)}_{9}=$ $\displaystyle
i\mathcal{M}^{(u)}_{9}$ $\displaystyle+g_{\gamma
SV}\int\frac{d^{4}k}{(2\pi)^{4}}\times\frac{\overline{u_{a}}[g_{a,Fh}^{L^{*}}P_{R}+g_{a,Fh}^{R^{*}}P_{L}](m_{F}+\not{k})\gamma_{\alpha}[{g_{b,FV}^{L}}P_{L}+{g_{b,FV}^{R}}P_{R}]u_{b}(\varepsilon^{*}.k_{2})k_{2}^{\alpha}}{D_{0}D_{1}D_{2}m^{2}_{V}},$

where $D_{0}=k^{2}-m_{F}^{2}$, $D_{1}=k_{1}^{2}-m_{h}^{2}$,
$D_{2}=k_{2}^{2}-\xi m^{2}_{V}$, and $\mathcal{M}^{(u)}_{9}$ is exactly the
part given in Eq. (D), calculated in the unitary gauge. The results of the two
diagrams (1) and (3) are:

$\displaystyle i\Delta\mathcal{M}^{(\xi)}_{9}\equiv$ $\displaystyle
i\mathcal{M}^{(\xi)}_{9}-i\mathcal{M}^{(u)}_{9}$ $\displaystyle=$
$\displaystyle\frac{ig_{\gamma
SV}}{16\pi^{2}}\overline{u_{a}}\left\\{\frac{m_{F}}{m_{V}^{2}}\left[(\gamma^{\mu}\varepsilon^{*\nu})C_{\mu\nu}-(C_{\mu}\gamma^{\mu})(p_{2}.\varepsilon^{*})+\not{p}_{2}X_{0}(p_{1}.\varepsilon^{*})\right]\left[A_{2}\right]\right.$
$\displaystyle\left.+\frac{1}{m_{V}^{2}}\left[A_{1}\right]\left[C_{00}\not{\varepsilon}^{*}\not{p}_{2}+(p_{1}.\varepsilon^{*})\left(m_{F}^{2}X_{0}+X_{1}\not{p}_{1}\not{p}_{2}+m_{b}^{2}X_{2}\right)\right]\right\\}u_{b},$
(119) $\displaystyle i\mathcal{M}^{(\xi)}_{3}=$
$\displaystyle\frac{-ieQ_{H}}{16\pi^{2}}\overline{u_{a}}\left\\{-2p_{1}.\varepsilon^{*}\left[A_{1}\right]m_{F}X_{0}+\left[2C^{f}_{00}\not{\varepsilon}^{*}+\left(X_{1}^{f}\not{p}_{1}+X_{2}^{f}\not{p}_{2}\right)(2p_{1}.\varepsilon^{*})\right][A_{2}]\right\\}u_{b},$
(120)

where $C_{00}=C_{00}(m_{F}^{2},m_{h}^{2},\xi m_{V}^{2})$ and
$X_{0,i}=X_{0,i}(m_{F}^{2},m_{h}^{2},\xi m_{V}^{2})$.

As we showed clearly in Eq. (115), a propagator of an arbitrary internal gauge
boson always consists of two parts: i) the first part is exactly the unitary
propagator resulting in $\mathcal{M}^{(u)}_{9}$, and ii) the second is
proportional to the propagator of the respective Goldstone boson, in which the
parameter $\xi$ defines a new mass value in the denominator, which results in
$\Delta\mathcal{M}^{(\xi)}_{9}+\mathcal{M}^{(\xi)}_{3}$. Because $\xi$ is
arbitrary, the two one-loop contributions corresponding to the two mentioned
parts are independent. As a result, the WI violation of the contributions
relating to $\mathcal{M}^{(u)}_{9}$ is enough to guarantee that the
contributions from the FSV-type diagrams always violate the WI.

##  Appendix E Higgs gauge couplings in the Higgs triplet models

Here we summarize the HTM and derive precisely the Higgs gauge couplings. The
Higgs sector consists of a Higgs triplet $\Delta\sim(3,2)$ and a Higgs doublet
$\Phi\sim(2,1)$ in the electroweak gauge symmetry $SU(2)_{L}\times U(1)_{Y}$
corresponding to the electric operator $Q=T^{3}+Y/2$. Here we will use the
notations from Ref. Ashanujjaman:2021txz ; Aoki:2011pz , the Higgs sector is

$\displaystyle\Phi=\begin{pmatrix}\varphi^{+}\\\
\frac{1}{\sqrt{2}}\left(\varphi+v_{\Phi}+i\chi\right)\end{pmatrix},\;\Delta=\begin{pmatrix}\frac{\Delta^{+}}{\sqrt{2}}&\Delta^{++}\\\
\Delta^{0}&-\frac{\Delta^{+}}{\sqrt{2}}\end{pmatrix}\;\mathrm{with}\;\Delta^{0}=\frac{\delta+v_{\Delta}+i\eta}{\sqrt{2}},$
(121)

where $v_{\Phi}$ and $v_{\Delta}$ are the vacuum expectation values (VEV) of
the neutral Higgs components. Because $v_{d}$ has the lepton number 2,
$v_{d}\ll v_{\Delta}$.

The Higgs gauge couplings appear in the following kinetic terms:

$\displaystyle\mathcal{L}_{k,H}$
$\displaystyle=\left(D_{\mu}\Phi\right)^{\dagger}\left(D^{\mu}\Phi\right)+\mathrm{Tr}\left[\left(D_{\mu}\Delta\right)^{\dagger}\left(D^{\mu}\Delta\right)\right],$
(122)

where

$\displaystyle
D_{\mu}\Phi=\left(\partial_{\mu}+i\frac{g}{2}\tau^{a}W^{a}_{\mu}+i\frac{g^{\prime}}{2}B_{\mu}\right)\Phi,\;D_{\mu}\Delta=\partial_{\mu}\Delta+i\frac{g}{2}\left[\tau^{a}W^{a}_{\mu},\Delta\right]+i\frac{g^{\prime}}{2}B_{\mu}\Delta.$
(123)

The masses and mixing parameters of the gauge bosons are derived from the Eq.
(122), with VEVs of $\Phi$ and $\Delta$. A detailed calculation shows that the
physical states $W^{\pm}$, neutral $Z$ and photon $A_{\mu}$ are:

$\displaystyle W^{\pm}_{\mu}=\frac{W^{1}_{\mu}\mp
iW^{2}_{\mu}}{\sqrt{2}},\;W^{3}_{\mu}=c_{W}Z_{\mu}+s_{W}A_{\mu},\;B_{\mu}=-s_{W}Z_{\mu}+c_{W}A_{\mu}.$
(124)

The respective masses are $m_{W}^{2}=g^{2}(v^{2}_{\Phi}+2v^{2}_{\Delta})/4$,
$m_{Z}^{2}=g^{2}(v^{2}_{\Phi}+2v^{2}_{\Delta})/(4c^{2}_{W})$ and the photon is
massless. The relation $g^{\prime}/g=t_{W}$ is well-known, the same as that in
the SM. The covariant derivatives in Eq. (123) are written in the mass
eigenstates as follows:

$\displaystyle D_{\mu}\Delta$
$\displaystyle=\frac{ig}{2}\begin{pmatrix}v_{\Delta}W^{+}_{\mu}+\sqrt{2}t_{W}B_{\mu},&-2\Delta^{+}W^{+}_{\mu}\\\
2\Delta^{+}W^{-}_{\mu}-\frac{\sqrt{2}v_{\Delta}Z_{\mu}}{c_{W}},&-v_{\Delta}W^{+}_{\mu}-\sqrt{2}t_{W}\Delta^{+}B_{\mu}\end{pmatrix}+\dots,$
$\displaystyle D_{\mu}\Phi$
$\displaystyle=\frac{ig}{2}\begin{pmatrix}\left(\frac{c^{2}_{W}-s^{2}_{W}}{c_{W}}Z_{\mu}+2s_{W}A_{\mu}\right)\varphi^{+}+v_{\Phi}W^{+}_{\mu}\\\
\sqrt{2}W^{-}_{\mu}\varphi^{+}-\frac{v_{\Phi}}{\sqrt{2}c_{W}}Z_{\mu}\end{pmatrix}+\dots,$
(125)

where we just focus on the couplings $SVV$ relating to the vertex
$H^{\pm}W^{\mp}\gamma$. Therefore, the relevant parts in the kinetic term are:

$\displaystyle\mathcal{L}_{k,H}$
$\displaystyle=\frac{g^{2}}{4}\left(2s_{W}A_{\mu}\varphi^{-}+v_{\Phi}W^{-}_{\mu}\
\right)\left(2s_{W}A^{\mu}\varphi^{+}+v_{\Phi}W^{+\mu}\ \right)$
$\displaystyle+\frac{g^{2}}{2}\left(v_{\Delta}W^{-}_{\mu}+\sqrt{2}t_{W}\Delta^{-}B_{\mu}\right)\left(v_{\Delta}W^{+\mu}+\sqrt{2}t_{W}\Delta^{+}B^{\mu}\right)+\dots$
$\displaystyle=\frac{g^{2}s_{W}}{2}\left[\left(v_{\Phi}\varphi^{-}+\sqrt{2}v_{\Delta}\Delta^{-}\right)W^{+\mu}+\mathrm{h.c.}\right]A_{\mu}+\dots$
(126)

The Higgs potential of all Higgs multiplets was investigated previously, for
example, Ashanujjaman:2021txz ; Aoki:2011pz . The results of masses and mixing
parameters of all Higgs bosons are confirmed by our careful cross-check. We
focus on the Higgs gauge couplings of the singly charged Higgs boson in this
model, the mixing parameter $\beta_{\pm}$ relating to mass eigenstates and the
original ones are:

$\displaystyle\begin{pmatrix}\varphi^{\pm}\\\
\Delta^{\pm}\end{pmatrix}=\begin{pmatrix}c_{\beta_{\pm}}&-s_{\beta_{\pm}}\\\
s_{\beta_{\pm}}&c_{\beta_{\pm}}\end{pmatrix}\begin{pmatrix}G^{\pm}_{W}\\\
H^{\pm}\end{pmatrix},\;t_{\beta_{\pm}}=\frac{\sqrt{2}v_{\Delta}}{v_{\Phi}}.$
(127)

Here $G^{\pm}_{W}$ is the Goldstone bosons of $W^{\pm}$, while $H^{\pm}$ is
the only singly charged Higgs boson predicted by the HTM. Then the couplings
$H^{\pm}W^{\mp}\gamma\sim\sqrt{2}c_{\beta_{\pm}}v_{\Delta}-s_{\beta_{\pm}}v_{\Phi}=0$,
and the couplings with $G^{\pm}_{W}$ are
$(em_{W})\left[W^{+}_{\mu}G^{-}_{W}+\mathrm{h.c.}\right]A_{\mu}$, consistent
with the SM. In contrast, Ref. Yu:2021suw  seems to take into account only the
contribution of $\Delta^{\pm}$ to $H^{\pm}$, and ignored that of
$\varphi^{\pm}$, although they have the same amplitude but opposite signs.

It is noted that the results derived from our calculation are consistent with
those in recent works discussing all tree-level decays of Higgs and gauge
bosons predicted by the HTM at LHC Ashanujjaman:2021txz ; Aoki:2011pz . The
decays $H^{\pm}\to W^{\pm}\gamma$ do not appear in the decay lists of these
works.

## References

  * (1) B. Abi et al. [Muon g-2], Phys. Rev. Lett. 126, no.14, 141801 (2021)
  * (2) T. Aoyama, N. Asmussen, M. Benayoun, J. Bijnens, T. Blum, M. Bruno, I. Caprini, C. M. Carloni Calame, M. Cè and G. Colangelo, et al. Phys. Rept. 887, 1 (2020)
  * (3) A. Keshavarzi, D. Nomura and T. Teubner, Phys. Rev. D 97, no.11, 114025 (2018) [arXiv:1802.02995 [hep-ph]].
  * (4) G. Colangelo, M. Hoferichter and P. Stoffer, JHEP 02, 006 (2019) [arXiv:1810.00007 [hep-ph]].
  * (5) M. Hoferichter, B. L. Hoid and B. Kubis, JHEP 08, 137 (2019) [arXiv:1907.01556 [hep-ph]].
  * (6) M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, Eur. Phys. J. C 80, no.3, 241 (2020) [erratum: Eur. Phys. J. C 80, no.5, 410 (2020)] [arXiv:1908.00921 [hep-ph]].
  * (7) A. Keshavarzi, D. Nomura and T. Teubner, Phys. Rev. D 101, no.1, 014029 (2020) [arXiv:1911.00367 [hep-ph]].
  * (8) A. Kurz, T. Liu, P. Marquard and M. Steinhauser, Phys. Lett. B 734, 144-147 (2014) [arXiv:1403.6400 [hep-ph]].
  * (9) K. Melnikov and A. Vainshtein, Phys. Rev. D 70, 113006 (2004) [arXiv:hep-ph/0312226 [hep-ph]].
  * (10) P. Masjuan and P. Sanchez-Puertas, Phys. Rev. D 95, no.5, 054026 (2017) [arXiv:1701.05829 [hep-ph]].
  * (11) G. Colangelo, M. Hoferichter, M. Procura and P. Stoffer, JHEP 04, 161 (2017) [arXiv:1702.07347 [hep-ph]].
  * (12) M. Hoferichter, B. L. Hoid, B. Kubis, S. Leupold and S. P. Schneider, JHEP 10, 141 (2018) [arXiv:1808.04823 [hep-ph]].
  * (13) A. Gérardin, H. B. Meyer and A. Nyffeler, Phys. Rev. D 100, no.3, 034520 (2019) [arXiv:1903.09471 [hep-lat]].
  * (14) J. Bijnens, N. Hermansson-Truedsson and A. Rodríguez-Sánchez, Phys. Lett. B 798, 134994 (2019) [arXiv:1908.03331 [hep-ph]].
  * (15) G. Colangelo, F. Hagelstein, M. Hoferichter, L. Laub and P. Stoffer, JHEP 03, 101 (2020) [arXiv:1910.13432 [hep-ph]].
  * (16) G. Colangelo, M. Hoferichter, A. Nyffeler, M. Passera and P. Stoffer, Phys. Lett. B 735, 90-91 (2014) [arXiv:1403.7512 [hep-ph]].
  * (17) T. Blum, N. Christ, M. Hayakawa, T. Izubuchi, L. Jin, C. Jung and C. Lehner, Phys. Rev. Lett. 124, no.13, 132002 (2020) [arXiv:1911.08123 [hep-lat]].
  * (18) T. Aoyama, M. Hayakawa, T. Kinoshita and M. Nio, Phys. Rev. Lett. 109, 111808 (2012) [arXiv:1205.5370 [hep-ph]].
  * (19) T. Aoyama, T. Kinoshita and M. Nio, Atoms 7, no.1, 28 (2019)
  * (20) A. Czarnecki, W. J. Marciano and A. Vainshtein, Phys. Rev. D 67, 073006 (2003) [erratum: Phys. Rev. D 73, 119901 (2006)] [arXiv:hep-ph/0212229 [hep-ph]].
  * (21) C. Gnendiger, D. Stöckinger and H. Stöckinger-Kim, Phys. Rev. D 88, 053005 (2013) [arXiv:1306.5546 [hep-ph]].
  * (22) M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, Eur. Phys. J. C 77, no.12, 827 (2017) [arXiv:1706.09436 [hep-ph]].
  * (23) M. Davier, A. Hoecker, B. Malaescu and Z. Zhang, Eur. Phys. J. C 71, 1515 (2011) [erratum: Eur. Phys. J. C 72, 1874 (2012)] [arXiv:1010.4180 [hep-ph]].
  * (24) L. Morel, Z. Yao, P. Cladé and S. Guellati-Khélifa, Nature 588, no.7836, 61-65 (2020)
  * (25) A. M. Baldini et al. [MEG], Eur. Phys. J. C 76, no.8, 434 (2016) [arXiv:1605.05081 [hep-ex]].
  * (26) B. Aubert et al. [BaBar], Phys. Rev. Lett. 104, 021802 (2010) [arXiv:0908.2381 [hep-ex]].
  * (27) A. Crivellin, M. Hoferichter and P. Schmidt-Wellenburg, Phys. Rev. D 98 (2018) no.11, 113002 [arXiv:1807.11484 [hep-ph]].
  * (28) M. Lindner, M. Platscher and F. S. Queiroz, Phys. Rept. 731 (2018), 1-82 [arXiv:1610.06587 [hep-ph]].
  * (29) I. Doršner, S. Fajfer and S. Saad, Phys. Rev. D 102 (2020) no.7, 075007 [arXiv:2006.11624 [hep-ph]].