# General one-loop formulas for $H\rightarrow f\bar{f}\gamma$ and its
applications

Vo Van On<EMAIL_ADDRESS>Dzung Tri Tran Chi Linh Nguyen Khiem Hong Phan
<EMAIL_ADDRESS>Institute of Applied Technology, Thu Dau Mot
University, Thu Dau Mot City, Binh Duong Province 75000, Vietnam University of
Science Ho Chi Minh City, $227$ Nguyen Van Cu, District $5$, Ho Chi Minh City,
Vietnam Institute of Fundamental and Applied Sciences, Duy Tan University, Ho
Chi Minh City $700000$, Vietnam Faculty of Natural Sciences, Duy Tan
University, Da Nang City $550000$, Vietnam

###### Abstract

We present general one-loop contributions to the decay processes $H\rightarrow
f\bar{f}\gamma$ including all possible the exchange of the additional heavy
vector gauge bosons, heavy fermions, and charged (also neutral) scalar
particles in the loop diagrams. As a result, the analytic results are valid in
a wide class of beyond the standard models. Analytic formulas for the form
factors are expressed in terms of Passarino-Veltman functions in the standard
notations of LoopTools. Hence, the decay rates can be computed numerically by
using this package. The computations are then applied to the cases of the
standard model, $U(1)_{B-L}$ extension of the standard model as well as two
Higgs doublet model. Phenomenological results of the decay processes for all
the above models are studied. We observe that the effects of new physics are
sizable contributions and these can be probed at future colliders.

###### keywords:

Higgs phenomenology, One-loop Feynman integrals, Analytic methods for Quantum
Field Theory, Dimensional regularization.

##  1 Introduction

After discovering the standard model-like (SM-like) Higgs boson [1, 2], one of
the main purposes at future colliders like the high luminosity large hadron
collider (HL-LHC) [3, 4] as well as future lepton colliders [5] is to probe
the properties of this boson (mass, couplings, spin and parity, etc). In the
experimental programs, the Higgs productions and its decay rates should be
measured as precise as possible. Based on the measurements, we can verify the
nature of the Higgs sector. In other words we can understand deeply the
dynamic of the electroweak symmetry breaking. It is well-known that the Higgs
sector is selected as the simplest case in the standard model (SM) which there
is only a scalar doublet field. From theoretical viewpoints, there are no
reasons for this simplest choice. Many of beyond the standard models (BSMs)
have extended the Higgs sector (some of them have also expanded gauge sectors,
introduced mass terms of neutrinos, etc). In these models many new particles
are proposed, for examples, new heavy gauge bosons, charged and neutral scalar
Higgs as well as new heavy fermions. These new particles may also contribute
to the productions and decay of Higgs boson. It means that the more precise
data and new theoretical approaches on the Higgs productions and its decay
rates could provide us a crucial tool to answer the nature of the Higgs sector
and, more important, to extract the new physics contributions.

Among all the Higgs decay channels, the processes $H\rightarrow
f\bar{f}\gamma$ are great of interest at the colliders by following reasons.
Firstly, the decay channels can be measured at the large hadron collider [6,
7, 8, 9]. Therefore, the processes can be used to test the SM at the high
energy regions. Secondly, many of new particles as mentioned in the beginning
of this section may propagate in the loop diagrams of the decay processes.
Subsequently, the decay rates could provide a useful tool for constraining new
physic parameters. Last but not least, apart from the SM-like Higgs boson, new
neutral Higgs bosons in BSMs may be mixed with the SM-like one. These effects
can be also observed directly by measuring of the decay rates of $H\rightarrow
f\bar{f}\gamma$. As above reasons, the detailed theoretical evaluations for
one-loop contributions to the decay of Higgs to fermion pairs and a photon
within the SM and its extensions are necessary.

Theoretical implications for the decay $H\rightarrow f\bar{f}\gamma$ in the SM
at the LHC have studied in Refs. [10, 11, 12]. Moreover, there have been many
available computations for one-loop contributing to the decay processes
$H\rightarrow f\bar{f}\gamma$ within the SM framework [13, 14, 15, 16, 17, 18,
19, 20]. The same evaluations for the Higgs productions at $e\gamma$ colliders
have proposed in [21, 22]. While one-loop corrections to $H\rightarrow
f\bar{f}\gamma$ in the context of the minimal super-symmetric standard model
Higgs sector have computed in [23]. Furthermore, one-loop contributions for
CP-odd Higgs boson productions in $e\gamma$ collisions have carried out in
[24]. In this article, we present general formulas for one-loop contributing
to the decay processes $H\rightarrow f\bar{f}\gamma$. The analytic results
presented in the current paper are not only valid in the SM but also in many
of BSMs in which new particles are proposed such as heavy vector bosons, heavy
fermions, and charged (neutral) scalar particles that may propagate in the
loop diagrams of the decay processes. The analytic formulas for the form
factors are expressed in terms of Passarino-Veltman (PV) functions in standard
notations of LoopTools [46]. As a result, they can be evaluated numerically by
using this package. The calculations are then applied to the SM and many of
beyond the SM such as the $U(1)_{B-L}$ extension of the SM [25], two Higgs
doublet models (THDM) [27]. Phenomenological results of the decay processes
for these models are also studied.

We also stress that our analytical results in the present paper can be also
applied for many of BSMs framework. In particular, in the super-symmetry
models, many super-partners of fermions and gauge bosons are introduced.
Furthermore, with extending the Higgs sector, we encounter charged and neutral
Higgs bosons in this framework. There are exist the extra charged gauge bosons
in many electroweak gauge extensions, for examples, the left-right models (LR)
constructed from the $SU(2)_{L}\times SU(2)_{R}\times U(1)_{Y}$ [28, 29, 30],
the 3-3-1 models ($SU(3)_{L}\times U(1)_{X}$) [31, 32, 33, 34, 35, 36, 37],
the $3$-$4$-$1$ models ($SU(4)_{L}\times U(1)_{X}$) [37, 38, 39, 40, 41, 42],
etc. Analytic results in this paper already include the contributions these
mentioned particles which may also exchange in the loop diagrams of the
aforementioned decay processes. Phenomenological results for the decay
processes in the above models are great of interest in future. These topic
will be devoted in our future publications.

The layout of the paper is as follows: We first write down the general
Lagrangian and introduce the notation for the calculations in the section $2$.
We then present the detailed calculations for one-loop contributions to
$H\rightarrow f\bar{f}\gamma$ in section $3$. The applications of this work to
the SM, $U(1)_{B-L}$ extensions of the SM and THDM are also studied in this
section. Phenomenological results for these models are analysed at the end of
section $3$. Conclusions and outlook are devoted in the section $4$. In
appendices, the checks for the computations are shown. We then review briefly
$U(1)_{B-L}$ extensions of the SM and THDM in the appendices. Finally, Feynman
rules and all involving couplings in the decay processes are shown.

##  2 Lagrangian and notations

In order to write down the general form of Lagrangian for a wide class of the
BSMs, we start from the well-known contributions appear in the SM. We then
consider the extra forms that extended from the SM. For example, the two Higgs
doublet model [27] adding a new Higgs doublet that predict new charged and
neutral scalar Higgs bosons; a model with a gauge symmetry $U(1)_{B-L}$ which
proposes a neutral gauge boson $Z^{\prime}$ [25, 26], neutral Higgs; a minimal
left-right models with a new non-Abelian gauge symmetry for electroweak
interactions $SU(2)_{L}\times SU(2)_{R}\times U(1)_{B-L}$ [28, 29, 30]
introducing many new particles including charged gauge bosons, neutral gauge
bosons and charged Higgs bosons. The mentioned particles give one-loop
contributions to the decays under consideration.

In this section, Feynman rules for the decay channels $H\rightarrow
f\bar{f}\gamma$ are derived for the most general extension of the SM with
considering all possible contributions from the mentioned particles. In this
computation, we denote $V_{i},V_{j}$ for extra charged gauge bosons,
$V_{k}^{0}$ for neutral gauge bosons. Moreover, $S_{i},S_{j}\;(S_{k}^{0})$ are
charged (neutral) Higgs bosons respectively and $f_{i},f_{j}$ show for
fermions. In general, the classical Lagrangian contains the following parts:

$\displaystyle\mathcal{L}=\mathcal{L}_{f}+\mathcal{L}_{G}+\mathcal{L}_{\Phi}+\mathcal{L}_{Y}.$
(1)

Where the fermion sector is given

$\displaystyle\mathcal{L}_{f}=\bar{\psi}_{f}i\not{D}\psi_{f}$ (2)

with $D_{\mu}=\partial_{\mu}-igT^{a}V_{\mu}^{a}+\cdots$. In this formula,
$T^{a}$ is generator of the corresponding gauge symmetry. The gauge sector
reads

$\displaystyle\mathcal{L}_{G}=-\frac{1}{4}\sum\limits_{a}V^{a}_{\mu\nu}V^{a,\mu\nu}+\cdots$
(3)

where
$V^{a}_{\mu\nu}=\partial_{\mu}V^{a}_{\nu}-\partial_{\nu}V^{a}_{\mu}+gf^{abc}V^{b}_{\mu}V^{c}_{\mu}$
with $f^{abc}$ is structure constant of the corresponding gauge group. The
scalar sector is expressed as follows:

$\displaystyle\mathcal{L}_{\Phi}=\sum\limits_{\Phi}Tr[\left(D_{\mu}\Phi\right)^{{\dagger}}\left(D^{\mu}\Phi\right)]-V(\Phi).$
(4)

We then derive all the couplings from the full Lagrangian. The couplings are
parameterized in general forms and presented by following:

  * 1.

By expanding the fermion sector, we can derive the vertices of vector boson
$V$ with fermions. In detail, the interaction terms are parameterized as

$\displaystyle\mathcal{L}_{Vff}=\sum\limits_{f_{i},f_{j},V}\bar{f}_{i}\gamma^{\mu}(g_{Vff}^{L}P_{L}+g_{Vff}^{R}P_{R})f_{j}V_{\mu}+\cdots$
(5)

with $P_{L,R}=(1\mp\gamma_{5})/2$.

  * 2.

Trilinear gauge and quartic gauge couplings are expanded from the gauge
sector:

$\displaystyle\mathcal{L}_{VVV,VVVV}$ $\displaystyle=$
$\displaystyle\sum\limits_{V_{k}^{0},V_{i},V_{j}}g_{V_{k}^{0}V_{i}V_{j}}\Big{[}\partial_{\mu}V^{0}_{k,\nu}V_{i}^{\mu}V_{j}^{\nu}+V_{k,\nu}^{0}V_{i}^{\mu}\partial^{\nu}V_{j,\mu}+\cdots\Big{]}$
(6)
$\displaystyle+\sum\limits_{V_{k}^{0},V_{l}^{0},V_{i},V_{j}}g_{V_{k}^{0}V_{l}^{0}V_{i}V_{j}}\Big{[}V^{0}_{k,\mu}V_{l,\nu}^{0}V_{i}^{\mu}V_{j}^{\nu}+\cdots\Big{]}+\cdots$

  * 3.

Next, the couplings of scalar $S$ to fermions are taken from the Yukawa part
$\mathcal{L}_{Y}$. The interaction terms are expressed as follows:

$\displaystyle\mathcal{L}_{Sf_{i}f_{j}}=\sum\limits_{f_{i},f_{j},S}\bar{f}_{i}(g_{Sff}^{L}P_{L}+g_{Sff}^{R}P_{R})f_{j}S+\cdots$
(7)

  * 4.

The couplings of scalar $S$ to vector boson $V$ can be derived from the
kinematic term of the Higgs sector. In detail, we have the interaction terms

$\displaystyle\mathcal{L}_{SVV,SSV,SSVV}$ $\displaystyle=$
$\displaystyle\sum\limits_{S,V_{i},V_{j}}g_{SV_{i}V_{j}}SV_{i}^{\mu}V_{j,\mu}+\sum\limits_{S_{i},S_{j},V}g_{S_{i}S_{j}V}[(\partial_{\mu}S_{i})S_{j}-(\partial_{\mu}S_{j})S_{i}]V^{\mu}$
(8)
$\displaystyle+\sum\limits_{S_{i},S_{j},V_{k},V_{l}}g_{S_{i}S_{j}V_{k}V_{l}}S_{i}S_{j}V_{k}^{\mu}V_{l,\mu}+\cdots$

  * 5.

Finally, the trilinear scalar and quartic scalar interactions are from the
Higgs potential $V(\Phi)$. The interaction terms are written as

$\displaystyle\mathcal{L}_{SSS,SSSS}$ $\displaystyle=$
$\displaystyle\sum\limits_{S_{i},S_{j},S_{k}}g_{S_{i}S_{j}S_{k}}S_{i}S_{j}S_{k}+\sum\limits_{S_{i},S_{j},S_{k},S_{l}}g_{S_{i}S_{j}S_{k},S_{l}}S_{i}S_{j}S_{k}S_{l}+\cdots$
(9)

All of the Feynman rules corresponding to the above couplings giving one-loop
contributions to the SM-like Higgs decays $H\to f\bar{f}\gamma$ are collected
in appendix $D$. In detail, the propagators involving the decay processes in
the unitary gauge are shown in Table 6. All the related couplings involving
the decay channels are parameterized in general forms which are presented in
Table 7 (we refer appendices $B$ and $C$ for two typical models).

##  3 Calculations

In this section, one-loop contributions to the decay processes $H\rightarrow
f(q_{1})\bar{f}(q_{2})\gamma(q_{3})$ are calculated in detail. In the present
paper, we consider the computations in the limit of $m_{f}\rightarrow 0$. All
Feynman diagrams involving these processes can be grouped in to the following
classes (seen Fig. 1).

Figure 1:  Types of Feynman diagrams involving to the SM-like Higgs decays
$H\to f\bar{f}\gamma$

For on-shell photon, we confirm that the contributions of diagrams $(e+h)$
will be vanished. One can neglect the Yukawa coupling $y_{f}$ (since
$m_{f}\rightarrow 0$) in this computation. As a result, the contributions of
diagrams $(g+k+m)$ can be omitted. Furthermore, the diagrams $(a+f)$ are not
contributed to the amplitude. Hence, we only have the contributions of
$(b+c+d)$ which are separated into two kinds of the contributions. The first
one is to the topology $b$ which is called $V_{k}^{0*}$ pole contributions.
The second type (diagrams $c$ and $d$) belongs to the non-$V_{k}^{0*}$ pole
contributions. We remind that $V_{k}^{0*}$ can include both $Z,\gamma$ in the
SM and the arbitrary neutral vector boson $Z^{\prime}$ in many of the BSMs.
General one-loop amplitude which obeys the invariant Lorentz structure can be
decomposed as follows [20]:

$\displaystyle\mathcal{A}_{\text{loop}}$ $\displaystyle=$
$\displaystyle\sum\limits_{k=1}^{2}\Big{\\{}[q_{3}^{\mu}q_{k}^{\nu}-g^{\mu\nu}q_{3}\cdot
q_{k}]\bar{u}(q_{1})(F_{k,R}\gamma_{\mu}P_{R}+F_{k,L}\gamma_{\mu}P_{L})v(q_{2})\Big{\\}}\varepsilon^{*}_{\nu}(q_{3}).$
(10)

In this equation, all form factors are computed as follows:

$\displaystyle F_{k,L/R}$ $\displaystyle=$ $\displaystyle
F_{k,L/R}^{\text{$V_{k}^{0*}$-poles}}+F_{k,L/R}^{\text{Non-$V_{k}^{0*}$}}$
(11)

for $k=1,2$. Kinematic invariant variables involved in the decay processes are
included: $q^{2}=q_{12}=(q_{1}+q_{2})^{2}$, $q_{13}=(q_{1}+q_{3})^{2}$ and
$q_{23}=(q_{2}+q_{3})^{2}$.

We first write down all Feynman amplitudes for the above diagrams. With the
help of Package-X [43], all Dirac traces and Lorentz contractions in $d$
dimensions are handled. Subsequently, the amplitudes are then written in terms
of tensor one-loop integrals. By following tensor reduction for one-loop
integrals in [44] (the relevant tensor reduction formulas are shown in
appendix $A$), all tensor one-loop integrals are expressed in terms of PV-
functions.

###  3.1 $V_{k}^{0*}$ pole contributions

In this subsection, we first arrive at the $V_{k}^{0*}$ pole contributions
which are corresponding to the diagram $b$. In this group of Feynman diagrams,
it is easily to confirm that the form factors follow the below relation:

$\displaystyle
F_{k,L/R}^{\text{$V_{k}^{0*}$-poles}}=F_{1,L/R}^{\text{$V_{k}^{0*}$-poles}}=F_{2,L/R}^{\text{$V_{k}^{0*}$-poles}}$
(12)

Their analytic results are will be shown in the following subsections. All
possible particles exchanging in the loop diagrams are included. We emphasize
that analytic expressions for the form factors presented in this subsection
cover the results in Ref. [45]. It means that we can reduce to the results for
$H\to\nu_{l}\bar{\nu}_{l}\gamma$ of Ref. [45] by setting $f$ to $\nu_{l}$ and
replacing the corresponding couplings. Furthermore, all analytic formulas
shown in the following subsection cover all cases of $V_{k}^{0*}$ poles. For
instance, when $V_{k}^{0*}\rightarrow\gamma^{*}$, we then set
$M_{V^{0}_{k}}=0$, $\Gamma_{V^{0}_{k}}=0$. In addition, $V_{k}^{0*}$ becomes
$Z$ (or $Z^{\prime}$) boson, we should fix $M_{V^{0}_{k}}=M_{Z}$ and
$\Gamma_{V^{0}_{k}}=\Gamma_{Z}$ (or $Z^{\prime}$) respectively.

Figure 2:  One-loop triangle diagrams with exchanging vector boson $V_{i,j}$
particles in the loop. Figure 3:  One-loop triangle diagrams with exchanging
scalar boson $S_{i,j}$ particles in loop. Figure 4:  One-loop triangle
diagrams with exchanging two scalar bosons $S_{i}$ and a vector boson $V_{j}$
particles in the loop. Figure 5:  One-loop triangle diagrams with exchanging a
scalar boson $S_{j}$ and two vector bosons $V_{i}$ in the loop. Figure 6:
One-loop triangle diagrams with exchanging charged fermion $f_{i,j}$ particles
in loop.

We begin with one-loop triangle Feynman diagrams which all vector bosons
$V_{i,j}$ are in the loop (seen Fig. 2). One-loop form factors of this group
of Feynman diagrams are expressed in terms of the PV functions as follows:

$\displaystyle F_{k,L}^{\text{$V_{k}^{0*}$-poles}}|_{V_{i},V_{j}}$
$\displaystyle=$
$\displaystyle\dfrac{eQ_{V}}{16\pi^{2}}\sum\limits_{V_{i},V_{j},V^{0}_{k}}\dfrac{g_{HViVj}\;g_{V^{0}_{k}ff}^{L}}{M_{V_{i}}^{2}M_{V_{j}}^{2}(q^{2}-M_{V^{0}_{k}}^{2}+i\Gamma_{V^{0}_{k}}M_{V^{0}_{k}})}\times$
$\displaystyle\times\Bigg{\\{}\Big{[}2g_{V^{0}_{k}AV_{i}V_{j}}(M_{H}^{2}-M_{V_{j}}^{2})+g_{V^{0}_{k}V_{i}V_{j}}(M_{H}^{2}+M_{V_{i}}^{2}+M_{V_{j}}^{2})\Big{]}B_{11}(M_{H}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2})$
$\displaystyle+\Big{[}g_{V^{0}_{k}V_{i}V_{j}}(M_{H}^{2}+3M_{V_{i}}^{2}-M_{V_{j}}^{2})-2g_{V^{0}_{k}AV_{i}V_{j}}M_{V_{i}}^{2}\Big{]}B_{1}(M_{H}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2})$
$\displaystyle+2M_{V_{i}}^{2}(g_{V^{0}_{k}V_{i}V_{j}}-g_{V^{0}_{k}AV_{i}V_{j}})B_{0}(M_{H}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2})$
$\displaystyle+2g_{V^{0}_{k}AV_{i}V_{j}}\Big{[}M_{H}^{2}B_{111}+B_{00}+2B_{001}\Big{]}(M_{H}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2})$
$\displaystyle+4g_{V^{0}_{k}V_{i}V_{j}}M_{V_{i}}^{2}\Big{(}M_{V_{i}}^{2}+3M_{V_{j}}^{2}-q^{2}\Big{)}C_{0}(0,q^{2},M_{H}^{2},M_{V_{i}}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2})$
$\displaystyle+2g_{V^{0}_{k}V_{i}V_{j}}\Big{[}M_{H}^{2}(M_{V_{i}}^{2}+M_{V_{j}}^{2}-q^{2})+M_{V_{i}}^{4}+M_{V_{j}}^{4}+(4d-6)M_{V_{i}}^{2}M_{V_{j}}^{2}$
$\displaystyle\hskip
56.9055pt-q^{2}(M_{V_{i}}^{2}+M_{V_{j}}^{2})\Big{]}(C_{22}+C_{12})(0,q^{2},M_{H}^{2},M_{V_{i}}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2})$
$\displaystyle+2g_{V^{0}_{k}V_{i}V_{j}}\Big{[}M_{H}^{2}(M_{V_{i}}^{2}+M_{V_{j}}^{2}-q^{2})+3M_{V_{i}}^{4}-M_{V_{j}}^{4}+(4d-6)M_{V_{i}}^{2}M_{V_{j}}^{2}$
$\displaystyle\hskip
85.35826pt-q^{2}(3M_{V_{i}}^{2}-M_{V_{j}}^{2})\Big{]}C_{2}(0,q^{2},M_{H}^{2},M_{V_{i}}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2})\Bigg{\\}},$
$\displaystyle F_{k,R}^{\text{$V_{k}^{0*}$-poles}}|_{V_{i},V_{j}}$
$\displaystyle=$ $\displaystyle
F_{k,L}^{\text{$V_{k}^{0*}$-poles}}|_{V_{i},V_{j}}\Big{(}g_{V^{0}_{k}ff}^{L}\rightarrow
g_{V^{0}_{k}ff}^{R}\Big{)}.$ (14)

The results are written in terms of $B$\- and $C$-functions. We note that one-
loop amplitude for each diagram in Fig. 2 may decompose into tensor one-loop
up to rank $R=6$. However, after taking into account all diagrams, the
amplitude for this subset Feynman diagrams is only expressed in terms of the
tensor integrals up to rank $R=2$. As a result, we have up to
$C_{22}$-functions contributing to the form factors. Furthermore, some of them
may contain UV divergent but after summing all these functions, the final
results are finite. The topic will be discussed at the end of this section.

We next concern one-loop triangle Feynman diagrams with $S_{i},\;S_{j}$ in the
loop (as described in Fig. 3). The corresponding one-loop form factors are
given:

$\displaystyle F_{k,L}^{\text{$V_{k}^{0*}$-poles}}|_{S_{i},S_{j}}$
$\displaystyle=$
$\displaystyle\dfrac{eQ_{S}}{4\pi^{2}}\sum\limits_{S_{i},S_{j},V^{0}_{k}}\dfrac{g_{HS_{i}S_{j}}\,g_{V^{0}_{k}S_{i}S_{j}}\,g_{V^{0}_{k}ff}^{L}}{M_{H}^{2}q^{2}(M_{H}^{2}-q^{2})(q^{2}-M_{V^{0}_{k}}^{2}+i\Gamma_{V^{0}_{k}}M_{V^{0}_{k}})}\times$
$\displaystyle\times\Bigg{\\{}(q^{2}-M_{H}^{2})\Big{[}A_{0}(M_{S_{i}}^{2})-A_{0}(M_{S_{j}}^{2})\Big{]}$
$\displaystyle\hskip
14.22636pt+M_{H}^{2}(M_{S_{i}}^{2}-M_{S_{j}}^{2}-q^{2})B_{0}(q^{2},M_{S_{i}}^{2},M_{S_{j}}^{2})$
$\displaystyle\hskip
14.22636pt+q^{2}(M_{H}^{2}-M_{S_{i}}^{2}+M_{S_{j}}^{2})B_{0}(M_{H}^{2},M_{S_{i}}^{2},M_{S_{j}}^{2})$
$\displaystyle\hskip
14.22636pt+2M_{H}^{2}q^{2}(M_{H}^{2}-q^{2})C_{12}(0,q^{2},M_{H}^{2},M_{S_{i}}^{2},M_{S_{i}}^{2},M_{S_{j}}^{2})\Bigg{\\}},$
$\displaystyle F_{k,R}^{\text{$V_{k}^{0*}$-poles}}|_{S_{i},S_{j}}$
$\displaystyle=$ $\displaystyle
F_{k,L}^{\text{$V_{k}^{0*}$-poles}}|_{S_{i},S_{j}}\Big{(}g_{V^{0}_{k}ff}^{L}\rightarrow
g_{V^{0}_{k}ff}^{R}\Big{)}.$ (16)

Similarly, we have the contributions of one-loop triangle Feynman diagrams
with exchanging scalar boson $S_{i}$ and vector boson $V_{j}$ in the loop. The
Feynman diagrams are depicted as in Fig. 4. Applying the same procedure, one
has the form factors

$\displaystyle F_{k,L}^{\text{$V_{k}^{0*}$-poles}}|_{S_{i},V_{j}}$
$\displaystyle=$
$\displaystyle\dfrac{eQ_{V}}{8\pi^{2}}\sum\limits_{S_{i},V_{j},V^{0}_{k}}\dfrac{g_{HS_{i}V_{j}}\,g_{V^{0}_{k}S_{i}V_{j}}\,g_{V^{0}_{k}ff}^{L}}{M_{H}^{2}M_{V_{j}}^{2}q^{2}(q^{2}-M_{H}^{2})(q^{2}-M_{V^{0}_{k}}^{2}+i\Gamma_{V^{0}_{k}}M_{V^{0}_{k}})}\times$
$\displaystyle\times\Bigg{\\{}(q^{2}-M_{H}^{2})(M_{H}^{2}-M_{S_{i}}^{2}+M_{V_{j}}^{2})\Big{[}A_{0}(M_{S_{i}}^{2})-A_{0}(M_{V_{j}}^{2})\Big{]}$
$\displaystyle+M_{H}^{2}\Big{[}q^{2}(M_{S_{i}}^{2}+3M_{V_{j}}^{2}-M_{H}^{2})$
$\displaystyle\hskip
36.98866pt+(M_{S_{i}}^{2}-M_{V_{j}}^{2})(M_{H}^{2}-M_{S_{i}}^{2}+M_{V_{j}}^{2})\Big{]}B_{0}(q^{2},M_{S_{i}}^{2},M_{V_{j}}^{2})$
$\displaystyle+q^{2}\Big{[}M_{H}^{2}-(M_{S_{i}}+M_{V_{j}})^{2}\Big{]}$
$\displaystyle\hskip
56.9055pt\times\Big{[}M_{H}^{2}-(M_{S_{i}}-M_{V_{j}})^{2}\Big{]}B_{0}(M_{H}^{2},M_{S_{i}}^{2},M_{V_{j}}^{2})$
$\displaystyle+2M_{H}^{2}q^{2}(M_{H}^{2}-q^{2})(M_{H}^{2}-M_{S_{i}}^{2}+M_{V_{j}}^{2})$
$\displaystyle\hskip 142.26378pt\times
C_{12}(0,q^{2},M_{H}^{2},M_{S_{i}}^{2},M_{S_{i}}^{2},M_{V_{j}}^{2})\Bigg{\\}},$
$\displaystyle F_{k,R}^{\text{$V_{k}^{0*}$-poles}}|_{S_{i},V_{j}}$
$\displaystyle=$ $\displaystyle
F_{k,L}^{\text{$V_{k}^{0*}$-poles}}|_{S_{i},V_{j}}\Big{(}g_{V^{0}_{k}ff}^{L}\rightarrow
g_{V^{0}_{k}ff}^{R}\Big{)}.$ (18)

We also consider the contributions of one-loop triangle diagrams with
exchanging a scalar boson $S_{j}$ and two vector boson $V_{i}$ in the loop.
The Feynman diagrams are presented as in Fig. 5. The corresponding form
factors for the above diagrams are given:

$\displaystyle F_{k,L}^{\text{$V_{k}^{0*}$-poles}}|_{V_{i},S_{j}}$
$\displaystyle=$
$\displaystyle\dfrac{eQ_{V}}{8\pi^{2}}\sum\limits_{V_{i},S_{j},V^{0}_{k}}\dfrac{g_{HV_{i}S_{j}}\;g_{V^{0}_{k}V_{i}S_{j}}\,g_{V^{0}_{k}ff}^{L}}{M_{H}^{2}M_{V_{i}}^{2}q^{2}(M_{H}^{2}-q^{2})(q^{2}-M_{V^{0}_{k}}^{2}+i\Gamma_{V^{0}_{k}}M_{V^{0}_{k}})}\times$
$\displaystyle\times\Bigg{\\{}(M_{H}^{2}-q^{2})(M_{H}^{2}-M_{S_{j}}^{2}+M_{V_{i}}^{2})\Big{[}A_{0}(M_{S_{j}}^{2})-A_{0}(M_{V_{i}}^{2})\Big{]}$
$\displaystyle+q^{2}\Big{[}M_{H}^{2}(M_{H}^{2}+4M_{V_{i}}^{2})-(M_{S_{j}}^{2}-M_{V_{i}}^{2})^{2}\Big{]}B_{0}(M_{H}^{2},M_{V_{i}}^{2},M_{S_{j}}^{2})$
$\displaystyle-
M_{H}^{2}\Big{[}M_{H}^{2}(M_{S_{j}}^{2}-M_{V_{i}}^{2}+q^{2})+M_{S_{j}}^{2}(2M_{V_{i}}^{2}-M_{S_{j}}^{2}-q^{2})$
$\displaystyle\hskip
113.81102pt+M_{V_{i}}^{2}(5q^{2}-M_{V_{i}}^{2})\Big{]}B_{0}(q^{2},M_{V_{i}}^{2},M_{S_{j}}^{2})$
$\displaystyle+2M_{H}^{2}q^{2}(M_{H}^{2}-q^{2})(M_{H}^{2}-M_{S_{j}}^{2}+M_{V_{i}}^{2})C_{12}(0,q^{2},M_{H}^{2},M_{V_{i}}^{2},M_{V_{i}}^{2},M_{S_{j}}^{2})$
$\displaystyle+4M_{H}^{2}M_{V_{i}}^{2}q^{2}(M_{H}^{2}-q^{2})C_{0}(0,q^{2},M_{H}^{2},M_{V_{i}}^{2},M_{V_{i}}^{2},M_{S_{j}}^{2})\Bigg{\\}},$
$\displaystyle F_{k,R}^{\text{$V_{k}^{0*}$-poles}}|_{V_{i},S_{j}}$
$\displaystyle=$ $\displaystyle
F_{k,L}^{\text{$V_{k}^{0*}$-poles}}|_{V_{i},S_{j}}\Big{(}g_{V^{0}_{k}ff}^{L}\rightarrow
g_{V^{0}_{k}ff}^{R}\Big{)}.$ (20)

Finally, we have consider fermions exchanging in the one-loop triangle
diagrams (shown in Fig. 6). The form factors then read

$\displaystyle F_{k,L}^{\text{$V_{k}^{0*}$-poles}}|_{f_{i},f_{j}}$
$\displaystyle=$
$\displaystyle\dfrac{eQ_{f}}{4\pi^{2}}\sum\limits_{f_{i},f_{j},V^{0}_{k}}\dfrac{N_{C}^{f}\;g_{V^{0}_{k}ff}^{L}}{(q^{2}-M_{V^{0}_{k}}^{2}+i\Gamma_{V^{0}_{k}}M_{V^{0}_{k}})}\times$
$\displaystyle\times\Bigg{\\{}\Big{[}2m_{f_{i}}(g_{Hf_{i}f_{j}}^{L}g_{V^{0}_{k}f_{i}f_{j}}^{L}+g_{Hf_{i}f_{j}}^{R}g_{V^{0}_{k}f_{i}f_{j}}^{R})+2m_{f_{j}}(g_{Hf_{i}f_{j}}^{L}g_{V^{0}_{k}f_{i}f_{j}}^{R}+g_{Hf_{i}f_{j}}^{R}g_{V^{0}_{k}f_{i}f_{j}}^{L})\Big{]}\times$
$\displaystyle\times\Big{[}C_{22}+C_{12}\Big{]}(0,q^{2},M_{H}^{2},m_{f_{i}}^{2},m_{f_{i}}^{2},m_{f_{j}}^{2})$
$\displaystyle+\Big{[}3m_{f_{i}}(g_{Hf_{i}f_{j}}^{L}g_{V^{0}_{k}f_{i}f_{j}}^{L}+g_{Hf_{i}f_{j}}^{R}g_{V^{0}_{k}f_{i}f_{j}}^{R})+m_{f_{j}}(g_{Hf_{i}f_{j}}^{L}g_{V^{0}_{k}f_{i}f_{j}}^{R}+g_{Hf_{i}f_{j}}^{R}g_{V^{0}_{k}f_{i}f_{j}}^{L})\Big{]}\times$
$\displaystyle\times
C_{2}(0,q^{2},M_{H}^{2},m_{f_{i}}^{2},m_{f_{i}}^{2},m_{f_{j}}^{2})$
$\displaystyle+m_{f_{i}}(g_{Hf_{i}f_{j}}^{L}g_{V^{0}_{k}f_{i}f_{j}}^{R}+g_{Hf_{i}f_{j}}^{R}g_{V^{0}_{k}f_{i}f_{j}}^{R})C_{0}(0,q^{2},M_{H}^{2},m_{f_{i}}^{2},m_{f_{i}}^{2},m_{f_{j}}^{2})\Bigg{\\}},$
$\displaystyle F_{k,R}^{\text{$V_{k}^{0*}$-poles}}|_{f_{i},f_{j}}$
$\displaystyle=$ $\displaystyle
F_{k,L}^{\text{$V_{k}^{0*}$-poles}}|_{f_{i},f_{j}}\Big{(}g_{V^{0}_{k}ff}^{L}\rightarrow
g_{V^{0}_{k}ff}^{R}\Big{)}.$ (22)

###  3.2 Non-$V_{k}^{0*}$ pole contributions

We turn our attention to the non-$V_{k}^{0*}$ pole contributions, considering
all possible particles exchanging in the loop diagrams $(c+d)$. One first
arrives at the group of Feynman diagrams with vector boson $V_{i,j}$ internal
lines (as depicted in Fig. 7). Analytic formulas for the form factors are
given:

Figure 7:  One-loop triangle and box diagrams with exchanging vector bosons
$V_{i,j}$ in the loop. Figure 8:  One-loop triangle and box diagrams with
exchanging vector bosons $V^{0}_{i},V^{0}_{j}$ in the loop. Figure 9:  One-
loop diagrams with exchanging charged scalar bosons $S_{i,j}$ in the loop.
Figure 10:  One-loop box diagrams with exchanging vector boson $S^{0}_{i,j}$
particles in loop.

Figure 11:  One-loop diagrams with exchanging vector boson $V_{i}$ and scalar
boson $S_{j}$ particles in loop.

Figure 12:  One-loop diagrams with exchanging vector boson $V^{0}_{i}$ and
scalar boson $S_{j}^{0}$ particles in loop.

$\displaystyle F_{1,L}^{\text{Non-$V_{k}^{0*}$}}|_{V_{i},V_{j}}$
$\displaystyle=$
$\displaystyle\dfrac{eQ_{V}}{16\pi^{2}}\sum\limits_{V_{i},V_{j}}\dfrac{g_{HV_{i}V_{j}}\,g_{V_{i}f\nu_{f}}^{L}\,g_{V_{j}f\nu_{f}}^{L}}{M_{V_{i}}^{2}M_{V_{j}}^{2}}\times$
$\displaystyle\times\Bigg{\\{}-2M_{V_{i}}^{2}\Big{[}C_{0}(0,q_{12},M_{H}^{2},M_{V_{i}}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2})+C_{0}(q_{12},0,M_{H}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2},M_{V_{j}}^{2})\Big{]}$
$\displaystyle-(M_{H}^{2}+M_{V_{i}}^{2}+M_{V_{j}}^{2})\Big{\\{}\Big{[}C_{22}+C_{12}\Big{]}(0,q_{12},M_{H}^{2},M_{V_{i}}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2})$
$\displaystyle\hskip
142.26378pt+\Big{[}C_{22}+C_{12}\Big{]}(q_{12},0,M_{H}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2},M_{V_{j}}^{2})\Big{\\}}$
$\displaystyle-(M_{H}^{2}+3M_{V_{i}}^{2}-M_{V_{j}}^{2})\Big{[}C_{2}(0,q_{12},M_{H}^{2},M_{V_{i}}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2})$
$\displaystyle\hskip
156.49014pt+C_{2}(q_{12},0,M_{H}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2},M_{V_{j}}^{2})\Big{]}$
$\displaystyle+2(M_{V_{j}}^{2}-M_{V_{i}}^{2})C_{1}(q_{12},0,M_{H}^{2},M_{V_{i}}^{2},M_{V_{j}}^{2},M_{V_{j}}^{2})$
$\displaystyle+(8-2d)M_{V_{i}}^{2}M_{V_{j}}^{2}\Big{[}D_{3}(0,0,0,M_{H}^{2};q_{12},q_{13},M_{V_{i}}^{2},0,M_{V_{j}}^{2},M_{V_{j}}^{2})$
$\displaystyle\hskip
99.58464pt+D_{3}(0,0,0,M_{H}^{2};q_{23},q_{12},M_{V_{i}}^{2},M_{V_{i}}^{2},0,M_{V_{j}}^{2}))\Big{]}$
$\displaystyle+(4-2d)M_{V_{i}}^{2}M_{V_{j}}^{2}\Big{\\{}\Big{[}D_{33}+D_{23}+D_{13}\Big{]}(0,0,0,M_{H}^{2};q_{23},q_{12},M_{V_{i}}^{2},M_{V_{i}}^{2},0,M_{V_{j}}^{2})$
$\displaystyle\hskip
99.58464pt+\Big{[}D_{33}+D_{23}\Big{]}(0,0,0,M_{H}^{2};q_{12},q_{13},M_{V_{i}}^{2},0,M_{V_{j}}^{2},M_{V_{j}}^{2})\Big{\\}}$
$\displaystyle+4M_{V_{i}}^{2}M_{V_{j}}^{2}D_{2}(0,0,0,M_{H}^{2};q_{12},q_{13},M_{V_{i}}^{2},0,M_{V_{j}}^{2},M_{V_{j}}^{2})\Bigg{\\}},$
$\displaystyle F_{1,R}^{\text{Non-$V_{k}^{0*}$}}|_{V_{i},V_{j}}$
$\displaystyle=$ $\displaystyle
F_{1,L}^{\text{Non-$V_{k}^{0*}$}}|_{V_{i},V_{j}}\Big{(}g_{V_{i}f\nu_{f}}^{L}\rightarrow
g_{V_{i}f\nu_{f}}^{R};\;g_{V_{j}f\nu_{f}}^{L}\rightarrow
g_{V_{j}f\nu_{f}}^{R}\Big{)},$ (24) $\displaystyle
F_{2,L}^{\text{Non-$V_{k}^{0*}$}}|_{V_{i},V_{j}}$ $\displaystyle=$
$\displaystyle
F_{1,L}^{\text{Non-$V_{k}^{0*}$}}|_{V_{i},V_{j}}\Big{(}\\{q_{13},q_{23}\\}\rightarrow\\{q_{23},q_{13}\\}\Big{)},$
(25) $\displaystyle F_{2,R}^{\text{Non-$V_{k}^{0*}$}}|_{V_{i},V_{j}}$
$\displaystyle=$ $\displaystyle
F_{2,L}^{\text{Non-$V_{k}^{0*}$}}|_{V_{i},V_{j}}\Big{(}g_{V_{i}f\nu_{f}}^{L}\rightarrow
g_{V_{i}f\nu_{f}}^{R};\;g_{V_{j}f\nu_{f}}^{L}\rightarrow
g_{V_{j}f\nu_{f}}^{R}\Big{)}.$ (26)

We find that the result is presented in terms of $C$\- and $D$-functions up to
$D_{33}$-coefficients. The reason for this fact is explained as follows. Due
to the exchange of vector boson in the loop, we have to handle with tensor
one-loop integrals with rank $R\geq 3$ in the amplitude of each diagram.
However, they are cancelled out after summing all diagrams. As a result, the
total amplitude is only expressed in terms of tensor integrals with $R\leq 2$
that causes of leading the results.

For neutral vector boson $V^{0}_{i},V^{0}_{j}$ internal lines in the loop
diagrams (seen Fig. 8), the corresponding form factors are obtained

$\displaystyle F_{1,L}^{\text{Non-$V_{k}^{0*}$}}|_{V^{0}_{i},V^{0}_{j}}$
$\displaystyle=$
$\displaystyle\dfrac{eQ_{f}}{16\pi^{2}}\sum\limits_{V^{0}_{i},V^{0}_{j}}g_{HV^{0}_{i}V^{0}_{j}}\;g_{V^{0}_{i}ff}^{L}\;g_{V^{0}_{j}ff}^{L}\times$
$\displaystyle\times\Big{\\{}(4-2d)\big{[}D_{33}+D_{23}\big{]}+(8-2d)D_{3}\Big{\\}}(0,0,0,M_{H}^{2},q_{23},q_{13},M_{V^{0}_{i}}^{2},0,0,M_{V^{0}_{j}}^{2}),$
$\displaystyle F_{1,R}^{\text{Non-$V_{k}^{0*}$}}|_{V^{0}_{i},V^{0}_{j}}$
$\displaystyle=$ $\displaystyle
F_{1,L}^{\text{Non-$V_{k}^{0*}$}}|_{V^{0}_{i},V^{0}_{j}}\Big{(}g_{V^{0}_{i}ff}^{L}\rightarrow
g_{V^{0}_{i}ff}^{R};\;g_{V^{0}_{j}ff}^{L}\rightarrow
g_{V^{0}_{j}ff}^{R}\Big{)},$ (28) $\displaystyle
F_{2,L}^{\text{Non-$V_{k}^{0*}$}}|_{V^{0}_{i},V^{0}_{j}}$ $\displaystyle=$
$\displaystyle
F_{1,L}^{\text{Non-$V_{k}^{0*}$}}|_{V^{0}_{i},V^{0}_{j}}\Big{(}\\{q_{13},q_{23}\\}\rightarrow\\{q_{23},q_{13}\\}\Big{)},$
(29) $\displaystyle F_{2,R}^{\text{Non-$V_{k}^{0*}$}}|_{V^{0}_{i},V^{0}_{j}}$
$\displaystyle=$ $\displaystyle
F_{2,L}^{\text{Non-$V_{k}^{0*}$}}|_{V^{0}_{i},V^{0}_{j}}\Big{(}g_{V^{0}_{i}ff}^{L}\rightarrow
g_{V^{0}_{i}ff}^{R};\;g_{V^{0}_{j}ff}^{L}\rightarrow
g_{V^{0}_{j}ff}^{R}\Big{)}.$ (30)

Next, we also consider one-loop diagrams with having charged scalar bosons
$S_{i,j}$ internal lines (shown in Fig. 9). The results read

$\displaystyle F_{1,L}^{\text{Non-$V_{k}^{0*}$}}|_{S_{i},S_{j}}$
$\displaystyle=$
$\displaystyle\dfrac{eQ_{S}}{8\pi^{2}}\sum\limits_{S_{i},S_{j}}g_{HS_{i}S_{j}}\,g_{S_{i}f\nu_{f}}^{R}\,g_{S_{j}f\nu_{f}}^{R}\times$
$\displaystyle\times\Bigg{\\{}\Big{[}D_{33}+D_{23}+D_{3}\Big{]}(0,0,0,M_{H}^{2};q_{12},q_{13},M_{S_{i}}^{2},0,M_{S_{j}}^{2},M_{S_{j}}^{2})$
$\displaystyle\hskip
8.5359pt+\Big{[}D_{33}+D_{23}+D_{13}+D_{3}\Big{]}(0,0,0,M_{H}^{2};q_{23},q_{12},M_{S_{i}}^{2},M_{S_{i}}^{2},0,M_{S_{j}}^{2})\Bigg{\\}},$
$\displaystyle F_{1,R}^{\text{Non-$V_{k}^{0*}$}}|_{S_{i},S_{j}}$
$\displaystyle=$ $\displaystyle
F_{1,L}^{\text{Non-$V_{k}^{0*}$}}|_{S_{i},S_{j}}\Big{(}g_{S_{i}f\nu_{f}}^{R}\rightarrow
g_{S_{i}f\nu_{f}}^{L};\;g_{S_{j}f\nu_{f}}^{R}\rightarrow
g_{S_{j}f\nu_{f}}^{L}\Big{)},$ (32) $\displaystyle
F_{2,L}^{\text{Non-$V_{k}^{0*}$}}|_{S_{i},S_{j}}$ $\displaystyle=$
$\displaystyle
F_{1,L}^{\text{Non-$V_{k}^{0*}$}}|_{S_{i},S_{j}}\Big{(}\\{q_{13},q_{23}\\}\rightarrow\\{q_{23},q_{13}\\}\Big{)},$
(33) $\displaystyle F_{2,R}^{\text{Non-$V_{k}^{0*}$}}|_{S_{i},S_{j}}$
$\displaystyle=$ $\displaystyle
F_{2,L}^{\text{Non-$V_{k}^{0*}$}}|_{S_{i},S_{j}}\Big{(}g_{S_{i}f\nu_{f}}^{R}\rightarrow
g_{S_{i}f\nu_{f}}^{L};\;g_{S_{j}f\nu_{f}}^{R}\rightarrow
g_{S_{j}f\nu_{f}}^{L}\Big{)}.$ (34)

Furthermore, one also has the contributions of neutral scalar boson
$S^{0}_{i,j}$ exchanging in the loop diagrams (as described in Fig. 10).
Analytic results for the form factors then read

$\displaystyle F_{1,L}^{\text{Non-$V_{k}^{0*}$}}|_{S^{0}_{i},S^{0}_{j}}$
$\displaystyle=$
$\displaystyle-\dfrac{eQ_{f}}{8\pi^{2}}\sum\limits_{S^{0}_{i},S^{0}_{j}}\,g_{HS^{0}_{i}S^{0}_{j}}\,g_{S^{0}_{i}ff}^{L}\,g_{S^{0}_{j}ff}^{R}\times$
$\displaystyle\times\Big{[}D_{33}+D_{23}+D_{3}\Big{]}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{S^{0}_{i}}^{2},0,0,M_{S^{0}_{j}}^{2}),$
$\displaystyle F_{1,R}^{\text{Non-$V_{k}^{0*}$}}|_{S^{0}_{i},S^{0}_{j}}$
$\displaystyle=$ $\displaystyle
F_{1,L}^{\text{Non-$V_{k}^{0*}$}}|_{S^{0}_{i},S^{0}_{j}}\Big{(}g_{S^{0}_{i}ff}^{L}\rightarrow
g_{S^{0}_{i}ff}^{R};\;g_{S^{0}_{j}ff}^{R}\rightarrow
g_{S^{0}_{j}ff}^{L}\Big{)},$ (36) $\displaystyle
F_{2,L}^{\text{Non-$V_{k}^{0*}$}}|_{S^{0}_{i},S^{0}_{j}}$ $\displaystyle=$
$\displaystyle
F_{1,L}^{\text{Non-$V_{k}^{0*}$}}|_{S^{0}_{i},S^{0}_{j}}\Big{(}\\{q_{13},q_{23}\\}\rightarrow\\{q_{23},q_{13}\\}\Big{)},$
(37) $\displaystyle F_{2,R}^{\text{Non-$V_{k}^{0*}$}}|_{S^{0}_{i},S^{0}_{j}}$
$\displaystyle=$ $\displaystyle
F_{2,L}^{\text{Non-$V_{k}^{0*}$}}|_{S^{0}_{i},S^{0}_{j}}\Big{(}g_{S^{0}_{i}ff}^{L}\rightarrow
g_{S^{0}_{i}ff}^{R};\;g_{S^{0}_{j}ff}^{R}\rightarrow
g_{S^{0}_{j}ff}^{L}\Big{)}.$ (38)

We now consider non-$V_{k}^{*0}$ pole one-loop diagrams with mixing of scalar
$S_{j}$ (or $S_{j}^{0}$) and $V_{i}(V^{0}_{i})$ exchanging in the loop. The
diagrams are depicted in Figs. 11, 12. The calculations are performed as same
procedure. We finally find that these contributions are proportional to
$m_{f}$. As a result, in the limit of $m_{f}\rightarrow 0$, one confirms that

$\displaystyle
F_{k,L}^{\text{Non-$V_{k}^{0*}$}}|_{V_{i},S_{j}}=F_{k,R}^{\text{Non-$V_{k}^{0*}$}}|_{V_{i},S_{j}}$
$\displaystyle=$ $\displaystyle 0,$ (39) $\displaystyle
F_{k,L}^{\text{Non-$V_{k}^{0*}$}}|_{V^{0}_{i},S^{0}_{j}}=F_{k,R}^{\text{Non-$V_{k}^{0*}$}}|_{V^{0}_{i},S^{0}_{j}}$
$\displaystyle=$ $\displaystyle 0.$ (40)

for $k=1,2$.

We verify the ultraviolet finiteness of the results. We find that the UV-
divergent parts of all the above form factors come from all $B$-functions.
While $C$\- and $D$-functions in this papers are UV-finite. Higher rank tensor
$B$-functions can be reduced into $B_{0}$ and $A_{0}$. We verify that the sum
of all $B$-functions gives a UV-finite result. As a result, all the form
factors are UV-finite (seen our previous paper [45] for example).

Having the correctness form factors for the decay processes, the decay rate is
given by [20]:

$\displaystyle\dfrac{d\Gamma}{dq_{12}q_{13}}=\dfrac{q_{12}}{512\pi^{3}M_{H}^{3}}\Big{[}q_{13}^{2}(|F_{1,R}|^{2}+|F_{2,R}|^{2})+q_{23}^{2}(|F_{1,L}|^{2}+|F_{2,L}|^{2})\Big{]}.$
(41)

Taking the above integrand over $0\leq q_{12}\leq M_{H}^{2}$ and $0\leq
q_{13}\leq M_{H}^{2}-q_{12}$, one gets the total decay rates. In the next
subsection, we show typical examples which we apply our analytical results for
$H\rightarrow f\bar{f}\gamma$ to the Standard Model, the $U(1)_{B-L}$
extension of the SM, THDM. Phenomenological results for the decay channels of
the mentioned models also studied with using the present parameters at the
LHC.

##  4 Applications

We are going to apply the above results to the standard model and several BSMs
such as the $U(1)_{B-L}$ extension of the SM, THDM. For phenomenological
analyses, we use the following input parameters: $\alpha=1/137.035999084$,
$M_{Z}=91.1876$ GeV, $\Gamma_{Z}=2.4952$ GeV, $M_{W}=80.379$ GeV,
$M_{H}=125.1$ GeV, $m_{\tau}=1.77686$ GeV, $m_{t}=172.76$ GeV, $m_{b}=4.18$
GeV, $m_{s}=0.93$ GeV and $m_{c}=1.27$ GeV. Depending on the models under
consideration, the input values for new parameters are then shown.

###  4.1 Standard model

We first reduce our result to the case of the standard model. In this case, we
have $V_{i},V_{j}\rightarrow W^{+},W^{-}$, $V_{k}^{0}\rightarrow Z,\gamma$.
All couplings relating to the decay channels $H\rightarrow f\bar{f}\gamma$ in
the SM are replaced as in Table 1.

Vertices | Couplings
---|---
$g_{HV_{i}V_{j}}$ | $eM_{W}/s_{W}$
$g_{V^{0}_{k}V_{i}V_{j}}$ | $e\;c_{W}/s_{W}$
$g_{V^{0}_{k}AV_{i}V_{j}}$ | $e^{2}\;c_{W}/s_{W}$
$g_{V^{0}_{k}\nu_{f}\nu_{f}}^{L}$ | $e/(2s_{W}c_{W})$
$g_{V^{0}_{k}\nu_{f}\nu_{f}}^{R}$ | $0$
$g_{Hf_{i}f_{j}}^{L,R}$ | $e\;m_{f}/(2s_{W}\;M_{W})$
$g_{V^{0}_{k}f_{i}f_{j}}^{L}$ | $e(T_{3}^{f}-Q_{f}\;s^{2}_{W})/(s_{W}\;c_{W})$
$g_{V^{0}_{k}f_{i}f_{j}}^{R}$ | $-eQ_{f}s_{W}/c_{W}$
$g_{V_{i}f\nu_{f}}^{L}$ | $e/(\sqrt{2}\;s_{W})$
$g_{V_{i}f\nu_{f}}^{R}$ | $0$
Table 1:  All the couplings involving the decay processes $H\rightarrow
f\bar{f}\gamma$ in the SM.

In the SM, the form factors are obtained by taking the contributions of Eqs.
(3.1, 3.1, 3.2, 3.2). Using the above couplings, we then get a compact
expression for the form factors as follows:

$\displaystyle F_{2L}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}m_{t}^{2}}{3s_{W}M_{W}}\Bigg{[}\dfrac{16}{q_{12}}+\dfrac{2(8s_{W}^{2}-3)}{es_{W}c_{W}}\;\dfrac{g_{Zff}^{L}}{q_{12}-M_{Z}^{2}+i\Gamma_{Z}M_{Z}}\Bigg{]}\times$
$\displaystyle\hskip
19.91684pt\times\Bigg{\\{}4\Big{(}C_{22}+C_{12}+C_{2}\Big{)}+C_{0}\Bigg{\\}}(0,q_{12},M_{H}^{2},m_{t}^{2},m_{t}^{2},m_{t}^{2})$
$\displaystyle+\dfrac{\alpha^{2}}{s_{W}M_{W}^{3}}\Bigg{[}\dfrac{1}{q_{12}}-\dfrac{c_{W}}{es_{W}}\dfrac{g_{Zff}^{L}}{q_{12}-M_{Z}^{2}+i\Gamma_{Z}M_{Z}}\Bigg{]}\times$
$\displaystyle\hskip
19.91684pt\times\Bigg{\\{}\Big{[}-M_{H}^{2}\Big{(}B_{1}+3B_{11}+2B_{111}\Big{)}-2\Big{(}B_{00}+2B_{001}\Big{)}\Big{]}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\Big{[}4M_{W}^{2}\Big{(}q_{12}-M_{H}^{2}\Big{)}+2M_{H}^{2}q_{12}+8(1-d)M_{W}^{4}\Big{]}\times$
$\displaystyle\hskip
85.35826pt\times\Big{(}C_{22}+C_{12}+C_{2}\Big{)}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+4M_{W}^{2}(q_{12}-4M_{W}^{2})C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})\Bigg{\\}}$
$\displaystyle+\dfrac{\alpha}{4\pi
s_{W}M_{W}^{3}}\Big{(}g_{Wf\nu_{f}}^{L}\Big{)}^{2}\times$ $\displaystyle\hskip
19.91684pt\times\Bigg{\\{}(-M_{H}^{2}-2M_{W}^{2})\Big{[}\Big{(}C_{12}+C_{11}+C_{1}\Big{)}(M_{H}^{2},0,q_{12},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
122.34692pt+\Big{(}C_{12}+C_{11}+C_{1}\Big{)}(M_{H}^{2},q_{12},0,M_{W}^{2},M_{W}^{2},M_{W}^{2})\Big{]}$
$\displaystyle-2M_{W}^{2}\Big{[}C_{0}(M_{H}^{2},0,q_{12},M_{W}^{2},M_{W}^{2},M_{W}^{2})+C_{0}(M_{H}^{2},q_{12},0,M_{W}^{2},M_{W}^{2},M_{W}^{2})\Big{]}$
$\displaystyle+2M_{W}^{4}(2-d)\Big{[}\Big{(}D_{13}+D_{12}+D_{11}\Big{)}(M_{H}^{2},0,0,0;q_{12},q_{13},M_{W}^{2},M_{W}^{2},M_{W}^{2},0)$
$\displaystyle+\Big{(}D_{23}+D_{22}+D_{13}+2D_{12}+D_{11}\Big{)}(M_{H}^{2},0,0,0;q_{23},q_{12},M_{W}^{2},M_{W}^{2},0,M_{W}^{2})\Big{]}$
$\displaystyle-4M_{W}^{4}\Big{[}\Big{(}D_{3}+D_{2}+D_{0}\Big{)}(M_{H}^{2},0,0,0;q_{12},q_{13},M_{W}^{2},M_{W}^{2},M_{W}^{2},0)$
$\displaystyle\hskip
113.81102pt+D_{0}(M_{H}^{2},0,0,0;q_{23},q_{12},M_{W}^{2},M_{W}^{2},0,M_{W}^{2})\Big{]}$
$\displaystyle-2dM_{W}^{4}\Big{[}\Big{(}D_{2}+D_{1}\Big{)}(M_{H}^{2},0,0,0;q_{23},q_{12},M_{W}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle\hskip
113.81102pt+D_{1}(M_{H}^{2},0,0,0;q_{12},q_{13},M_{W}^{2},M_{W}^{2},M_{W}^{2},0)\Big{]}\Bigg{\\}}$
$\displaystyle+\dfrac{\alpha M_{W}}{2\pi
c_{W}^{2}s_{W}}\Big{(}g_{Zff}^{L}\Big{)}^{2}\Bigg{\\{}(d-2)\times$
$\displaystyle\hskip
28.45274pt\times\Big{[}D_{23}+D_{22}+D_{13}+2D_{12}+D_{11}\Big{]}(M_{H}^{2},0,0,0;q_{23},q_{13},M_{Z}^{2},M_{Z}^{2},0,0)$
$\displaystyle\hskip
28.45274pt+\Big{[}d\Big{(}D_{2}+D_{1}\Big{)}+2\Big{(}D_{3}+D_{0}\Big{)}\Big{]}(M_{H}^{2},0,0,0;q_{23},q_{13},M_{Z}^{2},M_{Z}^{2},0,0)\Bigg{\\}}.$

Where some coupling constants relate in this representation like
$g_{Wf\nu_{f}}^{L}=e/(\sqrt{2}s_{W})$,
$g_{Zff}^{L}=e(2s_{W}^{2}-1)/(2c_{W}s_{W})$ and $g_{Wf\nu_{f}}^{R}=0$,
$g_{Zff}^{R}=es_{W}/c_{W}$.

Other form factors can be obtained as follows.

$\displaystyle F_{1L}$ $\displaystyle=$ $\displaystyle
F_{2L}\Big{(}\\{q_{13},q_{23}\\}\rightarrow\\{q_{23},q_{13}\\}\Big{)},$ (43)
$\displaystyle F_{kR}$ $\displaystyle=$ $\displaystyle
F_{kL}\Big{(}g_{Wf\nu_{f}}^{L}\rightarrow
g_{Wf\nu_{f}}^{R};g_{Zff}^{L}\rightarrow g_{Zff}^{R}\Big{)}$ (44)

for $k=1,2$.

It is stress that we derive alternative results for the form factors of
$H\rightarrow f\bar{f}\gamma$ in the SM in comparison with previous works.
Because our analytic results in this paper are computed in the unitary. Thus,
our formulas may get different forms in comparison with the results in [20]
which have calculated in $R_{\xi}$-gauge. In this paper, we perform cross-
check our results with [20] by numerical tests. The numerical results for this
check are shown in Table 2. We find that our results are in good agreement
with the ones in [20] with more than $10$ digits.

$(q_{12},q_{13})$ | This work |  Ref. [20]
---|---|---
$(100,200)$ |  $9.62231539663501\cdot 10^{-8}$ | $9.62231539662956\cdot 10^{-8}$
|  $-3.501515874673991\cdot 10^{-10}\,i$ | $-3.501515874674078\cdot 10^{-10}\,i$
$(-100,200)$ |  $-9.95227151085161\cdot 10^{-8}$ | $-9.95227151084899\cdot 10^{-8}$
|  $-3.531494528007124\cdot 10^{-10}\,i$ | $-3.531494528006995\cdot 10^{-10}\,i$
$(100,-200)$ |  $9.62360254230002\cdot 10^{-8}$ | $9.62360254229717\cdot 10^{-8}$
|  $-3.597907189717628\cdot 10^{-10}\,i$ | $-3.597907189717582\cdot 10^{-10}\,i$
$(-100,-200)$ |  $-9.95098785515085\cdot 10^{-8}$ | $-9.95098785514946\cdot 10^{-8}$
|  $-3.622848558573573\cdot 10^{-10}\,i$ | $-3.622848558573332\cdot 10^{-10}\,i$
Table 2:  Numerical checks for form factor $F_{2L}$ in this work with $b_{1}$
in $A2$ of [20].

We also generate the decay widths for $H\rightarrow e\bar{e}\gamma$ and cross-
check our results with [13]. The results are presented in Table 3. For this
test, we adjust the input parameters and apply all cuts as same in [13]. In
the Table 3, the parameter $k$ is taken account which come from the
kinematical cuts of the invariant masses $m_{ff},\;m_{f\gamma}$ as follows:

$\displaystyle m_{ff}^{2},m_{f\gamma}^{2}\geq(kM_{H})^{2}.$ (45)

We find that our results are in good agreement with the ones in [13].

$k$ | This work |  Ref. [13]
---|---|---
$0$ |  $0.576865$ | $0.5782$
$0.1$ |  $0.242514$ | $0.245$
$0.2$ |  $0.184121$ | $0.1897$
$0.3$ |  $0.121368$ | $0.1242$
$0.4$ |  $0.0572478$ | $0.05844$
Table 3:  Cross-check the results of the decay widths in this work with [13]

###  4.2 $U(1)_{B-L}$ extension of the SM

We refer to the appendix $B$ for reviewing briefly this model. In the appendix
$B$, all couplings relating to the decay processes $H\rightarrow
f\bar{f}\gamma$ are derived in $U(1)_{B-L}$ extension of the SM (seen Table
4). Apart from all particles in the SM, two additional neutral Higgs and a
neutral gauge boson $Z^{\prime}$ which belongs to $U(1)_{B-L}$ gauge symmetry
are taken into account in this model.

For phenomenological study, we have to include three new parameters such as
the mixing angle $c_{\alpha}$, the $U(1)_{B-L}$ coupling $g^{\prime}_{1}$ and
the mass of new gauge boson $M_{Z}^{\prime}$. In all the below results, we set
$c_{\alpha}=0.3,0.7$ and $c_{\alpha}=1$ (this case is to the standard model).
The mass of $Z^{\prime}$ is in the range of $600$ GeV $\leq M_{Z}^{\prime}\leq
1000$ GeV. The coupling $g^{\prime}_{1}$ is in $0.05\leq g_{1}^{\prime}\leq
0.5$.

We study the impact of the $U(1)_{B-L}$ extension of the SM on the
differential decay widths as functions of $m_{ff}$ and $m_{f\gamma}$. The
results are shown in the Fig. 13 with fixing $M_{Z}^{\prime}=1000$ GeV. In
these figures, the solid line shows for the SM case by setting $c_{\alpha}=1$.
While the dashed line presents for $c_{\alpha}=0.7$ and the dashed dot line is
for $c_{\alpha}=0.3$. In the left figure, we observe photon pole at the lowest
region of $m_{ff}$. The decay rates decrease up to $m_{ff}\sim 60$ GeV. They
then grown up to $Z$-peak (the peak of $Z\rightarrow f\bar{f}$) which locates
around $m_{ff}\sim 90$ GeV. Beyond the peak, the decay rates decrease rapidly.
In the right figure, the decay widths increase up to peak of $m_{f\gamma}\sim
81$ GeV which is corresponding the photon recoil mass at $Z$-pole. They also
decrease rapidly beyond the peak. It is interested to find that the
contributions of $U(1)_{B-L}$ extension are sizable in both case of
$c_{\alpha}=0.7,0.3$. These effects can be probed clearly at the future
colliders.

$\begin{array}[]{cc}\dfrac{d\Gamma}{dm_{f\bar{f}}}&\dfrac{d\Gamma}{dm_{f\gamma}}\\\
&\\\
\includegraphics[width=227.62204pt,height=227.62204pt]{./U1/HFFG_U1_mll_Input_gBL_0.5_MZBL_1000_cA_SM_1.0_0.7_0.3.eps}&\includegraphics[width=227.62204pt,height=227.62204pt]{./U1/HFFG_U1_mlg_Input_gBL_0.5_MZBL_1000_cA_SM_1.0_0.7_0.3.eps}\\\
\hskip 142.26378ptm_{ff}\;\text{[GeV]}&\hskip
142.26378ptm_{f\gamma}\;\text{[GeV]}\end{array}$

Figure 13:  Differential of the decay width as functions of $m_{ff}$ and
$m_{f\gamma}$

We next examine the decay widths as a function of $M_{Z}^{\prime}$. In this
study, we change the mass of $Z^{\prime}$ boson as $800$ GeV $\leq
M_{Z^{\prime}}\leq 1000$ GeV and set $c_{\alpha}=0.7$ (see Fig. 14), the
dashed line shows for the case of $g^{\prime}_{1}=0.05$, the dot line presents
for $g^{\prime}_{1}=0.2$ and the dashed dot line is for $g^{\prime}_{1}=0.5$.
In all range of $M_{Z}^{\prime}$, we observe that the decay widths are
proportional to $g_{1}^{\prime}$ (it is also confirmed later analyses). They
also decrease with increasing $M_{Z}^{\prime}$. We conclude that the
contributions from the $U(1)_{B-L}$ extension are massive and can be probed
clearly at future colliders.

$\Gamma$ [KeV]

$M_{Z^{\prime}}$

Figure 14:  The decay widths which are functions of $M_{Z}^{\prime}$

We finally discuss how the effect of $g^{\prime}_{1}$ on the decay widths
(seen Fig. 15). In these figures, we set $M_{Z^{\prime}}=800$ GeV and
$c_{\alpha}=0.3$ ($c_{\alpha}=0.7$) for the left figure (for the right figure)
respectively. We find that the decay width are proportional to
$g_{1}^{\prime}$.

$\begin{array}[]{cc}\Gamma\text{[KeV]}&\Gamma\text{[KeV]}\\\
\includegraphics[width=227.62204pt,height=227.62204pt]{./U1/HFFG_U1_gBL_Input_cA_0.3_MZBL_800.eps}&\includegraphics[width=227.62204pt,height=227.62204pt]{./U1/HFFG_U1_gBL_Input_cA_0.7_MZBL_800.eps}\\\
\hskip 142.26378ptg_{1}^{\prime}&\hskip 142.26378ptg_{1}^{\prime}\end{array}$

Figure 15:  The decay widths are function of $g_{1}^{\prime}$

###  4.3 Two Higgs Doublet Model

For reviewing THDM, we refer the appendix $C$ for more detail. In this
framework, the gauge sector is same the SM case. It means that we have
$V_{i}=V_{j}=W$, $V_{k}^{0}=Z,\gamma$. In the Higgs sectors, one has
additional charged Higgs $H^{+}$ and two neutral scalar CP even Higgs bosons
$H_{1}^{0},H_{2}^{0}$, a CP odd Higgs $A$. All couplings relating to the decay
processes $H\rightarrow f\bar{f}\gamma$ are shown in Table 5.

For the phenomenological results, we take $1\leq\tan\beta\leq 30$,
$M_{H_{1}^{0}}=125.1$ GeV, $-900^{2}$ GeV2 $\leq m_{12}^{2}\leq 900^{2}$ GeV2,
$340$ GeV $\leq M_{H^{\pm}}\leq 900$ GeV, $-\pi/2\leq\alpha\leq\pi/2$. We
study the differential decay rates as a function of the invariant mass of
leptons $m_{ff}$. We select $M_{H^{\pm}}=700$ GeV, $m_{12}^{2}=800^{2}$ GeV2.
The results are shown in Fig. 16. In this figure, the solid line shows for the
SM case. The dashed line presents for the case of $\alpha=-0.4\pi$. The dashed
dot line is for the case of $\alpha=0$. The dot line is for the case of
$\alpha=0.4\pi$. From the top to bottom of the figure, we have the
corresponding figure to $\tan\beta=5,10,15,25$. The decay rates are same
behavior in previous cases, we observe photon pole at the lowest region of
$m_{ff}$. They decrease from photon pole to the region of $m_{ff}\sim 60$ GeV
and then grown up to $Z$-peak. Beyond the $Z$-peak, the decay rates decrease
rapidly. The effect of THDM are also visible in all cases.

$\begin{array}[]{cc}\includegraphics[width=227.62204pt,height=227.62204pt]{./thdm/HFFG_C2HDM_DR_mll_tanb_5_alpha.eps}&\includegraphics[width=227.62204pt,height=227.62204pt]{./thdm/HFFG_C2HDM_DR_mll_tanb_10_alpha.eps}\\\
\\\
\includegraphics[width=227.62204pt,height=227.62204pt]{./thdm/HFFG_C2HDM_DR_mll_tanb_15_alpha.eps}&\includegraphics[width=227.62204pt,height=227.62204pt]{./thdm/HFFG_C2HDM_DR_mll_tanb_25_alpha.eps}\end{array}$

Figure 16:  Differential decay rates as a function of the invariant mass of
leptons $m_{ff}$ with changing $\alpha=-0.4\pi,0,0.4\pi$

We next change charged Higgs masses $M_{H^{\pm}}=400,600,800$ GeV and take
$\alpha=0.4\pi$. From the top to bottom of the figure, we have the
corresponding figure to the case of $\tan\beta=5,10,15,25$ (seen Fig. 17). The
decay widths depend on $m_{ff}$ the same previous explanation. They are
inversely proportional to the charged Higgs masses. We find that the effect of
THDM are sizeable contributions in all the above cases. These effects can be
discriminated clearly at future colliders.

$\begin{array}[]{cc}\includegraphics[width=227.62204pt,height=227.62204pt]{./thdm/HFFG_C2HDM_DR_mll_tanb_5_MCH.eps}&\includegraphics[width=227.62204pt,height=227.62204pt]{./thdm/HFFG_C2HDM_DR_mll_tanb_10_MCH.eps}\\\
\\\
\includegraphics[width=227.62204pt,height=227.62204pt]{./thdm/HFFG_C2HDM_DR_mll_tanb_15_MCH.eps}&\includegraphics[width=227.62204pt,height=227.62204pt]{./thdm/HFFG_C2HDM_DR_mll_tanb_25_MCH.eps}\end{array}$

Figure 17: Differential decay rates as a function of the invariant mass of
leptons $m_{ff}$ with changing charged Higgs masses $M_{H^{\pm}}=400,600,800$
GeV.

##  5 Conclusions

We have performed the calculations for one-loop contributing to the decay
processes $H\rightarrow f\bar{f}\gamma$ in the limit of $m_{f}\rightarrow 0$.
In this computation, we have considered all possible contributions of
additional heavy vector gauge bosons, heavy fermions, and charged (and also
neutral) scalar particles which they may exchange in the loop diagrams. The
analytic formulas are written in terms of Passarino-Veltman functions which
they can be evaluated numerically by using the package LoopTools. The
evaluations have then applied standard model, $U(1)_{B-L}$ extension of the
SM, two Higgs doublet model. Phenomenological results of the decay processes
for the above models have studied in detail. We find that the effects of new
physics are sizable contributions and they can be probed at future colliders.

Acknowledgment: This research is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under the grant number
$103.01$-$2019.346$. K. H. Phan would like to thank Dr. L. T. Hue for helpful
discussions.

##  Appendix A Tensor one-loop reductions

In this appendix, tensor one-loop reduction method in [44] is discussed
briefly. First, the definition of one-loop one-, two-, three- and four-point
tensor integrals with rank $R$ are as follows:

$\displaystyle\\{A;B;C;D\\}^{\mu_{1}\mu_{2}\cdots\mu_{R}}=(\mu^{2})^{2-d/2}\int\frac{d^{d}k}{(2\pi)^{d}}\dfrac{k^{\mu_{1}}k^{\mu_{2}}\cdots
k^{\mu_{R}}}{\\{P_{1};P_{1}P_{2};P_{1}P_{2}P_{3};P_{1}P_{2}P_{3}P_{4}\\}}.$
(46)

In this formula, $P_{j}$ ($j=1,\cdots,4$) are the inverse Feynman propagators

$\displaystyle P_{j}=(k+q_{j})^{2}-m_{j}^{2}+i\rho.$ (47)

In this equation, we use $q_{j}=\sum\limits_{i=1}^{j}p_{i}$ with $p_{i}$ for
the external momenta, and $m_{j}$ for internal masses in the loops.
Dimensional regularization is performed in space-time dimension
$d=4-2\varepsilon$. The parameter $\mu^{2}$ plays role of a renormalization
scale. When the numerator of the integrands in Eq. (46) becomes $1$, one has
the corresponding scalar one-loop functions (they are noted as $A_{0}$,
$B_{0}$, $C_{0}$ and $D_{0}$). Explicit reduction formulas for one-loop one-,
two-, three- and four-point tensor integrals up to rank $R=3$ are written as
follows [44]. In particular, for two-point tensor integrals, the reduction
formulas are:

$\displaystyle A^{\mu}$ $\displaystyle=$ $\displaystyle 0,$ (48)
$\displaystyle A^{\mu\nu}$ $\displaystyle=$ $\displaystyle g^{\mu\nu}A_{00},$
(49) $\displaystyle A^{\mu\nu\rho}$ $\displaystyle=$ $\displaystyle 0,$ (50)
$\displaystyle B^{\mu}$ $\displaystyle=$ $\displaystyle q^{\mu}B_{1},$ (51)
$\displaystyle B^{\mu\nu}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}B_{00}+q^{\mu}q^{\nu}B_{11},$ (52) $\displaystyle B^{\mu\nu\rho}$
$\displaystyle=$
$\displaystyle\\{g,q\\}^{\mu\nu\rho}B_{001}+q^{\mu}q^{\nu}q^{\rho}B_{111},$
(53)

For three-point functions, one has

$\displaystyle C^{\mu}$ $\displaystyle=$ $\displaystyle
q_{1}^{\mu}C_{1}+q_{2}^{\mu}C_{2}=\sum\limits_{i=1}^{2}q_{i}^{\mu}C_{i},$ (54)
$\displaystyle C^{\mu\nu}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}C_{00}+\sum\limits_{i,j=1}^{2}q_{i}^{\mu}q_{j}^{\nu}C_{ij},$ (55)
$\displaystyle C^{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{2}\\{g,q_{i}\\}^{\mu\nu\rho}C_{00i}+\sum_{i,j,k=1}^{2}q^{\mu}_{i}q^{\nu}_{j}q^{\rho}_{k}C_{ijk},$
(56)

Similarly, tensor reduction formulas for four-point functions are given by

$\displaystyle D^{\mu}$ $\displaystyle=$ $\displaystyle
q_{1}^{\mu}D_{1}+q_{2}^{\mu}D_{2}+q_{3}^{\mu}D_{3}=\sum\limits_{i=1}^{3}q_{i}^{\mu}D_{i},$
(57) $\displaystyle D^{\mu\nu}$ $\displaystyle=$ $\displaystyle
g^{\mu\nu}D_{00}+\sum\limits_{i,j=1}^{3}q_{i}^{\mu}q_{j}^{\nu}D_{ij},$ (58)
$\displaystyle D^{\mu\nu\rho}$ $\displaystyle=$
$\displaystyle\sum_{i=1}^{3}\\{g,q_{i}\\}^{\mu\nu\rho}D_{00i}+\sum_{i,j,k=1}^{3}q^{\mu}_{i}q^{\nu}_{j}q^{\rho}_{k}D_{ijk}.$
(59)

The short notation [44] $\\{g,q_{i}\\}^{\mu\nu\rho}$ is used as follows:
$\\{g,q_{i}\\}^{\mu\nu\rho}=g^{\mu\nu}q^{\rho}_{i}+g^{\nu\rho}q^{\mu}_{i}+g^{\mu\rho}q^{\nu}_{i}$
in the above relations. The scalar coefficients $A_{00},B_{1},\cdots,D_{333}$
in the right hand sides of the above equations are so-called Passarino-Veltman
functions [44]. They have been implemented into LoopTools [46] for numerical
computations.

##  Appendix B Review of $U(1)_{B-L}$ extension

In this appendix, we review briefly $U(1)_{B-L}$ extension [25]. This model
follows gauge symmetry $SU(3)_{C}\otimes SU(2)_{L}\otimes U(1)_{Y}\otimes
U(1)_{B-L}$. By including complex scalar $S$, general scalar potential is
given by

$\displaystyle V(\Phi,S)$ $\displaystyle=$ $\displaystyle
m^{2}\Phi^{{\dagger}}\Phi+\lambda_{1}(\Phi^{{\dagger}}\Phi)^{2}+\mu^{2}|S|^{2}+\lambda_{2}|S|^{4}+\lambda_{3}\Phi^{{\dagger}}\Phi|S|^{2}.$
(60)

In order to find mass spectrum of the scalar sector, we expand the scalar
fields around their vacuum as follows:

$\displaystyle\Phi=\begin{pmatrix}\phi^{+}\\\
\dfrac{v_{\phi}+h+i\xi}{\sqrt{2}}\end{pmatrix},\quad
S=\dfrac{v_{s}+h^{\prime}+i\xi^{\prime}}{\sqrt{2}}.$ (61)

The Goldstone bosons $\phi^{\pm},\xi$ will give the masses of $W^{\pm}$ and
$Z$ bosons. In unitary gauge, the mass eigenvalues of neutral Higgs are given:

$\displaystyle\begin{pmatrix}h_{1}\\\
h_{2}\end{pmatrix}=\begin{pmatrix}c_{\alpha}&-s_{\alpha}\\\
s_{\alpha}&c_{\alpha}\\\ \end{pmatrix}\begin{pmatrix}h\\\
h^{\prime}\end{pmatrix}$ (62)

with mixing angle

$\displaystyle s_{2\alpha}=\sin
2\alpha=\dfrac{\lambda_{3}v_{\phi}v_{s}}{\sqrt{(\lambda_{1}v_{\phi}^{2}-\lambda_{2}v_{s}^{2})^{2}+(\lambda_{3}v_{\phi}v_{s})^{2}}}$
(63)

After this transformation, masses of scalar Higgs are given

$\displaystyle M^{2}_{h_{1}}$ $\displaystyle=$
$\displaystyle\lambda_{1}v_{\phi}^{2}+\lambda_{2}v_{s}^{2}-\sqrt{(\lambda_{1}v_{\phi}^{2}-\lambda_{2}v_{s}^{2})^{2}+(\lambda_{3}v_{\phi}v_{s})^{2}},$
(64) $\displaystyle M^{2}_{h_{2}}$ $\displaystyle=$
$\displaystyle\lambda_{1}v_{\phi}^{2}+\lambda_{2}v_{s}^{2}+\sqrt{(\lambda_{1}v_{\phi}^{2}-\lambda_{2}v_{s}^{2})^{2}+(\lambda_{3}v_{\phi}v_{s})^{2}}.$
(65)

For gauge boson masses, $W,Z$ and $Z^{\prime}$ bosons are obtained their
masses by expanding the following kinematic terms

$\displaystyle\mathcal{L}_{\text{gauge}}\rightarrow(D^{\mu}\Phi)^{+}D_{\mu}\Phi,\quad(D^{\mu}S)^{+}D_{\mu}S.$
(66)

As a result, mass of $Z^{\prime}$ is $M_{Z^{\prime}}=2v_{s}g_{1}^{\prime}$.

The Yukawa interaction involving right-handed neutrinos are given:

$\displaystyle\mathcal{L}_{Y}$ $\displaystyle=$
$\displaystyle-y^{d}_{jk}\bar{q}_{Lj}d_{Rk}\Phi-y^{u}_{jk}\bar{q}_{Lj}u_{Rk}i\sigma_{2}\Phi^{*}-y^{e}_{jk}\bar{l}_{Lj}e_{Rk}\Phi$
(67)
$\displaystyle-y^{\nu}_{jk}\bar{l}_{Lj}\nu_{Rk}i\sigma_{2}\Phi^{*}-y^{M}_{jk}\overline{(\nu_{R})}_{j}^{c}\nu_{Rk}\;S+\text{h.c}.$

for $j,k=1,2,3$. The last term in this equation is Majorana mass terms for
right-handed neutrinos. After the spontaneous breaking symmetry, the mass
matrix of neutrinos is

$\displaystyle
M=\begin{pmatrix}0&m_{D}=\dfrac{(y^{\nu})^{*}}{\sqrt{2}}v_{\phi}\\\
m_{D}^{T}&M=\sqrt{2}y^{M}v_{s}\\\ \end{pmatrix}.$ (68)

The diagonalization is obtained by the transformation

$\displaystyle\text{diag}\Big{(}-\dfrac{(m^{i}_{D})^{2}}{M^{i}},M^{i}\Big{)}=\begin{pmatrix}\cos\alpha_{i}&-\sin\alpha_{i}\\\
\sin\alpha_{i}&\cos\alpha_{i}\\\ \end{pmatrix}\begin{pmatrix}0&m_{D}^{i}\\\
m_{D}^{i}&M^{i}\\\
\end{pmatrix}\begin{pmatrix}\cos\alpha_{i}&\sin\alpha_{i}\\\
-\sin\alpha_{i}&\cos\alpha_{i}\\\ \end{pmatrix}$ (69)

for $i=1,2,3$ and $\alpha_{i}=\text{arcsin}(m_{D}^{i}/M^{i})$.

We show all relevant couplings in the decay under consider In the case of
$\alpha_{i}\rightarrow 0$, all the couplings are shown in Table 4

Vertices | Couplings
---|---
$h_{1}(h_{2})f\bar{f}$ | $-i\dfrac{c_{\alpha}(s_{\alpha})e\;m_{f}}{2M_{W}s_{W}}$
$Z^{\prime}_{\mu}f\bar{f}$ | $iQ_{f}\;g^{\prime}_{1}\gamma_{\mu}$
$h_{1}(h_{2})W^{+}_{\mu}W^{-}_{\nu}$ | $i\frac{eM_{W}}{s_{W}}\;c_{\alpha}(s_{\alpha})g_{\mu\nu}$
$h_{1}(h_{2})Z_{\mu}Z_{\nu})$ | $i\frac{eM_{W}}{c_{W}^{2}\;s_{W}}\;c_{\alpha}(s_{\alpha})g_{\mu\nu}$
$h_{1}(h_{2})Z^{\prime}_{\mu}Z^{\prime}_{\nu})$ | $-i4g^{\prime}_{1}M_{Z}^{\prime}\;s_{\alpha}(-c_{\alpha})g_{\mu\nu}$
Table 4:  All the couplings involving the decay processes $H\rightarrow
f\bar{f}\gamma$ in the $U(1)_{B-L}$ extension of the SM.

##  Appendix C Review of Two Higgs Doublet Model

Base on Ref. [27], we review briefly the two Higgs doublet model with breaking
softly $Z_{2}$-symmetry. In this model, there are two scalar doublets
$\Phi_{1},\Phi_{2}$ with hypercharge $Y=1/2$. Parts of Lagrangian extended
from the SM are presented as follows:

$\displaystyle\mathcal{L}=\mathcal{L}_{K}+\mathcal{L}_{Y}-V(\Phi_{1},\Phi_{2}).$
(70)

Where the kinematic term is $\mathcal{L}_{K}$, the Yukawa part is
$\mathcal{L}_{Y}$ and $V(\Phi_{1},\Phi_{2})$ is Higgs potential. First, the
kinematic term is taken the form of

$\displaystyle\mathcal{L}_{K}=\sum\limits_{k=1}^{2}\left(D_{\mu}\Phi_{k}\right)^{{\dagger}}\left(D^{\mu}\Phi_{k}\right)$
(71)

with $D_{\mu}=\partial_{\mu}-igT^{a}W^{a}_{\mu}-i\frac{g^{\prime}}{2}B_{\mu}$.
The Higgs potential with breaking the $Z_{2}$-symmetry is expressed:

$\displaystyle V(\Phi_{1},\Phi_{2})$ $\displaystyle=$
$\displaystyle\frac{1}{2}m_{11}^{2}\Phi_{1}^{{\dagger}}\Phi_{1}-m_{12}^{2}(\Phi_{1}^{{\dagger}}\Phi_{2}+\Phi_{2}^{{\dagger}}\Phi_{1})+\frac{1}{2}m_{22}^{2}\Phi_{2}^{{\dagger}}\Phi_{2}+\dfrac{\lambda_{1}}{2}\left(\Phi_{1}^{{\dagger}}\Phi_{1}\right)^{2}+\dfrac{\lambda_{2}}{2}\left(\Phi_{2}^{{\dagger}}\Phi_{2}\right)^{2}$
(72)
$\displaystyle+\lambda_{3}\left(\Phi_{1}^{{\dagger}}\Phi_{1}\right)\left(\Phi_{2}^{{\dagger}}\Phi_{2}\right)+\lambda_{4}\left(\Phi_{1}^{{\dagger}}\Phi_{2}\right)\left(\Phi_{2}^{{\dagger}}\Phi_{1}\right)+\frac{\lambda_{5}}{2}\left[\left(\Phi_{1}^{{\dagger}}\Phi_{2}\right)^{2}+\left(\Phi_{1}^{{\dagger}}\Phi_{2}\right)^{2}\right].$

In this potential, $m_{12}^{2}$ plays role of soft broken scale of the
$Z_{2}$-symmetry. The two scalar doublet fields can be parameterized as
follows:

$\displaystyle\Phi_{1}=\begin{pmatrix}\phi_{1}^{+}\\\
\dfrac{v_{1}+\eta_{1}+i\xi_{1}}{\sqrt{2}}\end{pmatrix},\quad\Phi_{2}=\begin{pmatrix}\phi_{2}^{+}\\\
\dfrac{v_{2}+\eta_{2}+i\xi_{2}}{\sqrt{2}}\end{pmatrix}.$ (73)

We get a system equation for these parameters from the stationary conditions
of the Higgs potential. The relations are shown as follows:

$\displaystyle
m_{11}^{2}-\mu^{2}\frac{v_{2}^{2}}{v^{2}}+\frac{\lambda_{1}}{2}v_{1}^{2}+\frac{\lambda_{345}}{2}v_{2}^{2}$
$\displaystyle=$ $\displaystyle 0,$ (74) $\displaystyle
m_{22}^{2}-\mu^{2}\frac{v_{1}^{2}}{v^{2}}+\frac{\lambda_{2}}{2}v_{2}^{2}+\frac{\lambda_{345}}{2}v_{1}^{2}$
$\displaystyle=$ $\displaystyle 0.$ (75)

Where $v^{2}=v_{1}^{2}+v_{2}^{2}$ is fixed at electroweak scale or
$v=(\sqrt{2}G_{F})^{-1/2}=246$ GeV and new parameter $\mu^{2}$ is defined as
$\mu^{2}=\frac{v^{2}}{v_{1}v_{2}}m_{12}^{2}$. The shorten notation is
$\lambda_{345}=\lambda_{3}+\lambda_{4}+\lambda_{5}$. The mixing angle is given
$t_{\beta}=\tan\beta=v_{2}/v_{1}$. The mass terms of the Higgs potential
$V_{mass}$ can be expressed as:

$\displaystyle V_{mass}$ $\displaystyle=$
$\displaystyle(\phi^{+}_{1},\phi^{+}_{2})R_{\beta}\begin{pmatrix}0&0\\\
0&M^{2}_{H^{+}}\\\ \end{pmatrix}R_{\beta}^{-1}\begin{pmatrix}\phi_{1}^{+}\\\
\phi_{2}^{+}\end{pmatrix}+\frac{1}{2}(\xi_{1},\xi_{2})R_{\beta}\begin{pmatrix}0&0\\\
0&M^{2}_{A}\\\ \end{pmatrix}R_{\beta}^{-1}\begin{pmatrix}\xi_{1}\\\ \xi_{2}\\\
\end{pmatrix}$ (76)
$\displaystyle+\frac{1}{2}(\eta_{1},\eta_{2})R_{\beta}\begin{pmatrix}M^{2}_{H_{2}^{0}}&0\\\
0&M^{2}_{H_{1}^{0}}\\\ \end{pmatrix}R_{\beta}^{-1}\begin{pmatrix}\eta_{1}\\\
\eta_{2}\end{pmatrix}.$

Here diagonalized matrix of neutral mass is defined as

$\displaystyle\text{diag}(M^{2}_{H_{2}^{0}},M^{2}_{H_{1}^{0}})=R_{\alpha}\mathcal{M}^{2}R_{\alpha}^{T},\quad\text{with}\quad(\mathcal{M}^{2})_{ij}=\dfrac{\partial^{2}V}{\partial{\eta_{i}}\partial{\eta_{j}}}.$
(77)

The mass eigenstates can be then expressed as follows:

$\displaystyle\begin{pmatrix}G^{+}\\\
H^{+}\end{pmatrix}=R_{\beta}^{-1}\begin{pmatrix}\phi_{1}^{+}\\\
\phi_{2}^{+}\end{pmatrix},\quad\begin{pmatrix}G^{0}\\\
A\end{pmatrix}=R_{\beta}^{-1}\begin{pmatrix}\xi_{1}\\\
\xi_{2}\end{pmatrix},\quad\begin{pmatrix}H_{2}^{0}\\\
H_{1}^{0}\end{pmatrix}=R_{\alpha}^{-1}R_{\beta}^{-1}\begin{pmatrix}\eta_{1}\\\
\eta_{2}\end{pmatrix}$ (78)

where

$\displaystyle R_{\beta}=\begin{pmatrix}c_{\beta}&s_{\beta}\\\
-s_{\beta}&c_{\beta}\\\ \end{pmatrix},\quad
R_{\alpha}=\begin{pmatrix}c_{\alpha}&s_{\alpha}\\\ -s_{\alpha}&c_{\alpha}\\\
\end{pmatrix}$ (79)

with $-\pi/2\leq\alpha\leq\pi/2$. In the unitary gauge, it is well-known that
$G^{+}$ and $G^{0}$ are massless Goldstone bosons will become the longitudinal
polarization of $W^{+}$ and $Z^{0}$. The remains $H^{\pm}$, $A$ and
$H^{0}_{1,2}$ become the charged Higgs bosons, a CP-odd Higgs boson and CP-
even Higgs bosons respectively. The masses of these scalar bosons are given by

$\displaystyle M^{2}_{H^{\pm}}$ $\displaystyle=$
$\displaystyle\mu^{2}-\frac{v^{2}}{2}(\lambda_{4}+\lambda_{5}),$ (80)
$\displaystyle M^{2}_{A}$ $\displaystyle=$
$\displaystyle\mu^{2}-v^{2}\lambda_{5},$ (81) $\displaystyle
M^{2}_{H^{0}_{1}}$ $\displaystyle=$ $\displaystyle
s_{\alpha}^{2}\mathcal{M}^{2}_{11}-2s_{\alpha}c_{\alpha}\mathcal{M}^{2}_{12}+c_{\alpha}^{2}\mathcal{M}^{2}_{22},$
(82) $\displaystyle M^{2}_{H^{0}_{2}}$ $\displaystyle=$ $\displaystyle
c_{\alpha}^{2}\mathcal{M}^{2}_{11}+2s_{\alpha}c_{\alpha}\mathcal{M}^{2}_{12}+s_{\alpha}^{2}\mathcal{M}^{2}_{22}.$
(83)

From the Higgs potential in Eq. (72) with the stationary conditions in (74),
we have $7$ parameters. They are

$\displaystyle\Big{\\{}\lambda_{1,2,3,4,5},t_{\beta},m_{12}^{2}\Big{\\}}.$
(84)

For phenomenological analyses, the above parameters are transferred to the
following parameters:

$\displaystyle\Big{\\{}M^{2}_{H^{+}},M^{2}_{A},M^{2}_{H^{0}_{1}},M^{2}_{H^{0}_{2}},\alpha,t_{\beta},m_{12}^{2}\Big{\\}}.$
(85)

All the couplings involving the decay processes $H\rightarrow f\bar{f}\gamma$
are derived in this appendix. In general, we can consider the lightest Higgs
boson $H_{1}^{0}$ is the SM like-Higgs boson. In Table 5, all the couplings
are shown in detail.

Vertices | Couplings
---|---
$H_{1}^{0}W^{+}_{\mu}W^{-}_{\nu}$ | $i\frac{2M_{W}^{2}}{v}s_{(\beta-\alpha)}g_{\mu\nu}$
$H_{1}^{0}Z_{\mu}Z_{\nu}$ | $i\frac{2M_{Z}^{2}}{v}s_{(\beta-\alpha)}g_{\mu\nu}$
$H_{1}^{0}(p)H^{\pm}(q)W^{\mp}_{\mu}$ | $\mp i\frac{M_{W}}{v}c_{(\beta-\alpha)}\;(p-q)_{\mu}$
$H^{+}(p)H^{-}(q)A_{\mu}$ | $i\frac{2M_{W}}{v}s_{W}\;(p-q)_{\mu}$
$H^{+}(p)H^{-}(q)Z_{\mu}$ | $i\frac{M_{Z}}{v}c_{2W}\;(p-q)_{\mu}$
$H_{1}^{0}H^{+}H^{-}$ | $\frac{i}{v}\Big{[}(2\mu^{2}-2M_{H^{\pm}}^{2}-M^{2}_{H^{0}_{1}})s_{(\beta-\alpha)}+2(\mu^{2}-M^{2}_{H^{0}_{1}})\text{cot}2\beta\;c_{(\beta-\alpha)}\Big{]}$
$H^{+}H^{-}A_{\mu}A_{\nu}$ | $2ie^{2}\;g_{\mu\nu}$
$H^{+}H^{-}Z_{\mu}A_{\nu}$ | $ieg\frac{c_{2W}}{c_{W}}\;g_{\mu\nu}$
Table 5:  All the couplings involving the decay processes $H\rightarrow
f\bar{f}\gamma$ in THDM.

For the Yukawa part, we refer [27] for more detail. Depend on the types of
THDMs, we then have the couplings of scalar fields and fermions.

$\displaystyle\mathcal{L}_{H_{1}^{0}t\bar{t}}$ $\displaystyle=$
$\displaystyle-\frac{m_{t}}{v}\frac{c_{\alpha}}{s_{\beta}}\bar{t}tH_{1}^{0}.$
(86)

##  Appendix D Feynman rules and couplings

In the below Tables, we use $P_{L/R}=(1\mp\gamma_{5})/2$,
$\Gamma^{\mu\nu\lambda}(p_{1},p_{2},p_{3})=g^{\mu\nu}(p_{1}-p_{2})^{\lambda}+g^{\lambda\nu}(p_{2}-p_{3})^{\mu}+g^{\mu\lambda}(p_{3}-p_{1})^{\nu}$
and
$S^{\mu\nu,\alpha\beta}=2g^{\mu\nu}g^{\alpha\beta}-g^{\mu\alpha}g^{\nu\beta}-g^{\mu\beta}g^{\nu\alpha}$
and $Q_{V}$ denotes the electric charge of the gauge bosons $V_{i},V_{j}$ and
$Q_{S}$ is charge of the charged Higgs bosons $S_{i},S_{j}$. Moreover, the
factor $\xi_{V_{k}^{0}}$ is included for covering all possible cases of
neutral gauge boson. It can be $0$ and $1$ for the case of the pole of photon
and Z-boson ($Z^{\prime}$-boson) respectively.

Particle types | Propagators
---|---
Fermions $f$ | $i\dfrac{\not{k}+m_{f}}{k^{2}-m_{f}^{2}}$
Charged (neutral) gauge bosons $V_{i}(V^{0}_{i})$ | $\dfrac{-i}{p^{2}-M_{V_{i}}^{2}(M_{V^{0}_{i}}^{2})}\Bigg{[}g^{\mu\nu}-\dfrac{p^{\mu}p^{\nu}}{M_{V_{i}}^{2}(M_{V^{0}_{i}}^{2})}\Bigg{]}$
Gauge boson $V^{0*}_{k}$ poles | $\dfrac{-i}{p^{2}-M_{V_{k}^{0}}^{2}+i\Gamma_{V^{0}_{k}}M_{V_{k}^{0}}}\Bigg{[}g^{\mu\nu}-\xi_{V_{k}^{0}}\dfrac{p^{\mu}p^{\nu}}{M_{V^{0}_{k}}^{2}}\Bigg{]}$
Charged (neutral) scalar bosons $S_{i}(S_{i}^{0})$ | $\dfrac{i}{p^{2}-M_{S_{i}}^{2}(M_{S^{0}_{i}}^{2})}$
Table 6:  Feynman rules involving the decay channels in the unitary gauge.

Vertices | Couplings
---|---
$H\cdot S_{i}\cdot S_{j}$ | $-i\,g_{HS_{i}S_{j}}$
$H\cdot S^{0}_{i}\cdot S^{0}_{j}$ | $-i\,g_{HS^{0}_{i}S^{0}_{j}}$
$H\cdot V_{i}^{\mu}\cdot V_{j}^{\nu}$ | $i\,g_{HV_{i}V_{j}}\,g^{\mu\nu}$
$H\cdot V_{i}^{0\,\mu}\cdot V_{j}^{0\,\nu}$ | $i\,g_{HV^{0}_{i}V^{0}_{j}}\,g^{\mu\nu}$
$H\cdot\bar{f_{i}}\cdot f_{j}$ | $-i\,\Big{(}g_{Hf_{i}f_{j}}^{L}P_{L}+g_{Hf_{i}f_{j}}^{R}P_{R}\Big{)}$
$S^{0}_{k}\cdot\bar{f}\cdot f$ | $-i\,\Big{(}g_{S^{0}_{k}ff}^{L}P_{L}+g_{S^{0}_{k}ff}^{R}P_{R}\Big{)}$
$A^{\mu}\cdot f_{i}\cdot\bar{f_{i}}$ | $ieQ_{f}\gamma^{\mu}$
$H(p)\cdot V_{i}^{\mu}\cdot S_{j}(q)$ | $i\,g_{HV_{i}S_{j}}\,(p-q)^{\mu}$
$H(p)\cdot V_{i}^{0\,\mu}\cdot S^{0}_{j}(q)$ | $i\,g_{HV^{0}_{i}S^{0}_{j}}\,(p-q)^{\mu}$
$V^{0\;\mu}_{k}\cdot V_{i}^{\nu}\cdot S^{\pm}_{j}$ | $\pm i\,g_{V^{0}_{k}V_{i}S_{j}}\,g^{\mu\nu}$
$V^{0\;\mu}_{k}(p_{1})\cdot V_{i}^{\nu}(p_{2})\cdot V_{j}^{\lambda}(p_{3})$ | $-i\,g_{V^{0}_{k}V_{i}V_{j}}\,\Gamma_{\mu\nu\lambda}(p_{1},p_{2},p_{3})$
$A^{\mu}(p_{1})\cdot V_{i}^{\nu}(p_{2})\cdot V_{i}^{\lambda}(p_{3})$ | $-ieQ_{V}\;\Gamma^{\mu\nu\lambda}(p_{1},p_{2},p_{3})$
$V^{0\;\mu}_{k}\cdot f_{i}\cdot\bar{f_{j}}$ | $i\gamma^{\mu}\Big{(}g_{V^{0}_{k}f_{i}f_{j}}^{L}P_{L}+g_{V^{0}_{k}f_{i}f_{j}}^{R}P_{R}\Big{)}$
$V^{0\;\mu}_{k}\cdot A^{\nu}\cdot V_{i}^{\alpha}\cdot V_{j}^{\beta}$ | $-i\,eQ_{V}\;g_{V^{0}_{k}AV_{i}V_{j}}\,S_{\mu\nu,\alpha\beta}$
$V^{0\;\mu}_{k}\cdot A^{\nu}\cdot S_{i}\cdot S_{j}$ | $-i\,eQ_{S}\;g_{V^{0}_{k}AS_{i}S_{j}}\,g_{\mu\nu}$
$V^{0\;\mu}_{k}\cdot S_{i}(p)\cdot S_{j}(q)$ | $i\,g_{V^{0}_{k}S_{i}S_{j}}\,(p-q)^{\mu}$
$A^{\mu}\cdot S_{i}\cdot S_{i}$ | $ieQ_{S}\,(p-q)^{\mu}$
$V^{\mu}_{i}\cdot f\cdot\nu_{f}$ | $i\gamma^{\mu}\Big{(}g_{V_{i}f\nu_{f}}^{L}P_{L}+g_{V_{i}f\nu_{f}}^{R}P_{R}\Big{)}$
$V^{0\,\mu}_{i}\cdot f\cdot f$ | $i\gamma^{\mu}\Big{(}g_{V^{0}_{i}ff}^{L}P_{L}+g_{V^{0}_{i}ff}^{R}P_{R}\Big{)}$
$S_{i}\cdot\bar{f}\cdot\nu_{f}$ | $ig_{S_{i}f\nu_{f}}^{L}P_{L}+ig_{S_{i}f\nu_{f}}^{R}P_{R}$
$S^{*}_{i}\cdot f\cdot\bar{\nu}_{f}$ | $ig_{S_{i}f\nu_{f}}^{R}P_{L}+ig_{S_{i}f\nu_{f}}^{L}P_{R}$
Table 7: All couplings involving the decay processes in unitary gauge.

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