# One-loop contributions to the decay $H\rightarrow\nu_{l}\bar{\nu}_{l}\gamma$
in standard model revisited

Dzung Tri Tran University of Science Ho Chi Minh City, $227$ Nguyen Van Cu,
District $5$, Ho Chi Minh City, Vietnam Khiem Hong Phan
<EMAIL_ADDRESS>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

One-loop contributions to the decay $H\rightarrow\nu_{l}\bar{\nu}_{l}\gamma$
with $l=e,\mu,\tau$ within standard model framework are revisited in this
paper. We derive two representations for the form factors in this calculation.
As a result, the computations are not only checked numerically by verifying
the ultraviolet finiteness of the results but also confirming the ward
identity of the amplitude. We find that the results are good stability with
varying ultraviolet cutoff parameters as well as satisfy the ward identity. In
phenomenological results, all the physical results are examined with the
present input parameters. Especially, we study the partial decay widths for
the decay channels in both cases of the detected photon and invisible photon.
Differential decay widths are also generated as a function of energy of final
photon.

###### keywords:

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

##  1 Introduction

Today one of the main purposes of experimental program at the High Luminosity
Large Hadron Collider (HL-LHC) [1, 2] and Future Lepton Colliders [3] is to
measure precisely the properties of standard model-like (SM-like) Higgs boson.
In this program, searching for all decay modes of Higgs boson are priority
tasks and have to be probed as precisely as possible. Because the partial
decay widths of Higgs boson provide an important information for answering the
nature of Higgs sector (for our deeper understanding of the dynamics of
electroweak symmetry breaking). In all decay channels of Higgs boson,
$H\rightarrow$ invisible particles and $H\rightarrow\gamma$ plus invisible
particles [4, 5, 6, 7, 8, 9, 10] are also great of interest by following
aspects: (i) these processes can be probed at future colliders [4, 5, 6, 8,
10] for testing the standard model (SM) at modern energy regions; (ii) many
new heavy particles such as new extra gauge bosons, charged (and neutral)
scalar particles may exchange in the loop diagrams of the aforementioned decay
channels. Thus, the decay processes could provide a useful tool for testing
the SM and investigating possible new physics.

We know that the precise evaluations for the standard model background play a
crucial role in searching for new physics. It means that the theoretical
calculations for one-loop contributing to
$H\rightarrow\nu_{l}\bar{\nu}_{l}\gamma$ are necessary. In Ref. [11], one-loop
formulas for $H\rightarrow\nu_{l}\bar{\nu}_{l}\gamma$ within SM framework have
provided. In view of the importance of the decay channels we revisit the
calculation for the decay processes with extending the previous computation
and updating numerical predictions using the present values of input
parameters. The analytic results for one-loop form factors are expressed in
terms of Passarino-Veltman functions (called as PV-functions) which they can
be evaluated numerically by using package LoopTools [14]. In comparison with
the previous work, the evaluations in the present paper are extended by
following points. Firstly, two representations for the form factors are
derived in this calculation. As a result, the computations are not only
checked numerically by verifying the ultraviolet finiteness of the results but
also confirming the ward identity of the amplitude. We find that the results
are good stability with varying ultraviolet cutoff parameters as well as
follow the ward identity of the amplitude. Secondly, we study the partial
decay widths for the decay channels in both cases of the detected photon and
invisible photon using the present values of input parameters. Lastly, all the
PV-functions are also reduced to scalar one-, two-, three- and four-point
functions in this paper. In further, we point out that one uses the
generalised hypergeometric functions in [15, 17, 16] at general space-time
dimensions $d$ for scalar one-loop integrals, the form factors will be valid
in general $d$. Therefore, the form factors can be performed at higher-power
$\varepsilon$-expansion which may be taken into account in two-loop and
higher-loop contributing to these channels.

The layout of the paper is as follows: In section 2, detailed calculations for
one-loop contributions to $H\rightarrow\nu_{l}\bar{\nu}_{l}\gamma$ are
presented. Numerical test for the computations and physical results for the
decay processes are also shown in this sections. Conclusions and outlook are
devoted in section $3$. In appendices, reduction formulas for scalar PV-
functions are presented.

##  2 Calculation

Detailed calculations for one-loop contributions to
$H\rightarrow\nu_{l}\bar{\nu}_{l}\gamma$ are presented in this section. Two
different representations for one-loop form factors are derived. Numerical
tests for the computation and physical results for the decay processes are
then shown in the next subsections.

###  2.1 Analytic results

We devote detailed the calculation for the decay channels
$H(p)\rightarrow\nu_{l}(q_{1})\bar{\nu}_{l}(q_{2})\gamma(q_{3})$ in this
subsection. Within the standard model framework the decay processes consist of
$W$ bosons and fermions exchanging in one-loop triangle diagrams (seen Fig. 1
and Fig. 2 respectively) as well as $6$ one-loop box diagrams (shown in Fig.
3) in unitary gauge.

Figure 1:  $W$-bosons exchanging in the one-loop triangle diagrams of
$H\rightarrow\nu_{l}\bar{\nu_{l}}\gamma$ in unitary gauge.

Figure 2:  Fermion $f$ exchanging in the one-loop triangle diagrams of
$H\rightarrow\nu_{l}\bar{\nu_{l}}\gamma$ in unitary gauge.

Figure 3:  $W$-boson and lepton $l$ particles exchanging in the one-loop box
diagrams of $H\rightarrow\nu_{l}\bar{\nu_{l}}\gamma$ in unitary gauge.

Figure 4:  Lepton $l$ and $W$-boson particles exchanging in the one-loop box
diagrams of $H\rightarrow\nu_{l}\bar{\nu_{l}}\gamma$ in unitary gauge.

In general, one-loop amplitude of the decay channels can be divided as
follows:

$\displaystyle\mathcal{A}=\mathcal{A}^{(W)}_{\text{tri}}+\mathcal{A}^{(f)}_{\text{tri}}+\mathcal{A}^{(W)}_{\text{box}}+\mathcal{A}^{(l)}_{\text{box}}.$
(1)

In this formula, $\mathcal{A}^{(W)}_{\text{tri}}$ presents for the
contributions of one-loop triangle Feynman diagrams with $W$ boson internal
lines (shown in Fig. 1). The term $\mathcal{A}^{(f)}_{\text{tri}}$ is for the
amplitude of one-loop triangle Feynman diagrams with exchanging fermions in
the loop (seen in Fig. 2). In the same notations,
$\mathcal{A}^{(W)}_{\text{box}}(\mathcal{A}^{(l)}_{\text{box}})$ is
corresponding to the amplitude of one-loop box diagrams in Fig. 3 (and in Fig.
4 respectively).

For on-shell external photon, the ward identity is implied. As a result, we
have the following relation: $q_{3}^{\nu}\epsilon_{\nu}^{*}=0$ where
$q_{3}^{\nu}$, $\epsilon_{\nu}^{*}$ are corresponding to momentum and
polarization vector of photon. Kinematic invariant variables involving to this
decay process are concerned:

$\displaystyle p^{2}$ $\displaystyle=$ $\displaystyle
M_{H}^{2},\;q_{1}^{2}=q_{2}^{2}=q_{3}^{2}=0,\;q_{12}=(q_{1}+q_{2})^{2}=2q_{1}\cdot
q_{2},\;q_{13}=2q_{1}\cdot q_{3},\;q_{23}=2q_{2}\cdot q_{3}.$ (2)

With this definition, one confirms that $q_{12}+q_{13}+q_{23}=M_{H}^{2}$.

We first write down all Feynman amplitudes for the above diagrams. With the
help of Package-X [13], all Dirac traces and Lorentz contractions in $d$
dimensions are handled. Subsequently, the amplitudes are then expressed in
terms of tensor one-loop integrals. By following tensor reduction for one-loop
integrals in [12], all tensor one-loop integrals are written in terms of PV-
functions (scalar $PV$-coefficient functions are labled as
$A_{ijk\cdots},B_{ijk\cdots},C_{ijk\cdots}$ and $D_{ijk\cdots}$ for
$i,j,k\cdots=1,2,3$ as in Ref. [12]). In this paper, we also reduce the PV-
functions to scalar one-loop integrals (labeled as $A_{0},B_{0},C_{0}$ and
$D_{0}$ [12]). Latter, these scalar functions $A_{0},B_{0},C_{0}$ and $D_{0}$
can be evaluated numerically by using LoopTools. Furthermore, one can use
hypergeometric representations in [15, 17, 16] for these scalar functions in
general $d$.

####  2.1.1 First expressions for form factors

We first arrive at one-loop three-point Feynman diagrams with $W$ boson
internal lines (as shown in Fig. 1). The corresponding amplitude for the above
Feynman diagrams is decomposed into Lorentz invariant structure as follows:

$\displaystyle\mathcal{A}^{(W)}_{\text{tri}}$ $\displaystyle=$
$\displaystyle\Big{[}F_{1}^{(W)}q_{3}^{\mu}q^{\nu}+F_{2}^{(W)}g^{\mu\nu}\Big{]}\bar{u}(q_{1})\gamma^{\mu}\;P_{L}v(q_{2})\epsilon^{*}_{\nu}(q_{3}).$
(3)

Where we have used $P_{L}=\dfrac{1-\gamma_{5}}{2}$. Applying the tensor
reduction method in [12], all form factors in this formula are computed.
Analytic results for the form factors are expressed in terms of $B_{0}$ and
$C_{0}$ functions as follows:

$\displaystyle F_{1}^{(W)}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}}{2M_{W}^{3}s_{W}^{3}(d-2)(M_{H}^{2}-q_{12})^{2}}\dfrac{1}{q_{12}-M_{Z}^{2}+i\Gamma_{Z}M_{Z}}\times$
$\displaystyle\times\Bigg{\\{}\Big{[}M_{H}^{2}(2M_{W}^{2}-q_{12})+4M_{W}^{4}(d-1)-2M_{W}^{2}q_{12}\Big{]}\times$
$\displaystyle\times\Big{[}\Big{(}M_{H}^{2}(d-4)+(2-d)q_{12}\Big{)}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})+2q_{12}B_{0}(q_{12},M_{W}^{2},M_{W}^{2})\Big{]}$
$\displaystyle+4M_{W}^{2}(M_{H}^{2}-q_{12})\Big{[}2M_{W}^{2}(M_{H}^{2}-q_{12})(2d-5)-4M_{W}^{4}(d-1)$
$\displaystyle\hskip
71.13188pt+q_{12}^{2}(d-2)-M_{H}^{2}q_{12}(d-3)\Big{]}C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})\Bigg{\\}}.$

Here $s_{W}=\sin\theta_{W}$, $\theta_{W}$ is weak mixing angle and
$d=4-2\varepsilon$ is space-time dimension. In this paper, we also compute the
form factor $F_{2}^{(W)}$ which its analytical result is shown in the next
subsection. In general, $F_{2}^{(W)}$ may take different form with
$F_{1}^{(W)}$. Following the ward identity, one confirms that

$\displaystyle F_{2}^{(W)}=\dfrac{q_{12}-M_{H}^{2}}{2}F_{1}^{(W)}$ (5)

It can be observed the relation in Eq. (5) analytically by performing
$\varepsilon$-expansion for the form factors. Both the expressions for
$F_{1}^{(W)}$ and $F_{2}^{(W)}$ give same result (seen appendix $B$). We also
note that the relation can be also verified numerically in next subsections.

We now consider Feynman amplitude for one-loop triangle diagrams with fermion
internal lines. Following the same procedure, the amplitude is casted into the
form of

$\displaystyle\mathcal{A}^{(f)}_{\text{tri}}$ $\displaystyle=$
$\displaystyle\Big{[}F_{1}^{(f)}q_{3}^{\mu}q^{\nu}+F_{2}^{(f)}g^{\mu\nu}\Big{]}\bar{u}(q_{1})\gamma^{\mu}P_{L}v(q_{2})\epsilon^{*}_{\nu}(q_{3}).$
(6)

All form factors in this equation are evaluated by using the tensor reduction
method in [12]. The results read:

$\displaystyle F_{1}^{(f)}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}N_{C}^{f}Q_{f}m_{f}^{2}\Big{(}T_{3}^{f}-2Q_{f}s_{W}^{2}\Big{)}}{M_{W}c_{W}^{2}s_{W}^{3}(d-2)(M_{H}^{2}-q_{12})^{2}}\dfrac{1}{q_{12}-M_{Z}^{2}+i\Gamma_{Z}M_{Z}}\times$
$\displaystyle\times\Bigg{\\{}\Big{[}2M_{H}^{2}(d-4)-2q_{12}(d-2)\Big{]}B_{0}(M_{H}^{2},m_{f}^{2},m_{f}^{2})+4q_{12}B_{0}(q_{12},m_{f}^{2},m_{f}^{2})$
$\displaystyle+\Big{[}(d-2)(M_{H}^{2}-q_{12})^{2}-8m_{f}^{2}(M_{H}^{2}-q_{12})\Big{]}C_{0}(0,q_{12},M_{H}^{2},m_{f}^{2},m_{f}^{2},m_{f}^{2})\Bigg{\\}}.$

Here $N_{C}^{f}\;(Q_{f})$ is number of color and charge quantum number of
fermion $f$. While $T_{3f}$ is hypercharge of fermion $f$. The form factor
$F_{2}^{(f)}$ is shown in next subsection. It is important to note that all
fermions with non-zero masses in the loop are considered in present paper.

We turn our attention to one-loop four-point Feynman diagrams contributing to
these decay processes. We have one-loop box diagrams that Higgs couple
directly to $W$ bosons (as shown in Fig. 3). The corresponding amplitude is
expressed in Lorentz invariant structure as follows:

$\displaystyle\mathcal{A}^{(W)}_{\text{box}}$ $\displaystyle=$
$\displaystyle\bar{u}(q_{1})\Big{\\{}F_{3}^{(W)}q_{1}^{\nu}\not{q_{3}}+F_{4}^{(W)}q_{2}^{\nu}\not{q_{3}}+F_{5}^{(W)}\gamma^{\nu}\Big{\\}}P_{L}v(q_{2})\epsilon^{*}_{\nu}(q_{3}).$
(8)

Where $F_{5}^{(W)}=-F_{3}^{(W)}(q_{1}\cdot q_{3})-F_{4}^{(W)}(q_{2}\cdot
q_{3})$. Analytic results for all form factors are shown in terms of PV-
functions as follows:

$\displaystyle F_{3}^{(W)}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}}{2M_{W}^{5}s_{W}^{3}}\times$ (9)
$\displaystyle\times\Big{\\{}(M_{H}^{2}M_{W}^{2}+2M_{W}^{4})\Big{[}(C_{22}+C_{12}+C_{2})(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
106.69783pt+(C_{22}+C_{12}+C_{2})(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip 7.11317pt-
M_{W}^{2}m_{l}^{2}\Big{[}(C_{22}+C_{12}+C_{2})(0,0,q_{13},m_{l}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+(C_{22}+C_{12}+C_{2})(0,0,q_{13},m_{l}^{2},M_{W}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+2M_{W}^{4}m_{l}^{2}\Big{[}D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},m_{l}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},m_{l}^{2},m_{l}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+\Big{[}M_{W}^{2}m_{l}^{2}(M_{H}^{2}+2M_{W}^{2})+2(d-2)M_{W}^{6}\Big{]}\times$
$\displaystyle\hskip
7.11317pt\times\Big{[}(D_{33}+D_{23})(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},m_{l}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+(D_{33}+D_{23})(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},m_{l}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+(D_{33}+D_{23}+D_{13})(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+\Big{[}M_{W}^{2}m_{l}^{2}(M_{H}^{2}+2M_{W}^{2})+2(d-4)M_{W}^{6}\Big{]}\times$
$\displaystyle\hskip
7.11317pt\times\Big{[}D_{3}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},m_{l}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+D_{3}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+D_{3}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},m_{l}^{2},m_{l}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+4M_{W}^{4}\Big{[}C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip 56.9055pt-
M_{W}^{2}D_{2}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},m_{l}^{2},M_{W}^{2},M_{W}^{2})\Big{]}\Big{\\}}.$

Analytic formula for another form factor is shown

$\displaystyle F_{4}^{(W)}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}}{2M_{W}^{5}s_{W}^{3}}\times$
$\displaystyle\times\Big{\\{}(M_{H}^{2}M_{W}^{2}+2M_{W}^{4})\Big{[}(C_{22}+C_{12}+C_{2})(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
113.81102pt+(C_{22}+C_{12}+C_{2})(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip 7.11317pt-
M_{W}^{2}m_{l}^{2}\Big{[}(C_{22}+C_{12}+C_{2})(0,0,q_{23},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
56.9055pt+(C_{22}+C_{12}+C_{2})(0,0,q_{23},M_{W}^{2},M_{W}^{2},m_{l}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+2M_{W}^{4}m_{l}^{2}\Big{[}D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},m_{l}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},m_{l}^{2},m_{l}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+\Big{[}2M_{W}^{2}m_{l}^{2}(M_{H}^{2}+2M_{W}^{2})+4(d-2)M_{W}^{6}\Big{]}\times$
$\displaystyle\hskip
7.11317pt\times\Big{[}D_{23}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+D_{23}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},m_{l}^{2},m_{l}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+\Big{[}M_{W}^{2}m_{l}^{2}(M_{H}^{2}+2M_{W}^{2})+2(d-2)M_{W}^{6}\Big{]}\times$
$\displaystyle\hskip
7.11317pt\times\Big{[}(D_{33}+D_{23}+D_{13})(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},m_{l}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
28.45274pt+(D_{33}+D_{22}+D_{13}+D_{12})(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
28.45274pt+(D_{33}+D_{22}+D_{13}+D_{12})(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},m_{l}^{2},m_{l}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+\Big{[}M_{W}^{2}m_{l}^{2}(M_{H}^{2}+2M_{W}^{2})+2dM_{W}^{6}\Big{]}\times$
$\displaystyle\hskip
7.11317pt\times\Big{[}D_{3}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},m_{l}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+(D_{3}+D_{2})(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+(D_{3}+D_{2})(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},m_{l}^{2},m_{l}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+4M_{W}^{6}\Big{[}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+(D_{0}+D_{1})(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},m_{l}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+(D_{2}+D_{1}+D_{0})(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},m_{l}^{2},M_{W}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+4M_{W}^{4}C_{0}(0,q_{12},M_{H}^{2};M_{W}^{2},M_{W}^{2},M_{W}^{2})\Big{\\}}.$

We have different kinds of $D$-coefficient functions relating to tensor
reduction for box diagrams appear in all the above form factors. In principle,
tensor one-loop box integrals with rank $P=4$ appear in the amplitude of each
diagram. However, these tensor integrals are cancelled out. As a results, we
have up to $D_{ij}$ (for $i,j=1,\cdots 3$) functions contributing to the
aforementioned form factors.

We finally consider all one-loop box diagrams in Fig. 4 in which Higgs couples
directly to leptons. Amplitude for the diagrams are given as the same form in
Eq. (8):

$\displaystyle\mathcal{A}^{(l)}_{\text{box}}$ $\displaystyle=$
$\displaystyle\bar{u}(q_{1})\Big{\\{}F_{3}^{(l)}q_{1}^{\nu}\not{q_{3}}+F_{4}^{(l)}q_{2}^{\nu}\not{q_{3}}+F_{5}^{(l)}\gamma^{\nu}\Big{\\}}P_{L}v(q_{2})\epsilon^{*}_{\nu}(q_{3})$
(11)

with $F_{5}^{(l)}=-F_{3}^{(l)}(q_{1}\cdot q_{3})-F_{4}^{(l)}(q_{2}\cdot
q_{3})$. All related form factors read as follows:

$\displaystyle F_{3}^{(l)}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}}{2M_{W}^{3}s_{W}^{3}}m_{l}^{2}\times$ (12)
$\displaystyle\times\Big{\\{}(C_{22}+C_{12}+C_{2})(0,0,q_{13},M_{W}^{2},m_{l}^{2},m_{l}^{2})+(C_{22}+C_{12}+C_{2})(0,0,q_{13},M_{W}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
7.11317pt+\Big{[}(6-d)M_{W}^{2}-m_{l}^{2}\Big{]}D_{2}(0,0,0,M_{H}^{2};q_{12},q_{13};m_{l}^{2},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
7.11317pt+\Big{[}2m_{l}^{2}+2(d-2)M_{W}^{2}\Big{]}\Big{[}(D_{33}+D_{23})(0,0,0,M_{H}^{2};q_{12},q_{13};m_{l}^{2},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
113.81102pt+(D_{33}+D_{23})(0,0,0,M_{H}^{2};q_{23},q_{13};m_{l}^{2},M_{W}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
113.81102pt+(D_{33}+D_{23}+D_{13})(0,0,0,M_{H}^{2};q_{23},q_{12};m_{l}^{2},m_{l}^{2},M_{W}^{2},m_{l}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+\Big{[}(d+2)M_{W}^{2}+m_{l}^{2}\Big{]}\Big{[}D_{3}(0,0,0,M_{H}^{2};q_{12},q_{13};m_{l}^{2},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
113.81102pt+D_{3}(0,0,0,M_{H}^{2};q_{23},q_{12};m_{l}^{2},m_{l}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
113.81102pt+D_{3}(0,0,0,M_{H}^{2};q_{23},q_{13};m_{l}^{2},M_{W}^{2},M_{W}^{2},m_{l}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+2M_{W}^{2}\Big{[}D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};m_{l}^{2},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
113.81102pt+D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};m_{l}^{2},m_{l}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
113.81102pt+D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};m_{l}^{2},M_{W}^{2},M_{W}^{2},m_{l}^{2})\Big{]}\Big{\\}}$

and

$\displaystyle F_{4}^{(l)}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}}{2M_{W}^{3}s_{W}^{3}}m_{l}^{2}\times$ (13)
$\displaystyle\times\Big{\\{}(C_{22}+C_{12}+C_{2})(0,0,q_{23};m_{l}^{2},m_{l}^{2},M_{W}^{2})+(C_{22}+C_{12}+C_{2})(0,0,q_{23};m_{l}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
7.11317pt+\Big{[}4m_{l}^{2}+4(d-2)M_{W}^{2}\Big{]}\Big{[}D_{23}(0,0,0,M_{H}^{2};q_{23},q_{12};m_{l}^{2},m_{l}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
128.0374pt+D_{23}(0,0,0,M_{H}^{2};q_{23},q_{13};m_{l}^{2},M_{W}^{2},M_{W}^{2},m_{l}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+\Big{[}2m_{l}^{2}+2(d-2)M_{W}^{2}\Big{]}\Big{[}(D_{33}+D_{23}+D_{13})(0,0,0,M_{H}^{2};q_{12},q_{13};m_{l}^{2},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
85.35826pt+(D_{33}+D_{22}+D_{13}+D_{12})(0,0,0,M_{H}^{2};q_{23},q_{12};m_{l}^{2},m_{l}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
85.35826pt+(D_{33}+D_{22}+D_{13}+D_{12})(0,0,0,M_{H}^{2};q_{23},q_{13};m_{l}^{2},M_{W}^{2},M_{W}^{2},m_{l}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+\Big{[}m_{l}^{2}+(d-4)M_{W}^{2}\Big{]}\Big{[}D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};m_{l}^{2},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
113.81102pt+D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};m_{l}^{2},m_{l}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
113.81102pt+D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};m_{l}^{2},M_{W}^{2},M_{W}^{2},m_{l}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+\Big{[}3m_{l}^{2}+(3d-10)M_{W}^{2}\Big{]}\Big{[}D_{3}(0,0,0,M_{H}^{2};q_{12},q_{13};m_{l}^{2},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
113.81102pt+(D_{3}+D_{2})(0,0,0,M_{H}^{2};q_{23},q_{12};m_{l}^{2},m_{l}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
113.81102pt+(D_{3}+D_{2})(0,0,0,M_{H}^{2};q_{23},q_{13};m_{l}^{2},M_{W}^{2},M_{W}^{2},m_{l}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+\Big{[}m_{l}^{2}+(d-6)M_{W}^{2}\Big{]}\Big{[}D_{1}(0,0,0,M_{H}^{2};q_{23},q_{13};m_{l}^{2},M_{W}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
113.81102pt+(D_{2}+D_{1})(0,0,0,M_{H}^{2};q_{12},q_{13};m_{l}^{2},M_{W}^{2},m_{l}^{2},m_{l}^{2})\Big{]}\Big{\\}}.$

We find that this contribution is much smaller than other artributions because
it is proportional $m_{l}^{2}$. It is enough to take $\tau$-lepton for this
contribution.

####  2.1.2 Second expression for form factors

Another expression for all form factors appearing in this paper are also
presented in this subsection.

$\displaystyle F_{2}^{(W)}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}}{2M_{W}^{3}s_{W}^{3}(q_{12}-M_{Z}^{2}+i\Gamma_{Z}M_{Z})}\Bigg{\\{}\dfrac{q_{12}}{(d-1)}\Big{[}2M_{W}^{2}B_{0}(0,M_{W}^{2},M_{W}^{2})-(d-2)A_{0}(M_{W}^{2})\Big{]}$
$\displaystyle+\dfrac{q_{12}}{(d-2)(M_{H}^{2}-q_{12})}\Big{[}M_{H}^{2}(q_{12}-2M_{W}^{2})-4(d-1)M_{W}^{4}+2M_{W}^{2}q_{12}\Big{]}B_{0}(q_{12},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{2(2-d)(M_{H}^{2}-q_{12})}\Big{\\{}(d-4)M_{H}^{4}(2M_{W}^{2}-q_{12})-2M_{W}^{2}q_{12}(d-2)\Big{[}2M_{W}^{2}(d-1)-q_{12}\Big{]}$
$\displaystyle+M_{H}^{2}\Big{[}4M_{W}^{4}(d-1)(d-4)-4M_{W}^{2}q_{12}(d-3)+(d-2)q_{12}^{2}\Big{]}\Big{\\}}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{2M_{W}^{2}}{(d-2)}\Big{\\{}M_{H}^{2}\Big{[}(10-4d)M_{W}^{2}+(d-3)q_{12}\Big{]}$
$\displaystyle+4M_{W}^{4}(d-1)+2M_{W}^{2}q_{12}(2d-5)-(d-2)q_{12}^{2}\Big{\\}}C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})\Bigg{\\}}.$

For fermion particles exchanging in one-loop triangle Feynman diagrams, we
also have the following form factor:

$\displaystyle F_{2}^{(f)}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}\,N_{C}^{f}Q_{f}m_{f}^{2}\Big{(}T_{3}^{f}-2Q_{f}s_{W}^{2}\Big{)}}{2M_{W}c_{W}^{2}s_{W}^{3}(d-2)(M_{H}^{2}-q_{12})(q_{12}-M_{Z}^{2}+i\Gamma_{Z}M_{Z})}\times$
$\displaystyle\times\Bigg{\\{}\Big{[}4M_{H}^{2}-2(d-2)(M_{H}^{2}-q_{12})\Big{]}B_{0}(M_{H}^{2},m_{f}^{2},m_{f}^{2})-4q_{12}B_{0}(q_{12},m_{f}^{2},m_{f}^{2})$
$\displaystyle\hskip
14.22636pt+(M_{H}^{2}-q_{12})\Big{[}8m_{f}^{2}-(d-2)(M_{H}^{2}-q_{12})\Big{]}C_{0}(0,q_{12},M_{H}^{2},m_{f}^{2},m_{f}^{2},m_{f}^{2})\Bigg{\\}}.$

Form factor $F_{5}^{(W)}$ of the box diagrams in Fig. (3) is expressed:

$\displaystyle F_{5}^{(W)}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}}{4M_{W}^{5}s_{W}^{3}}\times$
$\displaystyle\times\Big{\\{}2M_{W}^{2}A_{0}(M_{W}^{2})-4M_{W}^{2}B_{00}(0,M_{W}^{2},M_{W}^{2})-M_{H}^{2}M_{W}^{2}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
7.11317pt+M_{W}^{2}(2M_{W}^{2}+m_{l}^{2})B_{0}(0,M_{W}^{2},m_{l}^{2})+M_{W}^{2}(m_{l}^{2}-4M_{W}^{2})B_{0}(q_{13},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip 7.11317pt-
M_{W}^{2}m_{l}^{2}q_{23}(C_{2}+C_{1}+C_{0})(0,0,q_{23},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
7.11317pt-2M_{W}^{2}m_{l}^{2}\Big{[}C_{00}(0,0,q_{13},m_{l}^{2},m_{l}^{2},M_{W}^{2})+C_{00}(0,0,q_{13},m_{l}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+C_{00}(0,0,q_{23},M_{W}^{2},m_{l}^{2},m_{l}^{2})+C_{00}(0,0,q_{23},M_{W}^{2},M_{W}^{2},m_{l}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+\Big{[}2M_{W}^{6}(1-d)-2M_{H}^{2}M_{W}^{4}-M_{H}^{2}m_{l}^{2}M_{W}^{2}\Big{]}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
7.11317pt+2M_{W}^{4}\Big{\\{}\Big{[}(q_{13}-3M_{H}^{2})C_{2}+(q_{13}-M_{H}^{2})C_{1}\Big{]}(0,q_{13},M_{H}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+2(q_{12}-M_{H}^{2})C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})\Big{\\}}$
$\displaystyle\hskip
7.11317pt+2M_{W}^{2}(2M_{W}^{2}+M_{H}^{2})\Big{[}C_{00}(0,q_{12},M_{H}^{2};M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
142.26378pt+C_{00}(q_{12},0,M_{H}^{2};M_{W}^{2},M_{W}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt-\Big{[}2M_{W}^{4}q_{13}(2M_{W}^{2}+m_{l}^{2})+2M_{W}^{4}q_{23}(4M_{W}^{2}+m_{l}^{2})\Big{]}\times$
$\displaystyle\hskip 149.37697pt\times
D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},m_{l}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
7.11317pt-\Big{[}2M_{W}^{4}q_{13}m_{l}^{2}+2M_{W}^{4}q_{23}(2M_{W}^{2}+m_{l}^{2})\Big{]}\times$
$\displaystyle\hskip 149.37697pt\times
D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
7.11317pt+\Big{\\{}M_{W}^{2}q_{23}\Big{[}M_{H}^{2}m_{l}^{2}+2M_{W}^{4}(d-6)\Big{]}-2M_{W}^{4}q_{13}m_{l}^{2}\Big{\\}}\times$
$\displaystyle\hskip 149.37697pt\times
D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},m_{l}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
7.11317pt+M_{W}^{2}q_{23}\Big{[}2M_{W}^{4}(d-6)+m_{l}^{2}(2M_{W}^{2}+M_{H}^{2})\Big{]}\times$
$\displaystyle\hskip
85.35826pt\times(D_{3}+D_{2}+D_{1})(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},m_{l}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
7.11317pt+\Big{[}4M_{W}^{6}(d-2)+2M_{W}^{2}m_{l}^{2}(M_{H}^{2}+2M_{W}^{2})\Big{]}\times$
$\displaystyle\hskip
56.9055pt\times\Big{[}D_{00}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},m_{l}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+D_{00}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+D_{00}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},m_{l}^{2},m_{l}^{2},M_{W}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt-8M_{W}^{6}q_{23}\Big{[}(D_{3}+D_{2}+D_{1})(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},m_{l}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
56.9055pt+(D_{3}+D_{2})(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},m_{l}^{2},M_{W}^{2})\Big{]}\Big{\\}}.$

Similarly, form factor $F_{5}^{(l)}$ of one-loop box diagrams in Fig. 4 is as
follows:

$\displaystyle F_{5}^{(l)}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}}{4M_{W}^{3}s_{W}^{3}}m_{l}^{2}\times$
$\displaystyle\times\Big{\\{}B_{0}(M_{H}^{2},m_{l}^{2},m_{l}^{2})-3B_{0}(q_{13},M_{W}^{2},m_{l}^{2})-q_{23}C_{2}(0,0,q_{23},m_{l}^{2},m_{l}^{2},M_{W}^{2})$
$\displaystyle\hskip
7.11317pt+\Big{[}(q_{13}-3M_{H}^{2})C_{2}+(q_{13}-M_{H}^{2})C_{1}\Big{]}(0,q_{13},M_{H}^{2},m_{l}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
7.11317pt+\Big{[}(5-2d)M_{W}^{2}-M_{H}^{2}-3m_{l}^{2}\Big{]}C_{0}(0,q_{13},M_{H}^{2},m_{l}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
7.11317pt+2C_{00}(0,0,q_{13},M_{W}^{2},m_{l}^{2},m_{l}^{2})+2C_{00}(0,0,q_{13},M_{W}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
7.11317pt+2C_{00}(0,0,q_{23},m_{l}^{2},m_{l}^{2},M_{W}^{2})+2C_{00}(0,0,q_{23},m_{l}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle\hskip
7.11317pt+\Big{[}4M_{W}^{2}(d-2)+4m_{l}^{2}\Big{]}\Big{[}D_{00}(0,0,0,M_{H}^{2};q_{12},q_{13};m_{l}^{2},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
120.92421pt+D_{00}(0,0,0,M_{H}^{2};q_{23},q_{12};m_{l}^{2},m_{l}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
120.92421pt+D_{00}(0,0,0,M_{H}^{2};q_{23},q_{13};m_{l}^{2},M_{W}^{2},M_{W}^{2},m_{l}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+q_{13}\Big{[}(4-d)M_{W}^{2}-m_{l}^{2}\Big{]}D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};m_{l}^{2},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
7.11317pt+q_{23}\Big{[}(10-2d)M_{W}^{2}-2m_{l}^{2}\Big{]}D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};m_{l}^{2},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
7.11317pt+\Big{[}(4-d)M_{W}^{2}q_{23}-m_{l}^{2}q_{23}-2M_{W}^{2}q_{13}\Big{]}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};m_{l}^{2},m_{l}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
7.11317pt+2q_{23}\Big{[}(6-d)M_{W}^{2}-m_{l}^{2}\Big{]}\times$
$\displaystyle\hskip
56.9055pt\times\Big{[}(D_{3}+D_{2}+D_{1})(0,0,0,M_{H}^{2};q_{12},q_{13};m_{l}^{2},M_{W}^{2},m_{l}^{2},m_{l}^{2})$
$\displaystyle\hskip
64.01869pt+(D_{3}+D_{2})(0,0,0,M_{H}^{2};q_{23},q_{12};m_{l}^{2},m_{l}^{2},M_{W}^{2},m_{l}^{2})\Big{]}$
$\displaystyle\hskip
7.11317pt+8M_{W}^{2}q_{23}(D_{3}+D_{2}+D_{1})(0,0,0,M_{H}^{2};q_{23},q_{13};m_{l}^{2},M_{W}^{2},M_{W}^{2},m_{l}^{2})$
$\displaystyle\hskip
7.11317pt+2M_{W}^{2}(3q_{23}-q_{13})D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};m_{l}^{2},M_{W}^{2},M_{W}^{2},m_{l}^{2})\Big{\\}}.$

Having the form factors, we are going to calculate the decay widths.
Differential decay width is derived in detail in Appendix $C$. Several decay
width formulas are presented in terms of the above form factors. In this
subsection, we present one of the differential decay width expression which
written in terms of $F_{1}$, $F_{3}$ and $F_{4}$. The formula is as follows:

$\displaystyle\dfrac{d^{2}\Gamma(H\rightarrow\nu_{l}\bar{\nu}_{l}\gamma)}{dq_{12}dq_{13}}$
$\displaystyle=$
$\displaystyle\dfrac{q_{12}}{512\pi^{3}M_{H}^{3}}\Big{\\{}\big{(}|F_{1}|^{2}+|F_{3}|^{2}+2\,\text{Re}[F_{1}F_{3}^{*}]\big{)}q_{13}^{2}$
$\displaystyle\hskip
99.58464pt+\big{(}|F_{1}|^{2}+|F_{4}|^{2}+2\,\text{Re}[F_{1}F_{4}^{*}]\big{)}q_{23}^{2}\Big{\\}}.$

Where

$\displaystyle F_{i}$ $\displaystyle=$
$\displaystyle\sum\limits_{f}F_{i}^{(f)}+F_{i}^{(W)},\quad
F_{j}=\sum\limits_{l=e,\mu,\tau}F_{j}^{(l)}+F_{j}^{(W)}$ (19)

for $i=1,2$ and $j=3,4,5$. Where the integration region is

$\displaystyle 0\leq q_{12}\leq M_{H}^{2},\quad 0\leq q_{13}\leq
M_{H}^{2}-q_{12}.$ (20)

Other forms for the differential decay width which expressed in terms of
$F_{2}$, $F_{3}$ (or $F_{4}$) and $F_{5}$ are shown in appendix $C$. It is
important to note that all form factors appearing in this paper are used for
computing the decay widths.

In appendix $B$, all reduction formulas for PV-functions expressing in terms
of scalar one-loop integrals $A_{0},B_{0},C_{0}$ and $D_{0}$ are derived in
space-time dimensions $d$. Analytical results for these scalar integrals are
well-known in $d=4-2\varepsilon$ at $\varepsilon^{0}$-expansion and they can
be evaluated numerically by using LoopTools [14].

We are going to end up the analytical calculation with interesting point for
future prospect. Recently, we have derived hypergeometric representations for
scalar one-loop in general space-time dimensions $d$ in Refs. [15, 17, 16]. By
expressing scalar one-loop integrals $A_{0},B_{0},C_{0}$ and $D_{0}$ in terms
of hypergeometric functions in Refs. [15, 17, 16], we support that all form
factors in this paper will be valid in general $d$. Subsequently, they can be
performed at higher-power of $\varepsilon$-expansion which these terms may be
taken into account in general framework for two-loop and higher-loop
contributing to the aforementioned channels.

###  2.2 Phenomenological results

All the physical results of the decay channels are examined with the present
input parameters in [18]. In detail, we use 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.

Before we are going to discuss phenomenological results for these processes.
First, numerical checks for the computations are performed. The results must
be independent of ultraviolet cutoff ($C_{UV}=1/\varepsilon$) and $\mu^{2}$
parameters ($\mu^{2}$ plays role of a renormalization scale [14]). We take the
form factors $F_{1}^{(W)}$, $F_{3}^{(W)}$ and $F_{4}^{(W)}$ as typical
examples. Numerical results are shown at arbitrary sampling point in physical
region (seen Tables 1, 2, 3).

$\textbf{Diagrams}/(C_{UV},\mu^{2})$ | $(0,1)$ | $(10^{5},10^{7})$
---|---|---
diag. 1 | $3.822254271421409\cdot 10^{-9}$ | $-0.000025848850122985293$
|  \+ $1.1328167325068205\cdot 10^{-10}\,i$ | $-7.660926734863565\cdot 10^{-7}\,i$
diag. 2 | $-4.284907193808991\cdot 10^{-9}$ | $0.00005170105984731962$
| $-1.2699350230775137\cdot 10^{-10}\,i$ |  \+ $1.5322849168169064\cdot 10^{-6}\,i$
$2\times$ diag. 1 $+$ diag. 2 | $3.3596013490338273\cdot 10^{-9}$ | $3.3596013490338273\cdot 10^{-9}$
|  \+ $9.956984419361279\cdot 10^{-11}\,i$ |  \+ $9.956984419361279\cdot 10^{-11}\,i$
Table 1:  Numerical checks for $F_{1}^{(W)}$ of triangle diagrams with exchanging $W$ bosons $\textbf{Diagrams}/(C_{UV},\mu^{2})$ | $(0,1)$ | $(10^{5},10^{7})$
---|---|---
diag. 1 | $-1.510623096811763\cdot 10^{-10}$ | $-1.510623096811763\cdot 10^{-10}$
| $-1.703281388817785\cdot 10^{-24}\,i$ | $-1.703281388817785\cdot 10^{-24}\,i$
diag. 2 | $6.682548276674295\cdot 10^{-10}$ | $6.682548276674295\cdot 10^{-10}$
|  \+ $6.582748337547062\cdot 10^{-10}\,i$ |  \+ $6.582748337547062\cdot 10^{-10}\,i$
diag. 3 | $-1.5802922737417574\cdot 10^{-10}$ | $-1.5802922737417574\cdot 10^{-10}$
|  \+ $1.680875383453354\cdot 10^{-10}\,i$ |  \+ $1.680875383453354\cdot 10^{-10}\,i$
Sum | $3.591632904379816\cdot 10^{-10}$ | $3.591632904379816\cdot 10^{-10}$
|  \+ $8.263623719173174\cdot 10^{-10}\,i$ |  \+ $8.263623719173174\cdot 10^{-10}\,i$
Table 2:  Numerical checks for $F_{3}^{(W)}$ box diagrams with exchanging $W$ bosons $\textbf{Diagrams}/(C_{UV},\mu^{2})$ | $(0,1)$ | $(10^{5},10^{7})$
---|---|---
diag. 1 | $-2.8890520180530153\cdot 10^{-10}$ | $-2.8890520180530153\cdot 10^{-10}$
|  \+ $3.09465553621708\cdot 10^{-25}\,i$ |  \+ $3.09465553621708\cdot 10^{-25}\,i$
diag. 2 | $5.932051711362282\cdot 10^{-10}$ | $5.932051711362282\cdot 10^{-10}$
|  \+ $9.380790178780355\cdot 10^{-10}\,i$ |  \+ $9.380790178780355\cdot 10^{-10}\,i$
diag. 3 | $7.503951368583933\cdot 10^{-11}$ | $7.503951368583933\cdot 10^{-11}$
|  \+ $3.476774032077318\cdot 10^{-10}\,i$ |  \+ $3.476774032077318\cdot 10^{-10}\,i$
Sum | $3.793394830167515\cdot 10^{-10}$ | $3.793394830167515\cdot 10^{-10}$
|  \+ $1.2857564210858306\cdot 10^{-9}\,i$ |  \+ $1.2857564210858306\cdot 10^{-9}\,i$
Table 3:  Numerical checks for $F_{4}^{(W)}$ box diagrams with exchanging $W$
bosons

After verifying the ultraviolet finiteness and $\mu^{2}$ independent of the
form factors, we perform a further test for the results which is to the ward
identity check for the amplitudes. As we mentioned in previous sections, two
representations for form factors are derived in this paper. Their relations
are tested numerically. Numerical results are shown at arbitrary sampling
point in physical region:

$\displaystyle F_{1}^{(W)}$ $\displaystyle=$
$\displaystyle\left(\dfrac{2}{q_{12}-M_{H}^{2}}\right)F_{2}^{(W)}=4.417241820666953\cdot
10^{-8}+5.8717660992932236\cdot 10^{-8}\,i,$ $\displaystyle F_{1}^{(\tau)}$
$\displaystyle=$
$\displaystyle\left(\dfrac{2}{q_{12}-M_{H}^{2}}\right)F_{2}^{(\tau)}=-2.906788516027605\cdot
10^{-12}-1.660128209352433\cdot 10^{-12}\,i,$ $\displaystyle F_{5}^{(W)}$
$\displaystyle=$ $\displaystyle-F_{3}^{(W)}(q_{1}\cdot
q_{3})-F_{4}^{(W)}(q_{2}\cdot q_{3})=$ (23) $\displaystyle\hskip
93.89418pt=-8.31386693767833\cdot 10^{-7}+2.076359788449464\cdot 10^{-20}\,i,$
$\displaystyle F_{5}^{(\tau)}$ $\displaystyle=$ $\displaystyle-
F_{3}^{(\tau)}(q_{1}\cdot q_{3})-F_{4}^{(\tau)}(q_{2}\cdot q_{3})=$ (24)
$\displaystyle\hskip 93.89418pt=9.43776046706387\cdot
10^{-9}-3.509321200912556\cdot 10^{-9}\,i.$

We find that the results are good stability when varying $C_{UV},\mu^{2}$
parameters as well as following the ward identity checks.

The computations are confirmed successfully by numerical checks. The
phenomenological results for the decay channels are studied. In present paper
both event of Higgs decay to invisible particles and a photon plus invisible
particles are computed. We first arrive at the case of photon that may be
tested or not. In this case, we do not apply any energy cut for final photon.
The partial decay width for $H\rightarrow\nu_{l}\bar{\nu}_{l}\gamma$ in which
$l$ can be one of $e,\mu$ and $\tau$ is presented:

$\displaystyle\Gamma_{H\rightarrow\gamma\nu_{l}\bar{\nu}_{l}}^{\text{tot}}$
$\displaystyle=$ $\displaystyle 0.471109(5)\,\text{KeV},$
$\displaystyle\Gamma_{H\rightarrow\gamma\nu_{l}\bar{\nu}_{l}}^{\text{Trig}}$
$\displaystyle=$ $\displaystyle 0.464686(1)\,\text{KeV},$
$\displaystyle\Gamma_{H\rightarrow\gamma\nu_{l}\bar{\nu}_{l}}^{\text{Trig}\times\text{W-box}}$
$\displaystyle=$ $\displaystyle 0.006064(4)\,\text{KeV},$
$\displaystyle\Gamma_{H\rightarrow\gamma\nu_{l}\bar{\nu}_{l}}^{\text{Trig}\times\text{lepton-
box}}$ $\displaystyle=$ $\displaystyle 0.000042(5)\,\text{KeV}.$

$\Gamma_{H\rightarrow\gamma\nu_{l}\bar{\nu}_{l}}^{\text{tot}}$ is for the case
of taking all Feynman diagrams. It gives a perfect agreement with the result
in [11]. $\Gamma_{H\rightarrow\gamma\nu_{l}\bar{\nu}_{l}}^{\text{Trig}}$ is
noted for the contributions of all triangle diagrams. This gives dominant
contributions in comparison with other parts. The contributions of the
interference between three-point diagrams and box diagrams
$\Gamma_{H\rightarrow\gamma\nu_{l}\bar{\nu}_{l}}^{\text{Trig}\times\text{W-box}}$
(box diagrams in Fig. 3),
$\Gamma_{H\rightarrow\gamma\nu_{l}\bar{\nu}_{l}}^{\text{Trig}\times\text{lepton-
box}}$ (for box diagrams in Fig. 4) are shown in these equations. These
contributions are much smaller than the results of triangle diagrams.

We also concern the case of photon that can be tested. In this case, we apply
energy cuts for final photon. The results are shown in Table 4.

$\Gamma$ [KeV]/$E_{\gamma}^{\text{cut}}$ [GeV] | $5$ | $30$ | $50$
---|---|---|---
$\Gamma_{H\rightarrow\gamma\nu_{l}\bar{\nu}_{l}}^{\text{Trig}}$ | $0.464684(7)$ | $0.171761(3)$ | $0.004632(7)$
$\Gamma_{H\rightarrow\gamma\nu_{l}\bar{\nu}_{l}}^{\text{Trig}\times\text{W-box}}$ | $0.006064(2)$ | $0.005928(6)$ | $0.001541(7)$
$\Gamma_{H\rightarrow\gamma\nu_{l}\bar{\nu}_{l}}^{\text{Trig}\times\text{lepton-box}}$ | $0.000042(4)$ | $0.000006(2)$ | $0.000000(1)$
$\Gamma_{H\rightarrow\gamma\nu_{l}\bar{\nu}_{l}}^{\text{tot}}$ | $0.471108(1)$ | $0.178727(1)$ | $0.006398(1)$
Table 4:  Decay widths in the case of photon that can be tested.

In this Fig. 5, differential decay widths in respective to $E_{\gamma}$ are
presented. The solid line denotes for contributing of all diagrams. The dashed
line shows for the contributions of triangle diagrams. While the dot-dashed
line presents for the interference between three-point diagrams and box
diagrams in Fig. 3. The dotted line is for the amplitude of the interference
between three-point diagrams and box diagrams in Fig. 4.

$\begin{array}[]{lr}\dfrac{d\Gamma}{dE_{\gamma}}&\\\
\includegraphics[width=426.79134pt,height=170.71652pt]{DR_Logscale.eps}\\\
&E_{\gamma}[GeV]\end{array}$

Figure 5:  Differential decay width as a function of $E_{\gamma}$

We find that the dominant contributions of triangle diagrams. We observe the
peak which is corresponding to the $Z$-pole that $Z$ boson decay to neutrinos.
The position of this peak is at
$E_{\gamma}=\frac{M_{H}^{2}-M_{Z}^{2}}{2M_{H}}=29.3159$ GeV. Beyond this peak,
the contributions of the interference between three-point diagrams and box
diagrams are visible. We note that all results present here are for a family
of neutrino in final state. For all neutrinos, we simply multiply factor $3$
for all above results. The physical results are important for testing standard
model at modern energy and evaluating precisely for the standard model
background.

Furthermore, it is stress that general results for one-loop contributing to
the mentioned decay processes in arbitrary beyond the standard model have
presented in [19].

##  3 Conclusions

One-loop contributions to the decay $H\rightarrow\nu_{l}\bar{\nu}_{l}\gamma$
with $l=e,\mu,\tau$ within standard model framework have revisited in this
paper. Two representations for the form factors have computed in this
calculation. As a result, the calculations are not only checked numerically by
verifying the ultraviolet finiteness of the results but also confirming the
ward identity of the amplitude. We find that the results satisfy the ward
identity and are good stability with varying ultraviolet cutoff parameters. In
phenomenological results, all the physical results have examined with the
present input parameters. In detail, we have studied the partial decay widths
for the decay channels in both cases of the detected photon and invisible
photon. Differential decay widths have also generated as a function of energy
of final photon. The physical results are important for testing standard model
at modern energy and evaluating precisely for the standard model background.
We have also discussed that one applies hypergeometric functions for scalar
one-loop integrals at general space-time dimensions $d$, the results in this
paper can be perfromed higher-power of $\varepsilon$-expansion that may be
taken into account in two-loop and higher-loop contributing to the mentioned
channels.

Acknowledgment: This research is funded by Vietnam National Foundation for
Science and Technology Development (NAFOSTED) under the grant number
$103.01$-$2019.346$.

##  Appendix A Reduction for scalar PV-functions

In this appendix, we present all reduction formulas for scalar PV-functions
expressing in terms of scalars one-loop integrals $A_{0},B_{0},C_{0}$ and
$D_{0}$. We first show reduction for all $A,B$-functions:

$\displaystyle A_{00}(M_{W}^{2})$ $\displaystyle=$
$\displaystyle\dfrac{M_{W}^{2}}{d}A_{0}(M_{W}^{2}),$ (25) $\displaystyle
B_{1}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$ $\displaystyle=$
$\displaystyle-\dfrac{1}{2}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2}),$ (26)
$\displaystyle B_{00}(p^{2},M_{W}^{2},M_{W}^{2})$ $\displaystyle=$
$\displaystyle\dfrac{1}{4(d-1)}\Big{\\{}2A_{0}(M_{W}^{2})-(p^{2}-4M_{W}^{2})B_{0}(p^{2},M_{W}^{2},M_{W}^{2})\Big{\\}},$
(27) $\displaystyle B_{11}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$ $\displaystyle=$
$\displaystyle=\dfrac{1}{4M_{H}^{2}(d-1)}\Big{\\{}(dM_{H}^{2}-4M_{W}^{2})B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})+2(d-2)A_{0}(M_{W}^{2})\Big{\\}},$
$\displaystyle B_{001}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$ $\displaystyle=$
$\displaystyle\dfrac{1}{8(d-1)}\Big{\\{}(M_{H}^{2}-4M_{W}^{2})B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})-2A_{0}(M_{W}^{2})\Big{\\}},$
(29) $\displaystyle B_{111}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$ $\displaystyle=$
(30)
$\displaystyle=\dfrac{1}{8M_{H}^{2}(1-d)}\Big{\\{}\Big{[}(d+2)M_{H}^{2}-12M_{W}^{2}\Big{]}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})+6(d-2)A_{0}(M_{W}^{2})\Big{\\}}.$

Here $p^{2}=M_{H}^{2},0$. We next have the following relations for
$C$-functions:

$\displaystyle C_{1}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle=$
$\displaystyle\dfrac{1}{(M_{H}^{2}-q_{13})^{2}}\Big{[}(M_{H}^{2}-q_{13})B_{0}(0,M_{W}^{2},0)$
$\displaystyle+(M_{H}^{2}+q_{13})B_{0}(q_{13},0,M_{W}^{2})-2M_{H}^{2}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+M_{H}^{2}(2M_{W}^{2}-M_{H}^{2}+q_{13})C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})\Big{]},$
$\displaystyle C_{2}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle=$
$\displaystyle\dfrac{1}{M_{H}^{2}-q_{13}}\Big{[}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})-B_{0}(q_{13},0,M_{W}^{2})$
(32) $\displaystyle-
M_{W}^{2}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})\Big{]},$
$\displaystyle C_{2}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle=$
$\displaystyle\dfrac{1}{(M_{H}^{2}-q_{12})^{2}}\Big{\\{}(q_{12}-M_{H}^{2})B_{0}(0,M_{W}^{2},M_{W}^{2})$
$\displaystyle+(M_{H}^{2}+q_{12})B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})-2q_{12}B_{0}(q_{12},M_{W}^{2},M_{W}^{2})$
$\displaystyle+q_{12}(M_{H}^{2}-q_{12})C_{0}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})\Big{\\}},$
$\displaystyle C_{00}(0,q_{12},M_{H}^{2},M^{2},M^{2},M^{2})$ $\displaystyle=$
$\displaystyle\dfrac{1}{2(d-2)(M_{H}^{2}-q_{12})}\times$
$\displaystyle\times\Big{\\{}M_{H}^{2}B_{0}(M_{H}^{2},M^{2},M^{2})-q_{12}B_{0}(q_{12},M^{2},M^{2})$
$\displaystyle+2M^{2}(M_{H}^{2}-q_{12})C_{0}(0,q_{12},M_{H}^{2},M^{2},M^{2},M^{2})\Big{\\}},$
$\displaystyle C_{00}(q_{12},0,M_{H}^{2};M^{2},M^{2},M^{2})$ $\displaystyle=$
$\displaystyle C_{00}(0,q_{12},M_{H}^{2};M^{2},M^{2},M^{2}),$ (34)
$\displaystyle C_{12}(0,q_{12},M_{H}^{2},M^{2},M^{2},M^{2})$ $\displaystyle=$
$\displaystyle\dfrac{1}{2(d-2)(M_{H}^{2}-q_{12})^{2}}\Big{\\{}-2M_{H}^{2}B_{0}(M_{H}^{2},M^{2},M^{2})$
$\displaystyle+\Big{[}(d-2)M_{H}^{2}-(d-4)q_{12}\Big{]}B_{0}(q_{12},M^{2},M^{2})$
$\displaystyle+4M^{2}(q_{12}-M_{H}^{2})C_{0}(0,q_{12},M_{H}^{2},M^{2},M^{2},M^{2})\Big{\\}},$
$\displaystyle C_{22}(0,q_{12},M_{H}^{2},M^{2},M^{2},M^{2})$ $\displaystyle=$
$\displaystyle\dfrac{1}{2(M_{H}^{2}-q_{12})}\Big{\\{}B_{0}(q_{12},M^{2},M^{2})-B_{0}(M_{H}^{2},M^{2},M^{2})\Big{\\}}.$

Where $M^{2}=M_{W}^{2},m_{f}^{2}$. Further, we also have

$\displaystyle C_{12}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle=$
$\displaystyle\dfrac{1}{(d-2)(M_{H}^{2}-q_{12})^{3}}\Big{\\{}(d-2)(M_{H}^{2}-q_{12})A_{0}(M_{W})$
$\displaystyle+(d-2)(M_{H}^{2}+M_{W}^{2})(M_{H}^{2}-q_{12})B_{0}(0,M_{W}^{2},M_{W}^{2})$
$\displaystyle+\Big{[}(3-2d)M_{H}^{2}q_{12}-M_{H}^{4}\Big{]}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\Big{[}(2d-3)M_{H}^{2}q_{12}+q_{12}^{2}\Big{]}B_{0}(q_{12},M_{W}^{2},M_{W}^{2})$
$\displaystyle+(q_{12}-M_{H}^{2})\Big{[}2M_{H}^{2}M_{W}^{2}+M_{H}^{2}q_{12}(d-2)+2M_{W}^{2}q_{12}\Big{]}$
$\displaystyle\times
C_{0}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})\Big{\\}},$
$\displaystyle C_{22}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle=$
$\displaystyle\dfrac{1}{2(d-2)(M_{H}^{2}-q_{12})^{3}}\Big{\\{}(4-2d)(M_{H}^{2}-q_{12})A_{0}(M_{W}^{2})$
$\displaystyle+(4-2d)(M_{H}^{2}-q_{12})(M_{W}^{2}+q_{12})B_{0}(0,M_{W}^{2},M_{W}^{2})$
$\displaystyle+4q_{12}^{2}(1-d)B_{0}(q_{12},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\Big{[}(2-d)(M_{H}^{4}-3q_{12}^{2})+2dM_{H}^{2}q_{12}\Big{]}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+2q_{12}(M_{H}^{2}-q_{12})\Big{[}4M_{W}^{2}+(d-2)q_{12}\Big{]}\times$
$\displaystyle\times
C_{0}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})\Big{\\}}.$

We finally devote for all relations for $D$-functions. In this paper, we
consider reduction formulas for all $D$-functions involving the decay
processes in the limit of $m_{l}\rightarrow 0$. In detail, the relations are
shown as follows:

$\displaystyle
D_{1}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2})$
$\displaystyle=$ $\displaystyle\dfrac{1}{2q_{12}q_{13}q_{23}}\times$
$\displaystyle\times\Big{\\{}q_{13}(M_{H}^{2}-q_{12})C_{0}(0,0,q_{13};0,M_{W}^{2},M_{W}^{2})+q_{12}(q_{13}-q_{23})C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\Big{[}q_{12}(q_{12}+q_{23})-M_{H}^{2}(q_{12}-q_{23})\Big{]}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle-(M_{H}^{2}-q_{12})^{2}C_{0}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+(M_{H}^{2}-q_{12})\Big{[}M_{H}^{2}(M_{W}^{2}-q_{12})+q_{12}(-M_{W}^{2}+q_{12}+q_{23})\Big{]}\times$
$\displaystyle\times
D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2})\Big{\\}},$
$\displaystyle D_{1}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})$
$\displaystyle=$ $\displaystyle\dfrac{1}{2q_{12}q_{13}q_{23}}\times$
$\displaystyle\times\Bigg{\\{}q_{13}(q_{12}+q_{13})C_{0}(0,0,q_{13},0,0,M_{W}^{2})+q_{23}(q_{13}-q_{12})C_{0}(0,0,q_{23},M_{W}^{2},0,0)$
$\displaystyle+(M_{H}^{2}q_{12}-q_{13}q_{23})C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle-(q_{12}+q_{13})^{2}C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle-\Big{\\{}M_{H}^{4}(q_{23}-M_{W}^{2})+M_{H}^{2}\Big{[}M_{W}^{2}(q_{12}+q_{23})-q_{23}(q_{12}+2q_{23})\Big{]}$
$\displaystyle+q_{23}\Big{[}M_{W}^{2}q_{12}+q_{23}(q_{12}+q_{23})\Big{]}\Big{\\}}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})\Bigg{\\}},$
$\displaystyle
D_{2}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2})$
$\displaystyle=$ $\displaystyle\dfrac{1}{2q_{12}q_{13}q_{23}}\times$
$\displaystyle\times\Big{\\{}q_{13}(q_{12}-q_{23})C_{0}(0,0,q_{13};0,M_{W}^{2},M_{W}^{2})-(q_{12}+q_{23})^{2}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\Big{[}q_{12}(q_{12}+q_{23})-M_{H}^{2}(q_{12}-q_{23})\Big{]}C_{0}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+q_{12}(q_{12}+q_{23})C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\Big{[}M_{H}^{2}M_{W}^{2}(q_{12}-q_{23})-M_{H}^{2}q_{12}(q_{12}+q_{23})+q_{12}(q_{12}+q_{23})(-M_{W}^{2}+q_{12}+q_{23})\Big{]}\times$
$\displaystyle\times
D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2})\Big{\\}},$
$\displaystyle D_{2}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})$
$\displaystyle=$ $\displaystyle\dfrac{1}{2q_{12}q_{13}q_{23}}\times$
$\displaystyle\times\Bigg{\\{}-(q_{12}+q_{23})^{2}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})-q_{13}(q_{12}-q_{23})C_{0}(0,0,q_{13},0,0,M_{W}^{2})$
$\displaystyle+(M_{H}^{2}q_{12}-q_{13}q_{23})C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})+q_{23}(q_{12}+q_{23})C_{0}(0,0,q_{23},M_{W}^{2},0,0)$
$\displaystyle+\Big{\\{}(q_{12}+q_{23})\Big{[}M_{W}^{2}q_{12}+q_{23}(q_{12}+q_{23})\Big{]}-M_{H}^{2}\Big{[}M_{W}^{2}(q_{12}-q_{23})+q_{23}(q_{12}+q_{23})\Big{]}\Big{\\}}\times$
$\displaystyle\times
D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})\Bigg{\\}},$
$\displaystyle
D_{2}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle=$ $\displaystyle-\dfrac{1}{2q_{12}q_{13}q_{23}}\times$
$\displaystyle\times\Bigg{\\{}q_{12}(q_{13}-q_{23})C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})+q_{23}(q_{12}-M_{H}^{2})C_{0}(0,0,q_{23},M_{W}^{2},M_{W}^{2},0)$
$\displaystyle+(M_{H}^{2}-q_{12})^{2}C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+(q_{12}q_{23}-M_{H}^{2}q_{13})C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+(q_{12}-M_{H}^{2})\Big{[}M_{H}^{2}M_{W}^{2}-q_{12}(M_{W}^{2}+q_{23})\Big{]}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2})\Bigg{\\}},$
$\displaystyle
D_{3}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2})$
$\displaystyle=$ $\displaystyle\dfrac{1}{2q_{13}q_{23}}\times$
$\displaystyle\times\Big{\\{}(q_{12}+q_{23})C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})-q_{13}C_{0}(0,0,q_{13},0,M_{W}^{2},M_{W}^{2})$
$\displaystyle+(q_{13}-q_{23})C_{0}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})-q_{12}C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\Big{[}M_{W}^{2}(q_{12}+2q_{23})-M_{H}^{2}(M_{W}^{2}-q_{12})-2q_{12}(q_{12}+q_{23})\Big{]}\times$
$\displaystyle\times
D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2})\Big{\\}},$
$\displaystyle
D_{3}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle=$ $\displaystyle-\dfrac{1}{2q_{12}q_{13}}\times$
$\displaystyle\times\Bigg{\\{}(q_{13}-q_{12})C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})+q_{12}C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+q_{23}C_{0}(0,0,q_{23},M_{W}^{2},M_{W}^{2},0)-(M_{H}^{2}-q_{12})C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\Big{[}M_{W}^{2}(M_{H}^{2}-q_{12})-q_{12}q_{23}\Big{]}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2})\Bigg{\\}},$
$\displaystyle D_{3}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})$
$\displaystyle=$ $\displaystyle-\dfrac{1}{2q_{12}q_{13}}\times$
$\displaystyle\times\Big{\\{}q_{13}C_{0}(0,0,q_{13},0,0,M_{W}^{2})-(q_{12}+q_{23})C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+(q_{12}-q_{13})C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})+q_{23}C_{0}(0,0,q_{23},M_{W}^{2},0,0)$
$\displaystyle+\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{23}q_{13}\Big{]}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})\Big{\\}}.$

For one-loop tensor four-point with rank $P=2$, we have

$\displaystyle
D_{00}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2})$
$\displaystyle=$ $\displaystyle\dfrac{1}{4(3-d)q_{12}q_{13}q_{23}}\times$
$\displaystyle\times\Bigg{\\{}\Big{[}M_{W}^{2}q_{12}(q_{13}-q_{23})-M_{W}^{2}q_{23}(q_{13}+q_{23})-q_{12}q_{13}(q_{12}+q_{23})\Big{]}$
$\displaystyle\times C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+(q_{13}+q_{23})\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{12}q_{13}\Big{]}C_{0}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+q_{12}\Big{[}M_{W}^{2}(q_{23}-q_{13})+q_{12}q_{13}\Big{]}C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+q_{13}\Big{[}q_{12}q_{13}-M_{W}^{2}(q_{13}+q_{23})\Big{]}C_{0}(0,0,q_{13},0,M_{W}^{2},M_{W}^{2})$
$\displaystyle-\Big{[}M_{W}^{4}(q_{13}+q_{23})^{2}+2M_{W}^{2}q_{12}q_{13}(q_{23}-q_{13})+q_{12}^{2}q_{13}^{2}\Big{]}\times$
$\displaystyle\times
D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2})\Bigg{\\}},$
$\displaystyle
D_{00}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle=$ $\displaystyle\dfrac{1}{4(3-d)q_{12}q_{13}q_{23}}\times$
$\displaystyle\times\Big{\\{}(q_{13}+q_{23})\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{12}q_{23}\Big{]}C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle-\Big{[}M_{W}^{2}q_{12}(q_{13}-q_{23})+M_{W}^{2}q_{13}(q_{13}+q_{23})+q_{12}q_{23}(q_{12}+q_{13})\Big{]}$
$\displaystyle\times C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+q_{12}\Big{[}M_{W}^{2}(q_{13}-q_{23})+q_{12}q_{23}\Big{]}C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+q_{23}\Big{[}q_{12}q_{23}-M_{W}^{2}(q_{13}+q_{23})\Big{]}C_{0}(0,0,q_{23},M_{W}^{2},M_{W}^{2},0)$
$\displaystyle-\Big{[}M_{W}^{4}(q_{13}+q_{23})^{2}+2M_{W}^{2}q_{12}q_{23}(q_{13}-q_{23})+q_{12}^{2}q_{23}^{2}\Big{]}\times$
$\displaystyle\times
D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2})\Big{\\}},$
$\displaystyle D_{00}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})$
$\displaystyle=$ $\displaystyle\dfrac{1}{4(3-d)q_{12}q_{13}q_{23}}\times$
$\displaystyle\times\Bigg{\\{}\Big{[}M_{W}^{2}q_{12}(q_{23}-q_{13})+M_{W}^{2}q_{23}(q_{13}+q_{23})-q_{13}q_{23}(q_{12}+q_{23})\Big{]}\times$
$\displaystyle\times C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\Big{\\{}M_{W}^{2}\Big{[}q_{12}(q_{13}-q_{23})+q_{13}(q_{13}+q_{23})\Big{]}-q_{13}q_{23}(q_{12}+q_{13})\Big{\\}}\times$
$\displaystyle\times C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+q_{13}\Big{[}q_{13}q_{23}-M_{W}^{2}(q_{13}+q_{23})\Big{]}C_{0}(0,0,q_{13},0,0,M_{W}^{2})$
$\displaystyle+q_{23}\Big{[}q_{13}q_{23}-M_{W}^{2}(q_{13}+q_{23})\Big{]}C_{0}(0,0,q_{23},M_{W}^{2},0,0)$
$\displaystyle-\Big{[}q_{13}q_{23}-M_{W}^{2}(q_{13}+q_{23})\Big{]}^{2}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})\Big{\\}},$
$\displaystyle D_{12}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})$
$\displaystyle=$
$\displaystyle\dfrac{M_{H}^{2}}{q_{12}q_{13}q_{23}}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle-\dfrac{(q_{12}+q_{13})}{q_{12}q_{13}q_{23}}B_{0}(q_{13},0,M_{W}^{2})-\dfrac{(q_{12}+q_{23})}{q_{12}q_{13}q_{23}}B_{0}(q_{23},M_{W}^{2},0)$
$\displaystyle+\dfrac{1}{4(3-d)q_{12}^{2}q_{13}q_{23}^{2}}\Big{\\{}M_{W}^{2}q_{12}^{2}\Big{[}(d-2)q_{13}+(10-3d)q_{23}\Big{]}$
$\displaystyle+M_{W}^{2}q_{12}\Big{[}(d-2)(q_{13}^{2}+q_{23}^{2})-2q_{13}q_{23}(d-4)\Big{]}-(d-2)M_{W}^{2}q_{13}q_{23}(q_{13}+q_{23})$
$\displaystyle+q_{13}q_{23}\Big{[}(d-4)M_{H}^{2}q_{12}+(d-2)q_{13}q_{23}\Big{]}\Big{\\}}C_{0}(0,0,q_{13},0,0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(3-d)q_{12}^{2}q_{13}^{2}q_{23}}\Big{\\{}M_{W}^{2}q_{12}^{2}\Big{[}(d-2)q_{23}+(10-3d)q_{13}\Big{]}$
$\displaystyle+M_{W}^{2}q_{12}\Big{[}(d-2)(q_{13}^{2}+q_{23}^{2})-2q_{13}q_{23}(d-4)\Big{]}-(d-2)M_{W}^{2}q_{13}q_{23}(q_{13}+q_{23})$
$\displaystyle+q_{13}q_{23}\Big{[}(d-4)M_{H}^{2}q_{12}+(d-2)q_{13}q_{23}\Big{]}\Big{\\}}C_{0}(0,0,q_{23},M_{W}^{2},0,0)$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}^{2}}\Big{\\{}q_{13}q_{23}(q_{12}+q_{23})\Big{[}(d-4)M_{H}^{2}q_{12}+(d-2)q_{13}q_{23}\Big{]}$
$\displaystyle-
M_{W}^{2}q_{12}^{3}(q_{13}-q_{23})(d-2)-M_{W}^{2}q_{12}^{2}\Big{[}(d-2)(q_{13}^{2}-2q_{23}^{2})+(3d-10)q_{13}q_{23}\Big{]}$
$\displaystyle-
M_{W}^{2}q_{12}q_{23}\Big{[}2q_{13}^{2}(d-4)+(3d-10)q_{13}q_{23}-(d-2)q_{23}^{2}\Big{]}$
$\displaystyle-
M_{W}^{2}q_{13}q_{23}^{2}(q_{13}+q_{23})(d-2)\Big{\\}}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}^{2}}\Big{\\{}(d-2)M_{W}^{2}q_{12}q_{13}(q_{12}+q_{13})^{2}$
$\displaystyle-
M_{H}^{2}M_{W}^{2}q_{23}\Big{[}(d-2)(q_{12}^{2}+q_{13}^{2})+2q_{12}q_{13}(d-4)\Big{]}$
$\displaystyle+q_{13}q_{23}(q_{12}+q_{13})\Big{[}(d-4)M_{H}^{2}q_{12}+(d-2)q_{13}q_{23}\Big{]}\Big{\\}}C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}^{2}}\Big{\\{}(d-2)M_{W}^{4}q_{13}q_{23}(q_{13}+q_{23})^{2}$
$\displaystyle-
M_{H}^{2}M_{W}^{4}q_{12}\Big{[}(d-2)(q_{13}^{2}+q_{23}^{2})-2q_{13}q_{23}(d-4)\Big{]}$
$\displaystyle+2M_{W}^{2}q_{13}q_{23}\Big{[}q_{12}^{2}(q_{13}+q_{23})+q_{12}\Big{(}q_{13}^{2}-2(d-4)q_{13}q_{23}+q_{23}^{2}\Big{)}$
$\displaystyle-(d-2)q_{13}q_{23}(q_{13}+q_{23})\Big{]}+q_{13}^{2}q_{23}^{2}\Big{[}(d-4)M_{H}^{2}q_{12}+(d-2)q_{13}q_{23}\Big{]}\Big{\\}}\times$
$\displaystyle\times
D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2}),$ $\displaystyle
D_{12}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle=$
$\displaystyle=\dfrac{1}{q_{12}q_{23}}B_{0}(0,M_{W}^{2},0)-\dfrac{(q_{13}+q_{23})}{q_{12}q_{13}q_{23}}B_{0}(q_{23},M_{W}^{2},0)$
$\displaystyle+\dfrac{M_{H}^{2}}{q_{12}q_{13}q_{23}}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})-\dfrac{(q_{12}+q_{13})}{q_{12}q_{13}q_{23}}B_{0}(q_{12},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}q_{13}^{2}q_{23}^{2}}\Big{\\{}q_{12}M_{W}^{2}\Big{[}(d-2)(q_{13}^{2}+q_{23}^{2})+2q_{13}q_{23}(d-4)\Big{]}$
$\displaystyle-q^{2}_{12}q_{23}\Big{[}(d-4)q_{13}+(d-2)q_{23}\Big{]}-(d-4)q_{12}q_{23}q_{13}(q_{13}+q_{23})$
$\displaystyle+q_{13}M_{W}^{2}\Big{[}(d-2)(q_{13}^{2}-q_{23}^{2})+4q_{13}q_{23}(d-3)\Big{]}\Big{\\}}C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(3-d)q_{12}^{2}q_{13}^{2}q_{23}}\Big{\\{}(d-2)M_{W}^{2}(q_{13}+q_{23})\Big{[}q_{12}(q_{13}-q_{23})+q_{13}(q_{13}+q_{23})\Big{]}$
$\displaystyle+q^{2}_{12}q_{23}\Big{[}(d-4)q_{13}+(d-2)q_{23}\Big{]}+(d-4)q_{12}q_{13}q_{23}(q_{13}+q_{23})\Big{\\}}\times$
$\displaystyle\times C_{0}(0,0,q_{23},M_{W}^{2},M_{W}^{2},0)$
$\displaystyle+\dfrac{(q_{13}+q_{23})}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}^{2}}\Bigg{\\{}(d-2)M_{W}^{2}(q_{13}+q_{23})\Big{[}q_{12}(q_{13}-q_{23})+q_{13}(q_{13}+q_{23})\Big{]}$
$\displaystyle+q_{12}q_{23}\Big{[}(d-4)q_{13}+(d-2)q_{23}\Big{]}+(d-4)q_{13}q_{12}q_{23}(q_{13}+q_{23})\Bigg{\\}}$
$\displaystyle\times C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}^{2}}\Bigg{\\{}q_{12}q_{23}(q_{12}+q_{13})\Bigg{[}q_{12}\Big{\\{}(d-4)q_{13}+(d-2)q_{23}\Big{]}$
$\displaystyle+(d-4)q_{13}(q_{13}+q_{23})\Big{]}$ $\displaystyle-
M_{W}^{2}\Big{\\{}q_{12}^{2}\Big{[}(d-2)(q_{13}^{2}+q_{23}^{2})+2(d-4)q_{13}q_{23}\Big{]}+(d-2)q_{13}^{2}(q_{13}+q_{23})^{2}$
$\displaystyle+2q_{12}q_{13}(q_{13}+q_{23})\Big{[}(d-2)q_{13}+(d-4)q_{23}\Big{]}\Big{\\}}\Bigg{\\}}C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}^{2}}\Bigg{\\{}(2-d)M_{W}^{4}(q_{13}+q_{23})^{2}\Big{[}q_{12}(q_{13}-q_{23})+q_{13}(q_{13}+q_{23})\Big{]}$
$\displaystyle-2M_{W}^{2}q_{12}q_{23}\Big{\\{}q_{12}\Big{[}q_{13}^{2}+(d-5)q_{13}q_{23}+(d-2)q_{23}^{2}\Big{]}+q_{13}(q_{13}^{2}-q_{23}^{2})\Big{\\}}$
$\displaystyle+q_{12}^{2}q_{23}^{2}\Big{\\{}q_{12}\Big{[}(d-4)q_{13}+(d-2)q_{23}\Big{]}+(d-4)q_{13}(q_{13}+q_{23})\Big{\\}}\;\Bigg{\\}}\times$
$\displaystyle\times
D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2}),$
$\displaystyle D_{13}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})$
$\displaystyle=$
$\displaystyle-\dfrac{M_{H}^{2}}{q_{12}q_{13}(q_{12}+q_{23})}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{q_{12}(q_{12}+q_{23})}B_{0}(q_{13},0,M_{W}^{2})+\dfrac{1}{q_{12}q_{13}}B_{0}(q_{23},M_{W}^{2},0)+$
$\displaystyle\dfrac{1}{4(d-3)q_{12}^{2}q_{13}q_{23}}\Big{\\{}q_{13}q_{23}\Big{[}(d-4)q_{12}+(d-2)q_{13}\Big{]}$
$\displaystyle-
M_{W}^{2}q_{12}\Big{[}(d-4)q_{13}-(d-2)q_{23}\Big{]}-M_{W}^{2}q_{13}(q_{13}+q_{23})(d-2)\Big{\\}}C_{0}(0,0,q_{13},0,0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}}\Big{\\{}q_{13}q_{23}\Big{[}(d-4)q_{12}+(d-2)q_{13}\Big{]}-M_{W}^{2}q_{12}\Big{[}(d-4)q_{13}-(d-2)q_{23}\Big{]}$
$\displaystyle-
M_{W}^{2}q_{13}(q_{13}+q_{23})(d-2)\Big{\\}}C_{0}(0,0,q_{23},M_{W}^{2},0,0)$
$\displaystyle+\dfrac{1}{4(3-d)q_{12}^{2}q_{13}^{2}q_{23}(q_{12}+q_{23})}\Big{\\{}M_{W}^{2}q_{12}^{3}\Big{[}(d-4)q_{13}+(d-2)q_{23}\Big{]}$
$\displaystyle+(d-2)M_{W}^{2}q_{12}^{2}\Big{[}q_{13}^{2}-q_{13}q_{23}+2q_{23}^{2}\Big{]}-(d-2)M_{W}^{2}q_{13}q_{23}^{2}(q_{13}+q_{23})$
$\displaystyle+M_{W}^{2}q_{12}q_{23}\Big{[}4q_{13}^{2}(3-d)+(8-3d)q_{13}q_{23}+(d-2)q_{23}^{2}\Big{]}$
$\displaystyle+q_{13}q_{23}(q_{12}+q_{23})^{2}\Big{[}(d-4)q_{12}+(d-2)q_{13}\Big{]}\Big{\\}}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}}\Big{\\{}M_{W}^{2}q_{12}^{2}\Big{[}(d-4)q_{13}+(d-2)q_{23}\Big{]}$
$\displaystyle+2M_{W}^{2}q_{12}q_{13}\Big{[}(d-3)q_{13}+(d-4)q_{23}\Big{]}+(d-2)M_{W}^{2}q_{13}^{2}(q_{13}+q_{23})$
$\displaystyle-
q_{13}q_{23}(q_{12}+q_{13})\Big{[}(d-4)q_{12}+(d-2)q_{13}\Big{]}\Big{\\}}C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{q_{13}q_{23}-M_{W}^{2}(q_{13}+q_{23})}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}}\Big{\\{}M_{W}^{2}q_{12}\Big{[}(d-4)q_{13}-(d-2)q_{23}\Big{]}$
$\displaystyle+(d-2)M_{W}^{2}q_{13}(q_{13}+q_{23})-q_{13}q_{23}\Big{[}(d-4)q_{12}+(d-2)q_{13}\Big{]}\Big{\\}}$
$\displaystyle\times
D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2}),$ $\displaystyle
D_{13}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle=$
$\displaystyle-\dfrac{M_{H}^{2}}{q_{12}q_{13}(q_{13}+q_{23})}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{q_{13}(q_{13}+q_{23})}B_{0}(q_{12},M_{W}^{2},M_{W}^{2})+\dfrac{1}{q_{12}q_{13}}B_{0}(q_{23},M_{W}^{2},0)$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}q_{13}^{2}q_{23}}\Big{\\{}q_{12}q_{23}\Big{[}(d-2)q_{12}+(d-4)q_{13}\Big{]}-M_{W}^{2}q_{12}\Big{[}(d-4)q_{13}+(d-2)q_{23}\Big{]}$
$\displaystyle-(d-2)M_{W}^{2}q_{13}(q_{13}-q_{23})\Big{\\}}C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}}\Big{\\{}M_{W}^{2}q_{12}\Big{[}(d-4)q_{13}-(d-2)q_{23}\Big{]}+(d-2)M_{W}^{2}q_{13}(q_{13}+q_{23})$
$\displaystyle+q_{12}q_{23}\Big{[}(d-2)q_{12}+(d-4)q_{13}\Big{]}\Big{\\}}C_{0}(0,0,q_{23},M_{W}^{2},M_{W}^{2},0)$
$\displaystyle+\dfrac{q_{13}+q_{23}}{4(3-d)q_{12}^{2}q_{13}^{2}q_{23}}\Big{\\{}M_{W}^{2}q_{12}\Big{[}(d-4)q_{13}-(d-2)q_{23}\Big{]}+(d-2)M_{W}^{2}q_{13}(q_{13}+q_{23})$
$\displaystyle+q_{12}q_{23}\Big{[}(d-2)q_{12}+(d-4)q_{13}\Big{]}\Big{\\}}C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}}\Big{\\{}M_{W}^{2}q_{12}^{2}\Big{[}(d-4)q_{13}+(d-2)q_{23}\Big{]}$
$\displaystyle+2M_{W}^{2}q_{12}q_{13}\Big{[}(d-3)q_{13}+(d-4)q_{23}\Big{]}-q_{12}q_{23}(q_{12}+q_{13})\Big{[}(d-2)q_{12}+(d-4)q_{13}\Big{]}$
$\displaystyle+(d-2)M_{W}^{2}q_{13}^{2}(q_{13}+q_{23})\Big{\\}}C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}},\Bigg{\\{}M_{W}^{4}(q_{13}+q_{23})\Big{\\{}q_{12}\Big{[}(d-4)q_{13}-(d-2)q_{23}\Big{]}$
$\displaystyle+(d-2)q_{13}(q_{13}+q_{23})\Big{\\}}-2M_{W}^{2}q_{12}q_{23}\Big{\\{}q_{12}\Big{[}q_{13}-(d-2)q_{23}\Big{]}+q_{13}(q_{23}-q_{13})\Big{\\}}$
$\displaystyle-
q_{12}^{2}q_{23}^{2}\Big{[}(d-2)q_{12}+(d-4)q_{13}\Big{]}\Bigg{\\}}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2}),$
$\displaystyle
D_{13}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2})$
$\displaystyle=$
$\displaystyle-\dfrac{M_{H}^{2}}{q_{13}q_{23}(q_{12}+q_{23})}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{q_{13}q_{23}}B_{0}(q_{12},M_{W}^{2},M_{W}^{2})+\dfrac{1}{q_{23}(q_{12}+q_{23})}B_{0}(q_{13},0,M_{W}^{2})+$
$\displaystyle\dfrac{1}{4(d-3)q_{13}^{2}q_{23}^{2}}\Big{\\{}(d-2)\Big{[}q_{13}^{2}(q_{12}-M_{W}^{2})-M_{W}^{2}q_{23}^{2}\Big{]}$
$\displaystyle+(d-4)q_{13}q_{23}(q_{12}-2M_{W}^{2})\Big{\\}}C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}q_{13}q_{23}^{2}}\Big{\\{}q_{12}q_{13}\Big{[}(d-2)q_{13}+(d-4)q_{23}\Big{]}$
$\displaystyle-(d-2)M_{W}^{2}(q_{13}^{2}-q_{23}^{2})\Big{\\}}C_{0}(0,0,q_{13},0,M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}q_{13}^{2}q_{23}^{2}(q_{12}+q_{23})}\Big{\\{}M_{W}^{2}\Big{[}q_{12}^{2}[(d-2)(q_{13}^{2}+q_{23}^{2})+2q_{13}q_{23}(d-4)]$
$\displaystyle+2q_{12}q_{23}[2q_{13}^{2}(d-3)+(d-4)q_{13}q_{23}+(d-2)q_{23}^{2}]+(d-2)q_{23}^{2}(q_{23}^{2}-q_{13}^{2})\Big{]}$
$\displaystyle-
q_{12}q_{13}(q_{12}+q_{23})^{2}\Big{[}(d-2)q_{13}+(d-4)q_{23}\Big{]}\Big{\\}}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}q_{13}^{2}q_{23}^{2}}\Big{\\{}(q_{13}+q_{23})\Big{[}(d-2)M_{W}^{2}(q_{13}^{2}-q_{23}^{2})$
$\displaystyle-
q_{12}q_{13}[(d-2)q_{13}+(d-4)q_{23}]\Big{]}\Big{\\}}C_{0}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(3-d)q_{12}q_{13}^{2}q_{23}^{2}}\Big{\\{}(d-2)M_{W}^{4}(q_{13}-q_{23})(q_{13}+q_{23})^{2}$
$\displaystyle-2M_{W}^{2}q_{12}q_{13}\Big{[}(d-2)q_{13}^{2}+(d-5)q_{13}q_{23}+q_{23}^{2}\Big{]}$
$\displaystyle+q_{12}^{2}q_{13}^{2}\Big{[}(d-2)q_{13}+(d-4)q_{23}\Big{]}\Big{\\}}D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2}),$
$\displaystyle
D_{22}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle=$
$\displaystyle=\dfrac{1}{q_{12}(q_{12}+q_{13})}B_{0}(0,0,M_{W}^{2})-\dfrac{1}{q_{12}q_{23}}B_{0}(0,M_{W}^{2},0)+\dfrac{(q_{13}-q_{23})}{q_{12}q_{13}q_{23}}B_{0}(q_{12},M_{W}^{2},M_{W}^{2})$
$\displaystyle-\dfrac{1}{q_{12}q_{13}q_{23}(q_{12}+q_{13})^{2}}\times$
$\displaystyle\times\Big{\\{}q_{12}^{2}(q_{13}-q_{23})+q_{12}\Big{[}2q_{13}^{2}+q_{13}q_{23}-q_{23}^{2}\Big{]}+q_{13}(q_{13}+q_{23})^{2}\Big{\\}}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{q_{12}q_{13}q_{23}(q_{12}+q_{13})^{2}}\times$
$\displaystyle\times\Big{\\{}q_{12}^{2}q_{13}+q_{12}\Big{[}2q_{13}^{2}+2q_{13}q_{23}-q_{23}^{2}\Big{]}+q_{13}(q_{13}+q_{23})^{2}\Big{\\}}B_{0}(q_{23},M_{W}^{2},0)$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}q_{13}^{2}q_{23}^{2}}\Big{\\{}q_{12}q_{23}\Big{[}(3d-10)q_{13}^{2}-(d-2)q_{23}(2q_{13}-q_{23})\Big{]}$
$\displaystyle-
M_{W}^{2}(q_{13}-q_{23})\Big{[}(d-2)(q_{13}^{2}+q_{23}^{2})+2q_{13}q_{23}(3d-8)\Big{]}\Big{\\}}C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{(d-2)(q_{13}+q_{23})^{2}}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}}\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{12}q_{23}\Big{]}\times$
$\displaystyle\times\Big{[}C_{0}(0,0,q_{23},M_{W}^{2},M_{W}^{2},0)-C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})\Big{]}$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}^{2}(q_{12}+q_{13})^{2}}\times$
$\displaystyle\times\Bigg{\\{}M_{W}^{2}\Big{[}q_{12}(q_{13}-q_{23})+q_{13}(q_{13}+q_{23})\Big{]}\Big{\\{}q_{12}^{2}\Big{[}(d-2)(q_{13}^{2}+q_{23}^{2})+2q_{13}q_{23}(3d-8)\Big{]}$
$\displaystyle+2q_{12}q_{13}(q_{13}+q_{23})\Big{[}(d-2)q_{13}+(3d-8)q_{23}\Big{]}+(d-2)q_{13}^{2}(q_{13}+q_{23})^{2}\Big{\\}}$
$\displaystyle-
q_{12}q_{23}(q_{12}+q_{13})\Big{\\{}q_{12}^{2}\Big{[}(3d-10)q_{13}^{2}-(d-2)q_{23}(2q_{13}-q_{23})\Big{]}+(3d-10)q_{13}^{2}(q_{13}+q_{23})^{2}$
$\displaystyle+2q_{12}q_{13}(q_{13}+q_{23})\Big{[}(3d-10)q_{13}-(d-2)q_{23}\Big{]}\Big{\\}}\Bigg{\\}}\;C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{(q_{13}+q_{23})^{2}}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}^{2}}\Big{\\{}(d-2)\Big{[}M_{W}^{4}(q_{13}+q_{23})^{2}+q_{12}^{2}q_{23}^{2}\Big{]}$
$\displaystyle-2M_{W}^{2}q_{12}q_{23}\Big{[}(d-4)q_{13}+(d-2)q_{23}\Big{]}\Big{\\}}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2}),$
$\displaystyle D_{22}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})$
$\displaystyle=$
$\displaystyle=\dfrac{1}{q_{13}(q_{12}+q_{13})}B_{0}(0,0,M_{W}^{2})+\dfrac{(q_{12}-q_{23})}{q_{12}q_{13}q_{23}}B_{0}(q_{13},0,M_{W}^{2})$
$\displaystyle+\dfrac{M_{H}^{2}(q_{13}q_{23}-M_{H}^{2}q_{12})}{q_{12}q_{13}q_{23}(q_{12}+q_{13})^{2}}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})+\dfrac{(M_{H}^{4}q_{12}-q_{13}q_{23}^{2})}{q_{12}q_{13}q_{23}(q_{12}+q_{13})^{2}}B_{0}(q_{23},M_{W}^{2},0)$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}q_{23}^{2}}\Big{\\{}M_{W}^{2}q_{12}^{2}\Big{[}(d-2)q_{13}+(10-3d)q_{23}\Big{]}$
$\displaystyle+2M_{W}^{2}q_{12}q_{23}\Big{[}(d-2)q_{23}-(d-4)q_{13}\Big{]}+q_{13}q_{23}\Big{[}(3d-10)q_{12}^{2}-(d-2)q_{23}(2q_{12}+q_{23})\Big{]}$
$\displaystyle+(d-2)M_{W}^{2}q_{23}^{2}(q_{13}+q_{23})\Big{\\}}C_{0}(0,0,q_{13},0,0,M_{W}^{2})$
$\displaystyle+\dfrac{(q_{12}+q_{23})}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}}\Big{\\{}M_{W}^{2}q_{12}\Big{[}(10-3d)q_{13}+(d-2)q_{23}\Big{]}$
$\displaystyle+(d-2)q_{23}\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{13}(q_{12}+q_{23})\Big{]}\Big{\\}}C_{0}(0,0,q_{23},M_{W}^{2},0,0)$
$\displaystyle+\dfrac{(d-2)(q_{12}+q_{23})^{2}}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}^{2}}\Big{\\{}M_{W}^{2}\Big{[}q_{12}(q_{13}-q_{23})-q_{23}(q_{13}+q_{23})\Big{]}$
$\displaystyle+q_{13}q_{23}(q_{12}+q_{23})\Big{\\}}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(3-d)q_{12}^{2}q_{13}^{2}q_{23}^{2}(q_{12}+q_{13})^{2}}\times$
$\displaystyle\Bigg{\\{}M_{W}^{2}q_{23}^{3}(q_{13}-q_{12})\Big{[}(d-2)(q_{12}^{2}+q_{13}^{2})+2q_{12}q_{13}(3d-8)\Big{]}$
$\displaystyle-
M_{W}^{2}q_{12}q_{23}(q_{12}+q_{13})^{2}\Big{[}(d-2)q_{12}^{2}+(5d-18)q_{12}q_{13}+2q_{13}^{2}(d-4)\Big{]}$
$\displaystyle-
M_{W}^{2}q_{23}^{2}(q_{12}+q_{13})\Big{[}(d-2)(2q_{12}^{3}-4q_{12}q_{13}^{2}-q_{13}^{3})+(11d-34)q_{12}^{2}q_{13}\Big{]}$
$\displaystyle+q_{13}q_{23}^{3}(q_{12}+q_{13})\Big{[}(3d-10)q_{12}^{2}-(d-2)q_{13}(2q_{12}-q_{13})\Big{]}$
$\displaystyle+2q_{12}q_{13}q_{23}^{2}(q_{12}+q_{13})^{2}\Big{[}(3d-10)q_{12}-(d-2)q_{13}\Big{]}+(d-2)M_{W}^{2}q_{12}^{2}q_{13}(q_{12}+q_{13})^{3}$
$\displaystyle+q_{12}^{2}q_{13}q_{23}(q_{12}+q_{13})^{3}(3d-10)\Bigg{\\}}C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}^{2}}\Big{\\{}M_{W}^{4}q_{12}^{2}\Big{[}(d-2)(q_{13}^{2}+q_{23}^{2})-2q_{13}q_{23}(d-4)\Big{]}$
$\displaystyle+2M_{W}^{4}q_{12}q_{23}(q_{13}+q_{23})\Big{[}(d-2)q_{23}-(d-4)q_{13}\Big{]}$
$\displaystyle+2M_{W}^{2}q_{13}q_{23}(q_{12}+q_{23})\Big{[}(d-4)q_{12}q_{13}-M_{H}^{2}q_{23}(d-2)\Big{]}$
$\displaystyle+(d-2)q_{23}^{2}\Big{[}M_{W}^{4}(q_{13}+q_{23})^{2}+q_{13}^{2}(q_{12}+q_{23})^{2}\Big{]}\Big{\\}}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2}),$
$\displaystyle
D_{23}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2})$
$\displaystyle=$
$\displaystyle=-\dfrac{1}{q_{13}(q_{13}+q_{23})}B_{0}(0,M_{W}^{2},M_{W}^{2})+\dfrac{1}{q_{13}q_{23}}B_{0}(q_{13},0,M_{W}^{2})$
$\displaystyle+\dfrac{(q_{23}-q_{13})M_{H}^{2}}{q_{13}q_{23}(q_{13}+q_{23})^{2}}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})+\dfrac{q_{12}(q_{13}-q_{23})-q_{23}(q_{13}+q_{23})}{q_{13}q_{23}(q_{13}+q_{23})^{2}}B_{0}(q_{12},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{13}^{2}q_{23}^{2}}\Big{\\{}M_{W}^{2}q_{23}q_{13}(d-4)$
$\displaystyle+(d-2)\Big{[}M_{W}^{2}q_{23}^{2}-M_{W}^{2}q_{12}(q_{13}-q_{23})+q_{12}q_{13}(q_{12}+q_{23})\Big{]}\Big{\\}}C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}q_{13}q_{23}^{2}}\Big{\\{}M_{W}^{2}q_{12}\Big{[}(2-d)q_{13}+(3d-10)q_{23}\Big{]}$
$\displaystyle+M_{W}^{2}q_{23}\Big{[}(d-4)q_{13}-(d-2)q_{23}\Big{]}+q_{12}q_{13}(q_{12}+q_{23})(d-2)\Big{\\}}C_{0}(0,0,q_{13},0,M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{(q_{12}+q_{23})}{4(3-d)q_{12}q_{13}^{2}q_{23}^{2}}\Big{\\{}M_{W}^{2}q_{23}\Big{[}(d-2)q_{23}-(d-4)q_{13}\Big{]}$
$\displaystyle+(2-d)\Big{[}M_{W}^{2}q_{12}(q_{13}-q_{23})-q_{12}q_{13}(q_{12}+q_{23})\Big{]}\Big{\\}}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}q_{13}^{2}q_{23}^{2}(q_{13}+q_{23})}\Big{\\{}M_{W}^{2}(q_{13}+q_{23})\Big{[}(d-2)[q_{12}(q_{13}^{2}+q_{23}^{2})+q_{23}^{3}]$
$\displaystyle-(d-4)q_{13}q_{23}(2q_{12}+q_{13})+2q_{13}q_{23}^{2}\Big{]}-q_{12}q_{13}\Big{[}q_{12}[(d-2)q_{13}(q_{13}+2q_{23})+(10-3d)q_{23}^{2}]$
$\displaystyle+q_{23}(q_{13}+q_{23})[(d-2)q_{13}+(10-3d)q_{23}]\Big{]}\Big{\\}}C_{0}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(3-d)q_{12}q_{13}^{2}q_{23}^{2}}\Big{\\{}M_{W}^{4}\Big{[}(d-2)q_{12}q_{13}^{2}+q_{23}^{2}[(d-2)q_{12}+2q_{13}]$
$\displaystyle-(d-4)q_{13}q_{23}(2q_{12}+q_{13})+(d-2)q_{23}^{3}\Big{]}$
$\displaystyle-2M_{W}^{2}q_{12}q_{13}\Big{[}(d-2)q_{12}(q_{13}-q_{23})+q_{23}[q_{13}+(2-d)q_{23}]\Big{]}$
$\displaystyle+(d-2)q_{12}^{2}q_{13}^{2}(q_{12}+q_{23})\Big{\\}}D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2}),$
$\displaystyle D_{23}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})$
$\displaystyle=$
$\displaystyle-\dfrac{1}{q_{13}(q_{12}+q_{13})}B_{0}(0,0,M_{W}^{2})+\dfrac{1}{q_{12}q_{13}}B_{0}(q_{13},0,M_{W}^{2})$
$\displaystyle+\dfrac{M_{H}^{2}(q_{12}-q_{13})}{q_{12}q_{13}(q_{12}+q_{13})^{2}}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})+\dfrac{(q_{13}q_{23}-M_{H}^{2}q_{12})}{q_{12}q_{13}(q_{12}+q_{13})^{2}}B_{0}(q_{23},M_{W}^{2},0)$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}q_{23}}\Big{\\{}M_{W}^{2}q_{12}\Big{[}(d-4)q_{13}-(d-2)q_{23}\Big{]}$
$\displaystyle-(d-2)q_{23}\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{13}(q_{12}+q_{23})\Big{]}\Big{\\}}C_{0}(0,0,q_{13},0,0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}}\Big{\\{}M_{W}^{2}q_{12}\Big{[}(d-4)q_{13}-(d-2)q_{23}\Big{]}$
$\displaystyle-(d-2)q_{23}\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{13}(q_{12}+q_{23})\Big{]}\Big{\\}}C_{0}(0,0,q_{23},M_{W}^{2},0,0)$
$\displaystyle+\dfrac{(q_{12}+q_{23})}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}}\Big{\\{}M_{W}^{2}q_{12}\Big{[}(d-4)q_{13}+(d-2)q_{23}\Big{]}$
$\displaystyle+(d-2)q_{23}\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{13}(q_{12}+q_{23})\Big{]}\Big{\\}}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(3-d)q_{12}^{2}q_{13}^{2}q_{23}(q_{12}+q_{13})^{2}}\Big{\\{}(d-4)M_{W}^{2}q_{12}q_{13}(q_{12}+q_{13})^{3}$
$\displaystyle+M_{W}^{2}q_{23}^{2}(q_{12}-q_{13})\Big{[}(d-2)(q_{12}^{2}+q_{13}^{2})+2q_{12}q_{13}(3d-8)\Big{]}$
$\displaystyle+M_{W}^{2}q_{23}(q_{12}+q_{13})\Big{[}(d-2)(q_{12}^{3}-3q_{12}q_{13}^{2}-q_{13}^{3})+(7d-22)q_{12}^{2}q_{13}\Big{]}$
$\displaystyle+q_{13}q_{23}^{2}(q_{12}+q_{13})\Big{[}(10-3d)q_{12}^{2}+(d-2)q_{13}(2q_{12}+q_{13})\Big{]}$
$\displaystyle+q_{12}q_{13}q_{23}(q_{12}+q_{13})^{2}\Big{[}(10-3d)q_{12}+(d-2)q_{13}\Big{]}\Big{\\}}C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{q_{13}q_{23}-M_{W}^{2}(q_{13}+q_{23})}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}}\Big{\\{}M_{W}^{2}q_{12}\Big{[}(d-2)q_{23}-(d-4)q_{13}\Big{]}$
$\displaystyle+(d-2)q_{23}\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{13}(q_{12}+q_{23})\Big{]}\Big{\\}}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2}),$
$\displaystyle
D_{23}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle=$
$\displaystyle-\dfrac{1}{q_{12}(q_{12}+q_{13})}B_{0}(0,0,M_{W}^{2})+\dfrac{1}{q_{12}q_{13}}B_{0}(q_{12},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{M_{H}^{2}(q_{13}-q_{12})}{q_{12}q_{13}(q_{12}+q_{13})^{2}}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})-\dfrac{q_{12}(q_{13}-q_{23})+q_{13}(q_{13}+q_{23})}{q_{12}q_{13}(q_{12}+q_{13})^{2}}B_{0}(q_{23},M_{W}^{2},0)$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}q_{13}^{2}q_{23}}\Big{\\{}M_{W}^{2}\Big{[}(d-2)(q_{13}^{2}-q_{23}^{2})-4q_{13}q_{23}(d-3)\Big{]}$
$\displaystyle+(d-2)q_{12}q_{23}(q_{13}+q_{23})\Big{\\}}C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{(d-2)(q_{13}+q_{23})}{4(3-d)q_{12}^{2}q_{13}^{2}}\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{12}q_{23}\Big{]}C_{0}(0,0,q_{23},M_{W}^{2},M_{W}^{2},0)$
$\displaystyle+\dfrac{(d-2)(q_{13}+q_{23})^{2}}{4(d-3)q_{12}^{2}q_{13}^{2}q_{23}}\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{12}q_{23}\Big{]}C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(3-d)q_{12}^{2}q_{13}^{2}q_{23}(q_{12}+q_{13})^{2}}\Bigg{\\{}M_{W}^{2}q_{12}^{3}\Big{[}(d-2)(q_{13}^{2}-q_{23}^{2})-4q_{13}q_{23}(d-3)\Big{]}$
$\displaystyle+(d-2)M_{W}^{2}q_{13}^{3}(q_{13}+q_{23})^{2}+M_{W}^{2}q_{12}^{2}q_{13}\Big{[}(d-2)q_{13}(3q_{13}+2q_{23})+(14-5d)q_{23}^{2}\Big{]}$
$\displaystyle+M_{W}^{2}q_{12}q_{13}^{2}(q_{13}+q_{23})\Big{[}3q_{13}(d-2)+(5d-14)q_{23}\Big{]}$
$\displaystyle+q_{12}q_{23}(q_{12}+q_{13})\Big{[}(d-2)q_{12}^{2}(q_{13}+q_{23})+2q_{12}q_{13}\Big{[}(d-2)q_{23}-(d-4)q_{13}\Big{]}$
$\displaystyle-(3d-10)q_{13}^{2}(q_{13}+q_{23})\Big{]}\Bigg{\\}}C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{(q_{13}+q_{23})}{4(3-d)q_{12}^{2}q_{13}^{2}q_{23}}\Big{\\{}(d-2)M_{W}^{4}(q_{13}+q_{23})^{2}-2M_{W}^{2}q_{12}q_{23}\Big{[}(d-4)q_{13}+(d-2)q_{23}\Big{]}$
$\displaystyle+(d-2)q_{12}^{2}q_{23}^{2}\Big{\\}}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2}),$
$\displaystyle
D_{33}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle=$
$\displaystyle=\dfrac{1}{q_{12}(q_{12}+q_{13})}B_{0}(0,0,M_{W}^{2})-\dfrac{1}{q_{12}q_{13}}B_{0}(q_{12},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{(M_{H}^{2}q_{12}-q_{13}q_{23})}{q_{12}q_{13}(q_{12}+q_{13})^{2}}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})-\dfrac{q_{23}(q_{12}-q_{13})}{q_{12}q_{13}(q_{12}+q_{13})^{2}}B_{0}(q_{23},M_{W}^{2},0)$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}q_{13}^{2}}\Big{\\{}M_{W}^{2}\Big{[}(3d-10)q_{13}+(d-2)q_{23}\Big{]}$
$\displaystyle-(d-2)q_{12}q_{23}\Big{\\}}C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{(d-2)q_{23}}{4(d-3)q_{12}^{2}q_{13}^{2}}\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{12}q_{23}\Big{]}C_{0}(0,0,q_{23},M_{W}^{2},M_{W}^{2},0)$
$\displaystyle+\dfrac{(d-2)(q_{13}+q_{23})}{4(3-d)q_{12}^{2}q_{13}^{2}}\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{12}q_{23}\Big{]}C_{0}(0,q_{12},M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}(q_{12}+q_{13})^{2}}\Bigg{\\{}M_{W}^{2}q_{12}^{3}\Big{[}(10-3d)q_{13}-(d-2)q_{23}\Big{]}$
$\displaystyle+M_{W}^{2}q_{12}^{2}q_{13}\Big{[}(14-5d)q_{23}-(d-6)q_{13}\Big{]}$
$\displaystyle+M_{W}^{2}q_{12}q_{13}^{2}\Big{[}3q_{13}(d-2)+(5d-14)q_{23}\Big{]}+(d-2)M_{W}^{2}q_{13}^{3}(q_{13}+q_{23})$
$\displaystyle+q_{12}q_{23}(q_{12}+q_{13})\Big{[}(d-2)q_{12}(q_{12}+2q_{13})+(10-3d)q_{13}^{2}\Big{]}\Bigg{\\}}C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}}\Big{\\{}(d-2)\Big{[}M_{W}^{4}(q_{13}+q_{23})^{2}+q_{12}^{2}q_{23}^{2}\Big{]}$
$\displaystyle-2M_{W}^{2}q_{12}q_{23}\Big{[}(d-4)q_{13}+(d-2)q_{23}\Big{]}\Big{\\}}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{12};M_{W}^{2},M_{W}^{2},0,M_{W}^{2}),$
$\displaystyle
D_{33}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2})$
$\displaystyle=$
$\displaystyle=\dfrac{1}{q_{13}(q_{13}+q_{23})}B_{0}(0,M_{W}^{2},M_{W}^{2})-\dfrac{1}{q_{13}q_{23}}B_{0}(q_{13},0,M_{W}^{2})$
$\displaystyle+\dfrac{q_{12}(q_{13}-q_{23})+q_{13}(q_{13}+q_{23})}{q_{13}q_{23}(q_{13}+q_{23})^{2}}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})+\dfrac{q_{12}(q_{23}-q_{13})}{q_{13}q_{23}(q_{13}+q_{23})^{2}}B_{0}(q_{12},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{q_{12}(2-d)[M_{W}^{2}(q_{23}-q_{13})+q_{12}q_{13}]}{4(d-3)q_{13}^{2}q_{23}^{2}}C_{0}(0,0,q_{12},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{q_{13}(M_{W}^{2}-q_{12})(d-2)+M_{W}^{2}q_{23}(10-3d)}{4(d-3)q_{13}q_{23}^{2}}C_{0}(0,0,q_{13},0,M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{13}^{2}q_{23}^{2}}\Big{\\{}M_{W}^{2}q_{13}q_{23}(10-3d)$
$\displaystyle+(d-2)\Big{[}M_{W}^{2}q_{23}^{2}-M_{W}^{2}q_{12}(q_{13}-q_{23})+q_{12}q_{13}(q_{12}+q_{23})\Big{]}\Big{\\}}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{13}^{2}q_{23}^{2}(q_{13}+q_{23})}\Big{\\{}q_{12}q_{13}\Big{[}q_{13}(q_{13}+2q_{23})(d-2)+(10-3d)q_{23}^{2}\Big{]}$
$\displaystyle-
M_{W}^{2}(q_{13}+q_{23})\Big{[}(d-2)(q_{13}^{2}+q_{23}^{2})-2(d-4)q_{13}q_{23}\Big{]}\Big{\\}}C_{0}(q_{12},0,M_{H}^{2},M_{W}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{13}^{2}q_{23}^{2}}\Big{\\{}(d-2)\Big{[}M_{W}^{4}q_{23}^{2}+q_{13}^{2}(M_{W}^{2}-q_{12})^{2}\Big{]}$
$\displaystyle+2M_{W}^{2}q_{13}q_{23}\Big{[}(d-2)q_{12}-(d-4)M_{W}^{2}\Big{]}\Big{\\}}D_{0}(0,0,0,M_{H}^{2};q_{12},q_{13};M_{W}^{2},0,M_{W}^{2},M_{W}^{2}),$
$\displaystyle D_{33}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2})$
$\displaystyle=$
$\displaystyle=\dfrac{1}{q_{13}(q_{12}+q_{13})}B_{0}(0,0,M_{W}^{2})-\dfrac{1}{q_{12}q_{13}}B_{0}(q_{13},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{q_{12}q_{13}(q_{12}+q_{13})^{2}}\Big{[}q_{12}(q_{13}-q_{23})+q_{13}(q_{13}+q_{23})\Big{]}B_{0}(M_{H}^{2},M_{W}^{2},M_{W}^{2})$
$\displaystyle+\dfrac{q_{23}(q_{12}-q_{13})}{q_{12}q_{13}(q_{12}+q_{13})^{2}}B_{0}(q_{23},M_{W}^{2},0)$
$\displaystyle+\dfrac{(d-2)}{4(d-3)q_{12}^{2}q_{13}}\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{13}q_{23}\Big{]}C_{0}(0,0,q_{13},0,0,M_{W}^{2})$
$\displaystyle+\dfrac{(d-2)q_{23}}{4(d-3)q_{12}^{2}q_{13}^{2}}\Big{[}M_{W}^{2}(q_{13}+q_{23})-q_{13}q_{23}\Big{]}C_{0}(0,0,q_{23},M_{W}^{2},0,0)$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}}\Big{\\{}(d-2)q_{13}q_{23}(q_{12}+q_{23})-M_{W}^{2}q_{12}\Big{[}(3d-10)q_{13}+(d-2)q_{23}\Big{]}$
$\displaystyle-
M_{W}^{2}q_{23}(q_{13}+q_{23})(d-2)\Big{\\}}C_{0}(0,q_{13},M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{1}{4(d-3)q_{12}^{2}q_{13}^{2}(q_{12}+q_{13})^{2}}\Big{\\{}M_{W}^{2}q_{12}^{3}\Big{[}(3d-10)q_{13}+(d-2)q_{23}\Big{]}$
$\displaystyle+M_{W}^{2}q_{12}^{2}q_{13}\Big{[}(d-6)q_{13}+(5d-14)q_{23}\Big{]}$
$\displaystyle+M_{W}^{2}q_{12}q_{13}^{2}\Big{[}(14-5d)q_{23}-3(d-2)q_{13}\Big{]}-(d-2)M_{W}^{2}q_{13}^{3}(q_{13}+q_{23})$
$\displaystyle-
q_{13}q_{23}(q_{12}+q_{13})\Big{[}(3d-10)q_{12}^{2}-2(d-2)q_{12}q_{13}-(d-2)q_{13}^{2}\Big{]}\Big{\\}}\times$
$\displaystyle\times C_{0}(q_{23},0,M_{H}^{2},M_{W}^{2},0,M_{W}^{2})$
$\displaystyle+\dfrac{(d-2)}{4(d-3)q_{12}^{2}q_{13}^{2}}\Big{[}q_{13}q_{23}-M_{W}^{2}(q_{13}+q_{23})\Big{]}^{2}D_{0}(0,0,0,M_{H}^{2};q_{23},q_{13};M_{W}^{2},0,0,M_{W}^{2}).$

##  Appendix B Expansion for form factors of $\mathcal{A}^{(W)}_{\text{tri}}$
and $\mathcal{A}^{(f)}_{\text{tri}}$

Since the dominant contributions of triangle diagrams with $W$ boson and
fermion internal lines to the decay channels. It is worth to perform the
$\varepsilon$-expansion for the form factors of
$\mathcal{A}^{(W)}_{\text{tri}}$ and $\mathcal{A}^{(f)}_{\text{tri}}$. Both
representations of form factors give same results

$\displaystyle\mathcal{A}^{(W)}_{\text{tri}}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}\Big{[}(M_{H}^{2}-q_{12})g^{\mu\nu}-2q^{\nu}q_{3}^{\mu}\Big{]}}{4M_{H}^{2}M_{W}^{3}s_{W}^{3}(M_{H}^{2}-q_{12})^{2}(q_{12}-M_{Z}^{2}+i\Gamma_{Z}M_{Z})}\times$
$\displaystyle\times\Bigg{\\{}M_{H}^{2}M_{W}^{2}\Big{[}M_{H}^{2}(q_{12}-6M_{W}^{2})+12M_{W}^{4}+6M_{W}^{2}q_{12}-2q_{12}^{2}\Big{]}\times$
$\displaystyle\hskip
85.35826pt\times\ln^{2}\Bigg{[}\dfrac{-M_{H}^{2}+\sqrt{M_{H}^{4}-4M_{H}^{2}M_{W}^{2}}+2M_{W}^{2}}{2M_{W}^{2}}\Bigg{]}$
$\displaystyle+M_{H}^{2}\Big{[}2M_{H}^{4}M_{W}^{2}-M_{H}^{4}q_{12}+12M_{H}^{2}M_{W}^{4}-4M_{H}^{2}M_{W}^{2}q_{12}$
$\displaystyle+M_{H}^{2}q_{12}^{2}-12M_{W}^{4}q_{12}+2M_{W}^{2}q_{12}^{2}\Big{]}$
$\displaystyle+M_{H}^{2}\sqrt{q_{12}^{2}-4M_{W}^{2}q_{12}}\Big{[}M_{H}^{2}(q_{12}-2M_{W}^{2})+2M_{W}^{2}(q_{12}-6M_{W}^{2})\Big{]}\times$
$\displaystyle\hskip
78.24507pt\times\ln\Bigg{[}\dfrac{-q_{12}+\sqrt{q_{12}^{2}-4M_{W}^{2}q_{12}}+2M_{W}^{2}}{2M_{W}^{2}}\Bigg{]}$
$\displaystyle-
M_{W}^{2}M_{H}^{2}\Big{[}M_{H}^{2}(q_{12}-6M_{W}^{2})+12M_{W}^{4}+6M_{W}^{2}q_{12}-2q_{12}^{2}\Big{]}\times$
$\displaystyle\hskip
85.35826pt\times\ln^{2}\Bigg{[}\dfrac{-q_{12}+\sqrt{q_{12}^{2}-4M_{W}^{2}q_{12}}+2M_{W}^{2}}{2M_{W}^{2}}\Bigg{]}$
$\displaystyle+q_{12}\sqrt{M_{H}^{4}-4M_{H}^{2}M_{W}^{2}}\Big{[}M_{H}^{2}(2M_{W}^{2}-q_{12})+12M_{W}^{4}-2M_{W}^{2}q_{12}\Big{]}\times$
$\displaystyle\hskip
85.35826pt\times\ln\Bigg{[}\dfrac{-M_{H}^{2}+\sqrt{M_{H}^{4}-4M_{H}^{2}M_{W}^{2}}+2M_{W}^{2}}{2M_{W}^{2}}\Bigg{]}\Bigg{\\}}\times$
$\displaystyle\times\bar{u}(q_{1})\gamma^{\mu}P_{L}v(q_{2})\epsilon^{*}_{\nu}(q_{3}),$
$\displaystyle\mathcal{A}^{(f)}_{\text{tri}}$ $\displaystyle=$
$\displaystyle\dfrac{\alpha^{2}m_{f}^{2}N_{C}^{f}Q_{f}\Big{[}2q^{\nu}q_{3}^{\mu}+(q_{12}-M_{H}^{2})g^{\mu\nu}\Big{]}(T^{3}_{f}-2Q_{f}s_{W}^{2})}{4M_{H}^{2}M_{W}c_{W}^{2}s_{W}^{3}(M_{H}^{2}-q_{12})^{2}(q_{12}-M_{Z}^{2}+i\Gamma_{Z}M_{Z})}\times$
$\displaystyle\times\Bigg{\\{}M_{H}^{2}\Big{[}(4m_{f}^{2}-M_{H}^{2}+q_{12})\ln^{2}\Bigg{[}\dfrac{-q_{12}+\sqrt{q_{12}^{2}-4q_{12}m_{f}^{2}}+2m_{f}^{2}}{2m_{f}^{2}}\Bigg{]}$
$\displaystyle\hskip
1.42271pt+4\sqrt{q_{12}^{2}-4q_{12}m_{f}^{2}}\ln\Bigg{[}\dfrac{-q_{12}+\sqrt{q_{12}^{2}-4q_{12}m_{f}^{2}}+2m_{f}^{2}}{2m_{f}^{2}}\Bigg{]}-4M_{H}^{2}+4q_{12}\Big{]}$
$\displaystyle\hskip
1.42271pt+M_{H}^{2}(M_{H}^{2}-4m_{f}^{2}-q_{12})\ln^{2}\Bigg{[}\dfrac{-M_{H}^{2}+\sqrt{M_{H}^{4}-4m_{f}^{2}M_{H}^{2}}+2m_{f}^{2}}{2m_{f}^{2}}\Bigg{]}$
$\displaystyle\hskip
1.42271pt-4q_{12}\sqrt{M_{H}^{4}-4M_{H}^{2}m_{f}^{2}}\ln\Bigg{[}\dfrac{-M_{H}^{2}+\sqrt{M_{H}^{4}-4m_{f}^{2}M_{H}^{2}}+2m_{f}^{2}}{2m_{f}^{2}}\Bigg{]}\Bigg{\\}}\times$
$\displaystyle\times\bar{u}(q_{1})\gamma^{\mu}P_{L}v(q_{2})\epsilon^{*}_{\nu}(q_{3}).$

For $W$ bosons and fermions exchanging in the loop, their masses are included
the Feynman’s prescription as $M_{W}^{2}-i\rho$ ($m_{f}^{2}-i\rho$
respectively). As a result, all logarithmic functions in the above equations
are well-defined in complex plane.

##  Appendix C Decay rate

Differential decay width is given by

$\displaystyle\dfrac{d^{2}\Gamma(H\rightarrow\nu_{l}\bar{\nu}_{l}\gamma)}{dq_{12}dq_{13}}=\dfrac{1}{256\pi^{3}\;M_{H}^{3}}\sum\limits_{\text{spin}}|\mathcal{A}_{\text{tri}}+\mathcal{A}_{\text{box}}|^{2},$
(63)

We have used following notations:

$\displaystyle\mathcal{A}_{\text{tri}}$ $\displaystyle=$
$\displaystyle\Big{[}F_{1}q_{3}^{\mu}q^{\nu}+F_{2}g^{\mu\nu}\Big{]}\bar{u}(q_{1})\gamma^{\mu}\;P_{L}v(q_{2})\epsilon^{*}_{\nu}(q_{3}),$
(64) $\displaystyle\mathcal{A}_{\text{box}}$ $\displaystyle=$
$\displaystyle\bar{u}(q_{1})\Big{\\{}F_{3}q_{1}^{\nu}\not{q_{3}}+F_{4}q_{2}^{\nu}\not{q_{3}}+F_{5}\gamma^{\nu}\Big{\\}}P_{L}v(q_{2})\epsilon^{*}_{\nu}(q_{3}).$
(65)

After performing spin-sum for the above amplitude-squared, one gets several
representations (together with Eq. 2.1.2) for the differential decay width as
follows:

$\displaystyle\dfrac{d^{2}\Gamma(H\rightarrow\nu_{l}\bar{\nu}_{l}\gamma)}{dq_{12}dq_{13}}$
$\displaystyle=$
$\displaystyle\dfrac{q_{12}}{256\pi^{3}M_{H}^{3}(q_{13}+q_{23})^{2}}\times$
$\displaystyle\times\Big{\\{}2(q_{13}^{2}+q_{23}^{2})|F_{2}|^{2}+q_{13}^{2}(q_{13}+q_{23})^{2}|F_{3}|^{2}+2(q_{13}+q_{23})^{2}|F_{5}|^{2}$
$\displaystyle+4q_{23}(q_{13}+q_{23})\,\text{Re}[F_{2}F_{5}^{*}]+2q_{13}(q_{23}^{2}-q_{13}^{2})\,\text{Re}[F_{2}F_{3}^{*}]+2q_{13}(q_{13}+q_{23})^{2}\,\text{Re}[F_{3}F_{5}^{*}]\Big{\\}},$
$\displaystyle\dfrac{d^{2}\Gamma(H\rightarrow\nu_{l}\bar{\nu}_{l}\gamma)}{dq_{12}dq_{13}}$
$\displaystyle=$
$\displaystyle\dfrac{q_{12}}{256\pi^{3}M_{H}^{3}(q_{13}+q_{23})^{2}}\times$
$\displaystyle\times\Big{\\{}2(q_{13}^{2}+q_{23}^{2})|F_{2}|^{2}+q_{23}^{2}(q_{13}+q_{23})^{2}|F_{4}|^{2}+2(q_{13}+q_{23})^{2}|F_{5}|^{2}$
$\displaystyle+4q_{13}(q_{13}+q_{23})\,\text{Re}[F_{2}F_{5}^{*}]+2q_{23}(q_{13}^{2}-q_{23}^{2})\,\text{Re}[F_{2}F_{4}^{*}]+2q_{23}(q_{13}+q_{23})^{2}\,\text{Re}[F_{4}F_{5}^{*}]\Big{\\}}.$

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