# Hypergeometric presentation for one-loop contributing to $H\rightarrow
Z\gamma$

Khiem Hong Phan and Dzung Tri Tran<EMAIL_ADDRESS>1)University of
Science Ho Chi Minh City, 227 Nguyen Van Cu, District 5, HCM City, Vietnam
2)Vietnam National University Ho Chi Minh City, Linh Trung Ward, Thu Duc
District, HCM City, Vietnam

###### Abstract

In this paper, new analytic formulas for one-loop contributing to Higgs decay
channel $H\rightarrow Z\gamma$ are presented in terms of hypergeometric
functions. The calculations are performed by following the technique for
tensor one-loop reduction developed in [A. I. Davydychev, Phys. Lett. B 263
(1991) 107]. For the first time, one-loop form factors for the decay process
are shown which are valid at arbitrary space-time dimension $d$.

B87

##  1 Introduction

Among Higgs ($H$) decay processes, the decay channel $H\rightarrow Z\gamma$ is
the most important at the Large Hadron Collider (LHC) Abazov:2008wg ;
Chatrchyan:2013vaa ; Aaboud:2017uhw . Because the channel arises at first from
one-loop Feynman diagrams. As a result, the decay width of this channel is
sensitive to new physics in which we assume that new heavy particles may
exchange in one-loop diagrams. For this reason, theoretical evaluations for
one-loop and higher-loop decay amplitudes of $H\rightarrow Z\gamma$ play
crucial roles in controlling standard model (SM) background as well as
constraining physical parameters in many beyond standard models (BSM).

There have been many computations for one-loop contributions to $H\rightarrow
Z\gamma$ within SM and its extensions in Cahn:1978nz ; Bergstrom:1985hp ;
Martinez:1989kr ; Djouadi:1996yq ; Djouadi:1997yw ; Chiang:2012qz ;
Chen:2013vi ; Cao:2013ur ; Bonciani:2015eua ; Hammad:2015eca ;
Belanger:2014roa ; No:2016ezr ; Monfared:2016vwr ; Fontes:2014xva ;
Funatsu:2015xba ; Hue:2017cph ; Hung:2019jue ; Boradjiev:2017khm . The
calculations have performed following the method for tensor one-loop reduction
in Denner:1991kt . When one-loop contributions to $H\rightarrow Z\gamma$ are
evaluated in unitary gauge, the results may meet large numerical
cancellations. This is because higher-rank tensor one-loop integrals appears
from Feynman loop diagrams with exchanging by vector bosons. To avoid this
problem, many of the above references have considered the calculations in ’t
Hooft-Feynman gauge. In this gauge, we need to handle more Feynman diagrams
with exchanging by Goldstone bosons. As a result, the calculations are rather
complicated. Furthermore, when we consider two-loop or higher-loop corrections
to $H\rightarrow Z\gamma$, two-loop and higher-loop Feynman integrals may be
evaluated by applying methods IBP ; Laporta:2001dd ; Tarasov:1996br , the
resulting integrals may contain the one-loop integrals in general space-time
dimension. These integrals have been not available in previous papers.

In this paper, we apply an alternative approach for evaluating one-loop
contributions to $H\rightarrow Z\gamma$. In this calculation, we follow the
method for tensor one-loop reduction developed in Davydychev:1991va  in which
tensor integrals are decomposed into scalar functions with arbitrary
propagator indexes and at higher space-time dimension $d>4$. Using
integration-by-part method (IBP) IBP ; Laporta:2001dd , scalar one-loop
integrals are then expressed in terms of master integrals which can be solved
analytically via generalized hypergeometric series. For instant, analytic
formulas for the master integrals which are one-loop one-, two-, three-point
functions at general $d$ appearing in $H\rightarrow Z\gamma$ are provided in
this work. Therefore, our methods are easy to apply for $H\rightarrow Z\gamma$
and expect to be numerical stability in unitary gauge. Furthermore, our
analytic expressions for the form factors of the decay process are general as
well as valid at arbitrary space-time dimension.

The layout of the paper is as follows: In section 2, we present a general
method for evaluating one-loop Feynman integrals. Using the method, the
computations for one-loop contributions to Higgs decay to $Z$ photon are
reported in the section $3$. Conclusions are shown in section $4$. Several
useful formulas used in this calculation and detailed calculations for the
process amplitudes are given in the appendixes.

##  2 Method

In this section, we describe a general approach for evaluating one-loop
Feynman integrals. In general, tensor one-loop $N$-point Feynman integrals
with rank $M$ are defined as follows:

$\displaystyle
J_{N,\mu_{1}\mu_{2}\cdots\mu_{M}}(d;\\{\nu_{1},\nu_{2},\cdots,\nu_{N}\\})\equiv
J_{N,\mu_{1}\mu_{2}\cdots\mu_{M}}(d;\\{\nu_{1},\nu_{2},\cdots,\nu_{N}\\};\\{p_{i}p_{j};m_{i}^{2}\\})=$
$\displaystyle=$
$\displaystyle\int\frac{d^{d}k}{i\pi^{d/2}}\dfrac{k_{\mu_{1}}k_{\mu_{2}}\cdots
k_{\mu_{M}}}{[(k+q_{1})^{2}-m_{1}^{2}+i\rho]^{\nu_{1}}[(k+q_{2})^{2}-m_{2}^{2}+i\rho]^{\nu_{2}}\cdots[(k+q_{N})^{2}-m_{N}^{2}+i\rho]^{\nu_{N}}}.$

Where $p_{i}$ ($m_{i}$) for $i=1,2,\cdots,N$ are external momenta (internal
masses) respectively. In this convention,
$q_{1}=p_{1},q_{2}=p_{1}+p_{2},\cdots,q_{i}=\sum_{j=1}^{i}p_{j},$ and
$q_{N}=\sum_{j=1}^{N}p_{j}=0$ thanks to momentum conservation. The term
$i\rho$ is Feynman’s prescription and $d$ is space-time dimension. One of
physical interests is $d=4+2n-2\epsilon$ for $n\in\mathbb{N}$.

Following the method for tensor reduction in Ref. Davydychev:1991va , tensor
one-loop integrals can be reduced to scalar functions with the shifted space-
time dimension as follows:

$\displaystyle
J_{N,\mu_{1}\mu_{2}\cdots\mu_{M}}(d;\\{\nu_{1},\nu_{2},\cdots,\nu_{N}\\})=\sum\limits_{\lambda,\kappa_{1},\kappa_{2}\cdots,\kappa_{N}}\left(-\frac{1}{2}\right)^{\lambda}\Big{\\{}[g]^{\lambda}[q_{1}]^{\kappa_{1}}[q_{2}]^{\kappa_{2}}\cdots[q_{N}]^{\kappa_{N}}\Big{\\}}_{\mu_{1}\mu_{2}\cdots\mu_{M}}$
$\displaystyle\hskip
28.45274pt\times(\nu_{1})_{\kappa_{1}}(\nu_{2})_{\kappa_{2}}\cdots(\nu_{N})_{\kappa_{N}}\;\;J_{N}(d+2(M-\lambda);\\{\nu_{1}+\kappa_{1},\nu_{2}+\kappa_{2},\cdots,\nu_{N}+\kappa_{N}\\}).$

Here $\lambda,\kappa_{1},\kappa_{2},\cdots,\kappa_{N}$ satisfy the following
constrains $2\lambda+\kappa_{1}+\kappa_{2}+\cdots+\kappa_{N}=M$,
$0\leq\kappa_{1},\kappa_{2},\cdots,\kappa_{N}\leq M$ and
$0\leq\lambda\leq[M/2]$ (integer of $M/2$). The Pochhammer symbol is used as
$(a)_{\kappa}=\Gamma(a+\kappa)/\Gamma(a)$. The tensor
$\\{[g]^{\lambda}[q_{1}]^{\kappa_{1}}[q_{2}]^{\kappa_{2}}\cdots[q_{N}]^{\kappa_{N}}\\}_{\mu_{1}\mu_{2}\cdots\mu_{M}}$
is symmetric in regard to $\mu_{1},\mu_{2},\cdots,\mu_{M}$. It is formed from
$\lambda$ of metric $g_{\mu\nu}$, $\kappa_{1}$ of momentum $q_{1}$, $\cdots$,
$\kappa_{N}$ of momentum $q_{N}$. The
$J_{N}(d+2(M-\lambda);\\{\nu_{1}+\kappa_{1},\nu_{2}+\kappa_{2},\cdots,\nu_{N}+\kappa_{N}\\})$
with changing space-time dimension to $d+2(M-\lambda)$, raising powers of
propagators $\\{\nu_{i}+\kappa_{i}\\}$ for $i=1,2,\cdots,N$ are scalar one-
loop $N$-point functions.

In the next step, the scalar integrals
$J_{N}(d;\\{\nu_{1},\nu_{2},\cdots,\nu_{N}\\})$ are casted into subset of
master functions by using IBP IBP . In detail, applying the operator
$\frac{\partial}{\partial k}\cdot k$ to the integrand of
$J_{N}(d;\\{\nu_{1},\nu_{2},\cdots,\nu_{N}\\})$ and setting $k$ to be the
momentum of $N$ internal lines
($k\equiv\\{k+q_{1},k+q_{2},\cdots,k+q_{N}\\}$). As a result,
$J_{N}(d;\\{\nu_{1},\nu_{2},\cdots,\nu_{N}\\})$ can be expressed in terms of
$J_{N}(d;\\{1,1,\cdots,1\\})$ and
$J_{N-1}(d;\\{\nu^{\prime}_{1},\nu^{\prime}_{2},\cdots,\nu^{\prime}_{N-1}\\})$.
In this recurrence way Laporta:2001dd , we arrive at the master integrals
which can be solved analytically. For examples, they may be
$J_{N}(d;\\{1,1,\cdots,1\\})$ and
$J_{N-L}(d;\\{\nu^{\prime\prime}_{1},\nu^{\prime\prime}_{2},\cdots,\nu^{\prime\prime}_{N-L}\\})$
with $L<N$. Recently, scalar one-loop integrals at general $d$ have been
expressed in terms of generalized hypergeometric series Bluemlein:2015sia ;
Phan:2019qee ; Phan:2018cnz ; Bluemlein:2017rbi .

In the Appendix $B$, this method is demonstrated in detail for the case of
$H\rightarrow Z\gamma$. We show here all analytic results for the master
integrals involving the decay process. In particular, scalar one-loop one-
point functions with arbitrary propagator index $\nu$ are given tHooft:1978jhc
:

$\displaystyle J_{1}(d;\\{\nu\\};M^{2})$ $\displaystyle=$
$\displaystyle(-1)^{\nu}\dfrac{\Gamma(\nu-d/2)}{\Gamma(\nu)}\;(M^{2})^{d/2-\nu}.$
(3)

Scalar one-loop two-point functions with general propagator indexes
$\nu_{1},\nu_{2}$ in the case of $m_{1}^{2}=m_{2}^{2}=M^{2}$ read khiemcadj :

$\displaystyle J_{2}(d;\\{\nu_{1},\nu_{2}\\};p^{2},M^{2})$ $\displaystyle=$
$\displaystyle(-1)^{N_{2}}\dfrac{\Gamma(N_{2}-d/2)}{\Gamma(N_{2})}(M^{2})^{d/2-N_{2}}\;\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}\nu_{1},\nu_{2},N_{2}-d/2\,;\\\
\frac{N_{2}}{2},\frac{N_{2}+1}{2}\,;\end{array}\dfrac{p^{2}}{4M^{2}}\right].$
(6)

Here $N_{2}=\nu_{1}+\nu_{2}$, $p^{2}=0,M_{H}^{2},M_{Z}^{2}$ and
$M^{2}=m_{f}^{2},M_{W}^{2}$ in this calculation. Other master integrals which
are scalar one-loop three-point functions are given:

$\displaystyle\dfrac{J_{3}(d;\\{1,1,1\\};p_{2}^{2},M_{H}^{2},M^{2})}{\Gamma\left(2-d/2\right)}$
$\displaystyle=$
$\displaystyle\dfrac{(d-4)M_{H}^{2}}{4(M_{H}^{2}-p_{2}^{2})}(M^{2})^{d/2-3}\times$
(13)
$\displaystyle\times\left\\{\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M^{2}}\right]-\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{p_{2}^{2}}{4M^{2}}\right]\right\\},$
$\displaystyle\dfrac{J_{3}(d;\\{1,2,1\\};p_{2}^{2},M_{H}^{2},M^{2})}{\Gamma\left(2-d/2\right)}$
$\displaystyle=$
$\displaystyle\dfrac{\left(4-d\right)}{2(M_{H}^{2}-p_{2}^{2})}(M^{2})^{d/2-3}\times$
(19)
$\displaystyle\times\left\\{\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,2,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M^{2}}\right]-\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,2,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{p_{2}^{2}}{4M^{2}}\right]\right\\},$
$\displaystyle\dfrac{J_{3}(d;\\{1,3,1\\};p_{2}^{2},M_{H}^{2},M^{2})}{\Gamma\left(2-d/2\right)}$
$\displaystyle=$
$\displaystyle\dfrac{\left(6-d\right)\left(d-4\right)}{16(M_{H}^{2}-p_{2}^{2})}(M^{2})^{d/2-4}\times$
(25)
$\displaystyle\times\left\\{\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,2,4-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M^{2}}\right]-\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,2,4-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{p_{2}^{2}}{4M^{2}}\right]\right\\},$
$\displaystyle\dfrac{J_{3}(d;\\{2,2,1\\};p_{2}^{2},M_{H}^{2},M^{2})}{\Gamma\left(2-d/2\right)}$
$\displaystyle=$ $\displaystyle\left(4-d\right)(M^{2})^{d/2-4}\times$ (32)
$\displaystyle\times\Bigg{\\{}\hskip
2.84544pt\dfrac{\left(6-d\right)M_{H}^{2}\left(4M^{2}-M_{H}^{2}\right)}{16M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,2,4-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M^{2}}\right]$
$\displaystyle\left.+\dfrac{\left(6-d\right)\left[M_{H}^{2}p_{2}^{2}-2M^{2}(M_{H}^{2}+p_{2}^{2})\right]}{16M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,2,4-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{p_{2}^{2}}{4M^{2}}\right]\right.$
$\displaystyle-\dfrac{\left(6-d\right)M_{H}^{2}}{4(M_{H}^{2}-p_{2}^{2})^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,2,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M^{2}}\right]$ (35)
$\displaystyle-\dfrac{\left[2M_{H}^{2}(d-5)-2p_{2}^{2}\right]}{8(M_{H}^{2}-p_{2}^{2})^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,2,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{p_{2}^{2}}{4M^{2}}\right]$ (38)
$\displaystyle+\dfrac{\left(d-4\right)M_{H}^{2}}{8(M_{H}^{2}-p_{2}^{2})^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M^{2}}\right]+\dfrac{\left(4-d\right)p_{2}^{2}}{8(M_{H}^{2}-p_{2}^{2})^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{p_{2}^{2}}{4M^{2}}\right]\Bigg{\\}}.$ (43)

Where $p_{2}^{2}=M_{Z}^{2},0$ and $M^{2}=m_{f}^{2},M_{W}^{2}$ in the present
calculation.

We are going to apply this method for evaluating the Higgs decay processes.
The first results for one-loop contributions to $H\rightarrow\gamma\gamma$
have been published in khiemcadj . In the next section, we show new analytic
results for $H\rightarrow Z\gamma$ by mean of ${}_{3}F_{2}$ hypergeometric
series.

##  3 Hypergeometric presentation for one-loop contributing to $H\rightarrow
Z\gamma$

In unitary gauge, the decay process $H\rightarrow Z\gamma$ consists top loop
and $W$ boson loop as shown in Figs. 1, 2. In general, the total amplitude of
the decay $H\rightarrow Z\gamma$ is expressed in terms of form factors with
reflecting the Lorentz invariant structure and the content of gauge symmetry
as follows:

$\displaystyle i\mathcal{A}_{H\rightarrow Z\gamma}$ $\displaystyle=$
$\displaystyle
i\mathcal{A}_{\mu\nu}\,\varepsilon^{\mu*}_{1}(q_{1})\varepsilon^{\nu*}_{2}(q_{2})=$
$\displaystyle=$
$\displaystyle\Big{(}F_{00}\,g_{\mu\nu}+\sum\limits_{i,j=1}^{2}F_{ij}\,q_{i,\mu}q_{j,\nu}+F_{5}\times
i\epsilon_{\mu\nu\alpha\beta}q_{1}^{\alpha}q_{2}^{\beta}\Big{)}\,\varepsilon^{\mu*}_{1}(q_{1})\varepsilon^{\nu*}_{2}(q_{2}).$

Where $\varepsilon^{\mu*}_{1}$ and $\varepsilon^{\nu*}_{2}$ are the
polarization vectors of the $Z$ boson and the photon $\gamma$ respectively.
$\epsilon_{\mu\nu\alpha\beta}$ is the Levi-Civita tensor. Kinematic invariant
variables related to this process are

$\displaystyle q_{1}^{2}=M_{Z}^{2},\quad q_{2}^{2}=0,\quad
p^{2}=(q_{1}+q_{2})^{2}=M_{H}^{2}.$ (45)

We also have $\varepsilon^{\nu*}_{2}(q_{2})q_{2,\nu}=0$ for external photon.
Following Ward identity, we confirm that

$\displaystyle F_{11}=0,\quad F_{00}=-\left(q_{1}\cdot
q_{2}\right)F_{21}=\frac{M_{Z}^{2}-M_{H}^{2}}{2}\;F_{21}$ (46)

and $F_{12,22}$ do not contribute to the total amplitude. Summing all the top-
loop diagrams, the result shows that $F_{5}=0$. Detailed calculations for the
form factors at general $d$ are presented in the appendix $D$.

Figure 1: Feynman diagrams contributing to the $H\longrightarrow Z\gamma$
decay through top quark loop in unitary gauge. Figure 2: Feynman diagrams
contributing to the $H\rightarrow Z\gamma$ decay through W boson loop in
unitary gauge.

The total amplitude for this decay process is then casted in the form of

$\displaystyle i\mathcal{A}_{H\rightarrow
Z\gamma}=\dfrac{e^{3}}{\sin\theta_{W}M_{W}}\mathcal{F}_{H\rightarrow
Z\gamma}(d;M_{H}^{2},M_{Z}^{2},M_{W}^{2},m_{f}^{2})\left[q_{2,\mu}q_{1,\nu}-\left(q_{1}\cdot
q_{2}\right)\,g_{\mu\nu}\right]\varepsilon^{\mu*}_{1}(q_{1})\varepsilon^{\nu*}_{2}(q_{2}),$

where $\mathcal{F}_{H\rightarrow
Z\gamma}(d;M_{H}^{2},M_{Z}^{2},M_{W}^{2},m_{f}^{2})$ are form factors which
can be derived from $F_{00}$ or $F_{21}$. These form factors are decomposed in
terms of $W$-loop and top-loop (including fermion-loop) contributions as
follows:

$\displaystyle\mathcal{F}_{H\rightarrow
Z\gamma}(d;M_{H}^{2},M_{Z}^{2},M_{W}^{2},m_{f}^{2})$ $\displaystyle=$
$\displaystyle\cot\theta_{W}\,\mathcal{F}^{(W)}_{H\rightarrow
Z\gamma}(d;M_{H}^{2},M_{Z}^{2},M_{W}^{2})$
$\displaystyle+\sum\limits_{f}\dfrac{Q_{f}N_{C}}{e}\left(\lambda^{f}_{1}+\lambda^{f}_{2}\right)\mathcal{F}^{(f)}_{H\rightarrow
Z\gamma}(d;M_{H}^{2},M_{Z}^{2},m_{f}^{2}).$

Where $\theta_{W}$ is Weinberg angle, $I^{3}_{f}$, $Q_{f}$ and $m_{f}$ are
iso-spin, electric charge, mass of fermions $f$ in the loops respectively.
$N_{C}$ is a color factor for fermions. It becomes $1$ for leptons and $3$ for
quarks. We use the symbolic-manipulation Package-X Patel:2015tea  to handle
all Dirac and tensor algebra in $d$ dimension.

###  3.1 Form factors

We show two representations for the form factors in terms of ${}_{3}F_{2}$
hypergeometric functions in this subsection.

####  3.1.1 First representation

We first present the form factors which are derived from $F_{00}$ in (3) in
terms of ${}_{3}F_{2}$ hypergeometric functions as follows:

$\displaystyle\dfrac{\mathcal{F}_{H\rightarrow
Z\gamma}^{(W)}(d;M_{H}^{2},M_{Z}^{2},M_{W}^{2})}{\Gamma\left(2-d/2\right)}=\dfrac{(M_{W}^{2})^{d/2-2}}{(4\pi)^{d/2}\;M_{W}^{2}(M_{Z}^{2}-M_{H}^{2})^{2}}\;\times$
(49) $\displaystyle\times\Bigg{\\{}\hskip
2.84544pt\left(4-d\right)(M_{Z}^{2}-4M_{W}^{2})(M_{H}^{2}-M_{Z}^{2})\times$
$\displaystyle\hskip
19.91684pt\times\left(M_{H}^{2}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]-M_{Z}^{2}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4M_{W}^{2}}\right]\right)$ (54)
$\displaystyle\hskip
14.22636pt\left.+\Big{[}2M_{W}^{2}(M_{H}^{2}-M_{Z}^{2})-M_{H}^{2}M_{Z}^{2}+4M_{W}^{4}(d-1)\Big{]}\right.\times$
$\displaystyle\hskip
19.91684pt\times\Bigg{(}\;(M_{H}^{2}-M_{Z}^{2})\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]$ (57)
$\displaystyle\hskip
42.67912pt+M_{Z}^{2}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4M_{W}^{2}}\right]-M_{H}^{2}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]\Bigg{)}\hskip
2.84544pt\Bigg{\\}},$ (62)

and

$\displaystyle\dfrac{\mathcal{F}_{H\rightarrow
Z\gamma}^{(t)}(d;M_{H}^{2},M_{Z}^{2},m_{t}^{2})}{\Gamma\left(2-d/2\right)}=\dfrac{(m_{t}^{2})^{d/2-2}}{(4\pi)^{d/2}\;(M_{Z}^{2}-M_{H}^{2})^{2}}\times$
(63) $\displaystyle\times\Bigg{\\{}\hskip
5.69046pt\left(4-d\right)M_{H}^{2}(M_{H}^{2}-M_{Z}^{2})\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4m_{t}^{2}}\right]+8M_{H}^{2}m_{t}^{2}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4m_{t}^{2}}\right]$ (68)
$\displaystyle\hskip
19.91684pt\left.+\left(4-d\right)M_{Z}^{2}(M_{Z}^{2}-M_{H}^{2})\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4m_{t}^{2}}\right]-8M_{Z}^{2}m_{t}^{2}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4m_{t}^{2}}\right]\right.$ (73)
$\displaystyle\hskip
19.91684pt-8(M_{H}^{2}-M_{Z}^{2})m_{t}^{2}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,1\,;\end{array}\dfrac{M_{H}^{2}}{4m_{t}^{2}}\right]\hskip
2.84544pt\Bigg{\\}}.$ (76)

The form factors $\mathcal{F}_{H\rightarrow
Z\gamma}^{(f)}(d;M_{H}^{2},M_{Z}^{2},m_{f}^{2})$ are obtained by replacing
$m_{t}\rightarrow m_{f}$ in Eq. (63). For the form factors which have fermion
masses are smaller than $M_{H}/2$, the argument of hypergeometric functions
${}_{3}F_{2}$ is greater than $1$ (or $|M_{H}^{2}/4m_{f}^{2}|>1$). We
subsequently apply analytic continuation in Eq. (193) for ${}_{3}F_{2}$
appearing in the form factors $\mathcal{F}_{H\rightarrow
Z\gamma}^{(f)}(d;M_{H}^{2},M_{Z}^{2},m_{f}^{2})$. In the limit $d\rightarrow
4$, we confirm that the terms in curly brackets of right hand side results of
(49,63) tend to zero

$\displaystyle\Big{[}2M_{W}^{2}(M_{H}^{2}-M_{Z}^{2})-M_{H}^{2}M_{Z}^{2}+12M_{W}^{4}\Big{]}\Big{(}M_{Z}^{2}+(M_{H}^{2}-M_{Z}^{2})-M_{H}^{2}\Big{)}$
$\displaystyle=$ $\displaystyle 0,$ (77) $\displaystyle
8M_{H}^{2}m_{t}^{2}-8M_{Z}^{2}m_{t}^{2}-8(M_{H}^{2}-M_{Z}^{2})m_{t}^{2}$
$\displaystyle=$ $\displaystyle 0.$ (78)

It means that the form factors always stay finite in the limit.

####  3.1.2 Second representation

Another presentation for the form factors which are obtained from $F_{21}$ in
(3) are given:

$\displaystyle\dfrac{\mathcal{F}_{H\rightarrow
Z\gamma}^{(W)}(d;M_{H}^{2},M_{Z}^{2},M_{W}^{2})}{\Gamma\left(2-d/2\right)}=\dfrac{(M_{W}^{2})^{d/2-2}}{(4\pi)^{d/2}\;M_{W}^{4}(M_{Z}^{2}-M_{H}^{2})^{2}}\;\times$
(79) $\displaystyle\times\Bigg{\\{}\hskip
2.84544pt\left(4-d\right)M_{W}^{2}(M_{Z}^{2}-4M_{W}^{2})(M_{H}^{2}-M_{Z}^{2})\times$
$\displaystyle\hskip
14.22636pt\times\left(M_{H}^{2}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]-M_{Z}^{2}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4M_{W}^{2}}\right]\right)$ (84)
$\displaystyle\hskip
14.22636pt\left.+\Big{[}2M_{W}^{4}(M_{H}^{2}-M_{Z}^{2})-M_{H}^{2}M_{W}^{2}M_{Z}^{2}+4M_{W}^{6}(d-1)\Big{]}\right.\times$
$\displaystyle\hskip
14.22636pt\times\Bigg{[}\dfrac{M_{H}^{2}\left(6M_{W}^{2}-M_{H}^{2}\right)-2M_{W}^{2}M_{Z}^{2}}{2M_{W}^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]$ (87)
$\displaystyle\hskip
22.76228pt+\dfrac{M_{H}^{2}(M_{Z}^{2}-4M_{W}^{2})}{2M_{W}^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4M_{W}^{2}}\right]$ (90)
$\displaystyle\hskip
22.76228pt-2M_{H}^{2}\left(\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]-\dfrac{M_{H}^{2}}{6M_{W}^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}3,2,2-d/2\,;\\\
5/2,3\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]\right)$ (95)
$\displaystyle\hskip
22.76228pt+\dfrac{2M_{H}^{2}(d-1)-2M_{Z}^{2}}{\left(d-2\right)}\left(\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4M_{W}^{2}}\right]-\dfrac{M_{Z}^{2}}{6M_{W}^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}3,2,2-d/2\,;\\\
5/2,3\,;\end{array}\dfrac{M_{Z}^{2}}{4M_{W}^{2}}\right]\right)$ (100)
$\displaystyle\hskip
22.76228pt+\dfrac{d}{\left(2-d\right)}\left(M_{H}^{2}\,\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]-\dfrac{M_{H}^{4}}{12M_{W}^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,2,2-d/2\,;\\\
5/2,3\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]\right)$ (105)
$\displaystyle\hskip
22.76228pt+\dfrac{d}{\left(d-2\right)}\left(M_{Z}^{2}\,\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4M_{W}^{2}}\right]-\dfrac{M_{Z}^{4}}{12M_{W}^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,2,2-d/2\,;\\\
5/2,3\,;\end{array}\dfrac{M_{Z}^{2}}{4M_{W}^{2}}\right]\right)\Bigg{]}\hskip
5.69046pt\Bigg{\\}},$ (110)

and

$\displaystyle\dfrac{\mathcal{F}_{H\rightarrow
Z\gamma}^{(t)}(d;M_{H}^{2},M_{Z}^{2},m_{t}^{2})}{\Gamma\left(2-d/2\right)}=\dfrac{(m_{t}^{2})^{d/2-2}}{(4\pi)^{d/2}\;(M_{Z}^{2}-M_{H}^{2})^{2}}\times$
(111) $\displaystyle\times\Bigg{\\{}\hskip
5.69046pt\dfrac{4M_{H}^{2}\left(4m_{t}^{2}-M_{H}^{2}\right)}{3}\Bigg{(}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,2,2-d/2\,;\\\
5/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4m_{t}^{2}}\right]+2\;\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}3,1,2-d/2\,;\\\
5/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4m_{t}^{2}}\right]\Bigg{)}$ (116)
$\displaystyle\left.+\dfrac{4M_{H}^{2}M_{Z}^{2}-8m_{t}^{2}(M_{H}^{2}+M_{Z}^{2})}{3}\Bigg{(}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,2,2-d/2\,;\\\
5/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4m_{t}^{2}}\right]+2\;\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}3,1,2-d/2\,;\\\
5/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4m_{t}^{2}}\right]\Bigg{)}\right.$ (121)
$\displaystyle\left.-16M_{H}^{2}m_{t}^{2}\left(\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4m_{t}^{2}}\right]-\dfrac{M_{H}^{2}}{6m_{t}^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}3,2,2-d/2\,;\\\
5/2,3\,;\end{array}\dfrac{M_{H}^{2}}{4m_{t}^{2}}\right]\right)\right.$ (126)
$\displaystyle\left.+\dfrac{16M_{H}^{2}m_{t}^{2}(d-1)-16M_{Z}^{2}m_{t}^{2}}{\left(d-2\right)}\left(\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4m_{t}^{2}}\right]-\dfrac{M_{Z}^{2}}{6m_{t}^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}3,2,2-d/2\,;\\\
5/2,3\,;\end{array}\dfrac{M_{Z}^{2}}{4m_{t}^{2}}\right]\right)\right.$ (131)
$\displaystyle\left.+\dfrac{8M_{H}^{2}m_{t}^{2}\,d}{\left(2-d\right)}\left(\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4m_{t}^{2}}\right]-\dfrac{M_{H}^{2}}{12m_{t}^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,2,2-d/2\,;\\\
5/2,3\,;\end{array}\dfrac{M_{H}^{2}}{4m_{t}^{2}}\right]\right)\right.$ (136)
$\displaystyle\left.+\dfrac{8M_{Z}^{2}m_{t}^{2}\,d}{\left(d-2\right)}\left(\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4m_{t}^{2}}\right]-\dfrac{M_{Z}^{2}}{12m_{t}^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,2,2-d/2\,;\\\
5/2,3\,;\end{array}\dfrac{M_{Z}^{2}}{4m_{t}^{2}}\right]\right)\right.$ (141)
$\displaystyle\left.+8m_{t}^{2}(M_{H}^{2}-M_{Z}^{2})\Bigg{(}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4m_{t}^{2}}\right]-\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4m_{t}^{2}}\right]\Bigg{)}\right.$ (146)
$\displaystyle\left.+\left(d-4\right)(M_{H}^{2}-M_{Z}^{2})\Bigg{(}M_{H}^{2}\;\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4m_{t}^{2}}\right]-M_{Z}^{2}\;\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{Z}^{2}}{4m_{t}^{2}}\right]\Bigg{)}\right.\hskip
2.84544pt\Bigg{\\}}.$ (151)

In the limit $d\rightarrow 4$, we also confirm that the terms in curly bracket
of right hand side results of (79, 111) tend to zero. It means that the form
factors always stay in finite in the limit.

###  3.2 $H\rightarrow\gamma\gamma$ reduction

In order to reduce to $H\rightarrow\gamma\gamma$, we take
$M^{2}_{Z}\rightarrow 0$, and
$\lambda^{f}_{1}=eQ_{f},\lambda^{f}_{2},\lambda^{f}_{3}\rightarrow 0$, the
total amplitude of the decay $H\rightarrow Z\gamma$ is reduced to
$H\rightarrow\gamma\gamma$. In detail, the results read

$\displaystyle\mathcal{F}_{H\rightarrow\gamma\gamma}(d;M_{H}^{2},M_{W}^{2},m_{f}^{2})=\mathcal{F}^{(W)}_{H\rightarrow\gamma\gamma}(d;M_{H}^{2},M_{W}^{2})+\sum\limits_{f}N_{C}Q_{f}^{2}\mathcal{F}^{(f)}_{H\rightarrow\gamma\gamma}(d;M_{H}^{2},m_{f}^{2}).$
(152)

Where the form factors are given

$\displaystyle\dfrac{\mathcal{F}^{(t)}_{H\rightarrow\gamma\gamma}(d;M_{H}^{2},m_{t}^{2})}{\Gamma\left(2-d/2\right)}=\dfrac{(m_{t}^{2})^{d/2-2}}{(4\pi)^{d/2}}\Bigg{\\{}-\dfrac{8m_{t}^{2}}{M_{H}^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,1\,;\end{array}\dfrac{M_{H}^{2}}{4m^{2}_{t}}\right]$ (155)
$\displaystyle\hskip
56.9055pt+\left(4-d\right)\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4m^{2}_{t}}\right]+\dfrac{8m_{t}^{2}}{M_{H}^{2}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4m^{2}_{t}}\right]\Bigg{\\}},$ (160)

and

$\displaystyle\dfrac{\mathcal{F}^{(W)}_{H\rightarrow\gamma\gamma}(d;M_{H}^{2},M_{W}^{2})}{\Gamma\left(2-d/2\right)}=\dfrac{(M_{W}^{2})^{d/2-2}}{(4\pi)^{d/2}}\Bigg{\\{}4\left(4-d\right)\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]+$ (163)
$\displaystyle\hskip
17.07182pt+\Bigg{[}2+4\dfrac{M_{W}^{2}}{M_{H}^{2}}(d-1)\Bigg{]}\left(\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]-\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]\right)\hskip
2.84544pt\Bigg{\\}}$ (168) $\displaystyle\hskip
8.5359pt=\dfrac{(M_{W}^{2})^{d/2-2}}{(4\pi)^{d/2}}\Bigg{\\{}\hskip
5.69046pt\Big{(}2+\frac{M_{H}^{2}}{M_{W}^{2}}\Big{)}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}2,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]$ (171)
$\displaystyle\hskip
17.07182pt-\Bigg{[}4+\frac{M_{H}^{2}}{M_{W}^{2}}+4(d-1)\frac{M_{W}^{2}}{M_{H}^{2}}\Bigg{]}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,1\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]$ (174)
$\displaystyle\hskip
17.07182pt+\Bigg{[}2+4(d-1)\frac{M_{W}^{2}}{M_{H}^{2}}\Bigg{]}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,2-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]-4(d-4)\;\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}1,1,3-d/2\,;\\\
3/2,2\,;\end{array}\dfrac{M_{H}^{2}}{4M_{W}^{2}}\right]\Bigg{\\}},$ (179)

To arrive at the last line result, we have already used the transformation for
hypergeometric functions ${}_{3}F_{2}$ in Eq. (209). These agree with the
results in khiemcadj .

###  3.3 Numerical results

In numerical results, we set $M_{H}=125$ GeV, $M_{Z}=91.2$ GeV, $m_{t}=173.5$
GeV and $M_{W}=80.4$ GeV. Our results are generated by using package NumEXP
Huang:2012qz  for numerical $\epsilon$-expansions of hypergeometric functions.
We first confirm two representations for the form factors in (49, 63) and (79,
111) at general $d$. It means that we verify numerically the Ward identity at
general $d$. In Tables 1, 2, we show numerical checks for the form factors at
general $d$. Two representations for the form factors are perfect agreement up
to last digit for $3.5\leq d\leq 5.5$.

$d$ |  $\mathcal{F}_{H\rightarrow Z\gamma}^{(t)}(d;M_{H}^{2},M_{Z}^{2},m_{t}^{2})$ in Eq. (63)
---|---
|  $\mathcal{F}_{H\rightarrow Z\gamma}^{(t)}(d;M_{H}^{2},M_{Z}^{2},m_{t}^{2})$
in Eq. (111)
$3.5$ | $-0.00117666222408164570889597705142$
| $-0.00117666222408164570889597705142$
$4.5$ | $-0.0756076123635421878866551078159$
| $-0.0756076123635421878866551078159$
$5.0$ | $-0.754001360017782779626359989943$
| $-0.754001360017782779626359989943$
$5.5$ | $-10.6345811567309032438825219401$
| $-10.6345811567309032438825219401$
Table 1:  Numerical confirmations for two representations of the form factors involving to top-loop diagrams at arbitrary $d$. $d$ |  $\mathcal{F}_{H\rightarrow Z\gamma}^{(W)}(d;M_{H}^{2},M_{Z}^{2},M_{W}^{2})$ in Eq. (49)
---|---
|  $\mathcal{F}_{H\rightarrow Z\gamma}^{(W)}(d;M_{H}^{2},M_{Z}^{2},M_{W}^{2})$
in Eq. (79)
$3.5$ | $-0.00924203129694608232780754562475$
| $-0.00924203129694608232780754562475$
$4.5$ | $-0.211488266331639234594811276488$
| $-0.211488266331639234594811276488$
$5.0$ | $-1.26786296363083047430009124220$
| $-1.26786296363083047430009124220$
$5.5$ | $-10.8040444333273283701507434992$
| $-10.8040444333273283701507434992$
Table 2:  Numerical confirmations for two representations for the form factors
involving to $W$-loop diagrams at arbitrary $d$.

We next perform higher-order $\epsilon$-expansion for the form factors in this
work up to $\epsilon^{5}$. We also compare our results with Hue:2017cph
($F^{SM}_{21,W}$) at $\epsilon^{0}$-terms. Our numerical results are shown in
Eqs. (181, 183). We find a perfect agreement between two results at
$\epsilon^{0}$-expansion. It is important to note that higher-power
$\epsilon$-expansions for the form factors in this paper are our first
results.

$\displaystyle\hskip
145.10922ptF^{SM}_{21,W}=-0.0418477713507083034768633206537\;\epsilon^{0}$
$\displaystyle\hskip 189.21065pt+\;\mathcal{O}(\epsilon);$ (180)
$\displaystyle\mathcal{F}_{H\rightarrow
Z\gamma}^{(W)}(d=4-2\epsilon;M_{H}^{2},M_{Z}^{2},M_{W}^{2})=-0.0418477713507083034768633206537\;\epsilon^{0}$
$\displaystyle\hskip
189.21065pt+0.260913488721110921277821252790\;\epsilon^{1}$
$\displaystyle\hskip
189.21065pt-0.849415964842831522240990065525\;\epsilon^{2}$
$\displaystyle\hskip
189.21065pt+1.93196240724203383916822579654\;\epsilon^{3}$
$\displaystyle\hskip
189.21065pt-3.46717780533875010127157401115\;\epsilon^{4}$
$\displaystyle\hskip
189.21065pt+5.25914558345954670519178485415\;\epsilon^{5}$
$\displaystyle\hskip 189.21065pt+\mathcal{O}(\epsilon^{6}).$ (181)
$\displaystyle\hskip
145.10922ptF^{SM}_{21,t}=-0.00894937919735623466782637004746\;\epsilon^{0}$
$\displaystyle\hskip 189.21065pt+\;\mathcal{O}(\epsilon);$ (182)
$\displaystyle\mathcal{F}_{H\rightarrow
Z\gamma}^{(t)}(d=4-2\epsilon;M_{H}^{2},M_{Z}^{2},m_{t}^{2})=-0.00894937919735623466782637004746\;\epsilon^{0}$
$\displaystyle\hskip
182.09746pt+0.0742785979879735824790115497100\;\epsilon^{1}$
$\displaystyle\hskip
182.09746pt-0.315615957203796781182876228270\;\epsilon^{2}$
$\displaystyle\hskip
182.09746pt+0.917527446546694361353843959657\;\epsilon^{3}$
$\displaystyle\hskip
182.09746pt-2.05845003852606360149227809637\;\epsilon^{4}$
$\displaystyle\hskip
182.09746pt+3.81281647820690166355588887060\;\epsilon^{5}$
$\displaystyle\hskip 183.52019pt+\;\mathcal{O}(\epsilon^{6}).$ (183)

##  4 Conclusions

In this paper, we have discussed the alternative approach for evaluating one-
loop Feynman integrals. In this method, tensor one-loop integrals are reduced
to scalar one-loop functions with the shifted space-time dimension. Scalar
one-loop integrals are solved analytically with the help of generalized
hypergeometric series. We have applied this method for computing one-loop
contributions to Higgs decay to $Z\gamma$. For the first time, we have
presented the form factors that are valid in general space-time dimension. The
method can be extended to evaluate one-loop contributions to Higgs decay to
$Zf\bar{f}$, $f\bar{f}\gamma$, etc., within the SM and many BSMs.

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

## Appendix $A$: Hypergeometricsereis

Series of hypergeometric functions ${}_{3}F_{2}$ Slater  are defined:

$\displaystyle\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}a_{1},a_{2},a_{3}\,;\\\
b_{1},b_{2}\,;\end{array}z\right]=\sum\limits_{m=0}^{\infty}\dfrac{(a_{1})_{m}(a_{2})_{m}(a_{3})_{m}}{(b_{1})_{m}(b_{2})_{m}}\dfrac{z^{m}}{m!}.$
(186)

The Mellin-Barnes representation for ${}_{3}F_{2}$ is

$\displaystyle\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}a_{1},a_{2},a_{3}\,;\\\
b_{1},b_{2}\,;\end{array}z\right]=\frac{\Gamma(a_{1})\Gamma(a_{2})\Gamma(a_{3})}{\Gamma(b_{1})\Gamma(b_{2})}\dfrac{1}{2\pi
i}\int\limits_{-i\infty}^{i\infty}ds\Gamma(-s)\frac{\Gamma(s+a_{1})\Gamma(s+a_{2})\Gamma(s+a_{3})}{\Gamma(s+b_{1})\Gamma(s+b_{2})}\left(-z\right)^{s},$
(189)

provided that $|\mathrm{Arg}(-z)|<\pi$. The integration contour is chosen in
such a way that the poles of $\Gamma(-s)$ and $\Gamma(\cdots+s)$ are well-
separated. Analytic continuation of ${}_{3}F_{2}$ functions:

$\displaystyle\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}a_{1},a_{2},a_{3}\,;\\\
b_{1},b_{2}\,;\end{array}z\right]=\frac{\Gamma(b_{1})\Gamma(b_{2})}{\Gamma(a_{1})\Gamma(a_{2})\Gamma(a_{3})}\times$
(193) $\displaystyle\hskip
14.22636pt\times\Bigg{\\{}\frac{\Gamma(a_{2}-a_{1})\Gamma(a_{3}-a_{1})\Gamma(a_{1})}{\Gamma(b_{1}-a_{1})\Gamma(b_{2}-a_{1})(-z)^{a_{1}}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}a_{1},1-b_{1}+a_{1},1-b_{2}+a_{1}\,;\\\
1-a_{2}+a_{1},1-a_{3}+a_{1}\,;\end{array}\frac{1}{z}\right]$ (196)
$\displaystyle\hskip
28.45274pt+\frac{\Gamma(a_{1}-a_{2})\Gamma(a_{3}-a_{2})\Gamma(a_{2})}{\Gamma(b_{1}-a_{2})\Gamma(b_{2}-a_{2})~{}(-z)^{a_{2}}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}a_{2},1-b_{1}+a_{2},1-b_{2}+a_{2}\,;\\\
1-a_{1}+a_{2},1-a_{3}+a_{2}\,;\end{array}\frac{1}{z}\right]$ (199)
$\displaystyle\hskip
28.45274pt+\frac{\Gamma(a_{1}-a_{3})\Gamma(a_{2}-a_{3})\Gamma(a_{3})}{\Gamma(b_{1}-a_{3})\Gamma(b_{2}-a_{3})~{}(-z)^{a_{3}}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}a_{3},1-b_{1}+a_{3},1-b_{2}+a_{3}\,;\\\
1-a_{1}+a_{3},1-a_{2}+a_{3}\,;\end{array}\frac{1}{z}\right]\Bigg{\\}}.$ (202)

In this work, a useful transformation for $\,{}_{3}F_{2}$ functions is
mentioned:

$\displaystyle\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}a_{1},a_{2},a_{3}\,;\\\
b_{1},b_{2}\,;\end{array}z\right]=\dfrac{b_{1}-a_{1}}{b_{1}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}a_{1},a_{2},a_{3}\,;\\\
b_{1}+1,b_{2}\,;\end{array}z\right]+\dfrac{a_{1}}{b_{1}}\,{}_{3}F_{2}\\!\\!\left[\begin{array}[]{c}a_{1}+1,a_{2},a_{3}\,;\\\
b_{1}+1,b_{2}\,;\end{array}z\right].$ (209)

## Appendix $B$: Calculating master integrals

Tensor one-loop three-point Feynman integrals with rank $M$ appearing in the
process $H\rightarrow Z\gamma$ are given as follows:

$\displaystyle
J_{3,\mu_{1}\mu_{2}\ldots\mu_{M}}(d;\\{\nu_{1},\nu_{2},\nu_{3}\\})$
$\displaystyle\equiv
J_{3,\mu_{1}\mu_{2}\ldots\mu_{M}}(d;\\{\nu_{1},\nu_{2},\nu_{3}\\};p_{2}^{2},M_{H}^{2},M^{2})=\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;\dfrac{k_{\mu_{1}}k_{\mu_{2}}\ldots
k_{\mu_{M}}}{P_{1}^{\nu_{1}}P_{2}^{\nu_{2}}P_{3}^{\nu_{3}}}.$ (210)

Where the inverse Feynman propagators are

$\displaystyle P_{1}$ $\displaystyle=$
$\displaystyle\left(k+q_{2}\right)^{2}-M^{2}+i\rho,$ (211) $\displaystyle
P_{2}$ $\displaystyle=$ $\displaystyle\left(k+p\right)^{2}-M^{2}+i\rho,$ (212)
$\displaystyle P_{3}$ $\displaystyle=$ $\displaystyle k^{2}-M^{2}+i\rho.$
(213)

The related kinematic invariant are $q_{1}^{2}=M_{Z}^{2},q_{2}^{2}=0,$ and
$p^{2}=(q_{1}+q_{2})^{2}=M_{H}^{2}$. In this paper, $p_{2}^{2}=M_{Z}^{2},0$
and internal masses $M^{2}=m_{f}^{2},M_{W}^{2}$.

After presenting tensor one-loop three-point integrals to scalar functions, we
next apply IBP for scalar one-loop functions with the general propagator
indexes. We then arrive at the following system of equations

$\displaystyle\begin{cases}(d-2\nu_{1}-\nu_{2}-\nu_{3})\mathbf{1}-\nu_{2}\mathbf{1}^{-}\mathbf{2}^{+}-\nu_{3}\mathbf{1}^{-}\mathbf{3}^{+}=\nu_{1}(2M^{2})\mathbf{1}^{+}+\nu_{2}(2M^{2}-q_{1}^{2})\mathbf{2}^{+}+\nu_{3}(2M^{2}-q_{2}^{2})\mathbf{3}^{+},\\\
(d-\nu_{1}-2\nu_{2}-\nu_{3})\mathbf{1}-\nu_{1}\mathbf{1}^{+}\mathbf{2}^{-}-\nu_{3}\mathbf{2}^{-}\mathbf{3}^{+}=\nu_{1}(2M^{2}-q_{1}^{2})\mathbf{1}^{+}+\nu_{2}(2M^{2})\mathbf{2}^{+}+\nu_{3}(2M^{2}-p^{2})\mathbf{3}^{+},\\\
(d-\nu_{1}-\nu_{2}-2\nu_{3})\mathbf{1}-\nu_{1}\mathbf{1}^{+}\mathbf{3}^{-}-\nu_{2}\mathbf{2}^{+}\mathbf{3}^{-}=\nu_{1}(2M^{2}-q_{2}^{2})\mathbf{1}^{+}+\nu_{2}(2M^{2}-p^{2})\mathbf{2}^{+}+\nu_{3}(2M^{2})\mathbf{3}^{+}.\end{cases}$
(214)

Here, the standard notation for increasing and lowering operators is used

$\displaystyle\mathbf{j}^{\pm}J_{3}(d;\\{\nu_{j}\\})=J_{3}(d;\\{\nu_{j}\pm
1\\})$ (215)

for $j=1,2,3$.

In the following paragraphs, we consider master integrals
$J_{3}(d;\\{\nu_{1},\nu_{2},\nu_{3}\\})$ by solving the above system of
equations in several special cases. In conclusions, the master integrals shown
at Section 2 are presented in terms of hypergeometric functions ${}_{3}F_{2}$
in this paper.

####  Case 1: $\nu_{1}=\nu_{2}=\nu_{3}=1$

$\displaystyle
J_{3}(d;\\{1,2,1\\};p_{2}^{2},M_{H}^{2},M^{2})=\dfrac{2}{(p_{2}^{2}-M_{H}^{2})}\Big{[}J_{2}(d;\\{2,1\\},M_{H}^{2},M^{2})-J_{2}(d;\\{2,1\\},p_{2}^{2},M^{2})\Big{]},$
$\displaystyle J_{3}(d;\\{2,1,1\\};p_{2}^{2},M_{H}^{2},M^{2})=$ (217)
$\displaystyle=\dfrac{(d-4)M_{H}^{2}}{2M^{2}(M_{H}^{2}-p_{2}^{2})}J_{3}(d;\\{1,1,1\\};p_{2}^{2},M_{H}^{2},M^{2})+\dfrac{2}{(p_{2}^{2}-M_{H}^{2})}J_{2}(d;\\{2,1\\},0,M^{2})$
$\displaystyle+\dfrac{M_{H}^{2}(4M^{2}-M_{H}^{2})}{M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}J_{2}(d;\\{2,1\\},M_{H}^{2},M^{2})+\dfrac{p_{2}^{2}M_{H}^{2}-2M^{2}(p_{2}^{2}+M_{H}^{2})}{M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}J_{2}(d;\\{2,1\\},p_{2}^{2},M^{2}),$
$\displaystyle J_{3}(d;\\{1,1,2\\};p_{2}^{2},M_{H}^{2},M^{2})=$ (218)
$\displaystyle=\dfrac{(d-4)p_{2}^{2}}{2M^{2}(p_{2}^{2}-M_{H}^{2})}J_{3}(d;\\{1,1,1\\};p_{2}^{2},M_{H}^{2},M^{2})+\dfrac{2}{(M_{H}^{2}-p_{2}^{2})}J_{2}(d;\\{2,1\\},0,M^{2})$
$\displaystyle+\dfrac{p_{2}^{2}M_{H}^{2}-2M^{2}(M_{H}^{2}+p_{2}^{2})}{M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}J_{2}(d;\\{2,1\\},M_{H}^{2},M^{2})+\dfrac{p_{2}^{2}(4M^{2}-p_{2}^{2})}{M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}J_{2}(d;\\{2,1\\},p_{2}^{2},M^{2}).$

#### Case 2: $\nu_{1}=1,\nu_{2}=2,\nu_{3}=1$

$\displaystyle J_{3}(d;\\{1,3,1\\};p_{2}^{2},M_{H}^{2},M^{2})=$
$\displaystyle=$
$\displaystyle\dfrac{1}{2(p_{2}^{2}-M_{H}^{2})}\Big{[}J_{2}(d;\\{2,2\\},M_{H}^{2},M^{2})-J_{2}(d;\\{2,2\\},p_{2}^{2},M^{2})\Big{]}$
$\displaystyle+\dfrac{1}{(p_{2}^{2}-M_{H}^{2})}\Big{[}J_{2}(d;\\{3,1\\},M_{H}^{2},M^{2})-J_{2}(d;\\{3,1\\},p_{2}^{2},M^{2})\Big{]},$
$\displaystyle J_{3}(d;\\{2,2,1\\};p_{2}^{2},M_{H}^{2},M^{2})=$
$\displaystyle=$
$\displaystyle\dfrac{(4-d)}{2M^{2}(M_{H}^{2}-p_{2}^{2})}J_{3}(d;\\{1,1,1\\};p_{2}^{2},M_{H}^{2},M^{2})$
$\displaystyle+\dfrac{M_{H}^{2}\left(4M^{2}-M_{H}^{2}\right)}{2M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}\Big{[}J_{2}(d;\\{2,2\\},M_{H}^{2},M^{2})+2J_{2}(d;\\{3,1\\},M_{H}^{2},M^{2})\Big{]}$
$\displaystyle+\dfrac{p_{2}^{2}M_{H}^{2}-2M^{2}(M_{H}^{2}+p_{2}^{2})}{2M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}\Big{[}J_{2}(d;\\{2,2\\},p_{2}^{2},M^{2})+2J_{2}(d;\\{3,1\\},p_{2}^{2},M^{2})\Big{]}$
$\displaystyle+\dfrac{(6-d)M_{H}^{2}}{M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}J_{2}(d;\\{2,1\\},M_{H}^{2},M^{2})+\dfrac{(d-5)M_{H}^{2}-p_{2}^{2}}{M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}J_{2}(d;\\{2,1\\},p_{2}^{2},M^{2}),$
$\displaystyle J_{3}(d;\\{1,2,2\\};p_{2}^{2},M_{H}^{2},M^{2})=$ (221)
$\displaystyle=\hskip
2.84544pt\dfrac{(d-4)}{2M^{2}(M_{H}^{2}-p_{2}^{2})}J_{3}(d;\\{1,1,1\\};p_{2}^{2},M_{H}^{2},M^{2})$
$\displaystyle+\dfrac{p_{2}^{2}M_{H}^{2}-2M^{2}(M_{H}^{2}+p_{2}^{2})}{2M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}\Big{[}J_{2}(d;\\{2,2\\},M_{H}^{2},M^{2})+2J_{2}(d;\\{3,1\\},M_{H}^{2},M^{2})\Big{]}$
$\displaystyle+\dfrac{p_{2}^{2}\left(4M^{2}-p_{2}^{2}\right)}{2M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}\Big{[}J_{2}(d;\\{2,2\\},p_{2}^{2},M^{2})+2J_{2}(d;\\{3,1\\},p_{2}^{2},M^{2})\Big{]}$
$\displaystyle+\dfrac{(d-5)p_{2}^{2}-M_{H}^{2}}{M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}J_{2}(d;\\{2,1\\},M_{H}^{2},M^{2})+\dfrac{(6-d)p_{2}^{2}}{M^{2}(M_{H}^{2}-p_{2}^{2})^{2}}J_{2}(d;\\{2,1\\},p_{2}^{2},M^{2}).$

## Appendix $C$: One-loop amplitudes for $H\rightarrow Z\gamma$

We present detailed calculations for the decay amplitude of $H\rightarrow
Z\gamma$ in unitary gauge in this appendix. All couplings involving the decay
process are listed in Table 3.

Vertices | Couplings
---|---
$HW_{\mu}W_{\nu}$ | $igM_{W}g^{\mu\nu}$
$Z_{\mu}(k_{1})W_{\nu}(k_{2})W_{\lambda}(k_{3})$ | $-ig\cos\theta_{W}\left[(k_{1}-k_{2})_{\lambda}g^{\mu\nu}+(k_{2}-k_{3})_{\mu}g^{\nu\lambda}+(k_{3}-k_{1})_{\nu}g^{\lambda\mu}\right]$
$A_{\mu}(k_{1})W_{\nu}(k_{2})W_{\lambda}(k_{3})$ | $-ie\left[(k_{1}-k_{2})_{\lambda}g^{\mu\nu}+(k_{2}-k_{3})_{\mu}g^{\nu\lambda}+(k_{3}-k_{1})_{\nu}g^{\lambda\mu}\right]$
$A_{\mu}Z_{\nu}W_{\alpha}W_{\beta}$ | $-ieg\cos\theta_{W}\left[2g^{\mu\nu}g^{\alpha\beta}-g^{\mu\alpha}g^{\nu\beta}-g^{\mu\beta}g^{\nu\alpha}\right]$
$HZ_{\mu}Z_{\nu}$ | $i\,gM_{Z}g^{\mu\nu}/\cos\theta_{W}$
$Htt$ | $-i\,gm_{t}/2M_{W}$
$ttZ_{\mu}$ | $i\,\left(\lambda^{t}_{1}+\lambda^{t}_{2}\right)\gamma^{\mu}+i\,\lambda^{t}_{3}\gamma^{\mu}\gamma^{5}$
$ttA_{\mu}$ | $ieQ_{t}\gamma^{\mu}$
Table 3: Couplings involving the decay $H\rightarrow Z\gamma$. In our
notation,
$\lambda^{t}_{1}=eQ_{t},\;\lambda^{t}_{2}=\dfrac{g}{2\cos\theta_{W}}\left(I^{3}_{t}-2Q_{t}\sin^{2}\theta_{W}-\,2Q_{t}\sin\theta_{W}\cos\theta_{W}\right)$,
and $\lambda^{t}_{3}=-\dfrac{g}{2\cos\theta_{W}}I^{3}_{t}$. Where $I^{3}_{t}$
and $Q_{t}$ are iso-spin and electric charge of top quarks in the loops. The
term $i\,\lambda^{t}_{1}\gamma^{\mu}$ should be the coupling of photon to top
quarks.

The decay amplitude $H\rightarrow Z\gamma$ of top-loop diagrams is expressed
as follows:

$\displaystyle i\mathcal{A}^{(T)}_{H\rightarrow Z\gamma}=\hskip
2.84544pt-\dfrac{eQ_{t}gm_{t}^{2}}{(4\pi)^{d/2}M_{W}}\left(\lambda^{t}_{1}+\lambda^{t}_{2}\right)\int\frac{{\rm
d}^{d}k}{i\pi^{d/2}}\,\dfrac{\varepsilon^{\mu*}_{1}(q_{1})\varepsilon^{\nu*}_{2}(q_{2})}{P_{1}P_{2}P_{3}}\times$
(222)
$\displaystyle\times\Bigg{\\{}16k^{\mu}k^{\nu}+8k^{\nu}q_{1}^{\mu}+16k^{\nu}q_{2}^{\mu}+4q_{1}^{\nu}q_{2}^{\mu}-g^{\mu\nu}\Big{[}8(k\cdot
q_{2})+4k^{2}+\left(2M_{H}^{2}-2M_{Z}^{2}-4m_{t}^{2}\right)\Big{]}\Bigg{\\}}.$

Where the coefficient factors are written in terms of the master integrals:

$\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\,\dfrac{k^{\mu}k^{\nu}}{P_{1}P_{2}P_{3}}=\hskip
2.84544pt\dfrac{-g^{\mu\nu}}{2}J_{3}(d+2;\\{1,1,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2})$
(223) $\displaystyle\hskip
73.97733pt+\;q_{1}^{\mu}q_{1}^{\nu}J_{3}(d+4;\\{1,3,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2})$
$\displaystyle\hskip
73.97733pt+\;q_{2}^{\mu}q_{1}^{\nu}\Big{[}J_{3}(d+4;\\{2,2,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2})+J_{3}(d+4;\\{1,3,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2})\Big{]},$
$\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\,\dfrac{k^{\nu}q_{1}^{\mu}}{P_{1}P_{2}P_{3}}=\hskip
2.84544ptq_{1}^{\mu}q_{1}^{\nu}J_{3}(d+2;\\{1,2,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2}),$
(224) $\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\,\dfrac{k^{\nu}q_{2}^{\mu}}{P_{1}P_{2}P_{3}}=\hskip
2.84544ptq_{2}^{\mu}q_{1}^{\nu}J_{3}(d+2;\\{1,2,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2}),$
(225) $\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\,\dfrac{q_{1}^{\nu}q_{2}^{\mu}}{P_{1}P_{2}P_{3}}=\hskip
2.84544ptq_{2}^{\mu}q_{1}^{\nu}J_{3}(d;\\{1,1,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2}),$
(226) $\displaystyle\int\dfrac{{\rm d}^{d}k}{i\pi^{d/2}}\,\dfrac{k\cdot
q_{2}}{P_{1}P_{2}P_{3}}=\hskip
2.84544pt\frac{1}{2}\Big{[}J_{2}(d;\\{1,1\\};M_{H}^{2},m_{t}^{2})-J_{2}(d;\\{1,1\\};M_{Z}^{2},m_{t}^{2})\Big{]},$
(227) $\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\,\dfrac{k^{2}}{P_{1}P_{2}P_{3}}=\hskip
2.84544ptJ_{2}(d;\\{1,1\\};M_{Z}^{2},m_{t}^{2})+m_{t}^{2}J_{3}(d;\\{1,1,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2}).$
(228)

For the $W$ boson loops contributions, the decay amplitude is written

$\displaystyle i\mathcal{A}^{(W)}_{H\rightarrow Z\gamma}=\hskip
2.84544pt\dfrac{ieg^{2}\cos\theta_{W}}{(4\pi)^{d/2}M_{W}^{5}}\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;\varepsilon^{\mu*}_{1}(q_{1})\varepsilon^{\nu*}_{2}(q_{2})\times$
(229) $\displaystyle\hskip
34.14322pt\times\Big{[}i\mathcal{A}_{1}g^{\mu\nu}+i\mathcal{A}_{2}k^{\mu}k^{\nu}+i\mathcal{A}_{3}k^{\mu}q_{1}^{\nu}+i\mathcal{A}_{4}q_{2}^{\mu}k^{\nu}+i\mathcal{A}_{5}q_{2}^{\mu}q_{1}^{\nu}+i\mathcal{A}_{6}q_{1}^{\mu}k^{\nu}+i\mathcal{A}_{7}q_{1}^{\mu}q_{1}^{\nu}\Big{]}.$

Where the coefficient factors are presented in terms of the master integrals
in detail:

$\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;(i\mathcal{A}_{1}g^{\mu\nu})=\hskip
2.84544pt\int\dfrac{{\rm d}^{d}k}{i\pi^{d/2}}\;\Bigg{\\{}\hskip
2.84544pt\frac{1}{P_{1}P_{2}P_{3}}\Big{[}2M_{W}^{4}\Big{(}M_{Z}^{2}-4M_{W}^{2}\Big{)}\Big{(}M_{H}^{2}-M_{Z}^{2}\Big{)}\Big{]}$
(230) $\displaystyle\hskip
42.67912pt+\frac{1}{P_{2}P_{3}}\Big{[}2M_{W}^{4}\Big{(}M_{Z}^{2}-M_{H}^{2}\Big{)}+M_{H}^{2}M_{W}^{2}M_{Z}^{2}-\frac{M_{H}^{4}M_{W}^{2}}{2}\Big{]}$
$\displaystyle\hskip
42.67912pt+\frac{1}{P_{2}P_{3}}\Big{[}-P_{1}\Big{(}M_{H}^{2}M_{W}^{2}+2M_{W}^{4}\Big{)}+2M_{W}^{6}\Big{(}1-d\Big{)}\Big{]}+\frac{1}{P_{1}}\Big{(}-2M_{W}^{4}\Big{)}$
$\displaystyle\hskip
42.67912pt+\Big{(}\frac{1}{P_{2}}+\frac{1}{P_{3}}\Big{)}\Big{[}\frac{M_{W}^{2}}{2}\Big{(}2P_{1}-P_{2}-P_{3}\Big{)}+M_{W}^{2}\Big{(}2M_{W}^{2}+M_{H}^{2}-M_{Z}^{2}\Big{)}\Big{]}\hskip
5.69046pt\Bigg{\\}}\;g^{\mu\nu}$ $\displaystyle=\hskip
2.84544pt\Big{\\{}\hskip
2.84544pt2M_{W}^{4}\Big{(}M_{Z}^{2}-4M_{W}^{2}\Big{)}\Big{(}M_{H}^{2}-M_{Z}^{2}\Big{)}J_{3}(d;\\{1,1,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})$
(231) $\displaystyle\hskip
14.22636pt+\Big{[}2M_{W}^{6}\Big{(}1-d\Big{)}+M_{H}^{2}M_{W}^{2}M_{Z}^{2}+M_{W}^{4}\Big{(}2M_{Z}^{2}-M_{H}^{2}\Big{)}\Big{]}J_{2}(d;\\{1,1\\};M_{H}^{2},M_{W}^{2})$
$\displaystyle\hskip
14.22636pt+M_{W}^{2}M_{Z}^{2}\Big{(}M_{H}^{2}+2M_{W}^{2}\Big{)}J_{2}(d+2;\\{2,1\\};M_{H}^{2},M_{W}^{2})-M_{W}^{2}M_{Z}^{2}\;J_{1}(d;\\{1\\};M_{W}^{2})\hskip
2.84544pt\Big{\\}}\;g^{\mu\nu},$ $\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;(i\mathcal{A}_{2}k^{\mu}k^{\nu})=$ (232)
$\displaystyle=\hskip 2.84544pt\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;\Big{\\{}\frac{k^{\mu}k^{\nu}}{P_{1}P_{2}P_{3}}\Big{[}4M_{W}^{4}\Big{(}M_{H}^{2}-M_{Z}^{2}\Big{)}-2M_{H}^{2}M_{W}^{2}M_{Z}^{2}+8M_{W}^{6}\Big{(}d-1\Big{)}\Big{]}+\frac{k^{\mu}k^{\nu}}{P_{1}P_{3}}\Big{(}2M_{W}^{2}M_{Z}^{2}\Big{)}\Big{\\}}$
$\displaystyle=\hskip 2.84544pt\Big{\\{}\hskip
2.84544pt\Big{[}2M_{W}^{4}\Big{(}M_{Z}^{2}-M_{H}^{2}\Big{)}+M_{H}^{2}M_{W}^{2}M_{Z}^{2}+4M_{W}^{6}\Big{(}1-d\Big{)}\Big{]}J_{3}(d+2;\\{1,1,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})$
$\displaystyle-M_{W}^{2}M_{Z}^{2}\;J_{2}(d+2;\\{1,1\\};0,M_{W}^{2})\hskip
2.84544pt\Big{\\}}\;g^{\mu\nu}$ (233) $\displaystyle+\Big{\\{}\hskip
2.84544pt\Big{[}4M_{W}^{4}\Big{(}M_{H}^{2}-M_{Z}^{2}\Big{)}-2M_{H}^{2}M_{W}^{2}M_{Z}^{2}+8M_{W}^{6}\Big{(}d-1\Big{)}\Big{]}\times$
$\displaystyle\times\Big{[}J_{3}(d+4;\\{2,2,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})+J_{3}(d+4;\\{1,3,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})\Big{]}\hskip
2.84544pt\Big{\\}}\;q_{2}^{\mu}q_{1}^{\nu}$ $\displaystyle+\Big{\\{}\hskip
2.84544pt\Big{[}4M_{W}^{4}\Big{(}M_{H}^{2}-M_{Z}^{2}\Big{)}-2M_{H}^{2}M_{W}^{2}M_{Z}^{2}+8M_{W}^{6}\Big{(}d-1\Big{)}\Big{]}J_{3}(d+4;\\{1,3,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})\hskip
2.84544pt\Big{\\}}q_{1}^{\mu}q_{1}^{\nu},$ $\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;(i\mathcal{A}_{3}k^{\mu}q_{1}^{\nu})=$ $\displaystyle=$
$\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;\Big{\\{}\frac{k^{\mu}q_{1}^{\nu}}{P_{1}P_{2}}(4M_{W}^{4}-2M_{W}^{2}M_{Z}^{2})-\frac{k^{\mu}q_{1}^{\nu}}{P_{2}P_{3}}\left(\frac{M_{H}^{2}M_{W}^{2}}{2}+7M_{W}^{4}\right)+\frac{M_{W}^{2}}{2}\left(\frac{k^{\mu}q_{1}^{\nu}}{P_{2}}+\frac{k^{\mu}q_{1}^{\nu}}{P_{3}}\right)\Big{\\}}$
$\displaystyle=$
$\displaystyle\Big{\\{}\Big{(}2M_{W}^{2}M_{Z}^{2}-4M_{W}^{4}\Big{)}J_{2}(d;\\{1,1\\};M_{Z}^{2},M_{W}^{2})$
$\displaystyle-\left(\frac{M_{H}^{2}M_{W}^{2}}{2}+7M_{W}^{4}\right)J_{2}(d+2;\\{2,1\\};M_{H}^{2},M_{W}^{2})-\frac{M_{W}^{2}}{2}J_{1}(d;\\{1\\};M_{W}^{2})\Big{\\}}\;q_{2}^{\mu}q_{1}^{\nu}$
$\displaystyle+\Big{\\{}\Big{(}4M_{W}^{4}-2M_{W}^{2}M_{Z}^{2}\Big{)}J_{2}(d+2;\\{2,1\\};M_{Z}^{2},M_{W}^{2})$
$\displaystyle-\left(\frac{M_{H}^{2}M_{W}^{2}}{2}+7M_{W}^{4}\right)J_{2}(d+2;\\{2,1\\};M_{H}^{2},M_{W}^{2})-\left(\frac{M_{W}^{2}}{2}\right)J_{1}(d;\\{1\\};M_{W}^{2})\Big{\\}}\;q_{1}^{\mu}q_{1}^{\nu},$
$\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;(i\mathcal{A}_{4}q_{2}^{\mu}k^{\nu})=$ $\displaystyle=$
$\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;\Big{\\{}\frac{q_{2}^{\mu}k^{\nu}}{P_{1}P_{2}P_{3}}\Big{[}4M_{W}^{4}\Big{(}M_{H}^{2}-M_{Z}^{2}\Big{)}-2M_{H}^{2}M_{W}^{2}M_{Z}^{2}+8M_{W}^{6}\Big{(}d-1\Big{)}\Big{]}$
$\displaystyle+\frac{q_{2}^{\mu}k^{\nu}}{P_{1}P_{3}}\Big{(}2M_{W}^{2}M_{Z}^{2}-4M_{W}^{4}\Big{)}+\frac{q_{2}^{\mu}k^{\nu}}{P_{2}P_{3}}(\frac{M_{H}^{2}M_{W}^{2}}{2}+7M_{W}^{4})-\frac{M_{W}^{2}}{2}\Big{(}\frac{q_{2}^{\mu}k^{\nu}}{P_{3}}+\frac{q_{2}^{\mu}k^{\nu}}{P_{2}}\Big{)}\Big{\\}}$
$\displaystyle=$
$\displaystyle\Big{\\{}\Big{[}4M_{W}^{4}(M_{H}^{2}-M_{Z}^{2})-2M_{H}^{2}M_{W}^{2}M_{Z}^{2}+8M_{W}^{6}(d-1)\Big{]}J_{3}(d+2;\\{1,2,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})$
$\displaystyle+\left(\frac{M_{H}^{2}M_{W}^{2}}{2}+7M_{W}^{4}\right)J_{2}(d+2;\\{2,1\\};M_{H}^{2},M_{W}^{2})+\frac{M_{W}^{2}}{2}\;J_{1}(d;\\{1\\};M_{W}^{2})\Big{\\}}\;q_{2}^{\mu}q_{1}^{\nu},$
$\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;(i\mathcal{A}_{5}q_{2}^{\mu}q_{1}^{\nu})=\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;\Big{\\{}\frac{(16M_{W}^{6}-4M_{W}^{4}M_{Z}^{2})}{P_{1}P_{2}P_{3}}+\frac{(4M_{W}^{4}-2M_{W}^{2}M_{Z}^{2})}{P_{1}P_{2}}\Big{\\}}\;q_{2}^{\mu}q_{1}^{\nu}$
$\displaystyle\hskip
42.67912pt=\Big{\\{}(16M_{W}^{6}-4M_{W}^{4}M_{Z}^{2})J_{3}(d;\\{1,1,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})$
$\displaystyle\hskip
85.35826pt+(4M_{W}^{4}-2M_{W}^{2}M_{Z}^{2})J_{2}(d;\\{1,1\\};M_{Z}^{2},M_{W}^{2})\Big{\\}}\;q_{2}^{\mu}q_{1}^{\nu},$
(236) $\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;(i\mathcal{A}_{6}q_{1}^{\mu}k^{\nu})=$ $\displaystyle=$
$\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;\Big{\\{}\dfrac{q_{1}^{\mu}k^{\nu}}{P_{1}P_{2}P_{3}}[2M_{W}^{4}(M_{H}^{2}-M_{Z}^{2})-M_{H}^{2}M_{W}^{2}M_{Z}^{2}+4M_{W}^{6}(d-1)]+\frac{q_{1}^{\mu}k^{\nu}}{P_{2}}\left(\frac{M_{W}^{2}}{2}\right)$
$\displaystyle+\frac{q_{1}^{\mu}k^{\nu}}{P_{2}P_{3}}\left(3M_{W}^{4}-\frac{3M_{H}^{2}M_{W}^{2}}{2}\right)+\frac{q_{1}^{\mu}k^{\nu}}{P_{1}P_{3}}\Big{[}-M_{W}^{2}P_{2}+M_{W}^{2}\Big{(}M_{H}^{2}+M_{Z}^{2}\Big{)}\Big{]}+\frac{q_{1}^{\mu}k^{\nu}}{P_{3}}\left(\dfrac{3M_{W}^{2}}{2}\right)\Big{\\}}$
$\displaystyle=$
$\displaystyle\Big{\\{}\Big{[}2M_{W}^{4}(M_{H}^{2}-M_{Z}^{2})-M_{H}^{2}M_{W}^{2}M_{Z}^{2}+4M_{W}^{6}(d-1)\Big{]}J_{3}(d+2;\\{1,2,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})$
$\displaystyle+\left(3M_{W}^{4}-\frac{3M_{H}^{2}M_{W}^{2}}{2}\right)J_{2}(d+2;\\{2,1\\};M_{H}^{2},M_{W}^{2})$
$\displaystyle+\left(-\frac{M_{W}^{2}}{2}\right)\Big{[}J_{1}(d;\\{1\\};M_{W}^{2})-2J_{2}(d+2;\\{1,1\\};0,M_{W}^{2})\Big{]}\Big{\\}}\;q_{1}^{\mu}q_{1}^{\nu},$
$\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;(i\mathcal{A}_{7}q_{1}^{\mu}q_{1}^{\nu})=$
$\displaystyle=$ $\displaystyle\int\dfrac{{\rm
d}^{d}k}{i\pi^{d/2}}\;\Big{\\{}\frac{1}{P_{2}P_{3}}(-M_{H}^{2}M_{W}^{2}-2M_{W}^{4})+\dfrac{1}{P_{1}P_{2}}(2M_{W}^{4}-M_{W}^{2}M_{Z}^{2})+M_{W}^{2}\Big{(}\dfrac{1}{P_{1}}+\dfrac{1}{P_{3}}\Big{)}\Big{\\}}\;q_{1}^{\mu}q_{1}^{\nu}$
$\displaystyle=\Big{\\{}(-M_{H}^{2}M_{W}^{2}-2M_{W}^{4})J_{2}(d;\\{1,1\\};M_{H}^{2},M_{W}^{2})$
$\displaystyle+(2M_{W}^{4}-M_{W}^{2}M_{Z}^{2})J_{2}(d;\\{1,1\\};M_{Z}^{2},M_{W}^{2})+2M_{W}^{2}J_{1}(d;\\{1\\};M_{W}^{2})\Big{\\}}\;q_{1}^{\mu}q_{1}^{\nu}.$

## Appendix $D$: Form factors at general $d$

The form factors in Eq. (3) are

$\displaystyle
F^{(t)}_{00}=-\dfrac{eQ_{t}gm_{t}^{2}}{(4\pi)^{d/2}M_{W}}\left(\lambda^{(t)}_{1}+\lambda^{(t)}_{2}\right)\Big{[}\left(2M_{Z}^{2}-2M_{H}^{2}\right)J_{3}(d;\\{1,1,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2})$
(239) $\displaystyle\hskip
113.81102pt-8J_{3}(d+2;\\{1,1,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2})-4J_{2}(d;\\{1,1\\};M_{H}^{2},m_{t}^{2})\Big{]},$
$\displaystyle
F^{(t)}_{11}=-\dfrac{eQ_{t}gm_{t}^{2}}{(4\pi)^{d/2}M_{W}}\left(\lambda^{(t)}_{1}+\lambda^{(t)}_{2}\right)\Big{[}\hskip
5.69046pt16J_{3}(d+4;\\{1,3,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2})$ (240)
$\displaystyle\hskip
159.3356pt+8J_{3}(d+2;\\{1,2,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2})\hskip
5.69046pt\Big{]},$ $\displaystyle
F^{(t)}_{21}=-\dfrac{eQ_{t}gm_{t}^{2}}{(4\pi)^{d/2}M_{W}}\left(\lambda^{(t)}_{1}+\lambda^{(t)}_{2}\right)\Big{[}16J_{3}(d+4;\\{2,2,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2})$
(241) $\displaystyle\hskip
71.13188pt+8J_{3}(d+2;\\{1,2,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2})+4J_{3}(d;\\{1,1,1\\};M_{Z}^{2},M_{H}^{2},m_{t}^{2})\Big{]},$
$\displaystyle F^{(t)}_{5}=0,$ (242)

and

$\displaystyle
F^{(W)}_{00}=\dfrac{eg^{2}\cos\theta_{W}}{(4\pi)^{d/2}M_{W}^{5}}\Bigg{\\{}\hskip
2.84544pt2M_{W}^{4}(M_{Z}^{2}-4M_{W}^{2})(M_{H}^{2}-M_{Z}^{2})J_{3}(d;\\{1,1,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})$
(243) $\displaystyle\hskip
28.45274pt+\Big{[}2M_{W}^{4}(M_{Z}^{2}-M_{H}^{2})+M_{H}^{2}M_{W}^{2}M_{Z}^{2}-4M_{W}^{6}(d-1)\Big{]}J_{3}(d+2;\\{1,1,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})$
$\displaystyle\hskip
28.45274pt+\Big{[}2M_{W}^{4}(M_{H}^{2}-M_{Z}^{2})-M_{H}^{2}M_{W}^{2}M_{Z}^{2}+4M_{W}^{6}(d-1)\Big{]}J_{2}(d+2;\\{2,1\\};M_{H}^{2},M_{W}^{2})\hskip
8.5359pt\Bigg{\\}},$ $\displaystyle
F^{(W)}_{11}=\dfrac{eg^{2}\cos\theta_{W}}{(4\pi)^{d/2}M_{W}^{5}}\times$ (244)
$\displaystyle\times\Big{\\{}\hskip
2.84544pt\Big{[}4M_{W}^{4}(M_{H}^{2}-M_{Z}^{2})-2M_{H}^{2}M_{W}^{2}M_{Z}^{2}+8M_{W}^{6}(d-1)\Big{]}J_{3}(d+4;\\{1,3,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})$
$\displaystyle\hskip
14.22636pt+\Big{[}2M_{W}^{4}(M_{H}^{2}-M_{Z}^{2})-M_{H}^{2}M_{W}^{2}M_{Z}^{2}+4M_{W}^{6}(d-1)\Big{]}J_{3}(d+2;\\{1,2,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})\hskip
2.84544pt\Big{\\}},$ $\displaystyle
F^{(W)}_{21}=\dfrac{eg^{2}\cos\theta_{W}}{(4\pi)^{d/2}M_{W}^{5}}\Big{\\{}\hskip
2.84544pt4M_{W}^{4}\left(4M_{W}^{2}-M_{Z}^{2}\right)J_{3}(d;\\{1,1,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})$
(245) $\displaystyle\hskip
14.22636pt+\Big{[}2M_{W}^{4}(M_{H}^{2}-M_{Z}^{2})-M_{H}^{2}M_{W}^{2}M_{Z}^{2}+4M_{W}^{6}\,(d-1)\Big{]}J_{3}(d+2;\\{1,2,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})$
$\displaystyle\hskip
14.22636pt+\Big{[}4M_{W}^{4}(M_{H}^{2}-M_{Z}^{2})-2M_{H}^{2}M_{W}^{2}M_{Z}^{2}+8M_{W}^{6}\,(d-1)\Big{]}J_{3}(d+4;\\{2,2,1\\};M_{Z}^{2},M_{H}^{2},M_{W}^{2})\hskip
2.84544pt\Big{\\}}.$

## References

  * (1) V. M. Abazov et al. [D0 Collaboration], Phys. Lett. B 671 (2009) 349.
  * (2) S. Chatrchyan et al. [CMS Collaboration], Phys. Lett. B 726 (2013) 587.
  * (3) M. Aaboud et al. [ATLAS Collaboration], JHEP 1710 (2017) 112.
  * (4) R. N. Cahn, M. S. Chanowitz and N. Fleishon, Phys. Lett. 82B (1979) 113.
  * (5) L. Bergstrom and G. Hulth, Nucl. Phys. B 259 (1985) 137 Erratum: [Nucl. Phys. B 276 (1986) 744].
  * (6) R. Martinez, M. A. Perez and J. J. Toscano, Phys. Lett. B 234 (1990) 503.
  * (7) A. Djouadi, V. Driesen, W. Hollik and A. Kraft, Eur. Phys. J. C 1 (1998) 163.
  * (8) A. Djouadi, J. Kalinowski and M. Spira, Comput. Phys. Commun. 108 (1998) 56.
  * (9) C. W. Chiang and K. Yagyu, Phys. Rev. D 87 (2013) no.3, 033003.
  * (10) C. S. Chen, C. Q. Geng, D. Huang and L. H. Tsai, Phys. Rev. D 87 (2013) 075019.
  * (11) J. Cao, L. Wu, P. Wu and J. M. Yang, JHEP 1309 (2013) 043.
  * (12) R. Bonciani, V. Del Duca, H. Frellesvig, J. M. Henn, F. Moriello and V. A. Smirnov, JHEP 1508 (2015) 108.
  * (13) A. Hammad, S. Khalil and S. Moretti, Phys. Rev. D 92 (2015) no.9, 095008.
  * (14) G. Belanger, V. Bizouard and G. Chalons, Phys. Rev. D 89 (2014) no.9, 095023.
  * (15) J. M. No and M. Spannowsky, Phys. Rev. D 95 (2017) no.7, 075027.
  * (16) S. Taheri Monfared, S. Fayazbakhsh and M. Mohammadi Najafabadi, Phys. Lett. B 762 (2016) 301.
  * (17) D. Fontes, J. C. Rom$\tilde{\text{a}}$o and J. P. Silva, JHEP 1412 (2014) 043.
  * (18) S. Funatsu, H. Hatanaka and Y. Hosotani, Phys. Rev. D 92 (2015) no.11, 115003.
  * (19) L. T. Hue, A. B. Arbuzov, T. T. Hong, T. P. Nguyen, D. T. Si and H. N. Long, Eur. Phys. J. C 78 (2018) no.11, 885.
  * (20) H. T. Hung, T. T. Hong, H. H. Phuong, H. L. T. Mai and L. T. Hue, Phys. Rev. D 100 (2019) no.7, 075014.
  * (21) I. Boradjiev, E. Christova and H. Eberl, Phys. Rev. D 97 (2018) no.7, 073008.
  * (22) A. Denner, Fortsch. Phys. 41 (1993) 307.
  * (23) F.V. Tkachov, Phys. Lett. B100 (1981) 65; K.G. Chetyrkin and F.V. Tkachov, Nucl. Phys. B192 (1981) 159.
  * (24) S. Laporta, Int. J. Mod. Phys. A 15 (2000) 5087.
  * (25) O. V. Tarasov, Phys. Rev. D 54 (1996) 6479.
  * (26) A. I. Davydychev, Phys. Lett. B 263 (1991) 107.
  * (27) G. ’t Hooft and M. J. G. Veltman, Nucl. Phys. B 153 (1979) 365.
  * (28) J. Blümlein, K. H. Phan and T. Riemann, Nucl. Part. Phys. Proc. 270-272 (2016) 227.
  * (29) K. H. Phan and D. T. Tran, PTEP 2019 (2019) no.6, 063B01.
  * (30) K. H. Phan and T. Riemann, Phys. Lett. B 791 (2019) 257.
  * (31) J. Blümlein, K. H. Phan and T. Riemann, Acta Phys. Polon. B 48 (2017) 2313.
  * (32) Khiem Hong Phan, Can. J. Phys. 97 (2019) no.10, 1096.
  * (33) H. H. Patel, Comput. Phys. Commun. 197 (2015) 276.
  * (34) Z. W. Huang and J. Liu, Comput. Phys. Commun. 184 (2013) 1973.
  * (35) Lucy Joan Slater, Generalized Hypergeometric Functions. Cambridge University Press, 1966.


# Faraday rotation signal amplification using high-power lasers

P.-A. Gourdain  1,2  A. Bachmann1  I. N. Erez1,2  M. E. Evans1,2  F. Garrett1
J. Hraki1  H. R. Hasson1,2  S. McGaffigan1  I. West-Abdallah1,2  J. R.
Young1,2  1Department of Physics and Astronomy, University of Rochester,
Rochester NY 14627, USA
2Laboratory for Laser Energetics, University of Rochester, Rochester NY 14627,
USA<EMAIL_ADDRESS>
###### Abstract

Magnetic fields play an important role in plasma dynamics, yet it is a
quantity difficult to measure accurately with physical probes, whose presence
disturbs the very field they measure. The Faraday rotation of a polarized beam
of light provides a mechanism to measure the magnetic field without disturbing
the dynamics, and has been used with great success in astrophysics and high
energy density plasma science, where physical probes cannot be used. However,
the rotation is typically small, which degrades the accuracy of the
measurement. Paradoxically, the main source of error is the probe beam itself.
Since polarization cannot be measured directly, detectors rely on a polarizer
to measure a small change in beam intensity instead. In this work, we show how
suppress the beam intensity that is not part of the Faraday rotation signal by
taking forming an optical derivative. Since the rotation measurement is now
strictly proportional to the beam intensity, the system allows to amplify the
rotation measurement simply by increasing the laser power.

††journal: optcon††articletype: Research Article

##  1 Introduction

Faraday rotation (FR)[1], the change of light polarization caused by a
magneto-ionized gas, thereafter called a plasma, has found applications in a
wide range of disciplines over the years, from laser engineering [2] to
astrophysics[3]. The landmark experiment, done by Michael Faraday in 1845[4],
showed that the polarization of an electromagnetic wave can be affected by a
magnetic field parallel to the direction of propagation. The magnetic field
triggers a birefringence inside the material, which in turn changes the
polarization angle $\theta_{FR}$ as

$\theta_{FR}=VBL,$

where $B$ is the magnetic field strength, projected along the direction of
propagation of the electromagnetic wave, $L$ is the propagation length in the
birefringent medium, and $V$ is the Verdet constant. A similar effect caused
by magnetic fields transverse to the direction of propagation is called the
Cotton-Mouton effect. While the Verdet constant can be difficult to compute
for materials, it is relatively straightforward for a plasma[5],

$\theta_{FR}(x,y)=\frac{e^{3}\lambda^{2}}{8\pi^{2}\epsilon_{0}m_{e}^{2}c^{3}}\int_{L}{n_{e}(x,y,z)\mathbf{B}(x,y,z)\cdot
d\mathbf{z}},$ (1)

where $e$ is the fundamental charge, $\lambda$ is the laser wavelength,
$\epsilon_{0}$ the vacuum permittivity, $m_{e}$ the electron mass, $n_{e}$ the
plasma electron number density. We supposed here that the light travels along
the $z$-axis of our coordinate system, the $x$-axis being horizontal and the
$y$-axis vertical. This dependence allows to compute the magnetic field
$B_{z}$, provided that the electron density $n_{e}$ is already known. This
effect has been used intensively to measure the magnetic field strength of the
interstellar medium[6] by using background polarized sources, generating
synchrotron radiations across a variety of wavelengths in the radio range[7].
FR was also used actively to highlight how bipolar jets spawned from accretion
disks[8] are sensitive to the Hall effect[9], an effect [10] also found in
high energy density plasma jets produced in the laboratory[11, 12].

However, using FR to measure magnetic fields in the laboratory has proven
challenging when using a visible laser[13] rather than microwaves[14]. The
laser wavelength is a scaling factor in front of the integral of Eq. (1). As
the wavelength becomes smaller, the cut-off density becomes larger and deeper
regions of the plasma can be probed[13]. However, the Faraday rotation angle
is now greatly reduced and becomes difficult to measure. In practice, this
scaling factor acts as a attenuation factor.

In this paper we show that an S-waveplate combined with a neutral density
filter with radial variation scan be used to amplify the Faraday rotation
signal. In general, the S-waveplate[15] utilizes polarization to convert a
linearly polarized Gaussian beam into a radially or azimuthally polarized
beam[16]. It is the complementary of the vortex plate[17, 18, 19], which
affects the phase of the light rather than its polarization. The vortex plate
achieves this by using a helical cavity with increasing depth corresponding to
the azimuthal angle of the plate, changing the incoming phase of light
azimuthally from 0 to $2m\pi$, where $m$ is the plate charge. In contrast, the
S-waveplate uses nanogratings with their alignment rotating with the plate’s
azimuthal angle to change the incoming polarization azimuthally from 0 to
$2m\pi$, also using $m$ as the plate charge. The proposed system gives the
spatial derivative of the FR rotation signal, de facto removing the bias light
from the probe beam. With the bias gone, the laser intensity can be increased,
improving the signal-to-noise ratio of the measurement.

##  2 The optical derivative of a polarized beam

When a light beam travels through the experimental setup shown in Fig. 1, its
spatial constituents undergo a transformation into its spectral (i.e. spatial
frequency) constituents at the focal plane of the initial lens. This procedure
finds mathematical expression through the utilization of the Fourier
transform:

$U(f_{x},f_{y})=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty}u(x,y)e^{-i2\pi
xf_{x}}e^{-i2\pi yf_{y}}dxdy,$

where $U(f_{x},f_{y})$ is the intensity representing the Fourier transform of
the light on the focal plane of the lens[20].

Figure 1: Two-$f$ system uses two identical lenses of focal length $f$. The
$z$-axis is along the optical axis.

###  2.1 Description of the S-waveplate using Jones matrix

The description of the S-waveplate can be effectively realized through the
application of Jones matrices[21, 22]. This element can be divided into a
retarder introducing a phase delay of $\varphi=\pi$ and a rotation matrix that
imparts a rotation determined by the orientation $\phi$ of the nanogratings.
Mathematically, the explicit expression for $\mathscr{S}$ takes the form:

$\mathscr{S}=\text{e}^{\frac{-i\varphi}{2}}\left[{\begin{array}[]{*{20}{c}}{{{\cos}^{2}}\phi+{e^{i\varphi}}{{\sin}^{2}}\phi}&{({1-{e^{i\varphi}}})\cos\phi\sin\phi}\\\
{({1-{e^{i\varphi}}})\cos\phi\sin\phi}&{{e^{i\varphi}}{{\cos}^{2}}\phi+{{\sin}^{2}}\phi}\end{array}}\right].$

Given that $kR=\pi$, the expression can be simplified to:

$\mathscr{S}(x,y)=-i\left[{\begin{array}[]{*{20}{c}}{\cos\Theta(x,y)}&{\sin\Theta(x,y)}\\\
{\sin\Theta(x,y)}&{-\\!\cos\Theta(x,y)}\end{array}}\right],$ (2)

where the local polarizing angle $\Theta(x,y)$ is twice the nanograting angle
$\phi(x,y)$.

###  2.2 Beam with variable intensity and constant linear polarization

We start with a linearly polarized laser beam having a stable polarization
angle $\theta$ relative to the polarization of the S-plate. Any electric field
with a stable polarization in the object plane can be expanded into a discrete
summation of sine and cosine functions using the discrete Fourier transform as

$\mathbf{E}(x,y)=\frac{\mathbf{A}_{0}}{2}+\sum_{n=1}^{+\infty}\cos(\textbf{k}_{n}\cdot\textbf{r})\mathbf{A}_{n}+\sin(\textbf{k}_{n}\cdot\textbf{r})\mathbf{B}_{n},$
(3)

where $\textbf{k}_{n}\cdot\textbf{r}=k_{x,n}x+k_{y,n}y$,
$\mathbf{A}_{n}=A_{n}\cos\theta\,\mathbf{x}+A_{n}\sin\theta\,\mathbf{y}$, and
$\mathbf{B}_{n}=B_{n}\cos\theta\,\mathbf{x}+B_{n}\sin\theta\,\mathbf{y}$. Note
that $A_{n}$ and $B_{n}$ can be complex. To illustrate the operation done in
the Fourier plane by the S-waveplate, we examine each component of
$\mathbf{E}(x,y)$ individually. We denote the component of $\mathbf{E}$
corresponding to
$\mathbf{k}\in\\{\mathbf{k}_{1},\ldots,\mathbf{k}_{n},\ldots,+\infty\\}$ as
$\mathbf{E}_{\textbf{k}}(x,y)=\mathbf{A}_{\textbf{k}}\cos(k_{x}x+k_{y}y)+\mathbf{B}_{\textbf{k}}\sin(k_{x}x+k_{y}y)$.
The Fourier transform of $\mathbf{E}_{\textbf{k}}$ is given by

$\displaystyle\mathscr{E}_{\textbf{k}}(f_{x},f_{y})=$
$\displaystyle\left[\pi\delta\left(\frac{k_{x}}{2\pi}-f_{x}\right)\delta\left(\frac{k_{y}}{2\pi}-f_{y}\right)+\pi\delta\left(\frac{k_{x}}{2\pi}+f_{x}\right)\delta\left(\frac{k_{y}}{2\pi}+f_{y}\right)\right]\mathbf{A}_{\textbf{k}}$
$\displaystyle+\frac{1}{i}$
$\displaystyle\left[\pi\delta\left(\frac{k_{x}}{2\pi}-f_{x}\right)\delta\left(\frac{k_{y}}{2\pi}-f_{y}\right)-\pi\delta\left(\frac{k_{x}}{2\pi}+f_{x}\right)\delta\left(\frac{k_{y}}{2\pi}+f_{y}\right)\right]\mathbf{B}_{\textbf{k}}.$

When introducing an S-waveplate at the Fourier plane $(f_{x},f_{y})$ where the
orientation of the local polarizing angle $\Theta$ varies azimuthally from 0
to $2\pi$, the Jones matrix $\mathscr{S}$ of Eq. (2) can be expressed in term
of the frequencies $f_{x}$ and $f_{y}$ as

$\mathscr{S}(f_{x},f_{y})=-i\left[{\begin{array}[]{*{20}{c}}{\frac{f_{x}}{\sqrt{f_{x}^{2}+f_{y}^{2}}}}&{\frac{f_{y}}{\sqrt{f_{x}^{2}+f_{y}^{2}}}}\\\
{\frac{f_{y}}{\sqrt{f_{x}^{2}+f_{y}^{2}}}}&{-\frac{f_{x}}{\sqrt{f_{x}^{2}+f_{y}^{2}}}}\end{array}}\right].$

Now, we combine the S-waveplate with a neutral density filter whose
transmission varies linearly with the radius $\rho$ and is defined as[23]

$\rho(f_{x},f_{y})=\sqrt{f_{x}^{2}+f_{y}^{2}}.$ (4)

In practice, the transmission cannot exceed 1. While a scaling factor could be
used, it is omitted here. The S-waveplate and the neutral density filter
collectively constitute a new element denoted as $\mathscr{T}$, with a Jones
matrix given by

$\mathscr{T}(f_{x},f_{y})=-i\left[{\begin{array}[]{*{20}{c}}{f_{x}}&{f_{y}}\\\
{f_{y}}&{-f_{x}}\end{array}}\right].$

Upon performing the inverse Fourier transform of
$\mathscr{E}_{\mathbf{k}}\mathscr{T}$, we obtain on the image plane[23]

$\mathbf{E}_{\textbf{k}}^{\prime}(x,y)=-\left[\begin{array}[]{*{20}{c}}{\cos\theta}&{\sin\theta}\\\
{-\sin\theta}&{\cos\theta}\end{array}\right]\nabla\left[A_{\textbf{k}}\cos(\mathbf{k}\cdot\mathbf{r})+B_{\textbf{k}}\sin(\mathbf{k}\cdot\mathbf{r})\right]$
(5)

where $\nabla$ is the gradient operator. Since the main results of the paper
focus on intensities, we will drop the minus sign in Eq. (5) from later
equations. As a result, the total electric field on the image plane is given
by

$\mathbf{E}^{\prime}(x,y)=\nabla_{-\theta}E(x,y),$ (6)

where the electric field $E(x,y)$ is the electric field along the polarization
direction $\theta$, i.e.

$E(x,y)=\frac{A_{0}}{2}+\sum_{n=1}^{+\infty}A_{n}\cos(\textbf{k}_{n}\cdot\textbf{r})+B_{n}\sin(\textbf{k}_{n}\cdot\textbf{r}),$

and $\nabla_{\theta}$ is the rotated gradient operator given by

$\nabla_{\theta}=\left[\begin{array}[]{*{20}{c}}{\cos\theta}&{-\sin\theta}\\\
{\sin\theta}&{\cos\theta}\end{array}\right]\nabla\text{ or
}\nabla_{\theta}=\left[\begin{array}[]{*{20}{c}}{\cos\theta\,\partial_{x}-\sin\theta\,\partial_{y}}\\\
{\sin\theta\,\partial_{x}+\cos\theta\,\partial_{y}}\end{array}\right]$

The resulting electric field of Eq. (6) is the gradient of the initial
electric field along the polarization direction $\theta$. However, its
polarization is along the $-\theta$ direction now. Note that for a beam with a
polarization angle $\theta=0$ (polarization along the $x$-axis),
$\nabla_{\theta}$ becomes the usual gradient operator $\nabla$. In this case,
the S-waveplate/neutral density filter give the gradient along two
perpendicular polarization axes, allowing to measure $\partial_{x}E$ and
$\partial_{y}E$ independently. This is a departure from the vortex plate setup
[23] where the derivative along $x$ and $y$ were combined together.

###  2.3 Beam with variable intensity and polarization

We again start with a linearly polarized laser beam with constant initial
polarization angle $\theta_{i}$ with respect to the S-waveplate polarization.
However, the electric field $E(x,y)$ has now acquired a variable polarization
$\theta_{FR}(x,y)$ from a magnetoactive medium. We are dropping the
coordinates $x$ and $y$ hereafter for the terms on the RHS of most equations.
After exiting the medium, the electric field is given by

$\mathbf{E}(x,y)=E\left[\begin{array}[]{*{20}{c}}{\cos\theta}\\\
{\sin\theta}\end{array}\right],$ (7)

where $\theta=\theta_{i}+\theta_{FR}$. Since a spatial change in polarization
simply corresponds to a change in intensity of the $x$\- and $y$-polarization
components of the beam, we can use Eq. (6) for each polarization independently
to find the electric field at the exit of the augmented two-$f$ system

$\mathbf{E}^{\prime}(x,y)=\nabla_{0}(E\cos\theta)+\nabla_{-\pi/2}(E\sin\theta),$

which we can write explicitly as

$\mathbf{E}^{\prime}(x,y)=\left[\begin{array}[]{*{20}{c}}{\partial_{x}\left(E\cos\theta\right)+\partial_{y}\left(E\sin\theta\right)}\\\
{\partial_{y}\left(E\cos\theta\right)-\partial_{x}\left(E\sin\theta\right)}\end{array}\right]$
(8)

If $\theta$ is constant, then Eq. (8) is simply Eq. (6).

##  3 Implementation of a Faraday rotation measurement

Figure 2: Typical example of a Faraday Rotation measurement yielding an
intensity that is proportional to $(1+\theta_{FR})$ using a polarizer that is
rotated by $\pi/4$ compared to the initial laser beam polarization.

###  3.1 Standard Faraday rotation measurement

It is typical to start with a $y$-polarized (i.e. vertically polarized) beam.
The beam then passes through the magnetoactive medium, impinging a Faraday
rotation $\theta_{FR}$ to the polarization (see Fig. 2). At this point, a
polarizer at $\pi/4$ from the vertical direction yields a new electric field

$E_{\pi/4}(x,y)=E\cos(\alpha),$ (9)

where $\alpha=\pi/4-\theta_{FR}$ and $E(x,y)$ is the electric field strength.
Now Eq. (9) can be turned into

$E_{\pi/4}(x,y)=\frac{1}{\sqrt{2}}E\left[\cos(\theta_{FR})+\sin(\theta_{FR})\right].$
(10)

The intensity can be written as

$I_{\pi/4}(x,y)=\frac{1}{2}E^{2}\left[\cos^{2}(\theta_{FR})+2\cos(\theta_{FR})\sin(\theta_{FR})+\sin^{2}(\theta_{FR})\right].$
(11)

As $\theta_{FR}<<1$ in most low density magnetoactive media such as plasmas,
we can do a Taylor expansion of Eq. (11) up to the first order leading to

$I_{\pi/4}(x,y)\simeq\frac{1}{2}E^{2}\left(1+2\theta_{FR}\right).$ (12)

While this approach increases the measurement sensitivity compared to using a
cross-polarizer downstream of the magnetoactive medium, the bias light
(represented by the factor 1 in Eq. (12)) really reduces the signal-to-noise
ratio.

###  3.2 Faraday rotation measurement with homogeneous beams

We look first at the case where the laser intensity is homogeneous, i.e.
$E(x,y)\simeq E_{0}$. We can choose $\theta_{i}=0$. In this case, the electric
field of Eq. (8) has a component along the $x$ given by

$E^{\prime}_{x}(x,y)=E_{0}\left(\partial_{x}\cos\theta_{FR}+\partial_{y}\sin\theta_{FR}\right)$
(13)

for the $x$-polarization and

$E^{\prime}_{y}(x,y)=E_{0}\left(\partial_{y}\cos\theta_{FR}-\partial_{x}\sin\theta_{FR}\right).$
(14)

for the $y$-polarization. Since $\theta_{FR}<<1$ the zeroth order of Eq. (13)
is

$E^{\prime}_{x}(x,y)\simeq E_{0}\partial_{y}\theta_{FR}$ (15)

In this case, the zeroth order of the intensity of the $x$-polarization is

$I^{\prime}_{x}(x,y)\simeq|E_{0}|^{2}{\partial_{y}\theta_{FR}}^{2},$ (16)

Following the same line of thinking, Eq. (14) gives

$E^{\prime}_{y}(x,y)\simeq-E_{0}\partial_{x}\theta_{FR},$ (17)

with an intensity

$I^{\prime}_{y}(x,y)\simeq|E_{0}|^{2}{\partial_{x}\theta_{FR}}^{2}.$ (18)

We clearly see that the sensitivity of the measurement is reduced compared to
Eq. (12) as we are looking at the square of the Faraday rotation angle
derivative. However, while the Faraday rotation signal is still proportional
to $E^{2}$ in Eqs. (16) and (18), the bias signal of Eq. (12) has now
disappeared. So the sensitivity can be recovered simply by increasing the
power of the input laser.

###  3.3 Faraday rotation measurement using beam shaping

Most high power laser beams can have some degree of controlled heterogeneity
[24]. In this case, Eq. (8) yields

$\mathbf{E}^{\prime}(x,y)=E\left[\begin{array}[]{*{20}{c}}{\cos\theta\partial_{x}E/E-\sin\theta\partial_{x}\theta+\sin\theta\partial_{y}E/E+\cos\theta\partial_{y}\theta}\\\
{\cos\theta\partial_{y}E/E-\sin\theta\partial_{y}\theta-\sin\theta\partial_{x}E/E-\cos\theta\partial_{x}\theta}\end{array}\right].$
(19)

To recover the sensitivity obtained by the setup of Fig. 2, we take
$\theta_{i}=0$. In this case, again supposing $\theta_{FR}<<1$, Eq. (19)
simplifies to

$\mathbf{E}^{\prime}(x,y)\simeq
E\left[\frac{\partial_{x}E}{E}+\partial_{y}\theta_{FR}\right]\mathbf{x}+E\left[\frac{\partial_{y}E}{E}-\partial_{x}\theta_{FR}\right]\mathbf{y},$
(20)

providing that $\partial_{y}\theta_{FR}/\theta_{FR}>>\partial_{y}E/E$. This
condition is fulfilled when the magnitude of E is excessively large compared
to its derivative. Using a polarizing beam splitter, we can record the
intensity of the $x$-polarization given by

$I_{x}^{\prime}(x,y)\simeq|E\partial_{x}E\left[\partial_{x}E/E+2\partial_{y}\theta_{FR}\right]|,$
(21)

We see that the measurement of the Faraday rotation derivative is now similar
to the standard measurement of section 3.1. The bias is $\partial_{x}E/E$
rather than the "1" seen in Eq. (12) and the Faraday rotation has been
replaced by its derivative. So, a high-power beam with small spatial
dependence will amplify the Faraday rotation $\partial_{y}\theta_{FR}$ while
reducing the inherent bias $\partial_{x}E/E$. Both will improve the signal-to-
noise ratio substantially. Since the intensity of the $y$-polarization is

$I_{y}^{\prime}(x,y)\simeq|E\partial_{y}E\left[\partial_{y}E/E-2\partial_{x}\theta_{FR}\right]|,$
(22)

we can draw similar conclusions.

##  4 Numerical simulations

In the rest of the paper, we use ray tracing[25] with vector Rayleigh-
Sommerfeld (VRS) diffraction [26] to compute the effect of each optical
element on the intensity and polarization of Faraday rotation measurements. We
will use two identical lenses with a diameter of 2 cm and a focal length of 25
cm.

###  4.1 Quasi-homogeneous beam

We use a laser beam profile in a large region of the ray-tracing domain that
is mostly homogeneous and given by:

$E(x,y)=\frac{E_{0}}{4}\left[1+\tanh\left(x_{0}-\frac{|x-x_{c}|}{\sigma}\right)\right]\left[1+\tanh\left(y_{0}-\frac{|y-y_{c}|}{\sigma}\right)\right],$
(23)

where $x_{c}$ and $y_{c}$ correspond to the beam center and $\sigma$ the
transition width to 0. With the given parameters the full-width-half-maximum
is 2$x_{0}$ along the $x$-axis and 2$y_{0}$ along the $y$-axis. Due to the
effect of boundary conditions, we keep 2$x_{0}$ and 2$y_{0}$ on the order of
75% of the computational domain width along the respective directions. The
spatial dependence of E is shown in Figs. 3 and 4. The Faraday rotation given
by

$\theta_{FR}(x,y,M)=M\pi\left(\left[\frac{x-x_{min}}{x_{max}-x_{min}}-\frac{1}{2}\right]^{2}+\left[\frac{y-y_{min}}{y_{max}-y_{min}}-\frac{1}{2}\right]^{3}\right),$
(24)

where the scaling factor $M$ allows to switch between large and small
rotation, $x_{min}$ and $x_{max}$ are the domain limits along the $x$-axis,
while $y_{min}$ and $y_{max}$ are the domain limits along the $y$-axis. This
analytical expression will help to identify the FR feature unequivocally in
the downstream beam and verify Eq. (8).

Figure 3: The electric field of the laser beam downstream of the setup of Fig
1 along the $x$-axis for the $y$-polarization (left) and along the $y$-axis
for the $x$-polarization (right) for M=1 computed using VRS diffraction. The
analytic values of the field are plotted using dotted lines. The magnitude of
the initial electric field is indicated in blue.

For M small (i.e. 1), the downstream electric field should follow Eqs. (13)
and (14), as plotted on Fig. 3. For small rotation angle (i.e. M=$10^{-3}$) we
expect a signal given by Eqs. (15) and (17). Indeed we recover the linear
dependence of the derivative along the $x$-axis, shown on the left hand side
of Fig. 4, and the quadratic dependence of the derivative along the $y$-axis,
shown on the right hand side of Fig. 4.

Figure 4: The electric field of the laser beam downstream of the setup of Fig
1 along the $x$-axis for the $y$-polarization (left) and along the $y$-axis
for the $x$-polarization (right) computed using VRS diffraction. The analytic
values are represented with dotted lines for small rotation angle(i.e.
M=$10^{-3}$), they are approximately $\partial_{x}\theta_{FR}$ for the
$y$-polarization (left) and $-\partial_{y}\theta_{FR}$ for the
$x$-polarization (right). The magnitude of the initial electric field is
indicated in blue.

Here we scaled the analytical value of the solution given by Eqs. (13) and
(14) to the $x$-polarization of the output beam profile of Fig. 3 and kept
this scaling constant throughout the simulation section.

###  4.2 Shaped beam

As Eqs. (21), (22) show, it is possible to recover the sensitivity of the
standard method given by Eq. (12). However, this can only be achieved if the
beam has well-defined spatial variations. In fact, these variations should be
as simple as possible to allow for an easy measurement of the rotation. While
we are using a laser for getting large intensities allowing to amplify the
rotation signal, the coherence of the beam is not primordial as the
S-waveplate affects the phase signal only by adding a constant. As a result,
the beam profile can be controlled using engineered diffusers [24] with an
extremely high level of precision. We need to use here a diffuser that
scrambles the phase rather than the polarization.

Figure 5: The electric field of the laser beam downstream of the setup of Fig
1 along the $x$-axis for the $y$-polarization (left) and along the $y$-axis
for the $x$-polarization (right) computed using VRS diffraction. The analytic
values are represented with dotted lines for a small rotation angle (i.e.
M=$10^{-3}$). The initial electric field is indicated in blue. The horizontal
dashed line are used to highlight the variation of the electric field of the
initial beam.

To this end, we now add a small, linear variation to the previous beam, e.g.

$\displaystyle E(x,y)=$
$\displaystyle\frac{E_{0}}{4}\left[1+\tanh\left(x_{0}-\frac{|x-x_{c}|}{\sigma}\right)\right]\left[1+\tanh\left(y_{0}-\frac{|y-y_{c}|}{\sigma}\right)\right]$
$\displaystyle\times\left[|\alpha_{x}x+x_{bias}|+|\alpha_{y}y+y_{bias}|\right].$
(25)

If the variation is too large, then we could be back with a case similar to
Eq. (12), where the bias decreases the signal to noise ratio. However, if this
is deemed necessary, we can reduce the impact on the Faraday rotation signal
simply by increasing the base electric field. In this case the bias terms
$\partial_{x}E/E$ and $\partial_{y}E/E$ in Eqs. (21) and (22) can be made
small compared to the Faraday rotation signal, while the amplification terms
$E\partial_{x}E$ and $E\partial_{y}E$ can be made large, de facto amplifying
the Faraday rotation signal. Using the same Faraday rotation profile of Eq.
(24), Fig. 5 shows that ray tracing confirms Eq. (20). While not plotted,
numerical simulations confirm both Eqs. (16) and (18).

###  4.3 Noisy shaped beam

It is now time to see what can be done about the beam noise. Clearly this is
an open problem so far, at least for the low frequency noise. Yet, several
techniques have shown good mitigation. For instance, low frequency noise can
be removed using spectral dispersion [27, 28] or micro-lens arrays [29, 30].
Polarization smoothing [31] cannot be used here since we are measuring
polarization.

High frequency intensity noise can be removed more easily. This is simply done
by placing an aperture in the Fourier plane of the setup shown in Fig. 1[32].
This can be combined directly inside the neutral density filter of Eq. (4),
setting the transmission to zero at the filter periphery. Fig. 6 shows that a
high frequency noise up to 10% has almost no impact on the Faraday rotation
measurement, once filtered optically.

Figure 6: The electric field of the laser beam downstream of the setup of Fig
1 along the $x$-axis for the $y$-polarization (left) and along the $y$-axis
for the $x$-polarization (right) using VRS. The expected values are plotted
using dotted lines for a small rotation angle (i.e. M=$10^{-3}$). The initial
electric field is indicated in blue and has a noise amplitude of 10%.

##  5 Conclusions

This work shows that the combination of an S-waveplate with a linearly varying
neutral density filter allows to compute the spatial derivative of the Faraday
rotation angle. The $y$-polarization exiting the setup will carry the partial
derivative along the $x$-axis, while the $x$-polarization will carry the
partial derivative along $y$. In a standard setup, where the Faraday rotation
signal appears on top of the beam carrier, increasing the beam energy does not
necessarily improve the Faraday rotation signal, except if the background
light from the medium (i.e. the continuum) is relatively bright. In this case,
larger intensities reduced the impact of the background noise. In the proposed
setup, the Faraday rotation derivative is proportional to the beam energy. So,
the laser acts as an amplifier, allowing to increase the signal simply by
increasing the beam energy. In real life situations, the spatial variation of
the beam now adds to the Faraday rotation signal, as shown by Eq. (8). So it
is imperative to smooth the beam to allow the signal. One possible method is
simply to expand the beam, possibly beyond the optical system, to reduce
unwanted gradients as much as possible near the optical axis. However, high
frequency noise can be removed using spatial filtering, providing that the
Faraday rotation signal varies much more slowly than the beam noise.

## Funding

This research was supported by the NSF CAREER Award PHY-1943939 and by the
Laboratory for Laser Energetics Horton Fellowships.

## Disclosures

The authors declare no conflicts of interest.

## Data availability

The data is available upon request to the first author.

## References

  * [1] J. M. Stone and W. Wintringham, “Radiation and optics,” (1964).
  * [2] L. Aplet and J. W. Carson, “A faraday effect optical isolator,” Applied Optics 3, 544–545 (1964).
  * [3] I. B. Zeldovich, A. A. Ruzmaikin, and D. D. Sokolov, “Magnetic fields in astrophysics,” New York 3 (1983).
  * [4] J. A. Crowther, _The life and discoveries of Michael Faraday_ (Society for promoting Christian knowledge, 1918).
  * [5] T. Pisarczyk, A. Rupasov, G. Sarkisov, and A. Shikanov, “Faraday-rotation method for magnetic-field diagnostics in a laser plasma,” Journal of Soviet Laser Research 11, 1–32 (1990).
  * [6] A. Gray, T. Landecker, P. Dewdney, and A. Taylor, “A large-scale, interstellar faraday-rotation feature of unknown origin,” Nature 393, 660–662 (1998).
  * [7] R. Manchester, “Pulsar rotation and dispersion measures and the galactic magnetic field.” Astrophysical Journal, vol. 172, p. 43 172, 43 (1972).
  * [8] R. Bachiller, “Bipolar molecular outflows from young stars and protostars,” Annual Review of Astronomy and Astrophysics 34, 111–154 (1996).
  * [9] A. Königl, “On the interpretation of the apparent existence of a preferred magnetic polarity in extragalactic jet sources,” Monthly Notices of the Royal Astronomical Society: Letters 407, L79–L83 (2010).
  * [10] P.-A. Gourdain and C. Seyler, “Impact of the hall effect on high-energy-density plasma jets,” Physical review letters 110, 015002 (2013).
  * [11] F. Suzuki-Vidal, S. V. Lebedev, A. Ciardi, S. Bland, J. P. Chittenden, G. N. Hall, A. Harvey-Thompson, A. Marocchino, C. Ning, C. Stehle _et al._ , “Formation of episodic magnetically driven radiatively cooled plasma jets in the laboratory,” Astrophysics and Space Science 322, 19–23 (2009).
  * [12] P.-A. Gourdain, I. Blesener, J. Greenly, D. Hammer, P. Knapp, B. Kusse, and P. Schrafel, “Initial experiments using radial foils on the cornell beam research accelerator pulsed power generator,” Physics of Plasmas 17 (2010).
  * [13] G. Swadling, S. Lebedev, G. Hall, S. Patankar, N. Stewart, R. Smith, A. Harvey-Thompson, G. Burdiak, P. de Grouchy, J. Skidmore _et al._ , “Diagnosing collisions of magnetized, high energy density plasma flows using a combination of collective thomson scattering, faraday rotation, and interferometry,” Review of Scientific Instruments 85 (2014).
  * [14] J. Zhang, N. Crocker, T. Carter, S. Kubota, and W. Peebles, “Interaction between faraday rotation and cotton–mouton effects in polarimetry modeling for nstx,” Review of Scientific Instruments 81 (2010).
  * [15] M. Beresna, M. Gecevičius, P. G. Kazansky, and T. Gertus, “Radially polarized optical vortex converter created by femtosecond laser nanostructuring of glass,” Applied Physics Letters 98 (2011).
  * [16] T. Bauer, S. Orlov, U. Peschel, P. Banzer, and G. Leuchs, “Nanointerferometric amplitude and phase reconstruction of tightly focused vector beams,” Nature Photonics 8, 23–27 (2014).
  * [17] Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photon. 1, 1–57 (2009).
  * [18] E. J. Galvez, S. Khadka, W. H. Schubert, and S. Nomoto, “Poincaré-beam patterns produced by nonseparable superpositions of Laguerre–Gauss and polarization modes of light,” Appl. Opt. 51, 2925–2934 (2012).
  * [19] R. W. Boyd and M. J. Padgett, “Quantum mechanical properties of light fields carrying orbital angular momentum,” in _Optics in Our Time,_ (Springer, Cham, 2016), pp. 435–454.
  * [20] J. Goodman, _Introduction to Fourier Optics_ (W.H. Freeman, New York, NY, 2017), 4th ed.
  * [21] P. S. Theocaris and E. E. Gdoutos, _Matrix theory of photoelasticity_ , vol. 11 (Springer, 1979).
  * [22] F. S. Roux, “Geometric phase lens,” JOSA A 23, 476–482 (2006).
  * [23] P.-A. Gourdain, I. Erez, M. Evans, H. Hasson, J. Nagasako, J. Young, and I. West-Abdallah, “True optical spatial derivatives for direct phase gradient measurements,” Optics Continuum 2, 838–846 (2023).
  * [24] F. M. Dickey, _Laser beam shaping: theory and techniques_ (CRC press, 2018).
  * [25] L. S. Brea, “Diffractio, python module for diffraction and interference optics,” (2019).
  * [26] F. Shen and A. Wang, “Fast-fourier-transform based numerical integration method for the Rayleigh-Sommerfeld diffraction formula,” Applied optics 45, 1102–1110 (2006).
  * [27] S. Skupsky, R. Short, T. Kessler, R. Craxton, S. Letzring, and J. Soures, “Improved laser-beam uniformity using the angular dispersion of frequency-modulated light,” Journal of Applied Physics 66, 3456–3462 (1989).
  * [28] S. P. Regan, J. A. Marozas, J. H. Kelly, T. R. Boehly, W. R. Donaldson, P. A. Jaanimagi, R. L. Keck, T. J. Kessler, D. D. Meyerhofer, W. Seka _et al._ , “Experimental investigation of smoothing by spectral dispersion,” JOSA B 17, 1483–1489 (2000).
  * [29] M. Zimmermann, N. Lindlein, R. Voelkel, and K. J. Weible, “Microlens laser beam homogenizer: from theory to application,” in _Laser Beam Shaping VIII,_ vol. 6663 (SPIE, 2007), pp. 9–21.
  * [30] J. Meng, J. He, M. Huang, Y. Li, Z. Mo, S. Dai, and Y. Liu, “Partially coherent beam smoothing using a microlens array,” Optics Express 29, 44045–44062 (2021).
  * [31] T. Boehly, V. Smalyuk, D. Meyerhofer, J. Knauer, D. Bradley, R. Craxton, M. Guardalben, S. Skupsky, and T. Kessler, “Reduction of laser imprinting using polarization smoothing on a solid-state fusion laser,” Journal of applied physics 85, 3444–3447 (1999).
  * [32] P. Elias, D. S. Grey, and D. Z. Robinson, “Fourier treatment of optical processes,” J. Opt. Soc. Am. 42, 127–134 (1952).