aainstitutetext: Zhejiang Institute of Modern Physics, Zhejiang University,
Hangzhou, 310027, P. R. China bbinstitutetext: Hangzhou Institute of Advanced
Study, UCAS, Hangzhou, 310027, P.R. Chinaccinstitutetext: Interdisciplinary
Center for Theoretical Study, University of Science and Technology of China,
Hefei, Anhui 230026, Chinaddinstitutetext: Peng Huanwu Center for Fundamental
Theory, Hefei, Anhui 230026, China

# One-loop Feynman Integral Reduction by Differential Operators

Chang Hu, a,1 Tingfei Li,111Corresponding author. c,d Xiaodi Li
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

For loop integrals, the standard method is reduction. A well-known reduction
method for one-loop integrals is the Passarino-Veltman reduction. Inspired by
the recent paper Feng:2021enk  where the tadpole reduction coefficients have
been solved, in this paper we show the same technique can be used to give a
complete integral reduction for any one-loop integrals. The differential
operator method is an alternative version of the PV-reduction method. Using
this method, analytic expressions of all reduction coefficients of the master
integrals can be given by algebraic recurrence relation easily. We demonstrate
our method explicitly with several examples.

###### Keywords:

One-loop Feynman integral, Integral Reduction, Differential Operator,
Recursion Relation

††preprint: USTC-ICTS/PCFT-21-30

##  1 Introduction

The calculation of scattering amplitude at higher loop level is always like a
chronic disease to block the evolution of High Energy Physics. Theoretical
physicists have made many prescriptions to cure this problem started in the
1970s. The most significant receipt is to reduce a loop amplitude into a
linear combination of some scalar master integrals (MIs) under dimensional
regularization Brown:1952eu ; Melrose:1965kb ; Passarino:1978jh ;
tHooft:1978jhc ; vanNeerven:1983vr ; Stuart:1987tt ; vanOldenborgh:1989wn ;
Bern:1992em ; Bern:1993kr ; Fleischer:1999hq ; Binoth:1999sp ; Denner:2002ii ;
Duplancic:2003tv ; Denner:2005nn ; Ellis:2007qk ; Ossola_2007 ; Bern:1994cg

$\displaystyle
I^{1-loop}=\sum_{i_{d_{0}+1}}C^{i_{d_{0}+1}}I_{d_{0}+1}^{i_{d_{0}+1}}+\sum_{i_{d_{0}}}C^{i_{d_{0}}}I_{d_{0}}^{i_{d_{0}}}+\cdots+\sum_{i_{1}}C^{i_{1}}I_{1}^{i_{1}},$
(1)

where $i_{s}$ is the set of propagators appearing in the master integrals. The
coefficient $C^{i_{s}}$ ($s=1,\cdots,d_{0}+1$) is simply a rational function
of some Lorentz invariant such like the scalar product of external momenta
while the terms $I_{s}^{i_{s}}$ are the $s$-gon scalar integral. With the
general expansion (1), the computation of general one-loop amplitudes has been
switched to determining those coefficients of $C^{i_{s}}$. Many tools have
been invented to shovel the brambles, such as integration-by-parts (IBP)
Chetyrkin:1981qh ; Tkachov:1981wb , PV reduction Passarino:1978jh , OPP
reduction Ossola:2006us ; Ossola:2007bb ; Ellis:2007br ; Ossola_2007 ,
Unitarity cut Bern:1994zx ; Bern:1994cg ; Britto:2004nc ; Britto:2005ha ;
Campbell:1996zw ; Bern:1997sc ; Denner:2005nn ; Anastasiou:2006gt ;
Britto:2010um  etc.

All these methods can be divided into two categories, i.e., the reduction at
the integrand level or the integral level. For reduction at the integrand
level, Ossola_2007  shows how to extract the coefficients of the 4-, 3-, 2-
and 1-point one-loop scalar integrals from the full one-loop integrand of
arbitrary scattering processes in an algebraical way. For the reduction at the
integral level, an efficient way is the unitarity cut method. The main idea is
to compare the imaginary part of two side of (1). However, since the loop-
integral is well-defined using the dimensional regularization, the unitarity
cut method in pure 4D need to generalize to $(4-2\epsilon)$-dimension, which
has been done in Anastasiou:2006jv ; Anastasiou:2006gt . Based on this
generalization, the analytic expressions for reduction coefficients (except
the tadpole coefficients) have been derived in a series of papers
Britto:2006fc ; Britto:2007tt ; Britto:2008vq ; Britto:2008sw ; Feng:2013sa .

In our previous work Feng:2021enk , we reconsider the problem by introducing
differential operators $\mathcal{D}$ and $\mathcal{T}$. We first introduce an
auxiliary vector $R^{\mu}$ and reduce it to master integrals, then applying
differential operators to the integrals with respect to $R$. By comparing two
sides of the expansion, We will achieve the recursion relations of those
coefficients of master integrals in differential form. With the knowledge of
the algebraical structure of the reduction coefficients, we transform those
differential equation form relations into algebraical form. In Feng:2021enk ,
we solve the remaining unsolved tadpole coefficients by this method. In this
paper, we will provide a general algorithm for calculating all reduction
coefficients for a general tensor one-loop integral and give the explicit
analytic results.

In section 2, we will review the derivation of the differential equations of
reduction coefficients and show how to get the recursion relations of
expansion coefficients. In section 3, we will solve the recursion relations of
reduction coefficients in the general case. In section 4, we provide some
examples and summarize the algorithm for calculating the reduction of a
general tensor one-loop Feynman integral. In appendix A, we list all reduction
coefficients for tensor triangles, boxes and pentagons with rank $1$ and $2$.

##  2 Differential equations and recursion relations

We will review the differential operators in Feng:2021enk  and show the way we
obtain the recursion relations of every reduction coefficients. Starting with
the following general one-loop $m$-rank tensor integral with $n+1$ propagators

$\displaystyle
I^{\mu_{1}\cdots\mu_{m}}_{n+1}=\int{d^{D}\ell\over(2\pi)^{D}}\frac{\ell^{\mu_{1}}\ell^{\mu_{2}}\cdots\ell^{\mu_{m}}}{P_{0}P_{1}\cdots
P_{n}},~{}$ (2)

where the $i$-th propagator is $P_{i}=(\ell-K_{i})^{2}-M_{i}^{2}$ with setting
$K_{0}=0$, we introduce an auxiliary vector $R^{\mu}$ and contract $\ell$ with
$R$ on (2) $m$ times to get

$\displaystyle I^{(m)}_{n+1}[R]\equiv$ $\displaystyle
2^{m}I^{\mu_{1}\cdots\mu_{m}}_{n+1}R_{\mu_{1}}\cdots
R_{\mu_{m}}=\int{d^{D}\ell\over(2\pi)^{D}}{(2\ell\cdot R)^{m}\over
P_{0}P_{1}\cdots P_{n}}.~{}$ (3)

The vector $R$ lies in the same dimension as the $\ell$ does. The (3) contains
all information as in (2) but with much simpler organization of tensor
structure. In this paper, we will focus on $D=(4-2\epsilon)$-dimensional
space, although our method can obviously be applied to arbitrary dimension.
With this assumption, the integral $I^{(m)}_{n+1}[R]$ is reduced to the linear
combination of pentagon, box, triangle, bubble and tadpole master integrals

$\displaystyle I^{(m)}_{n+1}[R]=$
$\displaystyle\sum_{a_{1},a_{2},\cdots,a_{5}}C^{(a_{1},a_{2},\cdots,a_{5})}(m|n)I_{5}[a_{1},a_{2},\cdots,a_{5}]$
$\displaystyle+\sum_{a_{1},a_{2},a_{3},a_{4}}C^{(a_{1},a_{2},a_{3},a_{4})}(m|n)I_{4}[a_{1},a_{2},a_{3},a_{4}]+\cdots+\sum_{a_{1}}C^{(a_{1})}(m|n)I_{1}[a_{1}].$
(4)

The reduction coefficients $C^{(a_{1},\cdots,a_{r})}(m|n),1\leq r\leq 5$ are
the rational functions of external momenta $K_{i}$, masses $M_{i}$, and vector
$R$. The summation in (4) covers all possible combinations of $r$ propagators
$\\{P_{a_{1}},,,P_{a_{r}}\\}\subseteq\\{P_{0},P_{1},...,P_{n}\\}$. We will use
the abbreviation $C^{i_{r}}(m|n)$ instead of $C^{(a_{1},...,a_{r})}(m|n)$ with
index set $i_{r}=\\{a_{1},...,a_{r}\\}\subseteq\\{0,1,2,...,n\\}$. It is easy
to see vector $R$ only appears in the numerator of $C^{i_{r}}$ with the form
$R\cdot R$ or $R\cdot K_{i},\ i=1,2,...,n$. We introduce the following two
operators

$\displaystyle\mathcal{D}_{i}\equiv K_{i}\cdot{\partial\over\partial
R},~{}~{}i=1,...,n;~{}~{}~{}~{}\mathcal{T}\equiv\eta^{\mu\nu}{\partial\over\partial
R^{\mu}}{\partial\over\partial R^{\nu}}.$ (5)

We take the derivative on the both sides of (4) by these two operators. The
left-hand side will be

$\displaystyle\mathcal{D}_{i}I^{(m)}_{n+1}[R]$
$\displaystyle=mI^{(m-1)}_{n+1;\widehat{0}}-mI^{(m-1)}_{n+1;\widehat{i}}+mf_{i}I^{(m-1)}_{n+1},$
$\displaystyle\mathcal{T}I^{(m)}_{n+1}[R]$
$\displaystyle=4m(m-1)M_{0}^{2}I^{(m-2)}_{n+1}+4m(m-1)I^{(m-2)}_{n+1;\widehat{0}},~{}~{}~{}$
(6)

where the constant $f_{i}\equiv M^{2}_{0}+K^{2}_{i}-M^{2}_{i}$, and

$I^{(m-1)}_{n+1;\widehat{i}}[R]=\int{d^{D}\ell\over(2\pi)^{D}}{(2\ell\cdot
R)^{m-1}\over P_{0}P_{1}\cdots P_{i-1}P_{i+1}\cdots P_{n}}$ (7)

i.e., the $i$-th propagator has been removed. For the right-hand side of (4),
since the master integrals contains no $R$, the operators will directly act on
coefficients $C^{i_{r}}(m|n)$. Therefore, We have the following equations

$\displaystyle\sum_{s=1}^{5}\sum_{i_{s}}\left(\mathcal{D}_{i}C^{i_{s}}(m|n)\right)I_{s}^{i_{s}}=$
$\displaystyle
m\sum_{s=1}^{5}\sum_{i_{s}}\left(C^{i_{s}}(m-1|n;\widehat{0})-C^{i_{s}}(m-1|n;\widehat{i})+f_{i}C^{i_{s}}(m-1|n)\right)I_{s}^{i_{s}},$
(8)

and

$\displaystyle\sum_{s=1}^{5}\sum_{i_{s}}\left(\mathcal{T}C^{i_{s}}(m|n)\right)I_{s}^{i_{s}}=$
$\displaystyle
4m(m-1)\sum_{s=1}^{5}\sum_{i_{s}}\left(C^{i_{s}}(m-2|n;\widehat{0})+M_{0}^{2}C^{i_{s}}(m-2|n)\right)I_{s}^{i_{s}},$
(9)

where $C^{i_{s}}(m-1;\widehat{i})$ is the coefficient of the master integrals
$I^{i_{s}}_{s+1}$ in the reduced expansion of tensor integral
$I^{(m-1)}_{n+1;\widehat{i}}[R]$. Assuming that all the reduction coefficient
of tensor integral $I^{(m^{\prime})}_{n^{\prime}+1}$ with either
$m^{\prime}<m$ or $n^{\prime}<n$ are known already, we can get a series of
differential equations of $C^{i_{r}}(m|n)$ by comparing the coefficients of
each master integral of both side of (8) and (9). Without loss of generality,
we choose $i_{r}=(0,1,...,r)$222For simplicity, we will consider the reduction
coefficients of master integrals with propagator $P_{0}$. Other cases can be
obtained either by permutations or by momentum shifting. The details are shown
in section 3.2. . Then we have

$\displaystyle\mathcal{T}C^{(0,1,\cdots,r)}(m|n)=$ $\displaystyle
4m(m-1)M_{0}^{2}C^{(0,1,\cdots,r)}(m-2|n),$ (10)

and

$\displaystyle\mathcal{D}_{i}C^{(0,1,\cdots,r)}(m|n)=$ $\displaystyle-
mC^{(0,1,\cdots,r)}(m-1|n;\widehat{i})+mf_{i}C^{(0,1,\cdots,r)}(m-1|n).$ (11)

In the equation (11), $C^{(0,1,\cdots,r)}(m-1|n;\widehat{i})$ is the reduction
coefficient of the master integral $I_{r+1}^{(0,1,...,r)}$. Since
$\widehat{i}$ means the propagator $P_{i}$ has been removed,
$C^{(0,1,\cdots,r)}(m-1|n;\widehat{i})$ is zero when $i\leq r$.

Similar to the idea used in Feng:2021enk , we do not solve the differential
equations directly, but expand the reduction coefficients according to its
tensor structure

$\displaystyle C^{(0,1,\cdots,r)}(m|n)=$
$\displaystyle\sum_{2a_{0}+\sum_{k=1}^{n}a_{k}=m}\Bigg{\\{}c^{(0,1,\cdots,r)}_{a_{0},a_{1},\cdots,a_{n}}(m)(M_{0}^{2})^{a_{0}+r-n}\prod_{k=0}^{n}s_{0k}^{a_{k}}\Bigg{\\}},$
(12)

where the notation $s_{00}\equiv(R\cdot R),s_{0i}\equiv(R\cdot K_{i})$. The
summation condition $2a_{0}+\sum_{k=1}^{n}a_{k}=m$ guarantees vector $R$
appears $m$ times. The exponent of $M^{2}_{0}$ makes
$c^{(0,1,\cdots,r)}_{a_{0},a_{1},\cdots,a_{n}}(m)$ dimensionless. The
expansion coefficients
$c^{(0,1,\cdots,r)}_{a_{0},a_{1},\cdots,a_{n}}(m)$333Note that we use capital
$C$ to represent reduction coefficient while use lower case letter $c$ to
represent expansion coefficient.  can only be a rational function of
$(K_{i}\cdot K_{j}),i,j\neq 0$ and $M_{i}^{2},(i=0,1,\cdots,n)$. Moreover,
$c^{(0,1,\cdots,r)}_{a_{0},a_{1},\cdots,a_{n}}(m)$ vanish if any
$a_{k}<0,k=0,1,\cdots,n$.

For $C^{(0,1,...,r)}(m-1|n;\widehat{i})$ in (11), the expansion is

$\displaystyle C^{(0,1,\cdots,r)}(m-1|n;\widehat{i})$ $\displaystyle=$
$\displaystyle\sum_{2a_{0}+\sum_{k=1,k\neq
i}^{n}a_{k}=m-1}c^{(0,1,\cdots,r)}_{a_{0},\cdots,a_{i-1},a_{i+1},\cdots,a_{n}}(m-1;\widehat{i})(M_{0}^{2})^{a_{0}+r-n}\prod_{k=0,k\neq
i}^{n}s_{0k}^{a_{k}}$ $\displaystyle=$
$\displaystyle\sum_{2a_{0}+\sum_{k=1}^{n}a_{k}=m-1}\delta_{0a_{i}}~{}c^{(0,1,\cdots,r)}_{a_{0},\cdots,\widehat{a_{i}},\cdots,a_{n}}(m-1;\widehat{i})~{}(M_{0}^{2})^{a_{0}+r-n}\prod_{k=0}^{n}s_{0k}^{a_{k}}.$
(13)

The absence of term $s_{0i}=(R\cdot K_{i})$ is because propagator $P_{i}$ has
been removed. In the last line of (13), we add a factor $\delta_{0a_{i}}$ to
make the expression simpler. The subscript $\widehat{a_{i}}$ means index
$a_{i}$ is absent.

To get the algebraic recursion relation for expansion coefficients, we need
rewrite $\mathcal{D}_{i}$ and $\mathcal{T}$ in terms of

$\displaystyle\mathcal{D}_{i}=$ $\displaystyle
K_{i}^{\mu}{\partial\over\partial R^{\mu}}=2s_{0i}{\partial\over\partial
s_{00}}+\sum_{j=1}^{n}s_{ij}{\partial\over\partial s_{0j}},$
$\displaystyle\mathcal{T}=$ $\displaystyle 2D{\partial\over\partial
s_{00}}+4s_{00}{\partial^{2}\over\partial
s_{00}^{2}}+4\sum_{i=1}^{n}s_{0i}{\partial\over\partial
s_{0i}}{\partial\over\partial
s_{00}}+\sum_{i=1}^{n}\sum_{j=1}^{n}s_{ij}{\partial\over\partial
s_{0i}}{\partial\over\partial s_{0j}}.~{}~{}~{}~{}$ (14)

With above explanation, putting (12) and (13) to (10) and (11), comparing the
expansion coefficients of $\prod_{k=0}^{n}s_{0k}^{a_{k}}$, the two types of
differential equations (10) and (11) become

$\displaystyle(m+1-\sum_{l=1}^{n}i_{l})c^{(0,1,\cdots,r)}_{a_{1},\cdots,a_{i}-1,\cdots,a_{n}}(m)+\sum_{l=1}^{n}(a_{l}+1)\beta_{il}c^{(0,1,\cdots,r)}_{a_{1},\cdots,a_{l}+1,\cdots,a_{n}}(m)$
$\displaystyle=m\alpha_{j}c^{(0,1,\cdots,r)}_{a_{1},\cdots,a_{n}}(m-1)-m\delta_{0a_{i}}c^{(0,1,\cdots,r)}_{a_{1},\cdots,\widehat{a_{i}},\cdots,a_{n}}(m-1;\widehat{i})$
(15)

for the ${\cal D}$-type and

$\displaystyle 4m(m-1)c^{(0,1,\cdots,r)}_{a_{1},\cdots,a_{n}}(m-2)=$
$\displaystyle(m-\sum_{k=1}^{n}a_{k})(D+m+\sum_{k=1}^{n}a_{k}-2)c^{(0,1,\cdots,r)}_{a_{1},\cdots,a_{n}}(m)$
$\displaystyle+\sum_{0<i<j}2(a_{i}+1)(a_{j}+1){\beta}_{ij}c^{(0,1,\cdots,r)}_{a_{1},\cdots,a_{i}+1,\cdots,a_{j}+1,\cdots,a_{n}}(m)$
$\displaystyle+\sum_{i=1}^{n}(a_{i}+1)(a_{i}+2){\beta}_{ii}c^{(0,1,\cdots,r)}_{a_{1},\cdots,a_{i}+2,\cdots,a_{n}}(m)$
(16)

for the ${\cal T}$-type, where $\alpha_{i}\equiv
f_{i}/M_{0}^{2},\beta_{il}\equiv s_{il}/M_{0}^{2}$ for simplicity. Again we
need to emphasize
$c^{(0,1,\cdots,r)}_{a_{1},\cdots,\widehat{a}_{i},\cdots,i_{n}}(m-1;\widehat{i})=0$
in the case $i\leq r$ for the same reason as discussed before. In (15) and
(16) we have ignored the subscript $a_{0}$ because it has been uniquely
determined by the restrictive condition $2a_{0}+\sum_{k=1}^{n}a_{n}=m$ in
(12).

##  3 Algorithm for recursion relations

The recurrence relations (15) and (16) are the key relations throughout the
whole paper. In this section we show how to solve expansion coefficients by
these two relations systematically.

###  3.1 Reduction coefficient of $I_{r+1}[0,1,\cdots,r]$

First, we choose the master basis contains propagator
$\\{P_{0},P_{1},\cdot,P_{r}\\}$. We start by rewriting $\mathcal{D}$-type
relations (15) in a compact form as

$\displaystyle\widetilde{\boldsymbol{G}}~{}\boldsymbol{T}~{}\boldsymbol{c}^{(0,1,\cdots,r)}(a_{1},\cdots,a_{n};m)=\boldsymbol{O}^{(0,1,\cdots,r)}(a_{1},\cdots,a_{n};m),~{}~{}~{}$
(17)

where $\widetilde{\boldsymbol{G}}=[\beta_{ij}]$ is the $n\times n$ rescaled
Gram matrix and $\boldsymbol{T}$ is a diagonal matrix

$\displaystyle\boldsymbol{T}=\text{diag}(a_{1}+1,a_{2}+1,\cdots,a_{n}+1).$
(18)

The
$\boldsymbol{c}^{(0,1,\cdots,r)}(a_{1},\cdots,a_{n};m),\boldsymbol{O}^{(0,1,\cdots,r)}(a_{1},\cdots,a_{n};m)$
are two vectors defined as

$\displaystyle[\boldsymbol{c}^{(0,1,\cdots,r)}(a_{1},\cdots,a_{n};m)]_{i}=c^{(0,1,\cdots,r)}_{a_{1},a_{2},\cdots,a_{i}+1,\cdots,a_{n}}(m),$
(19)

and

$\displaystyle[\boldsymbol{O}^{(0,1,\cdots,r)}(a_{1},\cdots,a_{n};m)]_{i}=$
$\displaystyle
m\alpha_{i}c^{(0,1,\cdots,r)}_{a_{1},\cdots,a_{n}}(m-1)-m\delta_{0a_{i}}c^{(0,1,\cdots,r)}_{a_{1},\cdots,\widehat{a}_{i},\cdots,a_{n}}(m-1;\widehat{i})$
$\displaystyle-(m+1-\sum_{l=1}^{n}a_{l})c^{(0,1,\cdots,r)}_{a_{1},\cdots,a_{i}-1,\cdots,a_{n}}(m).$
(20)

The definition of these two vectors are purposely for the recurrence
construction. The vector $\boldsymbol{c}$ contains coefficients with rank $m$
and subscript with the summation $1+\sum_{i}a_{i}$, while the vector
$\boldsymbol{O}$ contains coefficients of three different patterns: (1) the
first term with coefficients of rank $m-1$; (2) the second one with
coefficients of master integrals with one less propagator and lower rank $m-1$
rank; (3) the third one with coefficients of same rank $m$, but the summation
$-1+\sum_{i}a_{i}$ of subscript. By induction assumption, the first two terms
are considered to be known already. Thus, by rewriting (17) as

$\displaystyle\boldsymbol{c}^{(0,1,\cdots,r)}(a_{1},\cdots,a_{n};m)=\boldsymbol{T}^{-1}\widetilde{\boldsymbol{G}}^{-1}\boldsymbol{O}^{(0,1,\cdots,r)}(a_{1},\cdots,a_{n};m),$
(21)

we have established the recurrence relations between expansion coefficients
with higher summation of subscript and those of same rank but with the
summation of subscript reduced by two.

Iteratively using (21), we have left two kinds of unknown expansion
coefficients

$\displaystyle c_{0,0,\cdots,0}^{(0,1,\cdots,r)}(m),$ $\displaystyle m=2k,$
(22) $\displaystyle
c^{(0,1,\cdots,r)}_{1,0,\cdots,0};c^{(0,1,\cdots,r)}_{0,1,\cdots,0};\cdots;c^{(0,1,\cdots,r)}_{0,0,\cdots,1},$
$\displaystyle m=2k+1,$

depending on the parity of $m$. For the odd case $m=2k+1$, we solve
$c^{(0,1,\cdots,r)}_{1,0,\cdots,0}$’s by (21) again. To see it, setting
$a_{1}=\cdots=a_{n}=0$ in (21), the left hand side becomes

$\displaystyle(c^{(0,1,\cdots,r)}_{1,0,\cdots,0}(2k+1),c^{(0,1,\cdots,r)}_{0,1,\cdots,0}(2k+1),\cdots,c^{(0,1,\cdots,r)}_{0,0,\cdots,1}(2k+1))^{T},$
(23)

while the right-hand side is

$\displaystyle\boldsymbol{T}^{-1}\widetilde{\boldsymbol{G}}^{-1}\boldsymbol{O}^{(0,1,\cdots,r)}(0,\cdots,0;2k+1),$
(24)

where

$\displaystyle[\boldsymbol{O}^{(0,1,\cdots,r)}(0,\cdots,0;2k+1)]_{i}=$
$\displaystyle
m\alpha_{i}c^{(0,1,\cdots,r)}_{0,\cdots,0}(2k)-(2k+1)c^{(0,1,\cdots,r)}_{0,\cdots,\widehat{a}_{i},\cdots,0}(2k;\widehat{i})$
(25)

since the third term $c^{(0,1,\cdots,r)}_{0,\cdots,-1,\cdots,0}(2k+1)$
vanishes. Therefore, we reduced to the problem of solving
$c^{(0,1,\cdots,r)}_{0,\cdots,0}(2k)$.

Determining the value of $c_{0,0,\cdots,0}^{(0,1,\cdots,r)}(2k)$ requires
$\mathcal{T}$-type recursion relations. For $m=2k$ and
$a_{1}=a_{2}=\cdots=a_{n}=0$, $\mathcal{T}$-type recursion relation becomes

$\displaystyle 8k(2k-1)c^{(0,1,\cdots,r)}_{0,\cdots,0}(2k-2)=$ $\displaystyle
2k(D+2k-2)c^{(0,1,\cdots,r)}_{0,\cdots,0}(2k)+\sum_{0<i<j<n}2{\beta}_{ij}c^{(0,1,\cdots,r)}_{0,\cdots,1,\cdots,1,\cdots,0}(2k)$
$\displaystyle+\sum_{i=1}^{n}2{\beta}_{ii}c^{(0,1,\cdots,r)}_{0,\cdots,2,\cdots,0}(2k).$
(26)

In $c^{(0,1,\cdots,r)}_{0,\cdots,1,\cdots,1,\cdots,0}(2k)$, indices $1$ appear
in the both $i$-th and $j$-th positions, while in
$c^{(0,1,\cdots,r)}_{0,\cdots,2,\cdots,0}(2k)$ index $2$ appears in the $i$-th
position. For $c^{(0,1,\cdots,r)}_{0,\cdots,2,\cdots,0}(2k)$ and
$c^{(0,1,\cdots,r)}_{0,\cdots,1,\cdots,1,\cdots,0}(2k)$ in (26), we use (21)
again to reach $c^{(0,1,\cdots,r)}_{0,\cdots,0}(2k)$. Then we establish the
relation reduced rank $2k$ to $2k-2$

$\displaystyle c^{(0,1,\cdots,r)}_{\underbrace{0,\cdots,0}_{n\
times}}(2k)={2k-1\over
D+2k-n-2}\left[(4-\boldsymbol{{\alpha}}^{T}\widetilde{\boldsymbol{G}}^{-1}\boldsymbol{{\alpha}})c_{\underbrace{0,\cdots,0}_{n\
times}}^{(0,1,\cdots,r)}(2k-2)+\boldsymbol{{\alpha}}^{T}\widetilde{\boldsymbol{G}}^{-1}\boldsymbol{c}_{\underbrace{0,\cdots,0}_{n-1\
times}}^{(0,1,\cdots,r)}(2k-2)\right],$ (27)

where $\boldsymbol{{\alpha}}$ is a vector define as

$\displaystyle\left(\alpha\right)^{T}$ $\displaystyle=$
$\displaystyle({\alpha}_{1},{\alpha}_{2},\cdots,{\alpha}_{n})^{T}=\left(\frac{f_{1}}{M_{0}^{2}},\frac{f_{2}}{M_{0}^{2}},\cdots,\frac{f_{n}}{M^{2}_{0}}\right)^{T}.$
(28)

In the second term of right-hand side of (27),
$\boldsymbol{c}_{\underbrace{0,\cdots,0}_{n-1\ times}}^{(0,1,\cdots,r)}(m)$ is
a vector defined as

$\displaystyle\left(\boldsymbol{c}_{\underbrace{0,\cdots,0}_{n-1\
times}}^{(0,1,\cdots,r)}(m)\right)^{T}$ $\displaystyle=$
$\displaystyle\left(c_{\underbrace{0,\cdots,0}_{n-1\
times}}^{(0,1,\cdots,r)}(m;\widehat{1}),c_{\underbrace{0,\cdots,0}_{n-1\
times}}^{(0,1,\cdots,r)}(m;\widehat{2}),\cdots,c_{\underbrace{0,\cdots,0}_{n-1\
times}}^{(0,1,\cdots,r)}(m;\widehat{n})\right)^{T}$ (29) $\displaystyle=$
$\displaystyle\left(0,0,\cdots,0,c_{\underbrace{0,\cdots,0}_{n-1\
times}}^{(0,1,\cdots,r)}(m;\widehat{r+1}),\cdots,c_{\underbrace{0,\cdots,0}_{n-1\
times}}^{(0,1,\cdots,r)}(m;\widehat{n})\right)^{T}.$

The zero of first $r$ components has been explained under the Eq(11). Equation
(27) reduced rank $m$ by two. Furthermore, we see a propagator is removed in
the second term of Right-Hand-Side. Therefore, if we utilize the
$\mathcal{T}$-type recursion relation repeatedly, we will end with one of the
following two cases. (1) The rank $m$ is reduced to zero, which related to the
reduction coefficient of a master integral. So it is either 1 or 0. (2) One of
the propagator $P_{i},i\leq r$ has been removed. In this case the coefficients
must be zero because the master integral will not appear in the reduction.

To make a long story short, for $d_{0}=4$, we summarize the whole reduction
process below.

  * •

Step 1: For a tensor integral with more than $5$ propagators, we reduce it to
5-,4-,3-,2-,1-gon tensor integral.

  * •

Step 2: For an arbitrary rank $m_{0}$, we take each $m\leq m_{0}$ with
arrangement from small to large. If $m$ is even, we calculate the expansion
coefficients $c_{a_{1},\cdots,a_{n}}^{(0,1,\cdots,r)}(m)$ in the order
$\sum_{i=1}^{n}a_{i}=0,2,4,\cdots,m$ by using (21) and (26). If $m$ is odd, we
calculate in the order $\sum_{i=1}^{n}a_{i}=1,3,5,\cdots,m$ by (21).

  * •

Step 3: We continue the Step 2 until $m=m_{0}$.

  * •

Final step: Combining all expansion coefficients
$c_{a_{1},\cdots,a_{n}}^{(0,1,\cdots,r)}(m_{0})$, we obtain the reduction
coefficient by (12).

###  3.2 Calculate general $C^{(j_{0},j_{1},\cdots,j_{r})}(m|n)$ from
$C^{(0,1,\cdots,r)}(m|n)$

In this subsection, we will show how to obtain the reduction coefficients of
other MIs from the result of $C^{(0,1,\cdots,r)}(m|n)$. Let us begin with the
case that the Master Integral contains propagators $P_{0}$. It is obvious that
tensor integral $I_{n+1}^{(m)}$ is invariant under a permutation of labels
$\\{1,2,\cdots,n\\}$. Then the reduction coefficients
$C^{(0,j_{1}\cdots,j_{r})}(m|n)$ is simply given by a proper replacement
$\sigma:\\{M_{i},K_{i}\\}\to\\{M_{j_{i}},K_{j_{i}}\\},(i=1,2,\cdots,n)$

$\displaystyle C^{(0,j_{1}\cdots,j_{r})}(m|n)=\sigma C^{(0,1,\cdots,r)}(m|n).$
(30)

Now the remaining part is those MIs without $P_{0}$. Note that by a loop
momenta shift $\ell\to\ell+K_{j_{0}}$ we have

$\displaystyle\begin{aligned}
I_{n+1}^{(m)}[R]\to&\int{d^{D}\ell\over(2\pi)^{D}}{(2\ell\cdot
R+2K_{j_{0}}\cdot
R)^{m}\over(\ell^{2}-M_{j_{0}}^{2})\left[(\ell+K_{j_{0}})^{2}-M_{0}^{2}\right]\prod_{i=1,i\not=j_{0}}\left[(\ell-(K_{i}-K_{j_{0}})^{2}-M_{i}^{2})\right]}\\\
=&\sum_{k=0}^{m}\left(\begin{array}[]{c}m\\\ k\\\ \end{array}\right)(2R\cdot
K_{j_{0}})^{m-k}\times\\\ &\int{d^{D}\ell\over(2\pi)^{D}}{(2\ell\cdot
R)^{k}\over(\ell^{2}-M_{j_{0}}^{2})\left[(\ell+K_{j_{0}})^{2}-M_{0}^{2}\right]\prod_{i=1,i\not=j_{0}}\left[(\ell-(K_{i}-K_{j_{0}})^{2}-M_{i}^{2})\right]}.\end{aligned}$
(31)

By variable substitution $K_{j_{0}}\to-K_{j_{0}},K_{i}\to
K_{i}-K_{j_{0}},M_{j_{0}}\leftrightarrow M_{0}$ inside the integrand444Note
that we don’t substitute $K_{j_{0}}$ in $(2R\cdot K_{j_{0}})^{m-k}$., we
arrive the same form as (2) . Then we have

$\displaystyle C^{(j_{0},j_{1},\cdots,j_{r})}(m|n)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{m}\left(\begin{array}[]{c}m\\\ k\\\
\end{array}\right)(2R\cdot
K_{j_{0}})^{m-k}\left[\left.C^{(0,j_{1},\cdots,j_{r})}(k|n)\right|_{K_{j_{0}}\to-
K_{j_{0}},K_{i}\to K_{i}-K_{j_{0}},M_{j_{0}}\leftrightarrow M_{0}}\right]$
(34) $\displaystyle=$
$\displaystyle\left.\left[\sum_{k=0}^{m}\left(\begin{array}[]{c}m\\\ k\\\
\end{array}\right)(-2R\cdot
K_{j_{0}})^{m-k}C^{(0,j_{1},\cdots,j_{r})}(k|n)\right]\right|_{K_{j_{0}}\to-
K_{j_{0}},K_{i}\to K_{i}-K_{j_{0}},M_{j_{0}}\leftrightarrow M_{0}}.$ (37)

##  4 Examples

Having presented the general algorithm, in this section we will use various
examples to demonstrate the use of the algorithm. In the first subsection, we
will show how to reduce any tensor bubble to the basis of scalar bubble and
two scalar tadpoles. The reduction of tensor triangles, tensor boxes and
tensor pentagons of rank $1$ and $2$ has been given in the Appendix. In the
second subsection, we will show how to get the reduction coefficients of
tensor box with rank $1$ to scalar triangles without $P_{0}$ from the result
of $C^{(0,1,2)}(1|3)$ by (37).

###  4.1 The reduction of tensor bubble

The reduction of tensor bubble $I_{2}^{(m)}$ will contain three MIs as below

$\displaystyle Tadpoles:$ $\displaystyle I_{1}[0],I_{1}[1],$ $\displaystyle
Bubbles:$ $\displaystyle I_{2}[0,1],$ (39)

and we have the expansion

$\displaystyle
I_{2}^{(m)}=C^{(0)}(m|1)I_{1}[0]+C^{(1)}(m|1)I_{1}[1]+C^{(0,1)}(m|1)I_{2}[0,1].$
(40)

The way to achieve $C^{(0)}(m|1)$ have been given in Feng:2021enk . Here we
only provide how to calculate $C^{(1)}(m|1)$ and $C^{(0,1)}(m|1)$. The
coefficient of $I_{1}[1]$ can be obtained by (37) from $C^{(0)}(m|1)$. While
for $I_{2}[0,1]$, there is only one subscript in the expansion coefficients.
So the expansion of $C^{(0,1)}(m|1)$ is

$\displaystyle C^{(0,1)}(m|1)=\sum_{i}c_{i}^{(0,1)}(m)[M_{0}^{2}(R\cdot
R)]^{{m-i\over 2}}(R\cdot K_{1})^{i}.$ (41)

We have the corresponding $\mathcal{D}$-type recursion relation

$\displaystyle c^{(0,1)}_{i+2}(m)$ $\displaystyle=$
$\displaystyle{1\over(i+2)\beta_{11}}\left(m{\alpha}_{1}c^{(0,1)}_{i+1}(m-1)-m\delta_{0,i+1}c^{(0,1)}(m-1)-(m-i)c^{(0,1)}_{i}(m)\right)$
(42) $\displaystyle=$
$\displaystyle{1\over(i+2)\beta_{11}}\left(m{\alpha}_{1}c^{(0,1)}_{i+1}(m-1)-(m-i)c^{(0,1)}_{i}(m)\right),~{}$

and $\mathcal{T}$-type recursion relation

$\displaystyle c_{0}^{(0,1)}(2r)$ $\displaystyle=$
$\displaystyle\frac{2r-1}{2r+D-3}\left[\left(4-\frac{{\alpha}_{1}^{2}}{{\beta}_{11}}\right)c_{0}^{(0,1)}(2r-2)+\frac{{\alpha}_{1}}{{\beta}_{11}}c^{(0,1)}(2r-2)\right]$
(43) $\displaystyle=$
$\displaystyle\frac{2r-1}{2r+D-3}\left(4-\frac{{\alpha}_{1}^{2}}{{\beta}_{11}}\right)c_{0}^{(0,1)}(2r-2),$

where $c^{(0,1)}(m)$ and $c^{(0,1)}(2r-2)$ without subscript stands for
$c^{(0,1)}_{\widehat{a_{1}}}(m)$ and $c^{(0,1)}_{\widehat{a_{1}}}(2r-2)$.
These two terms vanish, because they come from the reduction coefficient of
bubble $I_{2}[0,1]$ for a tensor tadpole, i.e., the propagator $P_{1}$ has
been removed.

Now we show the result for rank $m\leq 4$. The rank $m=0$ is trivial. For
other ranks:

  * •

$m=1$

The reduction coefficients of tadpoles: $I_{1}[0],I_{1}[1]$

Using the result in Feng:2021enk , we have

$\displaystyle C^{(0)}(1|1)=-{R\cdot K_{1}\over K_{1}^{2}}.$ (44)

For $I_{1}[1]$, by choosing $j_{0}=1$ in (37), we have

$\displaystyle C^{(1)}(1|1)$ $\displaystyle=$
$\displaystyle\left.C^{(0)}(1|1)\right|_{K_{1}\to-K_{1},M_{0}\leftrightarrow
M_{1}}$ (45) $\displaystyle=$ $\displaystyle{R\cdot K_{1}\over K_{1}^{2}}.$

The reduction coefficients of bubble: $I_{2}[0,1]$

The expansion of $C^{(0,1)}(1|1)$ becomes

$\displaystyle C^{(0,1)}(1|1)=c_{1}^{(0,1)}(1)(R\cdot K_{1}).$ (46)

By (42), we have

$\displaystyle c_{1}^{(0,1)}(1)$ $\displaystyle=$
$\displaystyle{1\over\beta_{11}}\left({\alpha}_{1}c^{(0,1)}_{0}(0)-2c^{(0,1)}_{-1}(1)\right)$
(47) $\displaystyle=$ $\displaystyle{f_{1}\over s_{11}},$

where the boundary conditions are $c^{(0,1)}_{0}(0)=1,c^{(0,1)}_{-1}(1)=0$.
Then

$\displaystyle C^{(0,1)}(1|1)=c_{1}^{(0,1)}(1)R\cdot
K_{1}=\frac{\left(K_{1}\cdot K_{1}+M_{0}^{2}-M_{1}^{2}\right)R\cdot
K_{1}}{K_{1}\cdot K_{1}}.$ (48)

  * •

$m=2$

The reduction coefficients of tadpoles: $I_{1}[0],I_{1}[1]$

The reduction coefficient of tadpole $I_{1}[0]$ is

$\displaystyle C^{(0)}(2|1)=\frac{\left(K_{1}\cdot
K_{1}+M_{0}^{2}-M_{1}^{2}\right)\left(K_{1}\cdot K_{1}R\cdot R-D\left(R\cdot
K_{1}\right)^{2}\right)}{(D-1)\left(K_{1}\cdot K_{1}\right)^{2}}.$ (49)

For $I_{1}[1]$, by choosing $j_{0}=1$ in (37), we have

$\displaystyle C^{(1)}(2|1)$ $\displaystyle=$ $\displaystyle 2(2R\cdot
K_{1})\left[\left.C^{(0)}(1|1)\right|_{K_{1}\to-K_{1},M_{0}\leftrightarrow
M_{1}}\right]+\left.C^{(0)}(2|1)\right|_{K_{1}\to-K_{1},M_{0}\leftrightarrow
M_{1}}$ $\displaystyle=$ $\displaystyle{4(R\cdot K_{1})^{2}\over
K_{1}^{2}}+\frac{\left(K_{1}\cdot
K_{1}+M_{1}^{2}-M_{0}^{2}\right)\left(K_{1}\cdot K_{1}R\cdot R-D\left(R\cdot
K_{1}\right)^{2}\right)}{(D-1)\left(K_{1}\cdot K_{1}\right)^{2}}.$

The reduction coefficients of bubble: $I_{2}[0,1]$

The expansion of $C^{(0,1)}(2|1)$ is

$\displaystyle
C^{(0,1)}(2|1)=c_{0}^{(0,1)}(2)M_{0}^{2}s_{00}+c_{2}^{(0,1)}(2)s_{01}^{2}.$
(51)

By setting $r=1$ in (43) , we have

$\displaystyle c_{0}^{(0,1)}(2)$ $\displaystyle=$
$\displaystyle\frac{1}{D-1}\left(4-\frac{{\alpha}_{1}^{2}}{{\beta}_{11}}\right)c_{0}^{(0,1)}(0)$
(52) $\displaystyle=$ $\displaystyle{1\over
D-1}\left(4-\frac{{\alpha}_{1}^{2}}{{\beta}_{11}}\right)$ $\displaystyle=$
$\displaystyle\frac{4}{D-1}-\frac{f_{1}^{2}}{(D-1)M_{0}^{2}s_{11}},$

where the boundary conditions is $c^{(0,1)}_{0}(0)=1$.

By setting $i=0$ in (42), we have

$\displaystyle c^{(0,1)}_{2}(2)$ $\displaystyle=$ $\displaystyle{1\over
2\beta_{11}}\left(2{\alpha}_{1}c^{(0,1)}_{1}(1)-2c^{(0,1)}_{0}(2)\right)$ (53)
$\displaystyle=$ $\displaystyle{1\over\beta_{11}}\left(\alpha_{1}{f_{1}\over
s_{11}}-{1\over
D-1}\left(4-\frac{{\alpha}_{1}^{2}}{{\beta}_{11}}\right)\right)$
$\displaystyle=$
$\displaystyle\frac{Df_{1}^{2}}{(D-1)s_{11}^{2}}-\frac{4M_{0}^{2}}{(D-1)s_{11}},$

where $c^{(0,1)}_{1}(1)$ has been presented in the case $m=1$. Then

$\displaystyle C^{(0,1)}(2|1)$ $\displaystyle=$
$\displaystyle\frac{s_{01}^{2}\left(Df_{1}^{2}-4M_{0}^{2}s_{11}\right)+s_{00}s_{11}\left(4M_{0}^{2}s_{11}-f_{1}^{2}\right)}{(D-1)s_{11}^{2}}.$
(54)

  * •

$m=3$

The reduction coefficients of tadpoles: $I_{1}[0],I_{1}[1]$

The reduction coefficient of tadpole $I_{1}[0]$ is

$\displaystyle
C^{(0)}(3|1)=\frac{f_{1}^{2}\left(3s_{00}s_{01}s_{11}-(D+2)s_{01}^{3}\right)}{(D-1)s_{11}^{3}}+\frac{4M_{0}^{2}s_{01}\left(2s_{01}^{2}-3s_{00}s_{11}\right)}{Ds_{11}^{2}}.$
(55)

By choosing $j_{0}=1$ in (37), we have

$\displaystyle C^{(1)}(3|1)$ $\displaystyle=$ $\displaystyle 3(2R\cdot
K_{1})^{2}\left[\left.C^{(0)}(1|1)\right|_{K_{1}\to-K_{1},M_{0}\leftrightarrow
M_{1}}\right]+3(2R\cdot K_{1})\left[\left.C^{(0)}(2|1)\right|_{K_{1}\to-
K_{1},M_{0}\leftrightarrow M_{1}}\right]$ (56)
$\displaystyle+C^{(0)}(3|1)\left[\left.C^{(0)}(3|1)\right|_{K_{1}\to-
K_{1},M_{0}\leftrightarrow M_{1}}\right]$ $\displaystyle=$
$\displaystyle\frac{s_{01}\left(7D^{2}s_{01}^{2}+12DM_{1}^{2}s_{00}-10Ds_{01}^{2}-12M_{1}^{2}s_{00}\right)}{(D-1)Ds_{11}}+\frac{(D+2)\left(M_{0}^{2}-M_{1}^{2}\right)^{2}s_{01}^{3}}{(D-1)s_{11}^{3}}$
$\displaystyle+\frac{4\left(DM_{0}^{2}-DM_{1}^{2}-2M_{1}^{2}\right)s_{01}^{3}}{Ds_{11}^{2}}-\frac{3\left(M_{0}^{2}-M_{1}^{2}\right)^{2}s_{00}s_{01}}{(D-1)s_{11}^{2}}+\frac{3s_{00}s_{01}}{D-1}.$

The reduction coefficients of bubble: $I_{2}[0,1]$

The expansion of $C^{(0,1)}(3|1)$ is

$\displaystyle
C^{(0,1)}(3|1)=c_{1}^{(0,1)}(3)M_{0}^{2}s_{00}s_{01}+c_{3}^{(0,1)}(3)s_{01}^{3}.$
(57)

By setting $i=-1$ in (42), we have

$\displaystyle c_{1}^{(0,1)}(3)$ $\displaystyle=$
$\displaystyle{1\over\beta_{11}}\left(3{\alpha}_{1}c^{(0,1)}_{0}(2)-4c_{-1}^{(0,1)}(3)\right)$
(58) $\displaystyle=$
$\displaystyle{1\over\beta_{11}}\left[3{\alpha}_{1}\left(\frac{4}{D-1}-\frac{f_{1}^{2}}{(D-1)M_{0}^{2}s_{11}}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{12f_{1}}{(D-1)s_{11}}-\frac{3f_{1}^{3}}{(D-1)M_{0}^{2}s_{11}^{2}},$

where we have used $c_{-1}^{(0,1)}(3)=0$ and the result of expansion
coefficient $c^{(0,1)}_{0}(2)$ in the case $m=2$.

By setting $i=1$ in (42), we have

$\displaystyle c^{(0,1)}_{3}(3)$ $\displaystyle=$ $\displaystyle{1\over
3\beta_{11}}\left(3{\alpha}_{1}c^{(0,1)}_{2}(2)-2c^{(0,1)}_{1}(3)\right)$ (59)
$\displaystyle=$ $\displaystyle{1\over
3{\beta}_{11}}\left[3{\alpha}_{1}\left(\frac{Df_{1}^{2}}{(D-1)s_{11}^{2}}-\frac{4M_{0}^{2}}{(D-1)s_{11}}\right)-2\left(\frac{12f_{1}}{(D-1)s_{11}}-\frac{3f_{1}^{3}}{(D-1)M_{0}^{2}s_{11}^{2}}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{(D+2)f_{1}^{3}}{(D-1)s_{11}^{3}}-\frac{12f_{1}M_{0}^{2}}{(D-1)s_{11}^{2}}.$

Then the reduction coefficient is

$\displaystyle C^{(0,1)}(3|1)$ $\displaystyle=$
$\displaystyle\frac{f_{1}\left(s_{01}^{3}\left((D+2)f_{1}^{2}-12M_{0}^{2}s_{11}\right)+3s_{00}s_{11}s_{01}\left(4M_{0}^{2}s_{11}-f_{1}^{2}\right)\right)}{(D-1)s_{11}^{3}}.$
(60)

  * •

$m=4$

The reduction coefficients of tadpoles: $I_{1}[0],I_{1}[1]$

The reduction coefficient of tadpole $I_{1}[0]$ is

$\displaystyle C^{(0)}(4|1)$ $\displaystyle=$
$\displaystyle-\frac{3f_{1}s_{00}\left(8D^{2}M_{0}^{2}s_{01}^{2}+Df_{1}^{2}s_{00}+16DM_{0}^{2}s_{01}^{2}-16M_{0}^{2}s_{01}^{2}\right)}{(D-1)D(D+1)s_{11}^{2}}$
(61)
$\displaystyle+\frac{2(D+2)f_{1}s_{01}^{2}\left(3Df_{1}^{2}s_{00}+10DM_{0}^{2}s_{01}^{2}-8M_{0}^{2}s_{01}^{2}\right)}{(D-1)D(D+1)s_{11}^{3}}+\frac{12(2D-1)f_{1}M_{0}^{2}s_{00}^{2}}{(D-1)D(D+1)s_{11}}$
$\displaystyle-\frac{(D+2)(D+4)f_{1}^{3}s_{01}^{4}}{(D-1)(D+1)s_{11}^{4}}.$

By choosing $j_{0}=1$ in (37), we have

$\displaystyle C^{(1)}(4|1)$ $\displaystyle=$ $\displaystyle 4(2R\cdot
K_{1})^{3}\left[\left.C^{(0)}(1|1)\right|_{K_{1}\to-K_{1},M_{0}\leftrightarrow
M_{1}}\right]+6(2R\cdot K_{1})^{2}\left[\left.C^{(0)}(2|1)\right|_{K_{1}\to-
K_{1},M_{0}\leftrightarrow M_{1}}\right]$ (62) $\displaystyle+4(2R\cdot
K_{1})\left[\left.C^{(0)}(3|1)\right|_{K_{1}\to-K_{1},M_{0}\leftrightarrow
M_{1}}\right]+\left[\left.C^{(0)}(4|1)\right|_{K_{1}\to-
K_{1},M_{0}\leftrightarrow M_{1}}\right]$ $\displaystyle=$
$\displaystyle\widetilde{f}_{1}\left(\frac{4\left(5D^{2}+6D-8\right)M_{1}^{2}s_{01}^{4}}{D\left(D^{2}-1\right)s_{11}^{3}}+\frac{12s_{00}\left((2D-1)M_{1}^{2}s_{00}+2D(D+1)s_{01}^{2}\right)}{D\left(D^{2}-1\right)s_{11}}\right)$
$\displaystyle-\widetilde{f}_{1}\frac{24\left(\left(D^{2}+2D-2\right)M_{1}^{2}s_{00}s_{01}^{2}+D^{2}(D+1)s_{01}^{4}\right)}{D\left(D^{2}-1\right)s_{11}^{2}}$
$\displaystyle+\widetilde{f}_{1}^{3}\left(-\frac{\left(D^{2}+6D+8\right)s_{01}^{4}}{\left(D^{2}-1\right)s_{11}^{4}}+\frac{6(D+2)s_{00}s_{01}^{2}}{\left(D^{2}-1\right)s_{11}^{3}}-\frac{3s_{00}^{2}}{\left(D^{2}-1\right)s_{11}^{2}}\right)$
$\displaystyle+\widetilde{f}_{1}^{2}\left(\frac{8(D+2)s_{01}^{4}}{(D-1)s_{11}^{3}}-\frac{24s_{00}s_{01}^{2}}{(D-1)s_{11}^{2}}\right)-\frac{64M_{1}^{2}s_{01}^{4}}{Ds_{11}^{2}}+\frac{32s_{01}^{2}\left(\frac{3M_{1}^{2}s_{00}}{D}+s_{01}^{2}\right)}{s_{11}},$

where $\widetilde{f}_{1}=K_{1}^{2}+M_{1}^{2}-M_{0}^{2}$.

The reduction coefficients of bubble: $I_{2}[0,1]$

The expansion of $C^{(0,1)}(4|1)$ is

$\displaystyle
C^{(0,1)}(4|1)=c_{0}^{(0,1)}(4)M_{0}^{4}s_{00}^{2}+c_{2}^{(0,1)}(4)M_{0}^{2}s_{00}s_{01}^{2}+c_{4}^{(0,1)}(4)s_{01}^{4}.$
(63)

By setting $r=2$ in (43), we have

$\displaystyle c_{0}^{(0,1)}(4)$ $\displaystyle=$
$\displaystyle\frac{4}{D+1}\left(4-\frac{{\alpha}_{1}^{2}}{{\beta}_{11}}\right)c_{0}^{(0,1)}(2)$
(64) $\displaystyle=$ $\displaystyle{4\over
D+1}\left[\left(4-\frac{{\alpha}_{1}^{2}}{{\beta}_{11}}\right)\left(\frac{4}{D-1}-\frac{f_{1}^{2}}{(D-1)M_{0}^{2}s_{11}}\right)\right]$
$\displaystyle=$
$\displaystyle\frac{3f_{1}^{4}}{\left(D^{2}-1\right)M_{0}^{4}s_{11}^{2}}-\frac{24f_{1}^{2}}{\left(D^{2}-1\right)M_{0}^{2}s_{11}}+\frac{48}{D^{2}-1},$

By setting $i=0,2$ in (42), we calculate $c_{2}^{(0,1)}(4),c_{4}^{(0,1)}(4)$
iteratively

$\displaystyle c^{(0,1)}_{2}(4)$ $\displaystyle=$ $\displaystyle{1\over
2\beta_{11}}\left(4{\alpha}_{1}c^{(0,1)}_{1}(3)-4c^{(0,1)}_{0}(4)\right)$ (65)
$\displaystyle=$ $\displaystyle{1\over
2\beta_{11}}\Bigg{[}4{\alpha}_{1}\left(\frac{(D+2)f_{1}^{3}}{(D-1)s_{11}^{3}}-\frac{12f_{1}M_{0}^{2}}{(D-1)s_{11}^{2}}\right)$
$\displaystyle-4\left(\frac{3f_{1}^{4}}{\left(D^{2}-1\right)M_{0}^{4}s_{11}^{2}}-\frac{24f_{1}^{2}}{\left(D^{2}-1\right)M_{0}^{2}s_{11}}+\frac{48}{D^{2}-1}\right)\Bigg{]}$
$\displaystyle=$
$\displaystyle-\frac{6(D+2)f_{1}^{4}}{\left(D^{2}-1\right)M_{0}^{2}s_{11}^{3}}+\frac{24(D+3)f_{1}^{2}}{\left(D^{2}-1\right)s_{11}^{2}}-\frac{96M_{0}^{2}}{\left(D^{2}-1\right)s_{11}},$
$\displaystyle c_{4}^{(0,1)}(4)$ $\displaystyle=$ $\displaystyle{1\over
4\beta_{11}}\left(4{\alpha}_{1}c^{(0,1)}_{3}(3)-2c^{(0,1)}_{2}(4)\right)$ (66)
$\displaystyle=$ $\displaystyle{1\over
4\beta_{11}}\Bigg{[}4{\alpha}_{1}\left(\frac{(D+2)f_{1}^{3}}{(D-1)s_{11}^{3}}-\frac{12f_{1}M_{0}^{2}}{(D-1)s_{11}^{2}}\right)$
$\displaystyle-2\left(-\frac{6(D+2)f_{1}^{4}}{\left(D^{2}-1\right)M_{0}^{2}s_{11}^{3}}+\frac{24(D+3)f_{1}^{2}}{\left(D^{2}-1\right)s_{11}^{2}}-\frac{96M_{0}^{2}}{\left(D^{2}-1\right)s_{11}}\right)\Bigg{]}$
$\displaystyle=$
$\displaystyle-\frac{24(D+2)f_{1}^{2}M_{0}^{2}}{\left(D^{2}-1\right)s_{11}^{3}}+\frac{\left(D^{2}+6D+8\right)f_{1}^{4}}{\left(D^{2}-1\right)s_{11}^{4}}+\frac{48M_{0}^{4}}{\left(D^{2}-1\right)s_{11}^{2}}.$

Where we have used the results of the expansion coefficients with lower rank.
Then the reduction coefficient is

$\displaystyle C^{(0,1)}(4|1)$ $\displaystyle=$
$\displaystyle-\frac{24f_{1}^{2}M_{0}^{2}\left(s_{01}^{2}-s_{00}s_{11}\right)\left((D+2)s_{01}^{2}-s_{00}s_{11}\right)}{\left(D^{2}-1\right)s_{11}^{3}}+\frac{48M_{0}^{4}\left(s_{01}^{2}-s_{00}s_{11}\right)^{2}}{\left(D^{2}-1\right)s_{11}^{2}}$
(67)
$\displaystyle+\frac{f_{1}^{4}\left(\left(D^{2}+6D+8\right)s_{01}^{4}-6(D+2)s_{00}s_{11}s_{01}^{2}+3s_{00}^{2}s_{11}^{2}\right)}{\left(D^{2}-1\right)s_{11}^{4}}.$

###  4.2 Reduce tensor box to scalar triangles

We will consider the reduction coefficients of triangle MIs of tensor integral
$I^{(1)}_{4}$ as another example to illustrate the algorithm in section 3.2.
For simplicity, we denote
$G(i_{1},i_{2},\cdots,i_{s};j_{1},j_{2},\cdots,j_{r})$ as the determinant of
the Gram matrix $G$ with entries $G_{ab}=K_{a}\cdot K_{b}=s_{ab}$. Specially,
we denote $G(i_{1},i_{2},\cdots,i_{s})\equiv
G(i_{1},i_{2},\cdots,i_{s};i_{1},i_{2},\cdots,i_{s})$.

The reduction coefficient of the scalar triangle $I_{3}[0,1,2]$ is

$\displaystyle
C^{(0,1,2)}(1|3)=-\frac{G(2,3;1,2)s_{01}-G(1,3;1,2)s_{02}+G(1,2;1,2)s_{03}}{G(1,2,3)}.$
(68)

The reduction coefficients of $I_{3}[0,1,3],I_{3}[0,2,3]$ are easy to obtained
by simply changing labels $\\{1,2,3\\}\rightarrow\\{1,3,2\\}$ and
$\\{1,2,3\\}\rightarrow\\{2,1,3\\}$ respectively:

$\displaystyle C^{(0,1,3)}(1|3)$ $\displaystyle=$
$\displaystyle\left.C^{(0,1,2)}(1|3)\right|_{K_{2}\leftrightarrow
K_{3},M_{2}\leftrightarrow M_{3}}$ $\displaystyle=$
$\displaystyle-\frac{G(3,2;1,3)s_{01}-G(1,2;1,3)s_{03}+G(1,3;1,3)s_{02}}{G(1,2,3)},$
$\displaystyle C^{(0,2,3)}(1|3)$ $\displaystyle=$
$\displaystyle\left.C^{(0,1,2)}(1|3)\right|_{K_{1}\leftrightarrow
K_{2},M_{1}\leftrightarrow M_{2}}$ (69) $\displaystyle=$
$\displaystyle-\frac{G(2,1;3,2)s_{03}-G(3,1;3,2)s_{02}+G(3,2;3,2)s_{01}}{G(1,2,3)}.$

Now we consider the reduction coefficient of the triangle without $P_{0}$,
i.e., $I_{3}[1,2,3]$. In (37), choosing $j_{0}=3,j_{1}=1,j_{2}=2$, we have

$\displaystyle C^{(1,2,3)}(1|3)$ $\displaystyle=$ $\displaystyle
C^{(3,1,2)}(1|3)=(2R\cdot K_{3})\left.C^{(0,1,2)}(0|3)\right|_{K_{1}\to
K_{1}-K_{3},K_{2}\to K_{2}-K_{3},K_{3}\to-K_{3},M_{0}\leftrightarrow M_{3}}$
(70) $\displaystyle+\left.C^{(0,1,2)}(1|3)\right|_{K_{1}\to
K_{1}-K_{3},K_{2}\to K_{2}-K_{3},K_{3}\to-K_{3},M_{0}\leftrightarrow M_{3}}$
$\displaystyle=$
$\displaystyle\frac{G(K_{2}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3})(s_{01}-s_{03})}{G(K_{1}-K_{3},K_{2}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3},K_{3})}$
$\displaystyle+\frac{-G(K_{1}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3})(s_{02}-s_{03})}{G(K_{1}-K_{3},K_{2}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3},K_{3})}$
$\displaystyle+\frac{G(K_{1}-K_{3},K_{2}-K_{3};K_{1}-K_{3},K_{2}-K_{3})s_{03}}{G(K_{1}-K_{3},K_{2}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3},K_{3})},$

where $C^{(0,1,2)}(0|3)=0$ is the reduction coefficient of a triangle MI from
a box MI.

##  5 Discussion

In this paper, we show how to use the differential operators to get the
analytical expressions for the reduction coefficients of all master basis. By
these operators, one can establish the recurrence relations about reduction
coefficients in differential equation form. Another crucial step in this
method is that we use the information of tensor structure to avoid solving the
intricate differential equations directly.

As we have reviewed, in Britto:2006fc ; Britto:2007tt ; Britto:2008vq , the
analytical expressions for reduction coefficients can be solved by unitary
method. However, there are some differences between these two approaches.

  * •

The first difference is that the expression given by unitarity cut method is
written using the spinor formalism, while results in this paper use the
traditional Lorentz invariant contractions.

  * •

The second difference is that in unitarity cut method, we have assumed the
external momenta to be in pure $4D$ and only loop momentum is in general
$(4-2\epsilon)$ dimension. For our new method, there is no such a constraint
and the external momenta can be in $4D$ or in $(4-2\epsilon)$ dimension.

  * •

The third difference is that results in the paper are defined in an iterated
way, while expressions given by unitarity cut method are just one equation
(although the differentiation has the spirit of iteration).

  * •

The fourth difference is that expressions by unitarity cut method using the
input of arbitrary forms, while the one in this paper using the standard input
given in (3). The difference has a potential and huge impact on computation
efficiency. The reason is that with the development of on-shell program, it is
well known that tree-level amplitudes will be significantly simplified if we
use spinor variables with spurious poles, such as these given by the recursion
relation Britto:2004ap ; Britto:2005fq . Thus, it will be desirable to
cooperate these advantages of unitarity cut method to our current new
strategy.

Before ending this paper, let us emphasize that the purpose of the paper is to
establish an independent and complete reduction framework for one-loop
integrals using auxiliary vector $R$. In the previous work Feng:2021enk , we
have discussed the reduction coefficients of tadpoles. In this paper we have
complete the coefficients of other basis. However, for these two works, we
have assumed the power of each propagator is just one. To be a complete
reduction framework, we need to give the reduction of integrals with arbitrary
tensor structures and propagators having arbitrary powers. We will show in the
upcoming paper how to achieve this. After completing the framework of
reduction with auxiliary vector $R$, we can discuss various limit cases, like
the massless limit or the vanishing of Gram determinant, which will be
presented in another paper soon. Another direction is to apply our new
framework to higher loops. But unlike one-loop case, relations established by
differentiation over $R$ are usually not enough. Besides, the master integrals
are more complicated in higher-loop integrals. How to solve these difficulties
will be another future project.

## Acknowledgments

We are very grateful for Bo Feng’s collaboration in the previous related paper
and inspiring discussion. This work is supported by Qiu-Shi Funding and
Chinese NSF funding under Grant No.11935013, No.11947301, No.12047502 (Peng
Huanwu Center).

##  Appendix A More examples

In this appendix, we provide more examples to illustrate our method.555All
results have been checked with Fire6. Smirnov:2008iw ; Smirnov:2014hma ;
Smirnov:2013dia ; Smirnov:2019qkx ; Lee:2013mka  There are three points we
need to emphasize ahead.

  * •

For the tensor integral $I_{n+1}^{(m)}$, we only list the reduction
coefficient $C^{i_{r}}$ for $m\geq n-\left|i_{r}\right|$, because there are
not enough $\ell\cdot R$ in the nominator to cancel $n-\left|i_{r}\right|$
propagators for $m<n-\left|i_{r}\right|$.

  * •

We merely list the results of
$C^{(0,1,\cdots,r)}(m|n),C^{(1,2,\cdots,r+1)}(m|n),0\leq r\leq 4$ due to the
permutation symmetry:

$\displaystyle Tadpoles:C^{(i)}(m|n)$
$\displaystyle=\left.C^{(1)}(m|n)\right|_{1\leftrightarrow i}$ $\displaystyle
Bubbles:C^{(0,i)}(m|n)$
$\displaystyle=\left.C^{(0,1)}(m|n)\right|_{1\leftrightarrow i}$
$\displaystyle C^{(i,j)}(m|n)$
$\displaystyle=\left.C^{(1,2)}(m|n)\right|_{1\leftrightarrow
i,2\leftrightarrow j}$ $\displaystyle Triangles:C^{(0,i,j)}(m|n)$
$\displaystyle=\left.C^{(0,1,2)}(m|n)\right|_{1\leftrightarrow
i,2\leftrightarrow j}$ $\displaystyle C^{(i,j,k)}(m|n)$
$\displaystyle=\left.C^{(1,2,3)}(m|n)\right|_{1\leftrightarrow
i,2\leftrightarrow j,3\leftrightarrow k}$ $\displaystyle
Boxes:C^{(0,i,j,k)}(m|n)$
$\displaystyle=\left.C^{(0,1,2,3)}(m|n)\right|_{1\leftrightarrow
i,2\leftrightarrow j,3\leftrightarrow k}$ $\displaystyle C^{(i,j,k,l)}(m|n)$
$\displaystyle=\left.C^{(1,2,3,4)}(m|n)\right|_{1\leftrightarrow
i,2\leftrightarrow j,3\leftrightarrow k,4\leftrightarrow l}$
$\displaystyle\cdots$ (71)

where $0<i<j<k<l$.

  * •

There is a permutation symmetry about the expansion coefficient
$c^{(j_{0},\cdots,j_{r})}_{a_{1},\cdots,a_{n}}(m)$. If the Master Integral
$I_{r+1}[j_{0},j_{1},\cdots,j_{r}]$ is invariant under a label permutation
$\sigma:\\{1,2,\cdots,n\\}\to\\{\sigma(1),\sigma(2),\cdots,\sigma(n)\\}$, we
have $c^{(j_{0},\cdots,j_{r})}_{i_{1},\cdots,i_{n}}(m)=\sigma
c^{(j_{0},\cdots,j_{r})}_{i_{\sigma^{-1}(1)},\cdots,i_{\sigma^{-1}(n)}}(m)$.

For example, $I_{4}[0,1,2,3]$ is invariant under the label permutation
$\sigma:\\{1,2,3,4\\}\to\\{3,1,2,4\\}$, then we have

$\displaystyle c_{1,2,4,5}^{(0,1,2,3)}(14)$ $\displaystyle=$
$\displaystyle\left.c_{2,4,1,5}^{(0,1,2,3)}(14)\right|_{\\{1,2,3,4\\}\to\\{3,1,2,4\\}}.$
(72)

###  A.1 All reduction coefficients of tensor triangle with rank $m=1,2$

The MI of a tensor triangle $I_{3}^{(m)}$ are

$\displaystyle Tadpoles:$ $\displaystyle I_{1}[0],I_{1}[1],I_{1}[2],$
$\displaystyle Bubbles:$ $\displaystyle I_{2}[0,1],I_{2}[0,2],I_{2}[1,2],$
$\displaystyle Triangles:$ $\displaystyle I_{3}[0,1,2].$ (73)

Then the reduction of the tensor triangle is

$\displaystyle I_{3}^{(m)}$ $\displaystyle=$ $\displaystyle
C^{(0)}(m|2)I_{1}[0]+C^{(1)}(m|2)I_{1}[1]+C^{(2)}(m|2)I_{1}[2]+C^{(0,1)}(m|2)I_{2}[0,1]$
(74)
$\displaystyle+C^{(0,2)}I_{2}[0,2]+C^{(1,2)}I_{2}[1,2]+C^{(0,1,2)}I_{3}[0,1,2].$

  * •

$m=1$

Reduction coefficients of tadpoles

All reduction coefficients vanish.

Reduction coefficient of bubbles

$\displaystyle C^{(0,1)}(1|2)=\frac{s_{01}s_{12}-s_{02}s_{11}}{G(1,2)}.$ (75)

Choosing $j_{0}=1,j_{1}=2$ in (37), we have

$\displaystyle C^{(1,2)}(1|2)$ $\displaystyle=$
$\displaystyle\left.C^{(0,2)}(1|2)\right|_{M_{0}\leftrightarrow
M_{1},K_{1}\to-K_{1},K_{2}\to K_{2}-K_{1}}$ (76) $\displaystyle=$
$\displaystyle\left.\frac{s_{02}s_{12}-s_{01}s_{22}}{G(1,2)}\right|_{M_{0}\leftrightarrow
M_{1},K_{1}\to-K_{1},K_{2}\to K_{2}-K_{1}}$ $\displaystyle=$
$\displaystyle\frac{s_{02}\left(s_{11}-s_{12}\right)}{G(1,2)}+(1\leftrightarrow
2).$

Reduction coefficient of triangle

$\displaystyle
C^{(0,1,2)}(1|2)=\frac{s_{01}\left(f_{1}s_{22}-f_{2}s_{12}\right)}{G(1,2)}+(1\leftrightarrow
2).$ (77)

  * •

$m=2$

Reduction coefficients of tadpoles

$\displaystyle
C^{(0)}(2|2)=\frac{s_{11}s_{22}s_{01}s_{02}-s_{12}s_{22}s_{01}^{2}}{s_{11}s_{22}G(1,2)}+(1\leftrightarrow
2).$ (78)

Choose $j_{0}=1$ in (37), we have

$\displaystyle C^{(1)}(2|2)$ $\displaystyle=$
$\displaystyle\left.C^{(0)}(2|2)\right|_{M_{0}\leftrightarrow M_{1},K_{1}\to-
K_{1},K_{2}\to K_{2}-K_{1}}$ $\displaystyle=$
$\displaystyle\left.-\frac{s_{12}s_{22}s_{01}^{2}-2s_{02}s_{11}s_{22}s_{01}+s_{02}^{2}s_{11}s_{12}}{s_{11}s_{22}\left(s_{11}s_{22}-s_{12}^{2}\right)}\right|_{M_{0}\leftrightarrow
M_{1},K_{1}\to-K_{1},K_{2}\to K_{2}-K_{1}}$ $\displaystyle=$
$\displaystyle\frac{\left(-2s_{12}^{2}+s_{22}s_{12}+s_{11}s_{22}\right)s_{01}^{2}+2s_{02}s_{11}\left(s_{12}-s_{22}\right)s_{01}+s_{02}^{2}s_{11}\left(s_{12}-s_{11}\right)}{s_{11}\left(s_{11}-2s_{12}+s_{22}\right)G(1,2)}.$

Reduction coefficients of bubbles

$\displaystyle
C^{(0,1)}(2|2)=\frac{s_{01}^{2}c_{2,0}^{(0,1)}(2)}{M_{0}^{2}}+\frac{s_{02}s_{01}c_{1,1}^{(0,1)}(2)}{M_{0}^{2}}+\frac{s_{02}^{2}c_{0,2}^{(0,1)}(2)}{M_{0}^{2}}+s_{00}c_{0,0}^{(0,1)}(2),$
(80)

where

$\displaystyle
c^{(0,1)}_{0,0}(2)=\frac{f_{2}s_{11}-f_{1}s_{12}}{(D-2)G(1,2)},$ (81)
$\displaystyle
c^{(0,1)}_{1,1}(2)=\frac{2(D-1)f_{2}M_{0}^{2}s_{11}s_{12}}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right){}^{2}}-\frac{2f_{1}M_{0}^{2}\left((D-2)s_{11}s_{22}+s_{12}^{2}\right)}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right){}^{2}}$
(82) $\displaystyle
c^{(0,1)}_{2,0}(2)=-\frac{M_{0}^{2}\left(f_{2}s_{11}\left((D-2)s_{12}^{2}+s_{11}s_{22}\right)+f_{1}s_{12}\left((D-2)s_{12}^{2}+(3-2d)s_{11}s_{22}\right)\right)}{(D-2)s_{11}G(1,2)^{2}},$
$\displaystyle
c^{(0,1)}_{0,2}(2)=-\frac{(D-1)M_{0}^{2}s_{11}\left(f_{2}s_{11}-f_{1}s_{12}\right)}{(D-2)^{2}G(1,2)^{2}},$
(84) $\displaystyle C^{(1,2)}(2|2)$ $\displaystyle=$
$\displaystyle\left.2(2R\cdot
K_{2})\left[C^{(0,1)}(1,2)\right|_{\sigma}\right]+\left.C^{(0,1)}(2,2)\right|_{\sigma}$
$\displaystyle=$ $\displaystyle
4s_{02}\left[\frac{s_{01}\left(f_{1}s_{22}-f_{2}s_{12}\right)}{G(1,2)}+(1\leftrightarrow
2)\right]+\left.C^{(0,1)}(2,2)\right|_{\sigma},$

where

$\displaystyle\sigma=M_{0}\leftrightarrow M_{2},K_{2}\to-K_{2},K_{1}\to
K_{1}-K_{2}.$ (86)

Reduction coefficients of triangle

$\displaystyle
C^{(0,1,2)}(2|2)=c^{(0,1,2)}_{0,0}M_{0}^{2}s_{00}+c^{(0,1,2)}_{2,0}s_{01}^{2}+c^{(0,1,2)}_{0,2}s_{02}^{2}+c^{(0,1,2)}_{1,1}s_{01}s_{02}.$
(87)

where

$\displaystyle
c_{0,0}^{(0,1,2)}(2)=\frac{f_{1}^{2}s_{22}-2f_{2}f_{1}s_{12}+f_{2}^{2}s_{11}+4M_{0}^{2}\left(s_{12}^{2}-s_{11}s_{22}\right)}{(D-2)M_{0}^{2}\left(s_{12}^{2}-s_{11}s_{22}\right)},$
(88) $\displaystyle c_{2,0}^{(0,1,2)}(2)$ $\displaystyle=$
$\displaystyle\frac{f_{2}^{2}\left((D-2)s_{12}^{2}+s_{11}s_{22}\right)}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right)^{2}}-\frac{2(D-1)f_{1}f_{2}s_{12}s_{22}}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right)^{2}}$
(89)
$\displaystyle+\frac{(D-1)f_{1}^{2}s_{22}^{2}}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right)^{2}}+\frac{4M_{0}^{2}s_{22}}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right)},$
$\displaystyle c_{0,2}^{(0,1,2)}(2)$ $\displaystyle=$
$\displaystyle\frac{f_{1}^{2}\left((D-2)s_{12}^{2}+s_{11}s_{22}\right)}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right)^{2}}-\frac{2(D-1)f_{2}f_{1}s_{11}s_{12}}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right)^{2}}$
(90)
$\displaystyle+\frac{(D-1)f_{2}^{2}s_{11}^{2}}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right)^{2}}+\frac{4M_{0}^{2}s_{11}}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right)},$
$\displaystyle c_{1,1}^{(0,1,2)}(2)$ $\displaystyle=$
$\displaystyle-\frac{2(D-1)f_{1}^{2}s_{12}s_{22}}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right)^{2}}+\frac{2f_{2}f_{1}\left(Ds_{12}^{2}+(D-2)s_{11}s_{22}\right)}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right)^{2}}$
(91)
$\displaystyle-\frac{2(D-1)f_{2}^{2}s_{11}s_{12}}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right)^{2}}-\frac{8M_{0}^{2}s_{12}}{(D-2)\left(s_{12}^{2}-s_{11}s_{22}\right)}.$

###  A.2 All reduction coefficients of tensor box with rank $m=1,2$

The MIs of a tensor box $I_{4}^{(m)}$ are

$\displaystyle Tadpoles:$ $\displaystyle I_{1}[0],I_{1}[1],I_{1}[2],I_{1}[3],$
$\displaystyle Bubbles:$ $\displaystyle
I_{2}[0,1],I_{2}[0,2],I_{2}[0,3],I_{2}[1,2],I_{2}[1,3],I_{2}[2,3],$
$\displaystyle Triangles:$ $\displaystyle
I_{3}[0,1,2],I_{3}[0,1,3],I_{3}[0,2,3],I_{3}[1,2,3],$ $\displaystyle Box:$
$\displaystyle I_{4}[0,1,2,3].$ (92)

Then the reduction of the tensor box is

$\displaystyle I_{4}^{(m)}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{3}C^{(i)}(m|3)I_{1}[i]+\sum_{0\leq i_{1}<i_{2}\leq
3}C^{(i_{1},i_{2})}(m|3)I_{2}[i_{1},i_{2}]$ (93) $\displaystyle+\sum_{0\leq
i_{1}<i_{2}<i_{3}\leq
3}C^{(i_{1},i_{2},i_{3})}(m|3)I_{3}[i_{1},i_{2},i_{3}]+C^{(0,1,2,3)}I_{4}[0,1,2,3].$

  * •

$m=1$

Reduction coefficients of tadpoles, bubbles

All reduction coefficients vanish.

Reduction coefficients of triangles

$\displaystyle
C^{(0,1,2)}(1|3)=-\frac{G(2,3;1,2)s_{01}-G(1,3;1,2)s_{02}+G(1,2;1,2)s_{03}}{G(1,2,3)}.$
(94)

In (37), choosing $j_{0}=3,j_{1}=1,j_{2}=2$, we have

$\displaystyle C^{(1,2,3)}(1|3)$ $\displaystyle=$ $\displaystyle
C^{(3,1,2)}(1|3)=(2R\cdot K_{3})\left.C^{(0,1,2)}(0|3)\right|_{K_{1}\to
K_{1}-K_{3},K_{2}\to K_{2}-K_{3},K_{3}\to-K_{3},M_{0}\leftrightarrow M_{3}}$
(95) $\displaystyle+\left.C^{(0,1,2)}(1|3)\right|_{K_{1}\to
K_{1}-K_{3},K_{2}\to K_{2}-K_{3},K_{3}\to-K_{3},M_{0}\leftrightarrow M_{3}}$
$\displaystyle=$
$\displaystyle\frac{G(K_{2}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3})(s_{01}-s_{03})}{G(K_{1}-K_{3},K_{2}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3},K_{3})}$
$\displaystyle+\frac{-G(K_{1}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3})(s_{02}-s_{03})}{G(K_{1}-K_{3},K_{2}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3},K_{3})}$
$\displaystyle+\frac{G(K_{1}-K_{3},K_{2}-K_{3};K_{1}-K_{3},K_{2}-K_{3})s_{03}}{G(K_{1}-K_{3},K_{2}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3},K_{3})}.$

Reduction coefficients of box

$\displaystyle C^{(0,1,2,3)}(1|3)$ $\displaystyle=$
$\displaystyle\frac{f_{3}\left(s_{01}G(2,3;1,2)-s_{02}G(1,3;1,2)+s_{03}G(1,2;1,2)\right)}{G(1,2,3)}$
(96)
$\displaystyle-\frac{f_{2}\left(s_{01}G(2,3;1,3)-s_{02}G(1,3;1,3)+s_{03}G(1,3;1,2)\right)}{G(1,2,3)}$
$\displaystyle+\frac{f_{1}\left(s_{01}G(2,3;2,3)-s_{02}G(2,3;1,3)+s_{03}G(2,3;1,2)\right)}{G(1,2,3)}.$

  * •

$m=2$

Reduction coefficients of tadpoles

All reduction coefficients vanish.

Reduction coefficients of bubbles

$\displaystyle C^{(0,1)}(2|3)$ $\displaystyle=$
$\displaystyle(M_{0}^{2})^{-2}\Big{[}c_{0,0,0}^{(0,1)}(2)M_{0}^{2}s_{00}+c_{2,0,0}^{(0,1)}(2)s_{01}^{2}+c_{0,2,0}^{(0,1)}(2)s_{02}^{2}+c_{0,0,2}^{(0,1)}(2)s_{03}^{2}$
(97)
$\displaystyle+c_{1,1,0}^{(0,1)}(2)s_{01}s_{02}+c_{0,1,1}^{(0,1)}(2)s_{02}s_{03}+c_{1,0,1}^{(0,1)}(2)s_{01}s_{03}\Big{]}.$

where

$\displaystyle c_{0,0,0}^{(0,1)}(2)=0,$ (98) $\displaystyle
c_{2,0,0}^{(0,1)}(2)=\frac{M_{0}^{4}s_{13}G(2,3;1,3)}{G(1,3;1,3)G(1,2,3)}-\frac{M_{0}^{4}s_{12}G(2,3;1,2)}{G(1,2;1,2)G(1,2,3)},$
(99) $\displaystyle c_{0,2,0}^{(0,1)}(2)$ $\displaystyle=$
$\displaystyle-\frac{M_{0}^{4}s_{11}G(1,3;1,2)}{G(1,2;1,2)G(1,2,3)},$
$\displaystyle c_{0,0,2}^{(0,1)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{0,2,0}^{(0,1)}(2)\right|_{2\leftrightarrow 3},$ (100)
$\displaystyle c_{0,1,1}^{(0,1)}(2)=\frac{2M_{0}^{4}s_{11}}{G(1,2,3)},$ (101)
$\displaystyle c_{1,1,0}^{(0,1)}(2)$ $\displaystyle=$
$\displaystyle\frac{2M_{0}^{4}s_{11}G(2,3;1,2)}{G(1,2;1,2)G(1,2,3)},$
$\displaystyle c_{1,0,1}^{(0,1)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{1,1,0}^{(0,1)}(2)\right|_{2\leftrightarrow 3}.$ (102)

Choosing $j_{0}=2,j_{1}=1$ in (37) we have

$\displaystyle C^{(1,2)}(2|3)$ $\displaystyle=$
$\displaystyle\left[\left.2(-2R\cdot
K_{2})C^{(0,1)}(1|3)+C^{(0,1)}(2|3)\right]\right|_{K_{1}\to
K_{1}-K_{2},K_{2}\to-K_{2},K_{3}\to K_{3}-K_{2}M_{0}\leftrightarrow M_{2}}$
(103) $\displaystyle=$ $\displaystyle\left.C^{(0,1)}(2|3)\right|_{K_{1}\to
K_{1}-K_{2},K_{2}\to-K_{2},K_{3}\to K_{3}-K_{2}M_{0}\leftrightarrow M_{2}}.$

Reduction coefficients of triangles

$\displaystyle C^{(0,1,2)}(2|3)$ $\displaystyle=$ $\displaystyle
M_{0}^{-2}\Big{[}c_{0,0,0}^{(0,1,2)}(2)M_{0}^{2}s_{00}+c_{2,0,0}^{(0,1,2)}(2)s_{01}^{2}+c_{0,2,0}^{(0,1,2)}(2)s_{02}^{2}+c_{0,0,2}^{(0,1,2)}(2)s_{03}^{2}$
(104)
$\displaystyle+c_{1,1,0}^{(0,1,2)}(2)s_{01}s_{02}+c_{0,1,1}^{(0,1,2)}(2)s_{02}s_{03}+c_{1,0,1}^{(0,1,2)}(2)s_{01}s_{03}\Big{]},$

where

$\displaystyle c_{0,0,0}^{(0,1,2)}(2)$ $\displaystyle=$
$\displaystyle\frac{f_{1}G(2,3;1,2)}{(D-3)G(1,2,3)}-\frac{f_{2}G(1,3;1,2)}{(D-3)G(1,2,3)}+\frac{f_{3}G(1,2;1,2)}{(D-3)G(1,2,3)},$
(105) $\displaystyle c_{2,0,0}^{(0,1,2)}(2)$ $\displaystyle=$
$\displaystyle\frac{f_{2}M_{0}^{2}G(1,3;1,2)G(2,3;2,3)}{(D-3)G(1,2,3)^{2}}-\frac{f_{3}M_{0}^{2}G(1,2;1,2)G(2,3;2,3)}{(D-3)G(1,2,3)^{2}}$
(106)
$\displaystyle+G(2,3;1,2)\left(\frac{f_{2}M_{0}^{2}G(2,3;1,3)}{G(1,2,3)^{2}}-\frac{(D-2)f_{1}M_{0}^{2}G(2,3;2,3)}{(D-3)G(1,2,3)^{2}}\right)$
$\displaystyle+\frac{M_{0}^{2}\left(f_{2}s_{12}-f_{1}s_{22}\right)G(2,3;1,2)}{G(1,2;1,2)G(1,2,3)}-\frac{f_{3}M_{0}^{2}G(2,3;1,2)^{2}}{G(1,2,3)^{2}},$
$\displaystyle c_{0,2,0}^{(0,1,2)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{2,0,0}^{(0,1,2)}(2)\right|_{1\leftrightarrow 2},$ (107)
$\displaystyle c_{0,0,2}^{(0,1,2)}(2)$ $\displaystyle=$ $\displaystyle
G(1,2;1,2)\left[\frac{(D-2)f_{2}M_{0}^{2}G(1,3;1,2)}{(D-3)G(1,2,3;1,2,3)^{2}}-\frac{(D-2)f_{1}M_{0}^{2}G(2,3;1,2)}{(D-3)G(1,2,3;1,2,3)^{2}}\right]$
(108)
$\displaystyle-\frac{(D-2)f_{3}M_{0}^{2}G(1,2;1,2)^{2}}{(D-3)G(1,2,3;1,2,3)^{2}},$
$\displaystyle c_{1,1,0}^{(0,1,2)}(2)$ $\displaystyle=$
$\displaystyle-\frac{2(D-2)f_{2}M_{0}^{2}G(1,3;1,2)G(2,3;1,3)}{(D-3)G(1,2,3)^{2}}+\frac{2f_{3}M_{0}^{2}G(1,2;1,2)G(2,3;1,3)}{(D-3)G(1,2,3)^{2}}$
$\displaystyle+G(2,3;1,2)\left(\frac{2f_{1}M_{0}^{2}G(2,3;1,3)}{(D-3)G(1,2,3)^{2}}+\frac{2f_{3}M_{0}^{2}G(1,3;1,2)}{G(1,2,3)^{2}}\right)$
$\displaystyle+\frac{2M_{0}^{2}\left(f_{1}s_{12}-f_{2}s_{11}\right)G(2,3;1,2)}{G(1,2;1,2)G(1,2,3)}+\frac{2f_{1}M_{0}^{2}G(1,3;1,2)G(2,3;2,3)}{G(1,2,3)^{2}},$
$\displaystyle c_{0,1,1}^{(0,1,2)}(2)$ $\displaystyle=$
$\displaystyle\frac{2f_{1}M_{0}^{2}G(1,3;1,2)G(2,3;1,2)}{(D-3)G(1,2,3)^{2}}-\frac{2f_{2}M_{0}^{2}G(1,3;1,2)^{2}}{(D-3)G(1,2,3)^{2}}$
(110)
$\displaystyle+G(1,2;1,2)\left(\frac{2(D-2)f_{3}M_{0}^{2}G(1,3;1,2)}{(D-3)G(1,2,3)^{2}}+\frac{2f_{1}M_{0}^{2}G(2,3;1,3)}{G(1,2,3)^{2}}\right)$
$\displaystyle-\frac{2f_{2}M_{0}^{2}G(1,2;1,2)G(1,3;1,3)}{G(1,2,3)^{2}},$
$\displaystyle
c_{1,0,1}^{(0,1,2)}(2)=\left.c_{0,1,1}^{(0,1,2)}(2)\right|_{1\leftrightarrow
2}.$ (111)

Choosing $j_{0}=3,j_{1}=1,j_{2}=2$ in (37), we have

$\displaystyle C^{(1,2,3)}(1|3)$ $\displaystyle=$
$\displaystyle\left.\left[2(-2R\cdot
K_{3})C^{(0,1,2)}(1|3)+C^{(0,1,2)}(2|3)\right]\right|_{K_{1}\to
K_{1}-K_{3},K_{2}\to K_{2}-K_{3},K_{3}\to-K_{3},M_{0}\leftrightarrow M_{3}}$
(112) $\displaystyle=$ $\displaystyle
4s_{03}\Bigg{\\{}\frac{G(K_{2}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3})(s_{01}-s_{03})}{G(K_{1}-K_{3},K_{2}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3},K_{3})}$
$\displaystyle+\frac{-G(K_{1}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3})(s_{02}-s_{03})}{G(K_{1}-K_{3},K_{2}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3},K_{3})}$
$\displaystyle+\frac{G(K_{1}-K_{3},K_{2}-K_{3};K_{1}-K_{3},K_{2}-K_{3})s_{03}}{G(K_{1}-K_{3},K_{2}-K_{3},K_{3};K_{1}-K_{3},K_{2}-K_{3},K_{3})}\Bigg{\\}}$
$\displaystyle+\left.C^{(0,1,2)}(2,3)\right|_{K_{1}\to K_{1}-K_{3},K_{2}\to
K_{2}-K_{3},K_{3}\to-K_{3},M_{0}\leftrightarrow M_{3}}.$

Reduction coefficient of box

$\displaystyle C^{(0,1,2,3)}(2|3)$ $\displaystyle=$ $\displaystyle
c_{0,0,0}^{(0,1,2,3)}(2)M_{0}^{2}s_{00}+c_{2,0,0}^{(0,1,2,3)}(2)s_{01}^{2}+c_{0,2,0}^{(0,1,2,3)}(2)s_{02}^{2}+c_{0,0,2}^{(0,1,2,3)}(2)s_{03}^{2}$
(113)
$\displaystyle+c_{1,1,0}^{(0,1,2,3)}(2)s_{01}s_{02}+c_{0,1,1}^{(0,1,2,3)}(2)s_{02}s_{03}+c_{1,0,1}^{(0,1,2,3)}(2)s_{01}s_{03},$

where

$\displaystyle c_{0,0,0}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle-\frac{f_{1}^{2}G(2,3;2,3)}{(D-3)M_{0}^{2}G(1,2,3)}+\frac{2f_{2}f_{1}G(2,3;1,3)}{(D-3)M_{0}^{2}G(1,2,3)}$
(114)
$\displaystyle-\frac{2f_{3}f_{1}G(2,3;1,2)}{(D-3)M_{0}^{2}G(1,2,3)}-\frac{f_{3}^{2}G(1,2;1,2)}{(D-3)M_{0}^{2}G(1,2,3)}$
$\displaystyle+\frac{f_{2}\left(2f_{3}G(1,3;1,2)-f_{2}G(1,3;1,3)\right)}{(D-3)M_{0}^{2}G(1,2,3)}+\frac{4}{D-3},$
$\displaystyle c_{2,0,0}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\frac{f_{3}^{2}G(1,2;1,2)G(2,3;2,3)}{(D-3)G(1,2,3)^{2}}+\frac{2(D-2)f_{1}f_{3}G(2,3;1,2)G(2,3;2,3)}{(D-3)G(1,2,3)^{2}}$
(115)
$\displaystyle-\frac{2(D-2)f_{1}f_{2}G(2,3;1,3)G(2,3;2,3)}{(D-3)G(1,2,3)^{2}}+\frac{f_{2}^{2}G(2,3;1,3)^{2}}{G(1,2,3)^{2}}$
$\displaystyle+\frac{f_{2}\left(f_{2}G(1,3;1,3)G(2,3;2,3)-2f_{3}G(1,3;1,2)G(2,3;2,3)\right)}{(D-3)G(1,2,3)^{2}}$
$\displaystyle-\frac{4M_{0}^{2}G(2,3;2,3)}{(D-3)G(1,2,3)}+\frac{(D-2)f_{1}^{2}G(2,3;2,3)^{2}}{(D-3)G(1,2,3)^{2}}$
$\displaystyle+\frac{f_{3}^{2}G(2,3;1,2)^{2}}{G(1,2,3)^{2}}-\frac{2f_{2}f_{3}G(2,3;1,2)G(2,3;1,3)}{G(1,2,3)^{2}},$
$\displaystyle
c_{1,1,0}^{(0,1,2,3)}(2)=\frac{8M_{0}^{2}G(2,3;1,3)}{(D-3)G(1,2,3)}$
$\displaystyle-\frac{2f_{2}\left((D-2)f_{2}G(1,3;1,3)G(2,3;1,3)-(D-1)f_{3}G(1,3;1,2)G(2,3;1,3)\right)}{(D-3)G(1,2,3)^{2}}$
$\displaystyle+\frac{2(D-1)f_{1}f_{2}G(2,3;1,3)^{2}}{(D-3)G(1,2,3)^{2}}-\frac{2f_{3}^{2}G(1,2;1,2)G(2,3;1,3)}{(D-3)G(1,2,3)^{2}}$
$\displaystyle+\frac{2f_{1}G(2,3;2,3)\left(f_{2}G(1,3;1,3)-f_{3}G(1,3;1,2)\right)}{G(1,2,3)^{2}}$
$\displaystyle-\frac{2(D-2)f_{1}^{2}G(2,3;1,3)G(2,3;2,3)}{(D-3)G(1,2,3)^{2}}-\frac{2(D-1)f_{1}f_{3}G(2,3;1,3)G(2,3;1,2)}{(D-3)G(1,2,3)^{2}}$
$\displaystyle+\frac{2f_{3}G(2,3;1,2)\left(f_{2}G(1,3;1,3)-f_{3}G(1,3;1,2)\right)}{G(1,2,3)^{2}}.$
(116)

Other expansion coefficients can be got by using the permutation symmetry:

$\displaystyle c_{0,2,0}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{2,0,0}^{(0,1,2,3)}(2)\right|_{1\leftrightarrow 2},$
$\displaystyle c_{0,0,2}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{2,0,0}^{(0,1,2,3)}(2)\right|_{1\leftrightarrow 3},$
$\displaystyle c_{1,0,1}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{1,1,0}^{(0,1,2,3)}(2)\right|_{2\leftrightarrow 3},$
$\displaystyle c_{0,1,1}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{1,1,0}^{(0,1,2,3)}(2)\right|_{1\leftrightarrow 3}.$
(117)

###  A.3 All reduction coefficients of tensor pentagon with rank $m=1,2$

Consider the reduction of a tensor pentagon $I_{5}^{(m)}$, the MIs are as
below

$\displaystyle Tadpoles:$ $\displaystyle
I_{1}[0],I_{1}[1],I_{1}[2],I_{1}[3],I_{1}[4],$ $\displaystyle Bubbles:$
$\displaystyle
I_{2}[0,1],I_{2}[0,2],I_{2}[0,3],I_{2}[0,4],I_{2}[1,2],I_{2}[1,3],I_{2}[1,4],I_{2}[2,3],I_{2}[2,4],I_{2}[3,4],$
$\displaystyle Triangles:$ $\displaystyle
I_{3}[0,1,2],I_{3}[0,1,3],I_{3}[0,1,4],I_{3}[0,2,3],I_{3}[0,2,4],I_{3}[0,3,4],I_{3}[1,2,3],$
$\displaystyle I_{3}[1,2,4],I_{3}[1,3,4],I_{3}[2,3,4],$ $\displaystyle Box:$
$\displaystyle I_{4}[0,1,2,3],I_{4}[0,2,3,4],I_{4}[0,1,2,4],I_{4}[1,2,3,4],$
$\displaystyle Pentagon:$ $\displaystyle I_{5}[0,1,2,3,4].$ (118)

Then the reduction of the tensor pentagon is given by

$\displaystyle I_{5}^{(m)}$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{4}C_{0}^{(i)}(m|4)I_{1}[i]+\sum_{0\leq
i_{1}<i_{2}\leq 4}C^{(i_{1},i_{2})}(m|4)I_{2}[i_{1},i_{2}]+\sum_{0\leq
i_{1}<i_{2}<i_{3}\leq 4}C^{(i_{1},i_{2},i_{3})}(m|4)I_{3}[i_{1},i_{2},i_{3}]$
(119) $\displaystyle+\sum_{0\leq i_{1}<i_{2}<i_{3}<i_{4}\leq
4}C^{(i_{1},i_{2},i_{3},i_{4})}(m|4)I_{4}[i_{1},i_{2},i_{3},i_{4}]+C^{(0,1,2,3,4)}(m|4)I_{5}[0,1,2,3,4].$

  * •

$m=1$

Reduction coefficients of tadpoles, bubbles, triangles

All reduction coefficients vanish.

Reduction coefficients of boxes

$\displaystyle C^{(0,1,2,3)}(1|4)$ $\displaystyle=$
$\displaystyle\frac{s_{01}G(2,3,4;1,2,3)-s_{02}G(1,3,4;1,2,3)+s_{03}G(1,2,4;1,2,3)}{G(1,2,3,4;1,2,3,4)}$
(120) $\displaystyle-\frac{s_{04}G(1,2,3)}{G(1,2,3,4;1,2,3,4)}.$

Choosing $j_{0}=4,j_{1}=1,j_{2}=2,j_{3}=3$ in (37), we have

$\displaystyle C^{(1,2,3,4)}(1|4)$ $\displaystyle=$
$\displaystyle\left.C^{(0,1,2,3)}(1|4)\right|_{K_{1}\to K_{1}-K_{4},K_{2}\to
K_{2}-K_{4},K_{3}\to K_{3}-K_{4},K_{4}\to-K_{4},M_{0}\leftrightarrow M_{4}}$
$\displaystyle=$
$\displaystyle\frac{(-s_{01}+s_{04})G(K_{2}-K_{4},K_{3}-K_{4},K_{4};K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4})}{G(K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4},K_{4};K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4},K_{4})}$
$\displaystyle+\frac{(s_{02}-s_{04})G(K_{1}-K_{4},K_{3}-K_{4},K_{4};K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4})}{G(K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4},K_{4};K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4},K_{4})}$
$\displaystyle-\frac{(s_{03}-s_{04})G(K_{1}-K_{4},K_{2}-K_{4},K_{4};K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4})}{G(K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4},K_{4};K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4},K_{4})}$
$\displaystyle+\frac{s_{04}G(K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4};K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4})}{G(K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4},K_{4};K_{1}-K_{4},K_{2}-K_{4},K_{3}-K_{4},K_{4})}.$

Reduction coefficients of pentagon

$\displaystyle
C^{(0,1,2,3,4)}(1|4)=c_{1,0,0,0}^{(0,1,2,3,4)}(1)s_{01}+c_{0,1,0,0}^{(0,1,2,3,4)}(1)s_{02}+c_{0,0,1,0}^{(0,1,2,3,4)}(1)s_{03}+c_{0,0,0,1}^{(0,1,2,3,4)}(1)s_{04},$

where

$\displaystyle c_{1,0,0,0}^{(0,1,2,3,4)}(1)$ $\displaystyle=$
$\displaystyle\frac{f_{1}G(2,3,4)-f_{2}G(2,3,4;1,3,4)+f_{3}G(2,3,4;1,2,4)}{G(1,2,3,4;1,2,3,4)}$
(123) $\displaystyle-\frac{f_{4}G(2,3,4;1,2,3)}{G(1,2,3,4;1,2,3,4)},$
$\displaystyle c_{0,1,0,0}^{(0,1,2,3,4)}(1)$ $\displaystyle=$
$\displaystyle\left.c_{1,0,0,0}^{(0,1,2,3,4)}(1)\right|_{1\leftrightarrow 2},$
$\displaystyle c_{0,0,1,0}^{(0,1,2,3,4)}(1)$ $\displaystyle=$
$\displaystyle\left.c_{1,0,0,0}^{(0,1,2,3,4)}(1)\right|_{1\leftrightarrow 3},$
$\displaystyle c_{0,0,0,1}^{(0,1,2,3,4)}(1)$ $\displaystyle=$
$\displaystyle\left.c_{1,0,0,0}^{(0,1,2,3,4)}(1)\right|_{1\leftrightarrow 4}.$
(124)

  * •

$m=2$

Reduction coefficients of tadpoles, bubbles

All reduction coefficients vanish.

Reduction coefficients of triangles

$\displaystyle C^{(0,1,2)}(2|4)$ $\displaystyle=$
$\displaystyle-\frac{s_{04}^{2}G(1,2;1,2)G(1,2,4;1,2,3)}{G(1,2,4;1,2,4)G(1,2,3,4;1,2,3,4)}-\frac{s_{03}^{2}G(1,2;1,2)G(1,2,4;1,2,3)}{G(1,2,3)G(1,2,3,4;1,2,3,4)}$
$\displaystyle-\frac{s_{01}G(2,3,4;1,2,3)\left(s_{01}G(2,3;1,2)-2s_{02}G(1,3;1,2)+2s_{03}G(1,2;1,2)\right)}{G(1,2,3)G(1,2,3,4;1,2,3,4)}$
$\displaystyle+\frac{s_{02}G(1,3,4;1,2,3)\left(2s_{03}G(1,2;1,2)-s_{02}G(1,3;1,2)\right)}{G(1,2,3)G(1,2,3,4;1,2,3,4)}$
$\displaystyle+\frac{s_{01}^{2}G(2,4;1,2)G(2,3,4;1,2,4)}{G(1,2,4;1,2,4)G(1,2,3,4;1,2,3,4)}$
$\displaystyle+\frac{s_{02}G(1,4;1,2)\left(s_{02}G(1,3,4;1,2,4)-2s_{01}G(2,3,4;1,2,4)\right)}{G(1,2,4;1,2,4)G(1,2,3,4;1,2,3,4)}$
$\displaystyle+\frac{2s_{04}G(1,2;1,2)\left(s_{01}G(2,3,4;1,2,4)-s_{02}G(1,3,4;1,2,4)+s_{03}G(1,2,4;1,2,4)\right)}{G(1,2,4;1,2,4)G(1,2,3,4;1,2,3,4)}.$

Choosing $j_{0}=3,j_{1}=1,j_{2}=2$ in (37), we have

$\displaystyle C^{(1,2,3)}(2|4)$ $\displaystyle=$
$\displaystyle\left.\left(C^{(0,1,2)}(2|4)\right)\right|_{K_{i}\to
K_{i}-K_{3},i\not=3;K_{3}\to-K_{3},M_{0}\leftrightarrow M_{3}}.$ (126)

Reduction coefficients of boxes

$\displaystyle C^{(0,1,2,3)}(2|4)$ $\displaystyle=$ $\displaystyle{1\over
M_{0}^{2}}\Bigg{[}c_{0,0,0,0}^{(0,1,2,3)}(2)M_{0}^{2}s_{00}+c_{2,0,0,0}^{(0,1,2,3)}(2)s_{01}^{2}+c_{0,2,0,0}^{(0,1,2,3)}(2)s_{02}^{2}+c_{0,0,2,0}^{(0,1,2,3)}(2)s_{03}^{2}$
(127)
$\displaystyle+c_{0,0,0,2}^{(0,1,2,3)}(2)s_{04}^{2}+c_{1,1,0,0}^{(0,1,2,3)}(2)s_{01}s_{02}+c_{0,1,1,0}^{(0,1,2,3)}(2)s_{02}s_{03}+c_{1,0,1,0}^{(0,1,2,3)}(2)s_{01}s_{03}$
$\displaystyle+c_{1,0,0,1}^{(0,1,2,3)}(2)s_{01}s_{04}+c_{0,1,0,1}^{(0,1,2,3)}(2)s_{02}s_{04}+c_{0,0,1,1}^{(0,1,2,3)}(2)s_{03}s_{04}\Bigg{]}.$

where

$\displaystyle c_{0,0,0,0}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\frac{f_{1}G(2,3,4;1,2,3)}{(4-D)G(1,2,3,4)}+\frac{f_{2}G(1,3,4;1,2,3)}{(D-4)G(1,2,3,4)}$
(128)
$\displaystyle+\frac{f_{3}G(1,2,4;1,2,3)}{(4-D)G(1,2,3,4)}+\frac{f_{4}G(1,2,3)}{(D-4)G(1,2,3,4)}$
$\displaystyle c_{2,0,0,0}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\frac{(D-3)f_{1}M_{0}^{2}G(2,3,4;1,2,3)G(2,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}$
(129)
$\displaystyle-\frac{M_{0}^{2}G(2,3,4)\left(f_{2}G(1,3,4;1,2,3)-f_{3}G(1,2,4;1,2,3)\right)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}$
$\displaystyle-\frac{f_{4}M_{0}^{2}G(1,2,3)G(2,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}-\frac{f_{4}M_{0}^{2}G(2,3,4;1,2,3)^{2}}{G(1,2,3,4;1,2,3,4)^{2}}$
$\displaystyle+G(2,3,4;1,2,3)\frac{M_{0}^{2}\left(f_{1}G(2,3;2,3)-f_{2}G(2,3;1,3)+f_{3}G(2,3;1,2)\right)}{G(1,2,3)G(1,2,3,4;1,2,3,4)}$
$\displaystyle-G(2,3,4;1,2,3)\frac{M_{0}^{2}\left(f_{2}G(2,3,4;1,3,4)-f_{3}G(2,3,4;1,2,4)\right)}{G(1,2,3,4;1,2,3,4)^{2}},$
$\displaystyle c_{0,0,0,2}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\frac{(D-3)M_{0}^{2}G(1,2,3)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}\Big{[}f_{1}(G(2,3,4;1,2,3))$
(130) $\displaystyle-
f_{2}G(1,3,4;1,2,3)+f_{3}G(1,2,4;1,2,3)-f_{4}G(1,2,3)\Big{]},$ $\displaystyle
c_{1,1,0,0}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\frac{2(D-3)f_{2}M_{0}^{2}G(1,3,4;1,2,3)G(2,3,4;1,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}$
$\displaystyle+\frac{2M_{0}^{2}G(2,3,4;1,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}\Big{[}f_{4}G(1,2,3)-f_{3}G(1,2,4;1,2,3)\Big{]}$
$\displaystyle+{2M_{0}^{2}G(2,3,4;1,2,3)\over
G(1,2,3,4;1,2,3,4)^{2}}\Big{[}f_{4}G(1,3,4;1,2,3)-\frac{f_{1}G(2,3,4;1,3,4)}{(D-4)}\Big{]}$
$\displaystyle-\frac{2M_{0}^{2}G(1,3,4;1,2,3)}{G(1,2,3,4;1,2,3,4)^{2}}\Big{[}f_{1}G(2,3,4)-f_{3}G(2,3,4;1,2,4)\Big{]}$
$\displaystyle-\frac{2M_{0}^{2}G(2,3,4;1,2,3)\left(f_{1}G(2,3;1,3)-f_{2}G(1,3;1,3)+f_{3}G(1,3;1,2)\right)}{G(1,2,3)G(1,2,3,4;1,2,3,4)},$
$\displaystyle c_{1,0,0,1}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle-\frac{2f_{1}M_{0}^{2}G(2,3,4;1,2,3)^{2}}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}$
(132)
$\displaystyle+{2M_{0}^{2}G(2,3,4;1,2,3)\over(D-4)G(1,2,3,4;1,2,3,4)^{2}}\Big{[}f_{2}G(1,3,4;1,2,3)-f_{3}G(1,2,4;1,2,3)\Big{]}$
$\displaystyle+{2M_{0}^{2}G(1,2,3)\over
G(1,2,3,4;1,2,3,4)^{2}}\Bigg{[}\frac{(D-3)f_{4}G(2,3,4;1,2,3)}{(D-4)}$
$\displaystyle-f_{1}G(2,3,4)+f_{2}G(2,3,4;1,3,4)-f_{3}G(2,3,4;1,2,4)\Bigg{]}.$

Other expansion coefficients can be got by using the permutation symmetry:

$\displaystyle c_{0,2,0,0}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{2,0,0,0}^{(0,1,2,3)}(2)\right|_{1\leftrightarrow 2},$
$\displaystyle c_{0,0,2,0}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{2,0,0,0}^{(0,1,2,3)}(2)\right|_{1\leftrightarrow 3},$
$\displaystyle c_{1,0,1,0}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{1,1,0,0}^{(0,1,2,3)}(2)\right|_{2\leftrightarrow 3},$
$\displaystyle c_{0,1,1,0}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{1,1,0,0}^{(0,1,2,3)}(2)\right|_{1\leftrightarrow 2},$
$\displaystyle c_{0,1,0,1}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{1,0,0,1}^{(0,1,2,3)}(2)\right|_{1\leftrightarrow 2},$
$\displaystyle c_{0,0,1,1}^{(0,1,2,3)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{1,0,0,1}^{(0,1,2,3)}(2)\right|_{1\leftrightarrow 3}.$
(133)

Choosing $j_{0}=4,j_{1}=1,j_{2}=2,j_{3}=3$ in (37), we have

$\displaystyle C^{(1,2,3,4)}(2|4)$ $\displaystyle=$ $\displaystyle
4s_{04}\left.\left(C^{(0,1,2,3)}(1|4)\right)\right|_{K_{i}\to
K_{i}-K_{4},i<4;K_{4}\to-K_{4},M_{0}\leftrightarrow M_{4}}$ (134)
$\displaystyle+\left.\left(C^{(0,1,2,3)}(2|4)\right)\right|_{K_{i}\to
K_{i}-K_{4},i<4;K_{4}\to-K_{4},M_{0}\leftrightarrow M_{4}}.$

The reduction coefficient of pentagon

$\displaystyle C^{(0,1,2,3,4)}(2|4)$ $\displaystyle=$
$\displaystyle\Bigg{[}c_{0,0,0,0}^{(0,1,2,3,4)}(2)M_{0}^{2}s_{00}+c_{2,0,0,0}^{(0,1,2,3,4)}(2)s_{01}^{2}+c_{0,2,0,0}^{(0,1,2,3,4)}(2)s_{02}^{2}+c_{0,0,2,0}^{(0,1,2,3,4)}(2)s_{03}^{2}$
(135)
$\displaystyle+c_{0,0,0,2}^{(0,1,2,3,4)}(2)s_{04}^{2}+c_{1,1,0,0}^{(0,1,2,3,4)}(2)s_{01}s_{02}+c_{0,1,1,0}^{(0,1,2,3,4)}(2)s_{02}s_{03}+c_{1,0,1,0}^{(0,1,2,3,4)}(2)s_{01}s_{03}$
$\displaystyle+c_{1,0,0,1}^{(0,1,2,3,4)}(2)s_{01}s_{04}+c_{0,1,0,1}^{(0,1,2,3,4)}(2)s_{02}s_{04}+c_{0,0,1,1}^{(0,1,2,3,4)}(2)s_{03}s_{04}\Bigg{]}.$

where

$\displaystyle c_{0,0,0,0}^{(0,1,2,3,4)}(2)$ $\displaystyle=$
$\displaystyle-\frac{f_{1}^{2}G(2,3,4)}{(D-4)M_{0}^{2}G(1,2,3,4;1,2,3,4)}+\frac{2f_{2}f_{1}G(2,3,4;1,3,4)}{(D-4)M_{0}^{2}G(1,2,3,4;1,2,3,4)}$
(136)
$\displaystyle-\frac{2f_{3}f_{1}G(2,3,4;1,2,4)}{(D-4)M_{0}^{2}G(1,2,3,4;1,2,3,4)}+\frac{2f_{4}f_{1}G(2,3,4;1,2,3)}{(D-4)M_{0}^{2}G(1,2,3,4;1,2,3,4)}$
$\displaystyle-\frac{f_{2}^{2}G(1,3,4)}{(D-4)M_{0}^{2}G(1,2,3,4;1,2,3,4)}-\frac{f_{3}^{2}G(1,2,4)}{(D-4)M_{0}^{2}G(1,2,3,4;1,2,3,4)}$
$\displaystyle-\frac{f_{4}^{2}G(1,2,3)}{(D-4)M_{0}^{2}G(1,2,3,4;1,2,3,4)}+\frac{2f_{2}f_{3}G(1,3,4;1,2,4)}{(D-4)M_{0}^{2}G(1,2,3,4;1,2,3,4)}$
$\displaystyle-\frac{2f_{2}f_{4}G(1,3,4;1,2,3)}{(D-4)M_{0}^{2}G(1,2,3,4;1,2,3,4)}+\frac{2f_{3}f_{4}G(1,2,4;1,2,3)}{(D-4)M_{0}^{2}G(1,2,3,4;1,2,3,4)}$
$\displaystyle+\frac{4}{D-4},$ $\displaystyle c^{(0,1,2,3,4)}_{2,0,0,0}(2)$
$\displaystyle=$
$\displaystyle\frac{(D-3)f_{1}^{2}G(2,3,4)^{2}}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}-\frac{2(D-3)f_{2}f_{1}G(2,3,4;1,3,4)G(2,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}$
(137)
$\displaystyle+\frac{2(D-3)f_{3}f_{1}G(2,3,4;1,2,4)G(2,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}-\frac{2(D-3)f_{4}f_{1}G(2,3,4;1,2,3)G(2,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}$
$\displaystyle+f_{2}^{2}\left(\frac{G(1,3,4)G(2,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}+\frac{G(2,3,4;1,3,4)^{2}}{G(1,2,3,4;1,2,3,4)^{2}}\right)$
$\displaystyle+\frac{f_{3}^{2}\left((D-4)G(2,3,4;1,2,4)^{2}+G(1,2,4;1,2,4)G(2,3,4)\right)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}$
$\displaystyle+f_{4}^{2}\left(\frac{G(1,2,3)G(2,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}+\frac{G(2,3,4;1,2,3)^{2}}{G(1,2,3,4;1,2,3,4)^{2}}\right)$
$\displaystyle+f_{2}f_{3}\left(-\frac{2G(1,3,4;1,2,4)G(2,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}-\frac{2G(2,3,4;1,2,4)G(2,3,4;1,3,4)}{G(1,2,3,4;1,2,3,4)^{2}}\right)$
$\displaystyle+f_{2}f_{4}\left(\frac{2G(1,3,4;1,2,3)G(2,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}+\frac{2G(2,3,4;1,2,3)G(2,3,4;1,3,4)}{G(1,2,3,4;1,2,3,4)^{2}}\right)$
$\displaystyle+f_{3}f_{4}\left(-\frac{2G(1,2,4;1,2,3)G(2,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}-\frac{2G(2,3,4;1,2,3)G(2,3,4;1,2,4)}{G(1,2,3,4;1,2,3,4)^{2}}\right)$
$\displaystyle-\frac{4M_{0}^{2}G(2,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)},$
$\displaystyle c^{(0,1,2,3,4)}_{1,1,0,0}(2)$ $\displaystyle=$
$\displaystyle-\frac{2(D-3)G(2,3,4;1,3,4)G(2,3,4)f_{1}^{2}}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}+\frac{8G(2,3,4;1,3,4)M_{0}^{2}}{(D-4)G(1,2,3,4;1,2,3,4)}$
$\displaystyle+\left(\frac{2(D-2)G(2,3,4;1,3,4)^{2}}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}+\frac{2G(1,3,4)G(2,3,4)}{G(1,2,3,4;1,2,3,4)^{2}}\right)f_{2}f_{1}$
$\displaystyle+\left(-\frac{2(D-2)G(2,3,4;1,2,4)G(2,3,4;1,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}-\frac{2G(1,3,4;1,2,4)G(2,3,4)}{G(1,2,3,4;1,2,3,4)^{2}}\right)f_{3}f_{1}$
$\displaystyle+\left(\frac{2(D-2)G(2,3,4;1,2,3)G(2,3,4;1,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}+\frac{2G(1,3,4;1,2,3)G(2,3,4)}{G(1,2,3,4;1,2,3,4)^{2}}\right)f_{4}f_{1}$
$\displaystyle+\left(-\frac{2G(1,3,4;1,2,4)G(2,3,4;1,2,4)}{G(1,2,3,4;1,2,3,4)^{2}}-\frac{2G(1,2,4;1,2,4)G(2,3,4;1,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}\right)f_{3}^{2}$
$\displaystyle+\left(-\frac{2G(1,3,4;1,2,3)G(2,3,4;1,2,3)}{G(1,2,3,4;1,2,3,4)^{2}}-\frac{2G(1,2,3)G(2,3,4;1,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}\right)f_{4}^{2}$
$\displaystyle+\left(\frac{2G(1,3,4)G(2,3,4;1,2,4)}{G(1,2,3,4;1,2,3,4)^{2}}+\frac{2(D-2)G(1,3,4;1,2,4)G(2,3,4;1,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}\right)f_{2}f_{3}$
$\displaystyle+\left(-\frac{2G(1,3,4)G(2,3,4;1,2,3)}{G(1,2,3,4;1,2,3,4)^{2}}-\frac{2(D-2)G(1,3,4;1,2,3)G(2,3,4;1,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}\right)f_{2}f_{4}$
$\displaystyle+\left(\frac{2G(1,3,4;1,2,4)G(2,3,4;1,2,3)}{G(1,2,3,4;1,2,3,4)^{2}}+\frac{2G(1,3,4;1,2,3)G(2,3,4;1,2,4)}{G(1,2,3,4;1,2,3,4)^{2}}\right)f_{3}f_{4}$
$\displaystyle+\frac{4G(1,2,4;1,2,3)G(2,3,4;1,3,4)}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}f_{3}f_{4}-\frac{2(D-3)G(1,3,4)G(2,3,4;1,3,4)f_{2}^{2}}{(D-4)G(1,2,3,4;1,2,3,4)^{2}}.$

Other expansion coefficients can be got by using permutation symmetry:

$\displaystyle c_{0,2,0,0}^{(0,1,2,3,4)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{2,0,0,0}^{(0,1,2,3,4)}(2)\right|_{1\leftrightarrow 2},$
$\displaystyle c_{0,0,2,0}^{(0,1,2,3,4)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{2,0,0,0}^{(0,1,2,3,4)}(2)\right|_{1\leftrightarrow 3},$
$\displaystyle c_{0,0,0,2}^{(0,1,2,3,4)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{2,0,0,0}^{(0,1,2,3,4)}(2)\right|_{1\leftrightarrow 4},$
$\displaystyle c_{1,0,1,0}^{(0,1,2,3,4)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{1,1,0,0}^{(0,1,2,3,4)}(2)\right|_{2\leftrightarrow 3},$
$\displaystyle c_{0,1,1,0}^{(0,1,2,3,4)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{1,1,0,0}^{(0,1,2,3,4)}(2)\right|_{1\leftrightarrow 2},$
$\displaystyle c_{0,1,0,1}^{(0,1,2,3,4)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{1,0,0,1}^{(0,1,2,3,4)}(2)\right|_{1\leftrightarrow 2},$
$\displaystyle c_{0,0,1,1}^{(0,1,2,3,4)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{1,0,0,1}^{(0,1,2,3,4)}(2)\right|_{1\leftrightarrow 3},$
$\displaystyle c_{1,0,0,1}^{(0,1,2,3,4)}(2)$ $\displaystyle=$
$\displaystyle\left.c_{1,0,0,1}^{(0,1,2,3,4)}(2)\right|_{1\leftrightarrow 4}.$
(139)

## References

  * (1) B. Feng, T. Li, and X. Li, Analytic tadpole coefficients of one-loop integrals, JHEP 09 (2021) 081, [arXiv:2107.03744].
  * (2) L. M. Brown and R. P. Feynman, Radiative corrections to Compton scattering, Phys. Rev. 85 (1952) 231–244.
  * (3) D. B. Melrose, Reduction of Feynman diagrams, Nuovo Cim. 40 (1965) 181–213.
  * (4) G. Passarino and M. J. G. Veltman, One Loop Corrections for e+ e- Annihilation Into mu+ mu- in the Weinberg Model, Nucl. Phys. B 160 (1979) 151–207.
  * (5) G. ’t Hooft and M. J. G. Veltman, Scalar One Loop Integrals, Nucl. Phys. B 153 (1979) 365–401.
  * (6) W. L. van Neerven and J. A. M. Vermaseren, LARGE LOOP INTEGRALS, Phys. Lett. B 137 (1984) 241–244.
  * (7) R. G. Stuart, Algebraic Reduction of One Loop Feynman Diagrams to Scalar Integrals, Comput. Phys. Commun. 48 (1988) 367–389.
  * (8) G. J. van Oldenborgh and J. A. M. Vermaseren, New Algorithms for One Loop Integrals, Z. Phys. C 46 (1990) 425–438.
  * (9) Z. Bern, L. J. Dixon, and D. A. Kosower, Dimensionally regulated one loop integrals, Phys. Lett. B 302 (1993) 299–308, [hep-ph/9212308]. [Erratum: Phys.Lett.B 318, 649 (1993)].
  * (10) Z. Bern, L. J. Dixon, and D. A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751–816, [hep-ph/9306240].
  * (11) J. Fleischer, F. Jegerlehner, and O. V. Tarasov, Algebraic reduction of one loop Feynman graph amplitudes, Nucl. Phys. B 566 (2000) 423–440, [hep-ph/9907327].
  * (12) T. Binoth, J. P. Guillet, and G. Heinrich, Reduction formalism for dimensionally regulated one loop N point integrals, Nucl. Phys. B 572 (2000) 361–386, [hep-ph/9911342].
  * (13) A. Denner and S. Dittmaier, Reduction of one loop tensor five point integrals, Nucl. Phys. B 658 (2003) 175–202, [hep-ph/0212259].
  * (14) G. Duplancic and B. Nizic, Reduction method for dimensionally regulated one loop N point Feynman integrals, Eur. Phys. J. C 35 (2004) 105–118, [hep-ph/0303184].
  * (15) A. Denner and S. Dittmaier, Reduction schemes for one-loop tensor integrals, Nucl. Phys. B 734 (2006) 62–115, [hep-ph/0509141].
  * (16) R. K. Ellis and G. Zanderighi, Scalar one-loop integrals for QCD, JHEP 02 (2008) 002, [arXiv:0712.1851].
  * (17) G. Ossola, C. G. Papadopoulos, and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nuclear Physics B 763 (Feb, 2007) 147–169.
  * (18) Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys. B 435 (1995) 59–101, [hep-ph/9409265].
  * (19) K. G. Chetyrkin and F. V. Tkachov, Integration by Parts: The Algorithm to Calculate beta Functions in 4 Loops, Nucl. Phys. B 192 (1981) 159–204.
  * (20) F. V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65–68.
  * (21) G. Ossola, C. G. Papadopoulos, and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys. B 763 (2007) 147–169, [hep-ph/0609007].
  * (22) G. Ossola, C. G. Papadopoulos, and R. Pittau, Numerical evaluation of six-photon amplitudes, JHEP 07 (2007) 085, [arXiv:0704.1271].
  * (23) R. K. Ellis, W. T. Giele, and Z. Kunszt, A Numerical Unitarity Formalism for Evaluating One-Loop Amplitudes, JHEP 03 (2008) 003, [arXiv:0708.2398].
  * (24) Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217–260, [hep-ph/9403226].
  * (25) R. Britto, F. Cachazo, and B. Feng, Generalized unitarity and one-loop amplitudes in N=4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275–305, [hep-th/0412103].
  * (26) R. Britto, E. Buchbinder, F. Cachazo, and B. Feng, One-loop amplitudes of gluons in SQCD, Phys. Rev. D 72 (2005) 065012, [hep-ph/0503132].
  * (27) J. M. Campbell, E. W. N. Glover, and D. J. Miller, One loop tensor integrals in dimensional regularization, Nucl. Phys. B 498 (1997) 397–442, [hep-ph/9612413].
  * (28) Z. Bern, L. J. Dixon, and D. A. Kosower, One loop amplitudes for e+ e- to four partons, Nucl. Phys. B 513 (1998) 3–86, [hep-ph/9708239].
  * (29) C. Anastasiou, R. Britto, B. Feng, Z. Kunszt, and P. Mastrolia, Unitarity cuts and Reduction to master integrals in d dimensions for one-loop amplitudes, JHEP 03 (2007) 111, [hep-ph/0612277].
  * (30) R. Britto and E. Mirabella, Single Cut Integration, JHEP 01 (2011) 135, [arXiv:1011.2344].
  * (31) C. Anastasiou, R. Britto, B. Feng, Z. Kunszt, and P. Mastrolia, D-dimensional unitarity cut method, Phys. Lett. B 645 (2007) 213–216, [hep-ph/0609191].
  * (32) R. Britto and B. Feng, Unitarity cuts with massive propagators and algebraic expressions for coefficients, Phys. Rev. D 75 (2007) 105006, [hep-ph/0612089].
  * (33) R. Britto and B. Feng, Integral coefficients for one-loop amplitudes, JHEP 02 (2008) 095, [arXiv:0711.4284].
  * (34) R. Britto, B. Feng, and P. Mastrolia, Closed-Form Decomposition of One-Loop Massive Amplitudes, Phys. Rev. D 78 (2008) 025031, [arXiv:0803.1989].
  * (35) R. Britto, B. Feng, and G. Yang, Polynomial Structures in One-Loop Amplitudes, JHEP 09 (2008) 089, [arXiv:0803.3147].
  * (36) B. Feng and H. Wang, Analytic structure of one-loop coefficients, JHEP 05 (2013) 104, [arXiv:1301.7510].
  * (37) R. Britto, F. Cachazo, and B. Feng, New recursion relations for tree amplitudes of gluons, Nucl. Phys. B 715 (2005) 499–522, [hep-th/0412308].
  * (38) R. Britto, F. Cachazo, B. Feng, and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602, [hep-th/0501052].
  * (39) A. V. Smirnov, Algorithm FIRE – Feynman Integral REduction, JHEP 10 (2008) 107, [arXiv:0807.3243].
  * (40) A. V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun. 189 (2015) 182–191, [arXiv:1408.2372].
  * (41) A. V. Smirnov and V. A. Smirnov, FIRE4, LiteRed and accompanying tools to solve integration by parts relations, Comput. Phys. Commun. 184 (2013) 2820–2827, [arXiv:1302.5885].
  * (42) A. V. Smirnov and F. S. Chuharev, FIRE6: Feynman Integral REduction with Modular Arithmetic, Comput. Phys. Commun. 247 (2020) 106877, [arXiv:1901.07808].
  * (43) R. N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059, [arXiv:1310.1145].