Scalar $1$-loop Feynman integrals as meromorphic functions in space-time dimension $d$, $II$: Special kinematics

Scalar one-loop integrals in general $d$ are important for several reasons. In general framework for computing two-loop and higher-loop corrections, higher-terms in the $\epsilon$-expansion ($\epsilon=2-d/2$) for one-loop integrals are necessary for building blocks. For example, they are used for building counterterms. Furthermore, in the evaluations for multi-loop Feynman integrals, we may combine several methods in [1, 2] to optimize the master integrals. As a result, the resulting integrals may include of one-loop functions in arbitrary space-time dimensions. Last but not least, scalar one-loop functions at $d=4+2n\pm 2\epsilon$ with $n\in\mathbb{N}$ may be taken into account in the reduction for tensor one-loop Feynman integrals [3].

One-loop functions in space-time dimensions $d$ have been performed in [4, 5, 6, 7, 8, 9]. However, not all of the calculations have covered at general configurations of external momenta and internal masses. Recently, scalar one-loop three-point functions have been performed by applying multiple unitarity cuts for Feynman diagrams [10]. In Ref. [10], analytic results have been presented in terms of hypergeometric functions in special cases of external invariants and internal masses. In more general cases, the results have only presented $\epsilon^{0}$-terms in space-time $d=4-2\epsilon$. The algebraic structure of cut Feynman integrals, the diagrammatic coaction and its applications have proposed in [11, 12, 13]. However, the detailed analytic results for one-loop Feynman integrals have not been shown in the mentioned papers. More recently, detailed analytic results for one-loop three-point functions which are expressed in terms of Appell $F_{1}$ hypergeometric functions have been reported in [15].

A recurrence relation in $d$ for Feynman loop integrals has proposed by Tarasov [2, 14, 16]. By solving the differential equation in $d$, analytic results for scalar one-loop integrals up to four-points have been expressed in terms of generalized hypergeometric series such as Gauss ${}_{2}F_{1}$, Appell $F_{1}$, Lauricella-Saran $F_{S}$ functions. In [16], boundary terms have obtained by applying asymptotic theory of complex Laplace-type integrals. These terms are only valid in sub-domain of external momentum and mass configurations which the theory are applicable. Hence, the general solutions for one-loop integrals in arbitrary kinematics have not been found [16], as pointed out in [17, 18]. General solutions for this problem have been derived in [19] which proposed a different kind of recursion relation for one-loop integrals in comparison with [2, 14, 16]. In the scope of this paper, based on the method in Ref. [19], detailed analytic results for scalar one-loop two-, three- and four-point functions in general $d$-dimensions are presented. The calculations are considered all external kinematic configurations and internal mass assignments. Thus, we go beyond the material presented in [19].

The layout of the paper is as follows: In section $2$, we discuss briefly the method for evaluating one-loop integrals. We then apply this method for computing scalar one-loop two-, three- and four-point functions in sections $3,4,5$. Conclusions and plans for future work are presented in section $6$.

In this section, we describe briefly the method for evaluating scalar one-loop $N$-point Feynman integrals. Detailed description for this method can be found in Ref. [19]. A general recursion relation between scalar one-loop $N$-point and $(N$-$1)$-point Feynman integrals is shown in this section. From the representation, analytic formulas for scalar one-loop $N$-point functions can be constructed from basic integrals which are scalar one-point integrals. For illustrating, analytic expressions for scalar one-loop two-, three-, four-point functions are derived in detail in next sections.

The scalar one-loop $N$-point Feynman integrals are defined:

$\displaystyle J_{N}(d;\{p_{i}p_{j}\},\{m_{i}^{2}\})$

$\displaystyle=$

$\displaystyle\int\dfrac{d^{d}k}{i\pi^{d/2}}\dfrac{1}{P_{1}P_{2}\dots P_{N}}.$

(1)

The inverse Feynman propagators are given:

$\displaystyle P_{j}=$

$\displaystyle(k+q_{j})^{2}-m_{j}^{2}+i\rho,\;\text{for $j=1,2,\cdots,N$}.$

(2)

Where $p_{j}$ ($m_{j}$) for $j=1,2,\cdots,N$ are external momenta (internal masses) respectively. The momenta $q_{j}$ are given: $q_{1}=p_{1},q_{2}=p_{1}+p_{2},\cdots,q_{j}=\sum_{i=1}^{j}p_{i}$ and $q_{N}=\sum_{j=1}^{N}p_{j}=0$ thanks to momentum conservation. They are inward as described in Fig. 1. The term $i\rho$ is Feynman’s prescription and $d$ is space-time dimension. Several cases of physical interests for $d$ are $d=4+2n\pm 2\epsilon$ with $n\in\mathbb{N}$. In the complex-mass scheme [20], the internal masses take the form of $m_{j}^{2}=m_{0j}^{2}-im_{0j}\Gamma_{j}$ where $\Gamma_{j}\geqslant 0$ are decay widths of unstable particles.

The Cayley and Gram determinants [16] related to one-loop Feynman $N$-point topologies are defined as follows:

$\displaystyle Y_{N}\equiv Y_{12\cdots N}=\left|\begin{array}[]{cccc}Y_{11}&Y_{12}&\ldots&Y_{1N}\\ Y_{12}&Y_{22}&\ldots&Y_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ Y_{1N}&Y_{2N}&\ldots&Y_{NN}\end{array}\right|,$

$\displaystyle G_{N-1}\equiv G_{12\cdots N}=-2^{N}\;\left|\begin{array}[]{cccc}\!q_{1}^{2}&q_{1}q_{2}&\ldots&q_{1}q_{N-1}\\ \!q_{1}q_{2}&q_{2}^{2}&\ldots&q_{2}q_{N-1}\\ \vdots&\vdots&\ddots&\vdots\\ \!q_{1}q_{N-1}&q_{2}q_{N-1}&\ldots&q_{N-1}^{2}\end{array}\right|$

(11)

with $Y_{ij}=-(q_{i}-q_{j})^{2}+m_{i}^{2}+m_{j}^{2}$.

In this report, analytic solutions for one-loop integrals are expressed in terms of generalized hypergeometric with arguments given by ratios of the above determinants. Hence, it is worth to introduce the following kinematic variables

$\displaystyle R_{N}\equiv R_{12\cdots N}$

$\displaystyle=$

$\displaystyle-\frac{Y_{N}}{G_{N-1}}\quad\text{for}\quad G_{N-1}\neq 0.$

(13)

The kinematics $R_{N}$ play a role of the squared internal masses. In fact, when we shift $m_{j}^{2}\rightarrow m_{j}^{2}-i\rho$, one verifies easily that $R_{N}\rightarrow R_{N}-i\rho$ [16].

The recursion relation for $J_{N}$ [19] is given (master equation):

	$\displaystyle J_{N}(d;\{p_{i}p_{j}\},\{m_{i}^{2}\})$	$\displaystyle=$	$\displaystyle-\dfrac{1}{2\pi i}\int\limits_{-i\infty}^{+i\infty}ds\;\dfrac{\Gamma(-s)\;\Gamma(\frac{d-N+1}{2}+s)\Gamma(s+1)}{2\Gamma(\frac{d-N+1}{2})}\left(\frac{1}{R_{N}}\right)^{s}\times$
			$\displaystyle\hskip 28.45274pt\times\sum\limits_{k=1}^{N}\left(\frac{\partial_{k}R_{N}}{R_{N}}\right)\;{\bf k}^{-}\;J_{N}(d+2s;\{p_{i}p_{j}\},\{m_{i}^{2}\}),$

for $i,j=1,2,\cdots,N$ and $\partial_{k}=\partial/\partial m_{k}^{2}$. Here the operator ${\bf k}^{-}$ is defined as [19]

$\displaystyle{\bf k}^{-}J_{N}(d;\{p_{i}p_{j}\},\{m_{i}^{2}\})$

$\displaystyle=$

$\displaystyle\int\frac{d^{d}k}{i\pi^{d/2}}\frac{1}{P_{1}P_{2}\dots P_{k-1}P_{k+1}\dots P_{N-1}P_{N}}.$

(15)

The relation (2) indicates that the integral $J_{N}$ can be constructed by taking one-fold Mellin-Barnes (MB) integration over $J_{N-1}$ in $d+2s$. This representation has several advantages. First, analytic formulas for $J_{N}$ can be derived from basic functions which are scalar one-loop one-point functions. Second, $J_{N}$ is expressed as functions of kinematic variables such as $m_{j}^{2}$ for $j=1,2,\cdots,N$ and $R_{N}$. As a consequence of this fact, analytic expressions for $J_{N}$ reflect the symmetry as well as threshold behavior of the corresponding Feynman topologies. Two special cases of (2) are also mentioned as follows:

1. 1.

   $Y_{N}\rightarrow 0$ and $G_{N-1}\neq 0$: In this case, $R_{N}\rightarrow 0$ and we have [16] (deriving this equation for $N=4$ is shown in the appendix $D$)

   $\displaystyle J_{N}(d;\{p_{i}p_{j}\},\{m_{i}^{2}\})=\frac{1}{d-N-1}\sum\limits_{k=1}^{N}\left(\frac{\partial_{k}Y_{N}}{G_{N-1}}\right){\bf k^{-}}J_{N}(d-2;\{p_{i}p_{j}\},\{m_{i}^{2}\}).$

   (16)

2. 2.

   $G_{N-1}\rightarrow 0$ and $Y_{N}\neq 0$: In this case, $R_{N}\rightarrow\infty$. We close the integration contour in (2) to the right. Taking residue contributions from poles of $\Gamma(\cdots-s)$. In the limit $R_{N}\rightarrow\infty$, we find only the term with $s=0$ is non-zero. The result then reads

   $\displaystyle J_{N}(d;\{p_{i}p_{j}\},\{m_{i}^{2}\})=-\frac{1}{2}\sum\limits_{k=1}^{N}\left(\frac{\partial_{k}Y_{N}}{Y_{N}}\right)\;{\bf k}^{-}J_{N}(d;\{p_{i}p_{j}\},\{m_{i}^{2}\}).$

   (17)

   This equation is equivalent to ($65$) in [21] and ($3$) in [16].

We turn our attention to apply the method for evaluating scalar one-loop Feynman integrals. The detailed evaluations for scalar one-loop two-, three- and four-point functions are presented in next sections. As we pointed out in this section, the prescription $i\rho$ always follows with $R_{N}$ as $R_{N}-i\rho$. In order to simplify the notation, we omit $i\rho$ in $R_{N}$ in the next calculations. This term puts back into the final results when it is necessary.

The master equation for $J_{2}$ is obtained by setting $N=2$ in (2). MB representation for $J_{2}$ then reads

	$\displaystyle J_{2}\equiv J_{2}(d;p^{2},m_{1}^{2},m_{2}^{2})$	$\displaystyle=$	$\displaystyle\frac{1}{2\pi i}\int\limits_{-i\infty}^{+i\infty}ds\;\dfrac{\Gamma(-s)\Gamma\left(\frac{2-d}{2}-s\right)\Gamma\left(\frac{d-1}{2}+s\right)\Gamma(s+1)}{2\Gamma\left(\frac{d-1}{2}\right)}\times$
			$\displaystyle\hskip 28.45274pt\times\left(\dfrac{1}{R_{2}}\right)^{s}\left\{\left(\dfrac{\partial_{2}R_{2}}{R_{2}}\right)(m_{1}^{2})^{\frac{d-2}{2}+s}+(1\leftrightarrow 2)\right\}.$

Note that we used the analytic formula for $J_{1}$ in [22] with $d$ shifted to $d\rightarrow d+2s$. We write $J_{1}$ in $d+2s$ explicitly as follows:

$\displaystyle J_{1}(d+2s;m^{2})$

$\displaystyle=$

$\displaystyle-\Gamma\left(\frac{2-d}{2}-s\right)(m)^{\frac{d-2}{2}+s}.$

(19)

In order to evaluate the MB integrals in (3), we close the integration contour to the right. The residue contributions to $J_{2}$ at the sequence poles of $\Gamma(-s)$ and $\Gamma\left(\frac{2-d}{2}-s\right)$ are taken into account.

First, we calculate the residue at the poles of $\Gamma(-s)$. In this case, $s=m$ for $m=0,1,\cdots,\mathbb{N}$. Subsequently, we can apply the reflect formula for gamma functions (253) in the appendix $B$. In detail, it is implied that

$\displaystyle\Gamma\left(\frac{2-d}{2}-s\right)\Gamma\left(\frac{d}{2}+s\right)=-(-1)^{s}\Gamma\left(\frac{4-d}{2}\right)\Gamma\left(\frac{d-2}{2}\right).$

(20)

With the help of (20), the MB representation in (3) is casted into the form of

	$\displaystyle\dfrac{J_{2}}{\Gamma\left(\frac{4-d}{2}\right)}\Big{|}_{s=m}$	$\displaystyle=$	$\displaystyle-\dfrac{\Gamma\left(\frac{d-2}{2}\right)}{2\;\Gamma\left(\frac{d-1}{2}\right)}\;\frac{1}{2\pi i}\int\limits_{-i\infty}^{+i\infty}ds\;\dfrac{\Gamma(-s)\;\Gamma\left(\frac{d-1}{2}+s\right)\;\Gamma(s+1)}{\Gamma\left(\frac{d}{2}+s\right)}\;\times$
			$\displaystyle\hskip 56.9055pt\times\left\{\left(\frac{\partial_{2}R_{2}}{R_{2}}\right)(m_{1}^{2})^{\frac{d-2}{2}}\left(-\dfrac{m_{1}^{2}}{R_{2}}\right)^{s}+(1\leftrightarrow 2)\right\}.$

Using ($1.6.1.6$) in Ref. [25], these MB integrals are expressed in terms of Gauss hypergeometric series as follows:

$\displaystyle\dfrac{J_{2}}{\Gamma\left(\frac{4-d}{2}\right)}\Big{|}_{s=m}$

$\displaystyle=$

$\displaystyle-\dfrac{\Gamma\left(\frac{d-2}{2}\right)}{2\;\Gamma\left(\frac{d}{2}\right)}\;\left\{\left(\frac{\partial_{2}R_{2}}{R_{2}}\right)(m_{1}^{2})^{\frac{d-2}{2}}\;\,{}_{2}F_{1}\!\!\left[\begin{array}[]{c}1,\frac{d-1}{2}\,;\\ \frac{d}{2}\,;\end{array}\frac{m_{1}^{2}}{R_{2}}\right]+(1\leftrightarrow 2)\right\},$

(24)

provided that $\left|m_{1}^{2}/R_{2}\right|<1$, $\left|m_{2}^{2}/R_{2}\right|<1$ and $\mathcal{R}$e$(d-2)>0$. Using ($1.3.15$) in [25], we arrive at another representation for (24)

$\displaystyle\dfrac{J_{2}}{\Gamma\left(\frac{4-d}{2}\right)}\Big{|}_{s=m}$

$\displaystyle=$

$\displaystyle-\dfrac{\Gamma(\frac{d-2}{2})}{2\Gamma(\frac{d}{2})}\left\{\left(\frac{\partial_{2}R_{2}}{R_{2}}\right)\frac{(m_{1}^{2})^{\frac{d-2}{2}}}{\sqrt{1-m_{1}^{2}/R_{2}}}\;\,{}_{2}F_{1}\!\!\left[\begin{array}[]{c}\frac{d-2}{2},\frac{1}{2}\,;\\ \frac{d}{2}\,;\end{array}\dfrac{m_{1}^{2}}{R_{2}}\right]+(1\leftrightarrow 2)\right\},$

(27)

provided that $\left|m_{1}^{2}/R_{2}\right|<1$, $\left|m_{2}^{2}/R_{2}\right|<1$ and $\mathcal{R}$e$(d-2)>0$.

We next consider the residue at the second sequence poles of $\Gamma\left(\frac{2-d}{2}-s\right)$. In this case, $s=\frac{2-d}{2}+m$ for $m\in\mathbb{N}$. These contributions read

	$\displaystyle J_{2}\Big{|}_{s=\frac{2-d}{2}+m}$	$\displaystyle=$	$\displaystyle\sum_{m=0}^{\infty}\dfrac{(-1)^{m}}{m!}\dfrac{\Gamma(\frac{d-2}{2}-m)\Gamma(\frac{4-d}{2}+m)\Gamma(m+\frac{1}{2})}{2\Gamma(\frac{d-1}{2})}$		(30)
			$\displaystyle\hskip 19.91684pt\times(R_{2})^{\frac{d-2}{2}}\;\left[\left(\frac{\partial_{2}R_{2}}{R_{2}}\right)\left(\dfrac{m_{1}^{2}}{R_{2}}\right)^{m}+\left(\frac{\partial_{1}R_{2}}{R_{2}}\right)\left(\dfrac{m_{2}^{2}}{R_{2}}\right)^{m}\right]$
			$\displaystyle=\dfrac{\Gamma(\frac{d-2}{2})\Gamma(\frac{4-d}{2})}{2\Gamma(\frac{d-1}{2})}(R_{2})^{\frac{d-2}{2}}\sum_{m=0}^{\infty}\dfrac{\Gamma(m+\frac{1}{2})}{\Gamma(m+1)}\left[\left(\frac{\partial_{2}R_{2}}{R_{2}}\right)\left(\dfrac{m_{1}^{2}}{R_{2}}\right)^{m}+\left(\frac{\partial_{1}R_{2}}{R_{2}}\right)\left(\dfrac{m_{2}^{2}}{R_{2}}\right)^{m}\right]$
			$\displaystyle=\dfrac{\sqrt{\pi}}{2}\;\dfrac{\Gamma(\frac{4-d}{2})\Gamma(\frac{d-2}{2})}{2\Gamma(\frac{d-1}{2})}\left(R_{2}\right)^{\frac{d-4}{2}}\left[\dfrac{\partial_{2}R_{2}}{\sqrt{1-m_{1}^{2}/R_{2}}}+\dfrac{\partial_{1}R_{2}}{\sqrt{1-m_{2}^{2}/R_{2}}}\right].$

Noting that from (30) to (30), we have already applied the reflect formulas for gamma functions (see (253) in appendix $A$ for more detail). Summing all the above contributions in Eqs. (27, 30), we finally get

	$\displaystyle\dfrac{J_{2}}{\Gamma(\frac{4-d}{2})}$	$\displaystyle=$	$\displaystyle\dfrac{\sqrt{\pi}}{2}\dfrac{\Gamma(\frac{d-2}{2})}{\Gamma(\frac{d-1}{2})}\left(R_{2}\right)^{\frac{d-4}{2}}\left[\dfrac{\partial_{2}R_{2}}{\sqrt{1-m_{1}^{2}/R_{2}}}+\dfrac{\partial_{1}R_{2}}{\sqrt{1-m_{2}^{2}/R_{2}}}\right]$		(34)
			$\displaystyle-\dfrac{\Gamma(\frac{d-2}{2})}{2\Gamma(\frac{d}{2})}\left\{\left(\frac{\partial_{2}R_{2}}{R_{2}}\right)\dfrac{(m_{1}^{2})^{\frac{d-2}{2}}}{\sqrt{1-m_{1}^{2}/R_{2}}}\,{}_{2}F_{1}\!\!\left[\begin{array}[]{c}\frac{d-2}{2},\frac{1}{2}\,;\\ \frac{d}{2}\,;\end{array}\dfrac{m_{1}^{2}}{R_{2}}\right]+(1\leftrightarrow 2)\right\},$

provided that $\left|m_{1}^{2}/R_{2}\right|<1$, $\left|m_{2}^{2}/R_{2}\right|<1$ and $\mathcal{R}$e$(d-2)>0$. Eq. (34) is unchanged with exchanging $m_{1}^{2}\leftrightarrow m_{2}^{2}$. This reflects the symmetry of the scalar one-loop two-point Feynman diagrams. The result in (34) has shown in [18] and gives fully agreement with [16]. It is an important to remark that the solution for $J_{2}$ in (34) is also valid when $d\rightarrow d+2n$ for $n\in\mathbb{N}$. We would like to stress that one can perform the analytic continuation for $J_{2}$ in (34) to extend the kinematic regions for one-loop two-point functions. The analytic continuation formulas for Gauss hypergeometric functions are given from (271) to (286) in the appendix $B$. As an example, using (277), the result reads

$\displaystyle\dfrac{J_{2}}{\Gamma\left(\frac{4-d}{2}\right)}$

$\displaystyle=$

$\displaystyle\left(\frac{\partial_{2}R_{2}}{R_{2}}\right)(m_{1}^{2})^{\frac{d-2}{2}}\,{}_{2}F_{1}\!\!\left[\begin{array}[]{c}1,\frac{d-1}{2}\,;\\ \frac{3}{2}\,;\end{array}1-\frac{m_{1}^{2}}{R_{2}}\right]+(1\leftrightarrow 2).$

(37)

With (268), we arrive at

$\displaystyle\dfrac{J_{2}}{\Gamma\left(\frac{4-d}{2}\right)}$

$\displaystyle=$

$\displaystyle\left(\frac{\partial_{2}R_{2}}{R_{2}}\right)R_{2}^{\frac{d-2}{2}}\,{}_{2}F_{1}\!\!\left[\begin{array}[]{c}\frac{4-d}{2},\frac{1}{2}\,;\\ \frac{3}{2}\,;\end{array}1-\frac{m_{1}^{2}}{R_{2}}\right]+(1\leftrightarrow 2).$

(40)

We are going to consider special cases for scalar one-loop two-point Feynman integrals.

Setting $N=3$ in (2), master equation for $J_{3}$ reads

	$\displaystyle J_{3}\equiv J_{3}(d;\{p_{i}^{2}\},\{m_{i}^{2}\})$	$\displaystyle=$	$\displaystyle-\dfrac{1}{2\pi i}\int\limits_{-i\infty}^{+i\infty}ds\;\dfrac{\Gamma(-s)\;\Gamma(\frac{d-2}{2}+s)\Gamma(s+1)}{2\Gamma(\frac{d-2}{2})}\left(\frac{1}{R_{3}}\right)^{s}\times$
			$\displaystyle\hskip 28.45274pt\times\sum\limits_{k=1}^{3}\left(\frac{\partial_{k}R_{3}}{R_{3}}\right)\;{\bf k}^{-}J_{3}(d+2s;\{p_{i}^{2}\},\{m_{i}^{2}\}),$

for $i=1,2,3$. The term ${\bf k}^{-}J_{3}(d+2s;\{p_{i}p_{j}\},\{m_{i}^{2}\})$ becomes scalar one-loop two-point functions by shrinking an propagator $k$-th in the integrand of $J_{3}$. In the next steps, we take the contour integrals in (4). In order to understand how to take the contour integrals, we chose the term with $k=3$ in (4) for illustrating. This term is written explicitly as follows:

	$\displaystyle J_{3,(123)}$	$\displaystyle=$	$\displaystyle-\frac{1}{2\pi i}\int\limits_{-i\infty}^{+i\infty}ds\dfrac{\Gamma(-s)\;\Gamma(s+1)\Gamma\left(\frac{d-2}{2}+s\right)}{2\;\Gamma(\frac{d-2}{2})}\left(\frac{1}{R_{3}}\right)^{s}\left(\frac{\partial_{3}R_{3}}{R_{3}}\right)J_{2}(d+2s;p_{1}^{2},m_{1}^{2},m_{2}^{2})$		(54)
		$\displaystyle=$	$\displaystyle-\frac{1}{2\pi i}\int\limits_{-i\infty}^{+i\infty}ds\dfrac{\sqrt{\pi}\Gamma(-s)\Gamma(s+1)\Gamma\left(\frac{d-2}{2}+s\right)\Gamma(\frac{4-d}{2}-s)\Gamma(\frac{d-2}{2}+s)}{4\Gamma(\frac{d-2}{2})\Gamma(\frac{d-1}{2}+s)}\times$
			$\displaystyle\hskip 28.45274pt\times\left(\frac{\partial_{3}R_{3}}{R_{3}}\right)\left[\dfrac{\partial_{2}R_{12}}{\sqrt{1-m_{1}^{2}/R_{12}}}+\dfrac{\partial_{1}R_{12}}{\sqrt{1-m_{2}^{2}/R_{12}}}\right]\;\left(R_{12}\right)^{\frac{d-4}{2}}\left(-\frac{R_{12}}{R_{3}}\right)^{s}$
			$\displaystyle+\frac{1}{2\pi i}\int\limits_{-i\infty}^{+i\infty}ds\dfrac{\Gamma(-s)\;\Gamma(s+1)\Gamma\left(\frac{d-2}{2}+s\right)\Gamma(\frac{4-d}{2}-s)\Gamma(\frac{d-2}{2}+s)}{4\;\Gamma(\frac{d-2}{2})\Gamma(\frac{d}{2}+s)}\left(\frac{\partial_{3}R_{3}}{R_{3}}\right)\times$
			$\displaystyle\hskip 28.45274pt\times\left\{\left(\frac{\partial_{2}R_{12}}{R_{12}}\right)\dfrac{(m_{1}^{2})^{\frac{d-2}{2}}}{\sqrt{1-m_{1}^{2}/R_{12}}}\,{}_{2}F_{1}\!\!\left[\begin{array}[]{c}\frac{d-2}{2}+s,\frac{1}{2}\,;\\ \frac{d}{2}+s\,;\end{array}\dfrac{m_{1}^{2}}{R_{12}}\right]\left(-\dfrac{m_{1}^{2}}{R_{3}}\right)^{s}+(1\leftrightarrow 2)\right\}.$

For taking the MB integrals in (54), one closes the integration contour to the right. The residue contributions at the poles of the $\Gamma(\cdots-s)$ are taken into account.

First, the contributions of residua at the poles $s=m$ with $m\in\mathbb{N}$. In this case, one first applies the reflect formula (253) for gamma function:

$\displaystyle\Gamma\left(\frac{4-d}{2}-s\right)\;\Gamma\left(\frac{d-2}{2}+s\right)=(-1)^{s}\;\Gamma\left(\frac{4-d}{2}\right)\;\Gamma\left(\frac{d-2}{2}\right).$

(55)

Using this identity, the first MB integration reads

	$\displaystyle\dfrac{J_{3,(123)}^{\mathrm{1-term}}}{\Gamma\left(\frac{4-d}{2}\right)}\Big{|}_{s=m}$	$\displaystyle=$	$\displaystyle-\frac{\sqrt{\pi}}{4}\left(\frac{\partial_{3}R_{3}}{R_{3}}\right)\left[\frac{\partial_{2}R_{12}}{\sqrt{1-m_{1}^{2}/R_{12}}}+\frac{\partial_{1}R_{12}}{\sqrt{1-m_{2}^{2}/R_{12}}}\right]\left(R_{12}\right)^{\frac{d-4}{2}}\times$
			$\displaystyle\hskip 48.36958pt\times\frac{1}{2\pi i}\int\limits_{-i\infty}^{+i\infty}\;ds\;\frac{\Gamma(-s)\;\Gamma(s+1)\Gamma\Big{(}\frac{d-2}{2}+s\Big{)}}{\Gamma(\frac{d-1}{2}+s)}\left(-\frac{R_{12}}{R_{3}}\right)^{s}.$

This MB integral is then expressed in terms of hypergeometric ${}_{2}F_{1}$ as follows:

$\displaystyle\dfrac{J_{3,(123)}^{\mathrm{1-term}}}{\Gamma\left(\frac{4-d}{2}\right)}\Big{|}_{s=m}$

$\displaystyle=$

$\displaystyle-\frac{\sqrt{\pi}}{4}\frac{\Gamma\left(\frac{d-2}{2}\right)}{\Gamma\left(\frac{d-1}{2}\right)}\left(\frac{\partial_{3}R_{3}}{R_{3}}\right)\left[\frac{\partial_{2}R_{12}}{\sqrt{1-m_{1}^{2}/R_{12}}}+(1\leftrightarrow 2)\right]\left(R_{12}\right)^{\frac{d-4}{2}}\,{}_{2}F_{1}\!\!\left[\begin{array}[]{c}\frac{d-2}{2},1\,;\\ \frac{d-1}{2}\,;\end{array}\frac{R_{12}}{R_{3}}\right],$

(59)

provided that $\left|R_{12}/R_{3}\right|<1$ and $\mathcal{R}$e$\left(d-3\right)>0$. The second MB integral reads

	$\displaystyle\dfrac{J_{3,(123)}^{\mathrm{2-term}}}{\Gamma\left(\frac{4-d}{2}\right)}\Big{|}_{s=m}$	$\displaystyle=$	$\displaystyle\frac{1}{2\pi i}\int\limits_{-i\infty}^{+i\infty}\;ds\;\dfrac{\Gamma(-s)\;\Gamma(s+1)\Gamma\Big{(}\frac{d-2}{2}+s\Big{)}}{\Gamma(\frac{d}{2}+s)}\left(\frac{\partial_{3}R_{3}}{4R_{3}}\right)\times$
			$\displaystyle\times\left\{\left(\dfrac{\partial_{2}R_{12}}{R_{12}}\right)\dfrac{(m_{1}^{2})^{\frac{d-2}{2}}}{\sqrt{1-m_{1}^{2}/R_{12}}}\left(-\frac{m_{1}^{2}}{R_{3}}\right)^{s}\;\,{}_{2}F_{1}\!\!\left[\begin{array}[]{c}\frac{d-2}{2}+s,\frac{1}{2}\,;\\ \frac{d}{2}+s\,;\end{array}\dfrac{m_{1}^{2}}{R_{12}}\right]+(1\leftrightarrow 2)\right\}$
			$\displaystyle=\dfrac{\Gamma\left(\frac{d-2}{2}\right)}{4\Gamma\left(\frac{d}{2}\right)}\left(\frac{\partial_{3}R_{3}}{R_{3}}\right)\left[\left(\dfrac{\partial_{2}R_{12}}{R_{12}}\right)\dfrac{(m_{1}^{2})^{\frac{d-2}{2}}}{\sqrt{1-m_{1}^{2}/R_{12}}}F_{1}\left(\dfrac{d-2}{2};1,\frac{1}{2};\frac{d}{2};\frac{m_{1}^{2}}{R_{3}},\dfrac{m_{1}^{2}}{R_{12}}\right)+(1\leftrightarrow 2)\right],$

provided that $\left|m_{(1,2)}^{2}/R_{3}\right|$, $\left|m_{(1,2)}^{2}/R_{12}\right|<1$ and $\mathcal{R}$e$\left(d-2\right)>0$.

In the next steps, the residue contributions at the second sequence poles $s=\frac{4-d}{2}+m$ for $m\in\mathbb{N}$ are taken into account. The next MB integrations are considered as follows:

	$\displaystyle J_{3,(123)}^{\mathrm{1-term}}\Big{|}_{s=\frac{4-d+2m}{2}}$	$\displaystyle=$	$\displaystyle-\frac{\sqrt{\pi}}{2\pi i}\int\limits_{-i\infty}^{+i\infty}\;ds\;\dfrac{\Gamma(-s)\;\Gamma(s+1)\Gamma^{2}\Big{(}\frac{d-2}{2}+s\Big{)}\Gamma\left(\frac{4-d}{2}-s\right)}{4\;\Gamma\left(\frac{d-2}{2}\right)\Gamma(\frac{d-1}{2}+s)}$
			$\displaystyle\times\left(\frac{\partial_{3}R_{3}}{R_{3}}\right)\left[\dfrac{\partial_{2}R_{12}}{\sqrt{1-m_{1}^{2}/R_{12}}}+\dfrac{\partial_{1}R_{12}}{\sqrt{1-m_{2}^{2}/R_{12}}}\right]\left(R_{12}\right)^{\frac{d-4}{2}}\left(-\frac{R_{12}}{R_{3}}\right)^{s}$
		$\displaystyle=$	$\displaystyle\left(\frac{\partial_{3}R_{3}}{2\;R_{3}}\right)\left[\dfrac{\partial_{2}R_{12}}{\sqrt{1-m_{1}^{2}/R_{12}}}+\dfrac{\partial_{1}R_{12}}{\sqrt{1-m_{2}^{2}/R_{12}}}\right](R_{3})^{\frac{d-4}{2}}\;\,{}_{2}F_{1}\!\!\left[\begin{array}[]{c}1,1\,;\\ \frac{3}{2}\,;\end{array}\frac{R_{12}}{R_{3}}\right],$		(69)
	$\displaystyle\dfrac{J_{3,(123)}^{\mathrm{2-term}}}{\Gamma\left(\frac{4-d}{2}\right)}\Big{|}_{s=\frac{4-d}{2}+m}$	$\displaystyle=$	$\displaystyle-\left(\frac{\partial_{3}R_{3}}{2R_{3}}\right)\left[\left(\frac{\partial_{2}R_{12}}{2R_{12}}\right)\frac{m_{1}^{2}(R_{3})^{\frac{d-4}{2}}}{\sqrt{1-m_{1}^{2}/R_{12}}}F_{1}\left(1;1,\frac{1}{2};2;\frac{m_{1}^{2}}{R_{3}},\dfrac{m_{1}^{2}}{R_{12}}\right)+(1\leftrightarrow 2)\right],$

provided that $\left|m_{(1,2)}^{2}/R_{3}\right|$, $\left|m_{(1,2)}^{2}/R_{12}\right|<1$ and $\left|R_{12}/R_{3}\right|<1$.