One-loop three-point Feynman integrals with Appell $F_1$ hypergeometric functions

TL;DR

This paper derives new analytic formulas for one-loop three-point Feynman integrals in general dimensions, expressing them with Appell $F_1$ hypergeometric functions for broad configurations of masses and momenta.

Contribution

The paper introduces a novel analytic expression for these integrals using Appell $F_1$ functions applicable to general configurations, extending previous special-case results.

Findings

Analytic formulas expressed in hypergeometric functions for general configurations.

Cross-checked results with existing literature for special cases.

Provides a unified approach for different mass and momentum configurations.

Abstract

New analytic formulas for one-loop three-point Feynman integrals in general space-time dimension ($d$) are presented in this paper. The calculations are performed at general configurations for internal masses and external momenta. The analytic results are expressed in terms of hypergeometric series $_2F_1$, $_3F_2$ for special cases and Appell $F_1$ for general cases. Furthermore, we cross-check our analytic results with other references which have carried out the integrals in several special cases.

Equations

$$J_3(d;\{{\nu }_1,{\nu }_2,{\nu }_3\})\equiv J_3(d;\{{\nu }_1,{\nu }_2,{\nu }_3\};p_1^2,p_2^2,p_3^2;m_1^2,m_2^2,m_3^2)=$$

$$S_3$$

$$S_{12}$$

$$S_{13}$$

$$S_{23}$$

$$\displaystyle G_{3}=-8\left|\begin{array}[]{cc}p_{1}^{2}&p_{1}p_{2}\\ p_{1}p_{2}&p_{2}^{2}\end{array}\right|=2\lambda(p_{1}^{2},p_{3}^{2},p_{2}^{2}).$$

$$G_{12}=-4p_1^2,G_{13}=-4p_3^2,G_{23}=-4p_2^2.$$

$$M_3$$

$$M_{ij}$$

$$\frac{J_3}{\Gamma \left(\frac{6-d}{2}\right)}$$

$$A$$

$$B$$

$$C$$

$$J_3$$

$$\frac{J_3}{\Gamma \left(\frac{4-d}{2}\right)}$$

$$p_2^2{x}_1^2-(p_2^2+m_3^2-m_2^2)x_1+m_3^2=(m_2^2-m_1^2)x_1+m_1^2=M_3.$$

$$\frac{J_3}{\Gamma \left(\frac{4-d}{2}\right)}$$

$$J_3$$

$$=$$

$$\frac{J_3}{\Gamma \left(\frac{4-d}{2}\right)}$$

$$\left(p_2^2{y}^2-(p_2^2+m_2^2-m_1^2)y+m_2^2\right){|}_{y=(m_2^2-m_1^2)/p_2^2}=m_1^2.$$

$$\frac{J_3}{\Gamma \left(\frac{4-d}{2}\right)}$$

$$q^2{x}^2-(q^2+m^2)x+m^2{|}_{x=m^2/q^2}=0.$$

$$\frac{J_3}{\Gamma \left(\frac{4-d}{2}\right)}$$

$$=$$

$$\frac{J_3}{\Gamma \left(\frac{4-d}{2}\right)}$$

$$\frac{J_3}{\Gamma \left(\frac{6-d}{2}\right)}=-\underset{0}{\overset{1}{\int }}dx\underset{0}{\overset{1-x}{\int }}dy\frac{1}{(p_2^{2}xy+m^2-i\rho {)}^{\frac{6-d}{2}}}.$$

$$J_3=-\frac{1}{2\pi i}\underset{-i\infty }{\overset{i\infty }{\int }}ds\Gamma \left(-s\right)\Gamma \left(\frac{6-d+2s}{2}\right)\underset{0}{\overset{1}{\int }}dx\underset{0}{\overset{1-x}{\int }}dy\frac{[(p_2^2-i\rho )xy{]}^s}{(m^2{)}^{3-\frac{d}{2}+s}}.$$

$$J_3$$

Full text

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B87

1 Introduction

Future experimental programs at the High-Luminosity Large Hadron Collider (HL-LHC) ATLAS:2013hta ; CMS:2013xfa and the International Linear Collider (ILC) Baer:2013cma aim to measure precisely the properties of Higgs boson, top quark and vector bosons for discovering the nature of the Higgs sector as well as for finding the effects of physics beyond the standard model. In order to match the high-precision of experimental data in the near future, theoretical predictions including high-order corrections are required. In this framework, detailed evaluations of one-loop multi-leg and higher-loop at general scale and mass assignments are necessary.

One-loop Feynman integrals in general space-time dimension play a crucial role for several reasons. Within the general framework for computing two-loop or higher-loop corrections, higher-terms in the $\varepsilon$-expansion (with $\varepsilon=2-d/2$) from one-loop integrals are necessary for building blocks. Moreover, one-loop integrals in $d>4$ may be taken into account in a reduction for tensor one-loop Davydychev:1991va ; Fleischer:2010sq , two-loop and higher-loop integrals IBP . There have been available many calculations for scalar one-loop functions in general dimension $d$ Boos:1990rg ; Davydychev:1990cq ; Davydychev:1997wa ; Anastasiou:1999ui ; Suzuki:2003jn ; Abreu:2015zaa ; Phan:2017xsj . However, not all of the calculations cover general $\varepsilon$-expansion at general scale and internal mass assignments. Furthermore, a recurrence relation in $d$ for Feynman loop integrals has been proposed Tarasov:1996br and solved for scalar one-loop integrals which have been expressed in terms of generalized hypergeometric series Fleischer:2003rm . However, the general solutions for arbitrary kinematics have not been found, as pointed out in Bluemlein:2015sia . More recently, scalar 1-loop Feynman integrals as meromorphic functions in general space-time dimension, for arbitrary kinematics has been presented in Bluemlein:2017rbi ; Phan:2018cnz . In the present paper, new analytic formulas for one-loop three-point Feynman integrals in general space-time dimension are reported by following an alternative approach. The analytic results are expressed in terms of ${}_{2}F_{1}$, ${}_{3}F_{2}$ and Appell $F_{1}$ hypergeometric functions. The evaluations are performed in general configuration of internal masses and external momenta. Last but not least, our results are cross-checked to other papers which have been available in several special cases.

The layout of the paper is as follows: In section $2$, we present in detail the method for evaluating scalar one-loop three-point functions. In this section, we first introduce notations used for this work. We next consider the case of one-loop triangle diagram with two light-like external momenta and generalize the calculations for general case. Finally, tensor one-loop three-point integrals are discussed. Conclusions and outlooks are devoted in section $3$. Several useful formulas applied in this calculation can be found in the appendixes.

2 Analytic formulas

Detailed evaluations for one-loop three-point integrals are presented in this section.

2.1 Definitions

We arrive at notations for the calculations in this subsection. Feynman integrals of scalar one-loop three-point functions are defined:

[TABLE]

We refer hereafter $J_{3}\equiv J_{3}(d;\{1,1,1\})$. In this definition, the term $i\rho$ is Feynman’s prescription and $d$ is space-time dimension. The internal (loop) momentum is $k$ and the external momenta are $p_{1}$, $p_{2}$, $p_{3}$. They are inward as described in Fig. 1. We use the momenta $q_{1}=p_{1}$, $q_{2}=p_{1}+p_{2}$ and $q_{3}=p_{1}+p_{2}+p_{3}=0$ following momentum conservation. The internal masses are $m_{1}$, $m_{2}$ and $m_{3}$. $J_{3}$ is a function of $p_{1}^{2}$, $p_{2}^{2}$, $p_{3}^{2}$ and $m_{1}^{2}$, $m_{2}^{2}$, $m_{3}^{2}$.

It has known that an algebraically compact expression and numerically stable representation for Feynman diagrams can be obtained by using kinematic variables such as the determinants of Caylay and Gram matrices kajantie . The expression also reflects the symmetry of the corresponding topologies.

We hence review the mentioned kinematic variables in the following paragraphs. The determinant of Cayley matrix of one-loop triangle diagrams is given

[TABLE]

In the same manner, the Cayley determinants of one-loop two-point Feynman diagrams which are obtained by shrinking an propagator in the three-point integrals. The determinants are written explicitly as:

[TABLE]

Where $\lambda(x,y,z)=x^{2}+y^{2}+z^{2}-2xy-2xz-2yz$ is so-called Källen function. Next, the determinant of Gram matrix of one-loop three-point functions is given

[TABLE]

In the same above definitions, we also get the Gram determinants of two-point functions as follows

[TABLE]

In this work, the analytic formulas for scalar one-loop three-point integrals are expressed as functions with arguments of the ratio of the above kinematic determinants. Therefore, it is worth to introduce the following index variables

[TABLE]

By introducing Feynman parameters, we then integrate over the loop-momentum. The resulting integral after taking over one of Feynman parameters reads

[TABLE]

The corresponding coefficients $A,B,C,\;\cdots,F$ are shown:

[TABLE]

Detailed calculation for $J_{3}$ in (21) is presented in next subsections.

2.2 Two light-like momenta

We first consider the simple case which is two light-like external momenta. Without loss of the generality, we can take $p_{1}^{2}=0,~{}p_{3}^{2}=0$. The Feynman parameter integral in Eq. (21) is now casted into the simpler form

[TABLE]

It finds that the denominator function of the integrand in (25) depends linearly on both $x$ and $y$. Hence, the integral can be taken easily. The resulting integral reads after performing the $y$-integration

[TABLE]

Both the integrands are singularity at $x_{1}=(m_{3}^{2}-m_{1}^{2})/p_{2}^{2}$. However, it is verified that

[TABLE]

It means that the residue contributions at this pole of two integrations in Eq. (2.2) will be cancelled. As a result, $J_{3}$ stays finite at this point. The first integral in Eq. (2.2) can be formulated by mean of Appell $F_{1}$ functions. For the second integral in Eq. (2.2), we can apply the formula for master integral as Eq. (172). The result for $J_{3}$ then reads

[TABLE]

Another representation for $J_{3}$ can be obtained by using Eq. (171)

[TABLE]

The results are shown in Eqs. (2.2, 2.2) are new hypergeometric representations for scalar one-loop three-point functions in general space-time dimension. In the next paragraphs, we consider $J_{3}$ in several special cases.

$\underline{m_{1}^{2}=m_{2}^{2}=m_{3}^{2}=0:}$

One-loop triangle diagrams with all massless internal lines are considered. In this case, the integral $J_{3}$ in (25) becomes

[TABLE]

It is important to note that we use $-p_{2}^{2}xy-i\rho\equiv(-p_{2}^{2}-i\rho)xy$. Therefore, for analytic continuation the result when $p_{2}^{2}>0$, we should keep $i\rho$-term together with external momentum like $p_{2}^{2}\rightarrow p_{2}^{2}+i\rho$. The result in (31) shows full agreement with Eq. $(4.5)$ in Ellis:2007qk . 2. 2.

$\underline{m_{2}^{2}=m_{3}^{2}=0,m_{1}^{2}=m^{2}:}$

If internal mass configuration takes $m_{2}^{2}=m_{3}^{2}=0,m_{1}^{2}=m^{2}$, $J_{3}$ becomes

[TABLE]

This result is in agreement with Eq. ($B2$) in Ref. Abreu:2015zaa . In the limit of $p_{2}^{2}\rightarrow-m^{2}<0$, we arrive at

[TABLE]

When $\mathcal{R}$e$\left(d-6\right)>0$ and $m^{2}\rightarrow 0$, $J_{3}\rightarrow 0$. 3. 3.

$\underline{m_{1}^{2}=m_{3}^{2}:}$

We are also interested in the case of $m_{1}^{2}=m_{3}^{2}$. In such case, one should exchange the order of integration in Eq. (25). The $x$-integration can be taken first. Subsequently, we arrive at

[TABLE]

It is easy to find out that

[TABLE]

As previous explanation, the integral $J_{3}$ in this case also stays finite at $y=(m_{2}^{2}-m_{1}^{2})/p_{2}^{2}$. Using the analytical solution for master integral as Eq. (172) in appendix $A$, the result reads

[TABLE]

provided that $\left|\frac{M_{23}-m_{1,2}^{2}}{M_{3}-m_{1,2}^{2}}\right|<1$, $\left|1-\frac{M_{23}}{m_{1,2}^{2}}\right|<1$. 4. 4.

$\underline{m_{1}^{2}=m_{2}^{2}=0,m_{3}^{2}=m^{2},p_{2}^{2}=q^{2}:}$

This case has been calculated in Abreu:2015zaa . In this particular configuration, $J_{3}$ becomes

[TABLE]

It is confirmed that

[TABLE]

As a matter of this fact, $J_{3}$ stays finite at $x=m^{2}/q^{2}$. By shifting $x\rightarrow 1-x$, the resulting integral reads

[TABLE]

Following Eq. (179) in appendix $B$, we derive this integral in terms of Gauss hypergeometric functions

[TABLE]

provided that $\left|\frac{q^{2}}{m^{2}}\right|<1$ and $\mathcal{R}$e$\left(d-2\right)>0$. This gives full agreement result with Eq. ($B7$) in Ref Abreu:2015zaa . In the limit of $q^{2}\rightarrow m^{2}$, one arrives at

[TABLE]

provided that $\mathcal{R}$e$\left(d-4\right)>0$. In additional, if $\mathcal{R}$e$\left(d-6\right)>0$ and $m^{2}\rightarrow 0$, $J_{3}\rightarrow 0$. 5. 5.

$\underline{m_{1}^{2}=m_{2}^{2}=m_{3}^{2}=m^{2}:}$

This case has been performed in Ref Davydychev:2003mv . One confirms that $M_{3}=m^{2}$ in this kinematic configuration. We derive analytic formula for $J_{3}$ as follows

[TABLE]

Using Mellin-Barnes relation one has

[TABLE]

After taking over $x,y$-integrations, the contour integral is taken the form of

[TABLE]

By closing the integration contour to the right side of imaginary axis in the $s$-complex plane, we then take into account the residua of sequence poles from $\Gamma(-s)$. The result is presented in terms of series of generalized hypergeometric function

[TABLE]

provided that $\left|\frac{p_{2}^{2}-i\rho}{4m^{2}}\right|<1$. This result is in agreement with Ref Davydychev:2003mv .

2.3 One light-like momentum

We are going to proceed the method for one light-like momentum case. Without any loss of the generality, we can choose $p_{3}^{2}=0$. Let us use $q^{2}=-2p_{1}p_{3}$. Applying the same previous procedure, one obtains Feynman parameter integral

[TABLE]

We find that two integrands have same singularity pole at $x=(m_{1}^{2}-m_{3}^{2})/q^{2}$. It is easy to verify that the residue contributions from this pole will be cancelled out. As a result, $J_{3}$ stays finite at this point. The analytic result for $J_{3}$ can be presented as a compact form:

[TABLE]

Where the terms $J_{123}$, $J_{231}$ are obtained by using Eq. (172). These terms are written in terms of $F_{1}$ as follows

[TABLE]

and

[TABLE]

It is important to note that the result for $J_{3}$ in (63, 64) are only valid if the absolute value of the arguments of the Appell $F_{1}$ functions in this presentation are less than $1$. If these kinematic variables do not meet this condition. One has to perform analytic continuations for Appell $F_{1}$ functions olsson .

We can get another representation for $J_{123},J_{231}$ by using a transformation for $F_{1}$ (seen appendix $B$). Taking $J_{123}$ as a example, one has

[TABLE]

From Eqs. (63, 65), we can perform analytic continuation the result in the limits of $M_{ij}=0,m_{i}^{2},m_{j}^{2}$ (for $i,j=1,2,3$) and $M_{3}=0$, etc. This can be worked out by applying the transformations for Appell $F_{1}$ functions, seen appendix $B$.

The results in Eqs. (63, 65) are new hypergeometric representations for scalar one-loop three-point functions for this case in general $d$. Several special cases for $J_{3}$ are considered in the next paragraphs.

$\underline{m_{1}^{2}=m_{2}^{2}=m_{3}^{2}=0:}$

One first arrives at the case of all massless internal lines. In this case, Eq. (2.3) becomes

[TABLE]

We have already used $p_{2}^{2}x^{2}-p_{2}^{2}x-i\rho=(-p_{2}^{2}-i\rho)x(1-x)$. This result coincides with Eq. ($4.6$) in Ref. Ellis:2007qk . In the limit of $p_{2}^{2}\rightarrow p_{1}^{2}$, one arrives at

[TABLE]

If $\mathcal{R}$e$\left(d-6\right)>0$ and $p_{2}^{2}\rightarrow 0$, the integral $J_{3}\rightarrow 0$. 2. 2.

$\underline{m_{1}^{2}=m_{2}^{2}=0,m_{3}^{2}=m^{2}:}$

We are concerning the case in which the internal masses have $m_{1}^{2}=m_{2}^{2}=0,m_{3}^{2}=m^{2}$. Let us note that $q^{2}=2p_{1}p_{3}$, Eq. (2.3) now gets the form of

[TABLE]

We make a change variable like $x\rightarrow 1-x$ for the first integral. Subsequently, it is presented in terms of Appell $F_{1}$ functions. While the second integral is formulated by mean of Gauss hypergeometric functions. The result is shown in concrete as follows

[TABLE]

provided that $\left|q^{2}/m^{2}\right|<1$ and $\mathcal{R}$e$\left(d-2\right)>0$. One finds another representation for $J_{3}$ by applying Eq. (208) in appendix $B$

[TABLE]

This representation gives agreement result with Eq. ($C6$) in Abreu:2015zaa . In the limit of $q^{2}\rightarrow m^{2}$, one gets

[TABLE]

When $p_{2}^{2}\rightarrow m^{2}$, the result in (2) reads

[TABLE] 3. 3.

$\underline{m_{1}^{2}=m_{3}^{2}=0,m_{2}^{2}=m^{2}:}$

This case has been performed in Ref Fleischer:2003rm . The Feynman parameter integral for $J_{3}$ in this case reads

[TABLE]

This gives a perfect agreement with Ref Fleischer:2003rm . In the limit of $p_{2}^{2}\rightarrow p_{1}^{2}$, one first uses

[TABLE]

The result reads

[TABLE] 4. 4.

$\underline{m_{2}^{2}=m_{3}^{3}=0,m_{1}^{2}=m^{2}:}$

Let us note that $q^{2}=2p_{1}p_{3}$, the integral $J_{3}$ in (2.3) is written by

[TABLE]

Here we have already performed a shift $x\rightarrow 1-x$ for the second integral. It is then expressed in terms of Appell $F_{1}$ functions. While the first integral is presented in terms of Gauss hypergeometric functions. Combining all these terms, analytic result for $J_{3}$ reads

[TABLE]

Another representation for $J_{3}$ is derived by using Eq. (208) in appendix $B$. It is

[TABLE]

From this representation, one can perform analytic continuation this result in the limits of $q^{2}\rightarrow-m^{2}$ and $p_{1}^{2}\rightarrow m^{2}$. 5. 5.

$\underline{p_{1}^{2}=p_{2}^{2}\neq 0:}$

We are going to consider an interesting case in which is $p_{1}^{2}=p_{2}^{2}\neq 0$. In such the case, one recognizes that $G_{3}=0$. We can present $J_{3}$ in terms of two scalar one-loop two-point functions as follows

[TABLE]

Both terms in right hand side of Eq. (5) are determined as Feynman parameter integrals of scalar one-loop two-point functions. They are calculated in detail as follows. Let us consider $J_{231}$ as a example. We can rewrite $J_{231}$ in the following form:

[TABLE]

The integral will be worked out by applying Mellin-Barnes relation which is

[TABLE]

The $J_{3}$ now is casted into the form of

[TABLE]

provided that $\left|\frac{M_{12}-m_{1,3}^{2}}{M_{12}-i\rho}\right|<1$.

2.4 General case

We are going to generalize the method for the general case in which $p_{i}^{2}\neq 0$ for $i=1,2,3$. Following an idea in 'tHooft:1978xw , we first apply the Euler transformation like $y\rightarrow y-\beta x$, the polynomial written in terms of $x,y$ in $J_{3}$’s integrand will becomes

[TABLE]

By choosing $\beta$ is one of the roots of following equation

[TABLE]

the integral $J_{3}$ is casted into

[TABLE]

It is also important to note that the final result will be independent of $\beta$ in Eq. (127). It means that we are free to choice one of roots $\beta$ in Eq. (127). As a result, the integrand is linearized of $x$, hence the $x$-integration can be evaluated first. In order to work out the $x$-integration, we split the integration as follows

[TABLE]

To archive a more symmetric form we make a further transformations $y\rightarrow\beta(1-y)$ for the second integral and $y\rightarrow 1-(1-\beta)y$ for the third integral respectively. This brings some order in the arguments of the integrands. The denominators are all of the linear form $\sim(y-y_{i})$ for $i=1,2,3$. It is also easy to confirm that all $y_{i}$ follows the equations

[TABLE]

for $i,j,k=1,2,3$.

Finally, following an idea in 'tHooft:1978xw , we add extra terms which sum all of them is to zero for cancelling the residue of the pole at $y_{i}$. The result reads

[TABLE]

The analytic result for $J_{3}$ can be written in a compact form

[TABLE]

with

[TABLE]

for $i,j,k=1,2,3.$ Where the integrand’s poles are given

[TABLE]

Applying the formula for master integral in appendix $A$, we will present the result of $J_{ijk}$ in terms of Appell $F_{1}$ functions. For instant, one takes $J_{123}$ for an example. This term is expressed as follows

[TABLE]

provided that $\left|\frac{M_{12}-m_{1;2}^{2}}{M_{3}-m_{1;2}^{2}}\right|,\;\left|1-\frac{M_{12}}{m_{1;2}^{2}}\right|<1$. One finds another representation for $J_{123}$ is as (by applying Eq. (171) in appendix $A$)

[TABLE]

provided that $\left|\frac{M_{12}-m_{1;2}^{2}}{M_{12}-M_{3}}\right|,\;\left|1-\frac{m_{1;2}^{2}}{M_{12}}\right|<1$.

It is important to note that the results in (2.4, 2.4) are only valid if the absolute value of the arguments of the Appell $F_{1}$ functions in these representations are less than $1$. If these kinematic variables do not satisfy this condition. One has to perform analytic continuations, we refer the work of olsson for Appell $F_{1}$.

The results shown in Eqs. (2.4, 2.4) are also new hypergeometric representations for scalar one-loop three-point functions in general space-time dimension. From these representations, one can perform analytic continuation of the result in the cases of $M_{ij}=0,m_{i}^{2},m_{j}^{2}$, $M_{3}=0$ (for $i,j,k=1,2,3$), $G_{3}=0$ and massless case, etc. This can be done by applying transformations for Appell $F_{1}$ functions (seen appendix $B$).

For a example, we consider the case of $G_{3}=0$. In the case, $\beta=c/b$, repeating the calculation, we arrive at

[TABLE]

The operator ${\bf k^{-}}$ is defined in such a way that it will reduce the three-point integrals to two-point integrals by shrinking an propagator in the integrand of $J_{3}$. This equation equivalents with Eq. ($46$) in Ref. Devaraj:1997es . Noting that we have already arrived this relation in (5) of previous subsection.

2.5 Tensor one-loop three-point integrals

Following tensor reduction method in Ref. Davydychev:1991va , tensor one-loop three-point integrals with rank $M$ can be presented in terms of scalar ones with the shifted space-time dimension:

[TABLE]

In this formula, the condition for the indices $\lambda,\kappa_{1},\kappa_{2}$ and $\kappa_{3}$ is $2\lambda+\kappa_{1}+\kappa_{2}+\kappa_{3}=M$. Moreover, these indices also follow the constrain $0\leq\kappa_{1},\kappa_{2},\kappa_{3}\leq M$ and $0\leq\lambda\leq[M/2]$ (integer of $M/2$). The notation $(a)_{\kappa}=\Gamma(a+\kappa)/\Gamma(a)$ is the Pochhammer symbol. The structure of tensor $\{[g]^{\lambda}[q_{1}]^{\kappa_{1}}[q_{2}]^{\kappa_{2}}[q_{3}]^{\kappa_{3}}\}_{\mu_{1}\mu_{2}\cdots\mu_{M}}$ is symmetric with respect to $\mu_{1},\mu_{2},\cdots,\mu_{M}$. This tensor is constructed from $\lambda$ of metric $g_{\mu\nu}$, $\kappa_{1}$ of momentum $q_{1}$, $\cdots$, $\kappa_{3}$ of momentum $q_{3}$. The $J_{3}(d+2(M-\lambda);\{\nu_{1}+\kappa_{1},\nu_{2}+\kappa_{2},\nu_{3}+\kappa_{3}\})$ are scalar one-loop three-point functions with the shifted space-time dimension $d+2(M-\lambda)$, raising powers of propagators $\{\nu_{i}+\kappa_{i}\}$ for $i=1,2,3$.

For examples, we first take the simplest case $M=1$. In this case, one has $\lambda=0$. Subsequently, we get

[TABLE]

with $\delta_{jk}$ is the Kronecker symbol. We next consider $M=2$. One has

[TABLE]

In the next step, the scalar integrals J_{3}\Big{(}d+2(M-\lambda);\{\nu^{\prime}_{1},\nu^{\prime}_{2},\nu^{\prime}_{3}\}\Big{)} will be reduced to subset of master integrals by using integration-by-part method (IBP) IBP . By applying the operator $\frac{\partial}{\partial k}\cdot k$ to the integrand of $J_{3}\left(d;\{\nu_{1},\nu_{2},\nu_{3}\}\right)$ and choosing $k$ to be the momentum of three internal lines ($k\equiv\{k+q_{1},k+q_{2},k+q_{3}\}$). One arrives at the following system equations:

[TABLE]

Where we used the following notation:

[TABLE]

In order to solve system equations (142), one first considers the following matrix:

[TABLE]

If det$(Y_{3})\neq 0$, one then can present $J_{3}(d;\nu_{123}+1)$ in term of $J_{3}(d;\nu_{123})$ with $\nu_{123}=\nu_{1}+\nu_{2}+\nu_{3}$. In this recurrence way Laporta:2001dd , we can arrive at the following integrals: $J_{2}(d;\{\nu_{1},\nu_{2}\})$ and $J_{3}(d;\{1,1,1\})$. By applying IPB once again for the former integrals, we will arrive at master integrals such as: $J_{1}(d;\{\nu\})$, $J_{2}(d;\{1,1\})$ which can be found in Fleischer:2003rm and $J_{3}(d;\{1,1,1\})$ in this paper.

3 Conclusions

New analytic formulas for one-loop three-point Feynman integrals in general space-time dimension have presented in this paper. The results are expressed in terms of Appell $F_{1}$ functions, considered all cases of internal mass and external momentum assignments. We have also cross-checked the analytic results in this work with other paper in several special cases.

Acknowledgment: This research is funded by Vietnam National University (VNU-HCM), Ho Chi Minh City under grant number C$2019$-$18$-$06$???.

Appendix $A$: Evaluatingthe master integral

We are considering master integral

[TABLE]

with $p_{i}^{2}\neq 0$. Where $|x_{k}|>1$ or $-1\leq x_{k}<0$ and it follows the below equation

[TABLE]

for $i,j,k=1,2,3$.

We will discuss on the method to evaluate this integral under the above conditions. In the case of $0<x_{k}<1$, one will perform analytic continuation the result for all master integrals which are expected to appear in the general formula of $J_{3}$. This will be devoted in concrete in section $2$.

In order to work out the master integral one should write the polynomial of $x$ which appears in numerator of the $\mathcal{K}$’s integrand as follows

[TABLE]

We have introduced the kinematic variables

[TABLE]

for $p_{i}^{2}\neq 0$. From the conventions, we subsequently verify the below relations

[TABLE]

for $i,j=1,2,3$.

Using Mellin-Barnes relation MB we then decompose the integrand as

[TABLE]

With the help of this relation, the Feynman parameter integral will be casted into the simpler form. It will be calculated in terms of Gauss hypergeometric functions. In particular, we have

[TABLE]

One makes a shift $x=x_{ij}\;x^{\prime}$ (and $x=1-x_{ji}\;x^{\prime}$) for the first integral (second integral) respectively. The result reads

[TABLE]

We apply a further transformation like $\xi=(1-x)^{2}\geqslant 0$. The Jacobian of the shift is $dx=-\dfrac{d\xi}{2\sqrt{\xi}}$ and the integration domain is now $[1,0]$. As a result of this shift, we arrive at

[TABLE]

Following Eq. (179) in appendix $B$, we can present this integral in terms of Gauss hypergeometric functions

[TABLE]

provided that $\left|\frac{x_{ij}^{2}}{(x_{k}-x_{ij})^{2}}\right|<1$ for $i,j,k=1,2,3$ and $\mathcal{R}$e$\left(s+\frac{1}{2}\right)>0$.

Putting this result into Eq. (154), we are going to evaluate the following Mellin-Barnes integral

[TABLE]

with $|\mathrm{Arg}(-x)|<\pi$ and $|x|<1$ and $|y|<1$. Under these conditions, one could close the integration contour to the right side of the imaginary axis in the $s$-complex plane. Subsequently, we take into account the residua of the sequence poles of $\Gamma(-s)$. The result is presented as a series of Appell $F_{1}$ functions Slater

[TABLE]

provided that $|\mathrm{Arg}(-x)|<\pi$ and $|x|<1$ and $|y|<1$. Finally, the result for $\mathcal{K}$ reads

[TABLE]

provided that $\left|\frac{x_{ij}^{2}}{(x_{k}-x_{ij})^{2}}\right|<1$ and $\left|-\frac{p_{i}^{2}x_{ij}^{2}}{M_{ij}}\right|<1$ for $i,j,k=1,2,3$. It is confirmed that Arg$\left(\frac{x_{ij}^{2}}{(x_{k}-x_{ij})^{2}}\right)<\pi$ and Arg$\left(-\frac{p_{i}^{2}x_{ij}^{2}}{M_{ij}}\right)<\pi$ due to the $i\rho$-term. In other words, these kinematic variables are never on the negative real axis.

Applying the transformation for Appell $F_{1}$ functions (see Eq. (208) in appendix $B$, one finds another representation for integral $\mathcal{K}$ as

[TABLE]

provided that $\left|-\frac{x_{ij}^{2}}{(x_{k}-x_{ij})^{2}-x_{ij}^{2}}\right|<1$ and $\left|\frac{p_{i}^{2}x_{ij}^{2}}{p_{i}^{2}x_{ij}^{2}+M_{ij}}\right|<1$ for $i,j,k=1,2,3$.

Appendix $B$:

Generalized hypergeometric functions

The Gauss hypergeometric series are given (see Eq. ($1.1.1.4$) in Ref. Slater )

[TABLE]

provided that $|z|<1$. Here, the pochhammer symbol,

[TABLE]

is taken into account. The integral representation for Gauss hypergeometric functions is (see Eq. (1.6.6) in Ref. Slater )

[TABLE]

provided that $|z|<1$ and Re$(c)>$Re$(b)>0$.

The series of Appell $F_{1}$ functions are given (see Eq. (8.13) in Ref. Slater )

[TABLE]

provided that $|x|<1$ and $|y|<1$. The single integral representation for $F_{1}$ is (see Eq. (8.25) in Ref. Slater )

[TABLE]

provided that Re$(c)$ $>$ Re$(a)>0$ and $|x|<1$, $|y|<1$.

Transformations for Gauss ${}_{2}F_{1}$ hypergeometric functions

Basic linear transformation formulas for Gauss ${}_{2}F_{1}$ hypergeometric functions collected from Ref. Slater are listed as follows

[TABLE]

Transformations for Appell

$F_{1}$ hypergeometric functions

We collect all transformations for Appell $F_{1}$ functions from Refs. Slater . The first relation for $F_{1}$ is mentioned,

[TABLE]

If $b^{\prime}=0$, we arrive at the well-known Pfaff–Kummer transformation for the ${}_{2}F_{1}$. In detail, one has

[TABLE]

Furthermore, if $c=b+b^{\prime}$, one then obtains

[TABLE]

Similarly,

[TABLE]