Multi-Loop Techniques for Massless Feynman Diagram Calculations

Abstract

We review several multi-loop techniques for analytical massless Feynman diagram calculations in relativistic quantum field theories: integration by parts, the method of uniqueness, functional equations and the Gegenbauer polynomial technique. A brief, historically oriented, overview of some of the results obtained over the decades for the massless 2-loop propagator-type diagram is given. Concrete examples of up to 5-loop diagram calculations are also provided.

7 APPENDIX A.

GEGENBAUER POLYNOMIAL TECHNIQUE

This Appendix is devoted to a short presentation of the Gegenbauer polynomial technique. The latter should be considered as the effective (but rather cumbersome) method for calculating dimensionally regularized Feynman diagrams. In its modern form, it has been introduced by Chetyrkin, Kataev and Tkachov [46]. Later, subtle and important improvements were brought up by Kotikov [61] and we shall follow this reference in our brief review of the technique.

Hereafter we will use the variables $x,y,...,$ which are usually used in coordinate space. But we can also think about the variables $x,y,...$ as being some momenta. Thus, all formulae in this Appendix are also applicable in the momentum space. Such type of “duality” has already been considered in Sec. 4.5.

1.1 7.1 Presentation of the Method

The basic motivation for this technique lays in the fact that, in multi-loop computations, the complicated part of the integration is often the one over the angular variables. This task is considerably simplified by expanding some of the propagators in the integrand in terms of the Gegenbauer polynomials (the so-called multipole expansion):

$$\begin{gathered} \frac{1}{{{{{({{x}_{1}} - {{x}_{2}})}}^{{2\lambda }}}}}\sum\limits_{n = 0}^\infty {C_{n}^{\lambda }} ({{{\hat {x}}}_{1}} \cdot {{{\hat {x}}}_{2}}) \\ \times \,\,\left[ {\frac{{{{{(x_{1}^{2})}}^{{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2}}}}}}{{{{{(x_{2}^{2})}}^{{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2} + \lambda }}}}}\Theta (x_{2}^{2} - x_{1}^{2}) + (x_{1}^{2} \leftrightarrow x_{2}^{2})} \right], \\ \end{gathered} $$

((A1))

where $C_{n}^{\lambda }$ is the Gegenbauer polynomial of degree n and $\hat {x} = {x \mathord{\left/ {\vphantom {x {\sqrt {{{x}^{2}}} }}} \right. \kern-0em} {\sqrt {{{x}^{2}}} }},$ and then using the orthogonality relation of Gegenbauer polynomials on the unit $D$‑dimensional sphere:

$$\begin{gathered} \frac{1}{{{{\Omega }_{D}}}}\int {{{{\text{d}}}_{D}}} \hat {x}C_{n}^{\lambda }(\hat {z} \cdot \hat {x})C_{m}^{\lambda }(\hat {x} \cdot \hat {z}) \\ = {{\delta }_{{n,m}}}\frac{{\lambda \Gamma (n + 2\lambda )}}{{\Gamma (2\lambda )(n + \lambda )n!}},\,\,\,\,\lambda = \frac{D}{2} - 1, \\ \end{gathered} $$

((A2))

where ${{{\text{d}}}_{D}}\hat {x}$ is the surface element of the unit D-dimensional sphere and ${{\Omega }_{D}} = {{2{{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}} \mathord{\left/ {\vphantom {{2{{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}} {\Gamma ({D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2})}}} \right. \kern-0em} {\Gamma ({D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2})}}.$ The Gegenbauer polynomials can be defined from their generating function:

$$\begin{gathered} \frac{1}{{{{{(1 - 2xw + {{w}^{2}})}}^{\beta }}}} = \sum\limits_{k = 0}^\infty {C_{k}^{\beta }(x){{w}^{k}}} , \\ C_{n}^{\beta }(1) = \frac{{\Gamma (n + 2\beta )}}{{\Gamma (2\beta )n!}}, \\ \end{gathered} $$

((A3))

with some additional particular values given by:

$$\begin{gathered} C_{0}^{\lambda }(x) = 1,\,\,\,\,C_{1}^{\lambda }(x) = 2\lambda x, \\ C_{2}^{\lambda }(x) = 2\lambda (\lambda + 1){{x}^{2}} - \lambda . \\ \end{gathered} $$

((A4))

For our purpose, it is convenient to express the Gegenbauer polynomials in terms of traceless symmetric tensors [61]:

$$\begin{gathered} C_{n}^{\lambda }(\hat {x} \cdot \hat {z}){{({{x}^{2}}{{z}^{2}})}^{{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2}}}} = {{S}_{n}}(\lambda ){{x}^{{{{\mu }_{1}}{{\mu }_{2}} \cdots {{\mu }_{n}}}}}{{z}^{{{{\mu }_{1}}{{\mu }_{2}} \cdots {{\mu }_{n}}}}}, \\ {{S}_{n}}(\lambda ) = \frac{{{{2}^{n}}\Gamma (n + \lambda )}}{{n!\Gamma (\lambda )}}. \\ \end{gathered} $$

((A5))

From Eq. (A5) for $x = z$ and the last equation in (A3), we deduce the following equation for products of traceless tensors:

$${{S}_{n}}(\lambda ){{z}^{{{{\mu }_{1}}{{\mu }_{2}} \cdots {{\mu }_{n}}}}}{{z}^{{{{\mu }_{1}}{{\mu }_{2}} \cdots {{\mu }_{n}}}}} = \frac{{\Gamma (n + 2\lambda )}}{{\Gamma (2\lambda )n!}}{{z}^{{2n}}}.$$

((A6))

With the help of Eq. (A5), Eq. (A1) can be rewritten as:

$$\begin{gathered} \frac{1}{{{{{({{x}_{1}} - {{x}_{2}})}}^{{2\lambda }}}}}\sum\limits_{n = 0}^\infty {{{S}_{n}}(\lambda )} x_{1}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}x_{2}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}} \\ \times \,\,\left[ {\frac{1}{{{{{(x_{2}^{2})}}^{{n + \lambda }}}}}\Theta (x_{2}^{2} - x_{1}^{2}) + (x_{1}^{2} \leftrightarrow x_{2}^{2})} \right]. \\ \end{gathered} $$

((A7))

Notice that, for a propagator with arbitrary index, Eq. (A1) can be generalized as:

$$\begin{gathered} \frac{1}{{{{{({{x}_{1}} - {{x}_{2}})}}^{{2\beta }}}}}\sum\limits_{n = 0}^\infty {C_{n}^{\beta }({{{\hat {x}}}_{1}} \cdot {{{\hat {x}}}_{2}})} \\ \times \,\,\left[ {\frac{{{{{(x_{1}^{2})}}^{{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2}}}}}}{{{{{(x_{2}^{2})}}^{{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2} + \beta }}}}}\Theta (x_{2}^{2} - x_{1}^{2}) + (x_{1}^{2} \leftrightarrow x_{2}^{2})} \right], \\ \end{gathered} $$

((A8))

where $C_{n}^{\beta }(x)$ can then be related to $C_{{n - 2k}}^{\lambda }(x)$ ($0 \leqslant k \leqslant [{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2}]$) with the help of:

$$\begin{gathered} C_{n}^{\delta }(x) = \sum\limits_{k = 0}^{[{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2}]} {C_{{n - 2k}}^{\lambda }} (x)\frac{{(n - 2k + \lambda )\Gamma (\lambda )}}{{k!\Gamma (\delta )}} \\ \times \,\,\frac{{\Gamma (n + \delta - k)\Gamma (k + \delta - \lambda )}}{{\Gamma (n - k + \lambda + 1)\Gamma (\delta - \lambda )}}. \\ \end{gathered} $$

((A9))

Moreover, the series appearing upon expanding the propagators and after performing all integrations may sometimes be resummed in the form of a generalized hypergeometric function $_{3}{{F}_{2}}$ of unit argument. There is a very useful transformation property relating such hypergeometric functions. Even though not directly connected with Gegenbauer polynomials, we mention it here:

$$\begin{gathered} _{3}{{F}_{2}}(a,b,c;e,f;1) \\ = \frac{{\Gamma (1 - a)\Gamma (e)\Gamma (f)\Gamma (c - b)}}{{\Gamma (e - b)\Gamma (f - b)\Gamma (1 + b - a)\Gamma (c)}} \\ \times \,{{\,}_{3}}{{F}_{2}}(b,b - e + 1,b - f + 1; \\ 1 + b - c,1 + b - a;1) + (b \leftrightarrow c). \\ \end{gathered} $$

((A10))

Of peculiar importance is the case where $e = b + 1$ in which case the $_{3}{{F}_{2}}$ function can be expressed in terms of another $_{3}{{F}_{2}}$ plus a term involving only products of Gamma functions:

$$\begin{gathered} \sum\limits_{p = 0}^\infty {\frac{{\Gamma (p + a)\Gamma (p + c)}}{{p!\Gamma (p + f)}}} \frac{1}{{p + b}} \\ = \frac{{\Gamma (a)\Gamma (1 - a)\Gamma (b)\Gamma (c - b)}}{{\Gamma (f - b)\Gamma (1 + b - a)}} - \frac{{\Gamma (1 - a)\Gamma (a)}}{{\Gamma (f - c)\Gamma (1 + c - f)}} \\ \times \,\,\sum\limits_{p = 0}^\infty {\frac{{\Gamma (p + c - f + 1)\Gamma (p + c)}}{{p!\Gamma (p + 1 + c - a)}}} \frac{1}{{p + c - b}}. \\ \end{gathered} $$

((A11))

1.2 7.2 One-Loop Integral

Let’s consider some simple examples in order to illustrate the method. We start with the one-loop massless p-type diagram with two arbitrary indices in $x$-space (transformation rules between x-space and p‑space are provided in Sec. 4.5):

$$\begin{gathered} J(D,z,\alpha ,\beta ) = \int {\frac{{{{{\text{d}}}^{D}}x}}{{{{x}^{{2\alpha }}}{{{(x - z)}}^{{2\beta }}}}}} , \\ {{{\text{d}}}^{D}}x = \frac{1}{2}{{x}^{{2\lambda }}}d{{x}^{2}}{{d}_{D}}\hat {x}. \\ \end{gathered} $$

((A12))

Combining Eqs. (A8) and (A9), the integral can be separated into a radial and an angular part as follows:

$$\begin{gathered} J(D,z,\alpha ,\beta ) = \frac{1}{2}\sum\limits_{n = 0}^\infty {\sum\limits_{k = 0}^{[{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2}]} {\int\limits_0^\infty {d{{x}^{2}}} } } {{({{x}^{2}})}^{{\lambda - \alpha }}} \\ \times \,\,\left[ {\frac{{{{{({{x}^{2}})}}^{{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2}}}}}}{{{{{({{z}^{2}})}}^{{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2} + \beta }}}}}\Theta ({{z}^{2}} - {{x}^{2}}) + ({{x}^{2}} \leftrightarrow {{y}^{2}})} \right] \\ \times \,\,\underbrace {\int {{{{\text{d}}}_{D}}\hat {x}C_{{n - 2k}}^{\lambda }(\hat {x} \cdot \hat {z})} }_{{{\Omega }_{D}}{{\delta }_{{n,2k}}}}\frac{{(n - 2k + \lambda )\Gamma (\lambda )}}{{k!\Gamma (\beta )}} \\ \times \,\,\frac{{\Gamma (n + \beta - k)\Gamma (k + \beta - \lambda )}}{{\Gamma (n + \lambda + 1 - k)\Gamma (\beta - \lambda )}}, \\ \end{gathered} $$

((A13))

where the orthogonality relation, Eq. (A2) has been used to compute the angular part. It then follows that n must be an even integer: $n = 2p$ and $k = [{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2}] = p.$ The remaining radial integrals are easily performed. The resulting expression can be conveniently written as a sum of two one-fold series:

$$\begin{gathered} J(D,z,\alpha ,\beta ) = \frac{{{{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}}}{{{{{({{z}^{2}})}}^{{\alpha + \beta - \lambda - 1}}}}}\frac{1}{{\Gamma (\beta )\Gamma (\beta - \lambda )}} \\ \times \,\,\sum\limits_{p = 0}^\infty {\frac{{\Gamma (p + \beta )\Gamma (p + \beta - \lambda )}}{{p!\Gamma (p + \lambda + 1)}}} \\ \times \,\,\left[ {\frac{1}{{p + \alpha + \beta - 1 - \lambda }} + \frac{1}{{p - \alpha + \lambda + 1}}} \right]. \\ \end{gathered} $$

((A14))

This expression can be further simplified by transforming the first sum with the help of Eq. (A11) with $a = \beta - \lambda ,$$b = \alpha + \beta - 1 - \lambda ,$$c = \beta$ and $f = \lambda + 1.$ Indeed, this yields:

$$\begin{gathered} \sum\limits_{p = 0}^\infty {\frac{{\Gamma (p + \beta )\Gamma (p + \beta - \lambda )}}{{p!\Gamma (p + \lambda + 1)}}} \frac{1}{{p + \alpha + \beta - 1 - \lambda }} \\ \times \,\,\frac{{\Gamma (\beta - \lambda )\Gamma (1 + \lambda - \alpha )\Gamma (1 + \lambda - \beta )\Gamma (\alpha + \beta - 1 - \lambda )}}{{\Gamma (\alpha )\Gamma (2 + 2\lambda - \alpha - \beta )}} \\ - \,\,\sum\limits_{p = 0}^\infty {} \frac{{\Gamma (p + \beta )\Gamma (p + \beta - \lambda )}}{{p!\Gamma (p + \lambda + 1)}}\frac{1}{{p - \alpha + 1 + \lambda }}, \\ \end{gathered} $$

((A15))

and the sum on the lhs is simply the opposite of the second sum in $J(D,z,\alpha ,\beta ).$ Hence, the sum of the two one-fold series reduces to a product of Γ-functions and we recover the well-known result:

$$\begin{gathered} J(D,z,\alpha ,\beta ) = \frac{{{{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}}}{{{{{({{z}^{2}})}}^{{\alpha + \beta - \lambda - 1}}}}}G(D,\alpha ,\beta ), \hfill \\ G(D,\alpha ,\beta ) = \frac{{a(\alpha )a(\beta )}}{{a(\alpha + \beta - 1 - \lambda )}}, \hfill \\ \end{gathered} $$

((A16))

where $a(\alpha ) = \Gamma ({D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2} - {{\alpha )} \mathord{\left/ {\vphantom {{\alpha )} {\Gamma (\alpha )}}} \right. \kern-0em} {\Gamma (\alpha )}}$ and which was given in Eq. (2.17) in p-space.

1.3 7.3 One-Loop Integral with Traceless Products

We may next generalize this result to the case where a traceless product appears in the numerator:

$$\begin{gathered} {{J}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}(D,z,\alpha ,\beta ) = \int {{{{\text{d}}}^{D}}x} \frac{{{{x}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}}}{{{{x}^{{2\alpha }}}{{{(x - z)}}^{{2\beta }}}}}, \\ {{{\text{d}}}^{D}}x = \frac{1}{2}{{x}^{{2\lambda }}}d{{x}^{2}}{{d}_{D}}\hat {x}. \\ \end{gathered} $$

((A17))

Dimensional analysis suggests that this integral should have the form:

$${{J}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}(D,z,\alpha ,\beta ) = {{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}\frac{{{{z}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}}}{{{{{({{z}^{2}})}}^{{\alpha + \beta - \lambda - 1}}}}}{{G}^{{(n,0)}}}(\alpha ,\beta ),$$

((A18))

where the coefficient function, ${{G}^{{(n,0)}}}(\alpha ,\beta ),$ is yet to be determined. In order to do so, we consider the scalar function:

$$\begin{gathered} {{z}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}{{J}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}(D,z,\alpha ,\beta ) = {{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}\frac{{{{z}^{{2n}}}}}{{{{{({{z}^{2}})}}^{{\alpha + \beta - \lambda - 1}}}}} \\ \times \,\,\frac{{\Gamma (\lambda )\Gamma (n + 2\lambda )}}{{{{2}^{n}}\Gamma (2\lambda )\Gamma (n + \lambda )}}{{G}^{{(n,0)}}}(\alpha ,\beta ), \\ \end{gathered} $$

((A19))

where Eqs. (A18) and (A6) have been used. The corresponding integral can be evaluated by using the relation between traceless products and Gegenbauer polynomials, Eq. (A5):

$$\begin{gathered} {{z}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}{{J}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}(D,z,\alpha ,\beta ) = \int {{{{\text{d}}}^{D}}x} \frac{{{{z}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}{{x}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}}}{{{{x}^{{2\alpha }}}{{{(x - z)}}^{{2\beta }}}}} \\ = \frac{{n!\Gamma (\lambda )}}{{{{2}^{n}}\Gamma (n + \lambda )}}\int {{{{\text{d}}}^{D}}x} \frac{{{{C}_{n}}(\hat {z} \cdot \hat {x}){{{({{x}^{2}}{{z}^{2}})}}^{{{n \mathord{\left/ {\vphantom {n 2}} \right. \kern-0em} 2}}}}}}{{{{x}^{{2\alpha }}}{{{(x - z)}}^{{2\beta }}}}}, \\ \end{gathered} $$

((A20))

and then expanding the propagator in Gegenbauer polynomials as before. This yields:

$$\begin{gathered} {{z}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}{{J}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}(D,z,\alpha ,\beta ) = \frac{{n!\Gamma (\lambda )}}{{{{2}^{n}}\Gamma (n + \lambda )}}\frac{1}{2} \\ \times \,\,\sum\limits_{p = 0}^\infty {\sum\limits_{k = 0}^{[{p \mathord{\left/ {\vphantom {p 2}} \right. \kern-0em} 2}]} {\int {d{{x}^{2}}} } } {{({{x}^{2}})}^{{\lambda - \alpha }}} \\ \times \,\,\left[ {\frac{{{{{({{x}^{2}})}}^{{\tfrac{{p + n}}{2}}}}}}{{{{{({{y}^{2}})}}^{{\tfrac{{p - n}}{2} + \beta }}}}}\Theta ({{z}^{2}} - {{x}^{2}}) + ({{x}^{2}} \leftrightarrow {{y}^{2}})} \right] \\ \times \,\,\int {{{{\text{d}}}_{D}}\hat {x}{{C}_{n}}(\hat {z} \cdot \hat {x})} C_{{p - 2k}}^{\lambda }(\hat {x} \cdot \hat {z})\frac{{(p - 2k + \lambda )\Gamma (\lambda )}}{{k!\Gamma (\beta )}} \\ \times \,\,\frac{{\Gamma (p + \beta - k)\Gamma (k + \beta - \lambda )}}{{\Gamma (p + \lambda + 1 - k)\Gamma (\beta - \lambda )}}. \\ \end{gathered} $$

((A21))

The angular integral is non-zero for $2k = p - n$ which implies that p must have the same parity as n and $p \geqslant n.$ Separate analysis of the even and odd n cases yield, after some simple manipulations:

$$\begin{gathered} {{z}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}{{J}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}(D,z,\alpha ,\beta ) = {{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}\frac{{{{z}^{{2n}}}}}{{{{{({{z}^{2}})}}^{{\alpha + \beta - \lambda - 1}}}}} \\ \times \,\,\frac{{\Gamma (\lambda )\Gamma (n + 2\lambda )}}{{{{2}^{n}}\Gamma (2\lambda )\Gamma (n + \lambda )}}\sum\limits_{m = 0}^\infty {B(m,\left. n \right|\beta ,\lambda )} \\ \times \,\,\left( {\frac{1}{{m + \alpha + \beta - 1 - \lambda }} + \frac{1}{{m + n - \alpha + \lambda + 1}}} \right), \\ \end{gathered} $$

((A22))

where:

$$\begin{gathered} B(m,\left. n \right|\beta ,\lambda ) = \frac{{\Gamma (m + n + \beta )}}{{m!\Gamma (m + n + \lambda + 1)\Gamma (\beta )}} \\ \times \,\,\frac{{\Gamma (m + \beta - \lambda )}}{{\Gamma (\beta - \lambda )}}. \\ \end{gathered} $$

((A23))

Comparing Eqs. (A22) and (A18), we see that the coefficient function equals the sum of two one-fold series:

$$\begin{gathered} {{G}^{{(n,0)}}}(D,\alpha ,\beta ) = \sum\limits_{m = 0}^\infty {B(m,\left. n \right|\beta ,\lambda )} \\ \times \,\,\left( {\frac{1}{{m + \alpha + \beta - 1 - \lambda }} + \frac{1}{{m + n - \alpha + \lambda + 1}}} \right). \\ \end{gathered} $$

((A24))

Such a series representation reduces to a product of Γ-functions upon using the transformation properties of hypergeometric functions:

$$\begin{gathered} {{G}^{{(n,0)}}}(D,\alpha ,\beta ) = \frac{{{{a}_{n}}(\alpha ){{a}_{0}}(\beta )}}{{{{a}_{n}}(\alpha + \beta - \lambda - 1)}}, \\ {{a}_{n}}(\alpha ) = \frac{{\Gamma (n + {D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2} - \alpha )}}{{\Gamma (\alpha )}}, \\ \end{gathered} $$

((A25))

in accordance with Eq. (2.21).

1.4 7.4 One-Loop Integral with Heaviside Functions

The above results yield integration rules for Feynman integrals involving traceless products and Heaviside functions which were given in Ref. [61]. From Eq. (A22) we indeed recover the basic results of this reference:

$$\begin{gathered} \int {{{{\text{d}}}^{D}}x} \frac{{{{x}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}}}{{{{x}^{{2\alpha }}}{{{(x - y)}}^{{2\beta }}}}}\Theta ({{x}^{2}} - {{y}^{2}}) \\ = {{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}\frac{{{{y}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}}}{{{{{({{y}^{2}})}}^{{\alpha + \beta - \lambda - 1}}}}}\sum\limits_{m = 0}^\infty {\frac{{B(m,\left. n \right|\beta ,\lambda )}}{{m + \alpha + \beta - 1 - \lambda }}} \\ \mathop = \limits^{(\beta = \lambda )} {{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}\frac{{{{y}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}}}{{{{{({{y}^{2}})}}^{{ - 1}}}}}\frac{1}{{\Gamma (\lambda )}}\frac{1}{{(\alpha - 1)(n + \lambda )}}, \\ \end{gathered} $$

((A26))

and

$$\begin{gathered} \int {{{{\text{d}}}^{D}}x\frac{{{{x}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}}}{{{{x}^{{2\alpha }}}{{{(x - y)}}^{{2\beta }}}}}} \Theta ({{y}^{2}} - {{x}^{2}}) = {{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}\frac{{{{y}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}}}{{{{{({{y}^{2}})}}^{{\alpha + \beta - \lambda - 1}}}}} \\ \times \,\,\sum\limits_{m = 0}^\infty {\frac{{B(m,\left. n \right|\beta ,\lambda )}}{{m + n - \alpha + 1 + \lambda }}} \mathop = \limits^{(\beta = \lambda )} {{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}\frac{{{{y}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}}}{{{{{({{y}^{2}})}}^{{\alpha - 1}}}}} \\ \times \,\,\frac{1}{{\Gamma (\lambda )}}\frac{1}{{(n + \lambda + 1 - \alpha )(n + \lambda )}}, \\ \end{gathered} $$

((A27))

where the peculiar case $\beta = \lambda$ has been explicitly displayed. The following more complicated cases are also useful (see [61]):

$$\begin{gathered} \int {{{{\text{d}}}^{D}}x\frac{{{{x}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}}}{{{{x}^{{2\alpha }}}{{{(x - y)}}^{{2\beta }}}}}} \Theta ({{x}^{2}} - {{z}^{2}}) = {{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}{{y}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}} \\ \times \,\,\left[ {\frac{{\Theta ({{y}^{2}} - {{z}^{2}})}}{{{{{({{y}^{2}})}}^{{\alpha + \beta - \lambda - 1}}}}}{{G}^{{n,0}}}(\alpha ,\beta )} \right. + \sum\limits_{m = 0}^\infty {\frac{{B(m,\left. n \right|\beta ,\lambda )}}{{{{{({{z}^{2}})}}^{{\alpha + \beta - \lambda - 1}}}}}} \\ \times \,\,\left( {{{{\left( {\frac{{{{y}^{2}}}}{{{{z}^{2}}}}} \right)}}^{m}}\frac{{\Theta ({{z}^{2}} - {{y}^{2}})}}{{m + \alpha + \beta - 1 - \lambda }}} \right. \\ - \,\,{{\left( {\frac{{{{z}^{2}}}}{{{{y}^{2}}}}} \right)}^{{m + n + \beta }}}\left. {\left. {\frac{{\Theta ({{y}^{2}} - {{z}^{2}})}}{{m - \alpha + n + 1 + \lambda }}} \right)} \right] \\ \mathop = \limits^{(\beta = \lambda )} {{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}\frac{1}{{\Gamma (\lambda )}}{{y}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}\left[ {\frac{{\Theta ({{y}^{2}} - {{z}^{2}})}}{{{{{({{y}^{2}})}}^{{\alpha - 1}}}}}\frac{1}{{(\alpha - 1)(n + \lambda + 1 - \alpha )}}} \right. \\ \left. { + \,\,\frac{1}{{{{{({{z}^{2}})}}^{{\alpha - 1}}}}}\frac{1}{{n + \lambda }}\left( {\frac{{\Theta ({{z}^{2}} - {{y}^{2}})}}{{\alpha - 1}} - {{{\left( {\frac{{{{z}^{2}}}}{{{{y}^{2}}}}} \right)}}^{{n + \lambda }}}\frac{{\Theta ({{y}^{2}} - {{z}^{2}})}}{{n + 1 + \lambda - \alpha }}} \right)} \right], \\ \end{gathered} $$

((A28))

and

$$\begin{gathered} \int {{{{\text{d}}}^{D}}x} \frac{{{{x}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}}}{{{{x}^{{2\alpha }}}{{{(x - y)}}^{{2\beta }}}}}\Theta ({{z}^{2}} - {{x}^{2}}) = {{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}{{y}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}} \\ \times \,\,\left[ {\frac{{\Theta ({{z}^{2}} - {{y}^{2}})}}{{{{{({{y}^{2}})}}^{{\alpha + \beta - \lambda - 1}}}}}{{G}^{{n,0}}}(\alpha ,\beta )} \right. - \sum\limits_{m = 0}^\infty {\frac{{B(m,\left. n \right|\beta ,\lambda )}}{{{{{({{z}^{2}})}}^{{\alpha + \beta - \lambda - 1}}}}}} \\ \times \,\,\left( {{{{\left( {\frac{{{{y}^{2}}}}{{{{z}^{2}}}}} \right)}}^{m}}\frac{{\Theta ({{z}^{2}} - {{y}^{2}})}}{{m + \alpha + \beta - 1 - \lambda }}} \right. \\ - \,\,{{\left( {\frac{{{{z}^{2}}}}{{{{y}^{2}}}}} \right)}^{{m + n + \beta }}}\left. {\left. {\frac{{\Theta ({{y}^{2}} - {{z}^{2}})}}{{m - \alpha + n + 1 + \lambda }}} \right)} \right] \\ \mathop = \limits^{(\beta = \lambda )} {{\pi }^{{{D \mathord{\left/ {\vphantom {D 2}} \right. \kern-0em} 2}}}}\frac{1}{{\Gamma (\lambda )}}{{y}^{{{{\mu }_{1}} \cdots {{\mu }_{n}}}}}\left[ {\frac{{\Theta ({{z}^{2}} - {{y}^{2}})}}{{{{{({{y}^{2}})}}^{{\alpha - 1}}}}}\frac{1}{{(\alpha - 1)(n + \lambda + 1 - \alpha )}}} \right. \\ \left. { - \,\,\frac{1}{{{{{({{z}^{2}})}}^{{\alpha - 1}}}}}\frac{1}{{n + \lambda }}\left( {\frac{{\Theta ({{z}^{2}} - {{y}^{2}})}}{{\alpha - 1}} - {{{\left( {\frac{{{{z}^{2}}}}{{{{y}^{2}}}}} \right)}}^{{n + \lambda }}}\frac{{\Theta ({{y}^{2}} - {{z}^{2}})}}{{n + 1 + \lambda - \alpha }}} \right)} \right]. \\ \end{gathered} $$

((A29))

With these rules in hand, the Gegenbauer polynomials technique allows to compute the massless p-type two-loop master integral with up three arbitrary indices as a linear combination of up to four hypergeometric functions $_{3}{{F}_{2}}$ of argument 1, a result which can be found in Ref. [61]. In particular, the method provides an alternative representation for the integral $I(1 + a)$ found in the previous section with functional equations (see Eq. (4.138)).

1.5 7.5 Application to $I(1 + a)$

Here we reconsider the simple but very important example of: $I(1 + a)$ = $J(D,p,1,1,1,1,1 + a)$ (see Eq. (3.28)). Applying the rules of the previous paragraph, its coefficient function ${{{\text{C}}}_{D}}[I(1 + a)]$ can be expressed as:

$$\begin{gathered} {{{\text{C}}}_{D}}[I(a + 1)] = 2\frac{{{{\Gamma }^{2}}(1 - \varepsilon )\Gamma ( - \varepsilon - a)\Gamma (a + 2\varepsilon )}}{{\Gamma (2 - 2\varepsilon )\Gamma (1 - 3\varepsilon - a)}} \\ \times \,\,\left[ {\frac{{\pi \Gamma (1 - a - 3\varepsilon )\Gamma (1 - a - 2\varepsilon )\Gamma {{{(a + 2\varepsilon )}}^{{}}}}}{{\Gamma (1 - \varepsilon )\Gamma ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2} - a - 2\varepsilon )\Gamma ({1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-0em} 2} + 2\varepsilon + a)}}} \right. \\ - \,\,\left. {\sum\limits_{n = 1}^\infty {\frac{{\Gamma (n + 1 - 2\varepsilon )}}{{\Gamma (n + 1 + a)}}} \frac{1}{{n + a + 1 + \varepsilon }}} \right], \\ \end{gathered} $$

((A30))

which coincides with Eq. (3.58) after changing of variables.

So, using the method of Gegenbauer polynomials, the results for $I(1 + a)$ can be expressed as a combination of Γ-functions together with one hypergeometric function with the arguments “1”. Such result can be successfully used for an efficient ε-expansion of the diagram. Moreover, the combination of the two results (4.138) and (A30) provides the advertised relation (3.59) between two hypergeometric functions of argument “–1” and one hypergeometric function of argument “$1$”. Such a relation is absent in standard textbooks and was recently proven exactly in [137].