The Gegenbauer Polynomial Technique: the evaluation of a class of Feynman diagrams

TL;DR

The paper extends the Gegenbauer Polynomial technique to evaluate a class of master two-loop Feynman diagrams with a vertex containing two propagators of index $1$ or $\lambda= D/2-1$ in $D=4-2\varepsilon$. Using $x$-space methods, Fourier transforms, and dual diagrams, it expresses the coefficient functions as $_3F_2$-hypergeometric series with unit argument and derives a new transformation rule for $_3F_2$ with argument $-1$. A concrete result is obtained for $J(1,1,1,1,\alpha)$, yielding an explicit $I(\alpha)$, while the work demonstrates a powerful analytical route for higher-order radiative corrections. The findings provide new analytical tools and a potential for generalization within dimensional regularization frameworks.

Abstract

We extend Gegenbauer Polynomials technique to evaluate a class of complicated Feynman diagrams. New results in the form of $_3F_2$-hypergeometrical series of unit argument, are presented. As a by-product, we present a new transformation rule for $_3F_2$-hypergeometric series with argument $-1$.