The Gegenbauer Polynomial Technique: the evaluation of complicated Feynman integrals

TL;DR

The paper advances the Gegenbauer Polynomial technique (GP) for analytic evaluation of complex massless Feynman integrals in dimensional regularization. It develops a refined GP framework with the traceless product (TP) formalism, explicit integration rules, and hypergeometric reductions, enabling systematic treatment of propagator-type diagrams, including $\Theta$-function regions. A key result is the explicit evaluation of a broad class of two-loop master diagrams and an analysis of a concrete diagram $J(1,1,1,1,\alpha)$, leading to a transformation rule for ${}_3F_2$ with unit argument. The methods are applied to diverse problems, culminating in the $O(1/N^3)$ critical exponents of $\phi^4$ theory across dimensions, thereby providing precise analytic tools for high-order perturbative and critical phenomena calculations.

Abstract

We discuss a progress in calculation of Feynman integrals which has been done with help of the Gegenbauer Polynomial Technique and demonstrate the results for most complicated parts of O(1/N^3) contributions to critical exponents of φ^4 -theory, for any spacetime dimensionality D.