Some topics concerning the standard model, Feynman integrals and renormalization group methods: a review of some recent investigations and results

In this review, we present a comprehensive overview of some of our work carried out in numerous collaborations on important topics in the context of higher-order calculations in perturbative quantum field theories. The approach of this review is one where analytical methods are given prominence. Thus, we primarily concern ourselves with the study of multi-loop scalar Feynman integrals appearing in simplified and idealized models on the one hand, and on the other, to methods for obtaining analytic results for such integrals that are amenable to an implementation on Mathematica as the computer algebra software of choice. After a preliminary discussion of some of the commonly used parametric representations for Feynman integrals, we review (a) the construction of an algorithm and an automated program to find the ‘regions’ of Feynman integrals using Landau equations and power geometry, (b) the analysis of a non-trivial two-loop non-planar Feynman integral using Hopf algebras, (c) some basic aspects of multi-variable hypergeometric functions, namely, their regions of convergence and analytic continuations, (d) the interplay between the theory of multi-variable hypergeometric functions and Feynman integrals, including an algorithmic method for finding series representations for multi-fold Mellin-Barnes representations of Feynman integrals, the interpretation of Feynman integrals as GKZ hypergeometric functions and an automated program that uses this idea for obtaining series solutions, the ϵ-expansion for multi-variable hypergeometric functions arising from dimensionally regularized Feynman integrals, algebraic relations for products of propagators, and (e) the summation of large logarithms for renormalizable as well as non-renormalizable quantum field theories.