Studies of Feynman and Related Integrals and their Hypergeometric Connection using the Method of Brackets, Algebraic Relations and Singularity Analysis

Abstract

Feynman integrals are special kind of integrals that arise in perturbative quantum field theory and are often very difficult to evaluate in terms of known functions of mathematical physics. At times some of the methods used to evaluate other related integrals may be of use in the evaluation of Feynman integrals and vice versa. In this thesis work, we use one such method, the method of brackets to evaluate certain improper integrals of quadratic and quartic type. We further investigate integrals such as the Ising integrals, the Box integrals and the other related integrals. In the course of the study, we obtain multi-variable hypergeometric functions naturally and this leads us to carry out further analysis of these functions such as obtaining their connection formulas. We take Appell $F_2$ as an example and evaluate its analytic continuations. With the techniques developed for it, we further evaluate the analytic continuations of a more complicated function called Horn’s $H_1$. We then return to the evaluation of Feynman integrals and study the algebraic relations of the product of their propagators. We suitably modify the known technique so as to implement it in a computer algebra system for automated evaluation of such relations. In this investigation, we again find the presence of multi-variable hypergeometric functions and were also able to derive many elusive and non-trivial reduction formulas for the same. In connection with the numerical evaluation of the multi-variable hypergeometric series we also study the approximation theory focusing on the bi-variate series. We then further develop an automated \textsc{Mathematica} package for the evaluation of diagonal bi-variate approximants which are called the Chisholm approximants. We study their construction and their usage in the context of convergence acceleration and analytic continuation. We also show their utility using examples from various branches of physics such as particle physics and condensed matter physics. Finally, we return to the analysis of the Feynman integrals again and study their analytic properties. We study singularities associated with them without their explicit evaluation by compactifying the integrals in a projective space. We also extend the usage of the previous analysis in this regard, which was restricted to the singularities of second type, to other types of singularities.