On Hypergeometric solutions of Feynman integrals using Mellin-Barnes Integrals with Applications

Abstract

Experimental measurements of physical observables in particle colliders rely on high-precision theoretical calculations to validate the predictions of any theoretical model and detect signals of new physics. These theoretical calculations, in the perturbative framework of quantum field theory, require the computation of complicated integrals, commonly known as Feynman integrals. Therefore, the evaluation of Feynman integrals is the backbone of theoretical precision calculations. However, this is a highly non-trivial task as we go to higher order in the perturbative series expansion, where the number of scales and loops of the associated Feynman diagrams become large, and the number of Feynman diagrams grows factorially. Nevertheless, such computations are necessary to match the extraordinary precision level of modern and future colliders, and thus the study of Feynman integrals has become a very active field of research in recent years.

In this thesis, we focus on studying the mathematical aspects of Feynman integrals and devise techniques to compute them. This field of research is at the cutting edge of theoretical physics, mathematics and computer programming, with new efficient tools proposed every year. Some of the well-known techniques to compute Feynman integrals include the method of differential equations, Mellin-Barnes (MB) representation, hypergeometric functions, Gelfand-Kapranov-Zelevinsky (GKZ) system of equations etc.

Our primary focus in this thesis is the study of the Mellin-Barnes (MB) representation approach to solving Feynman integrals in terms of hypergeometric functions. In one of the projects reported in this thesis, we present the first systematic computational technique to solve N-fold MB integrals commonly appearing in the context of Feynman integrals. This technique is based on a surprising geometrical connection between MB integrals and the theory of conic hulls, and therefore we call this the conic hull method. This method yields the final solution in terms of hypergeometric series, which are useful for further analytic study as well as numerical computation. Along with this project, we also present a computer implementation of the conic hull method in the form of the package MBConicHulls.wl based on Mathematica.

In another project in this thesis, we apply the conic hull method to solve the previously unsolved dual-conformal hexagon and double box Feynman integrals. These integrals remained unsolved as they involve solving a nine-fold MB integral. However, the conic hull method is successful in solving both the above integrals as it is applicable to N-fold MB integrals. The solutions we obtain for the hexagon and double box are cross-checked numerically against direct integration of Feynman parametrization and analytically using the differential equation satisfied by the double box and hexagon integrals.

In another project, we show yet another application of the conic hull method by computing one-loop N-point massive conformal Feynman integrals. Here, we prove two conjectures, first proposed from the Yangian bootstrap approach, stating that any one-loop N-point massive conformal Feynman integral can be written as a single multi-fold hypergeometric series. To prove this, we show that there is always a conic hull, associated with the MB integral of the Feynman integrals, which does not intersect with any other conic hulls, and therefore yields a series solution with a single hypergeometric series.

Finally, in the last project of this thesis, we use the conic hull method to illustrate the limitations of the method of brackets. The latter is a computational technique based on the Ramanujan Master Theorem, and was devised originally to evaluate Feynman integrals but is also useful to evaluate certain definite integrals. However, the method is plagued with divergences whose origin is not from the ultraviolet divergence of the Feynman integral, but from the breakdown of one of the rules of the method of brackets. Studying this method in parallel with the conic hull method helps us show some of the fundamental issues of the method of brackets and point out the domain of validity of the method.