Asymptotic Analysis of Multi scale Multi loop Feynman Diagrams

Abstract

It is very challenging to solve multi-scale, multi-loop Feynman diagrams analytically. The presence of different kinematic scales makes the computation of Feynman diagrams very difficult, sometimes impossible to get the analytic results. One way to tackle this problem is to consider systematic approximations based on the hierarchies of the scales. The basic idea is to simplify the integral before the integration. The Method of Regions (MoR) is one of the powerful methods for handling the evaluation of multi-scale, multi-loop Feynman diagrams asymptotically. The whole loop momentum domain is divided into several regions and the integrand of the given Feynman diagram is expanded, in each of the regions, in a Taylor series based on a small expansion parameter, which is the ratio of low scale and the high scale. After the expansion, the sum of the contributions which are obtained from the integration of the expanded terms over the whole range of momentum, gives the result for the original Feynman diagram in an expanded form. It is a non-trivial task to identify the correct set of regions required for the asymptotic analysis of the Feynman integrals. In one of the projects reported in this thesis, we have designed an algorithm for unveiling the regions associated with the multi-scale multi-loop Feynman integrals in given limits. We show that the regions can be unveiled from the neighborhood of the singular surfaces of the Feynman diagrams. The associated singularities are known as the Landau singularities. The Feynman diagrams are characterized by two homogeneous polynomials, called the Symanzik polynomials. The location of the singularities of the Feynman diagrams are determined from the Landau equations, which are obtained by equating the Symanzik polynomial of second kind and all of its partial derivatives with respect to the Feynman parameters to zero. In our framework, we consider the set of the Landau equations for a given multi-loop, multi-scale Feynman diagram and express them via the Gröbner basis element...