| dc.description.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 elements. By equating the Gröbner basis elements of the Landau equations to zero, we derive a set of linear transformations which map the singular surfaces to the origin, co-ordinate axes or co-ordinate planes in the parametric space. The so obtained linear transformations are then applied to the sum of the Symanzik polynomials of first and second kind. The obtained set of polynomials are then analyzed within the framework of Power Geometry. The analysis consists of several steps. The first one is to find the support of the obtained polynomials which basically are the vector exponents for each of the terms of the considered polynomials. The second step is to find the convex hulls of the obtained supports, which are called the Newton polytopes. We then find the normal vector for each of the facets of the Newton polytopes based on certain rules. The set of unique normal vectors then give the set of required regions for the given Feynman diagram. We call our algorithm ASPIRE. Within our approach, we show that all the regions including the potential and Glauber can be unveiled. The algorithm ASPIRE thus provides a useful method for identifying the regions required for the asymptotic expansion of Feynman diagrams based on rigorous mathematical tools.
In another project of this thesis, we consider the notion of top facets of the ASPIRE program. The top facet scalings with equal components under the consideration of the Landau equations and the analysis of Power Geometry gives rise to a criterion which allows us to correlate the top facet scalings with equal components to the maximally cut Feynman diagrams. We use the method of residue calculus for one loop cases for finding out the maximally cut Feynman diagrams. The integrals for the top facets with equal components have been evaluated considering the parametric representation. The top facets with equal components have the property that the external momenta can be set to zero (neglected with respect the internal masses) for a given diagram and thus corresponds to the case of the large expansion parameter. We call this property to be the top facet condition. When the top facet condition is applied to the case of maximal cut of the given Feynman diagram, we show that the top facet with equal components is exactly proportional to the maximal cut. We also provide the results for the other top facets with non-equal components. In this work, we have connected two independent approaches via the consideration of Landau equations and the Power Geometry.
In another work, we analyze a two loop planar three point Feynman diagram asymptotically using the method of regions in the high energy limit. We consider the Schwinger parametric representation for Feynman diagrams with the consideration of the analytic regulators. The standard method of dimensional regularization is not sufficient to regularize the contributions of some of the regions. For the most generic consideration, we can take the help of extra analytic regulators for the regularization of the contributions of those regions. For the considered diagram, we obtain six regions. Out of six regions, we have solved the contribution from three regions. The contribution of two regions are expressed via Mellin-Barnes integrations and the contribution from the hard region remain to be evaluated. | en_US |
