On Different Parametrizations of Feynman Integrals
In this doctoral thesis, we discuss and apply advanced techniques for the calculations of scattering amplitudes which, on the one hand, allow us to compute cross sections and differential distributions at high precision and, on the other hand, give us deep mathematical insights on the mathematical structures of Feynman integrals. We start by presenting phenomenological calculations relevant for the experimental analyses at the Large Hadron Collider. We use the resummation of soft gluon emission corrections to study the associated production of a top pair and a Z boson to next-to-next-to-leading logarithmic accuracy, and compute the total cross section and differential distributions for a range of interesting observables. Our evaluations are currently the most precise predictions for this process available in the literature. Then we introduce various representations for Feynman integrals, namely Schwinger, Feynman, Lee-Pomeransky, and Baikov parametrizations. We show detailed derivations and discuss the relevant mathematical properties for these parametrizations. We also discuss Integration-By-Parts identities, illustrate the reduction process under Laporta's algorithm, and examine the boundary regions for Baikov parametrization. We finally present two novel applications of Schwinger-Feynman parametrizations. First, we consider Integration-By-Parts identities over Schwinger-Feynman parameters. Through this procedure, we touch upon techniques which are related to graph theory and complex analysis. Then we use Intersection Theory to compute the recurrence relations among classes of integrals. We confirm the consistency of these two new approaches using traditional Integration-By-Parts calculations.
Elementary Particles and Fields and String Theory, multi-loop Feynman integrals, Mellin transform, Feynman diagrams, scattering amplitudes
City University of New York - Graduate Center
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The Graduate School and University Center of The City University of New York