Show simple item record

dc.contributor.advisorAnanthanarayan, Balasubramanian
dc.contributor.authorBera, Souvik
dc.date.accessioned2024-02-16T06:20:26Z
dc.date.available2024-02-16T06:20:26Z
dc.date.submitted2023
dc.identifier.urihttps://etd.iisc.ac.in/handle/2005/6413
dc.description.abstractFeynman integrals play a fundamental role in perturbative calculations within the realm of quantum field theory. In this thesis, we study several important mathematical properties of multi-loop multi-leg Feynman Integrals in dimensional regularization and their realization in terms of well-known functions of mathematical physics, especially hypergeometric functions, including in their Laurent expansion in the dimensional parameter epsilon. We study the GKZ systems using the Gr\"obner deformation technique, which offers solutions for these integrals expressed in terms of hypergeometric functions with multiple variables. We implement this algorithm as a \textsc{Mathematica} package, which has become part of the publication \texttt{FeynGKZ.wl}. It is important to note that multivariate hypergeometric functions exhibit their validity within specific domains of convergence, which can be obtained by Horn's theorem. To extend our analysis beyond these defined domains and determine numerical values, the pursuit of analytic continuations becomes imperative. We explore the method of Olsson for this purpose, developing it into the \textsc{Mathematica} package\texttt{ Olsson.wl}, thereby streamlining the process of calculating these analytic continuations. In addition to this, we introduce the \texttt{ROC2.wl} package, which serves as a realization of Horn's theorem, capable of finding the domains of convergence for bi-variate hypergeometric functions. Our exploration further extends to the discovery of analytic continuations for Appell $F_2$ and Horn $H_5$ functions through the same method. We optimize the analytic continuations of Appell $F_2$ to facilitate numerical evaluation for generic values of parameters and for real values of its arguments except for its singular curves, giving rise to the foundational package, \texttt{AppellF2.wl}. Lastly, we present an algorithm designed to determine the series expansion of hypergeometric functions based on their series representation. This algorithm is implemented in the \textsc{Mathematica} package \texttt{MultiHypExp.wl}, offering a tool to expand certain hypergeometric functions in series around integer-valued parameters in terms of multiple polylogarithms. Our works demonstrate the importance of the impact of theory and computer algebra on modern Feynman Integral calculus and provide general-purpose open-source packages which are bound to be of great utility to the community.en_US
dc.language.isoen_USen_US
dc.relation.ispartofseries;ET00425
dc.rightsI grant Indian Institute of Science the right to archive and to make available my thesis or dissertation in whole or in part in all forms of media, now hereafter known. I retain all proprietary rights, such as patent rights. I also retain the right to use in future works (such as articles or books) all or part of this thesis or dissertationen_US
dc.subjectFeynman integralsen_US
dc.subjectMultivariate hypergeometric functionsen_US
dc.subjecthypergeometric functionsen_US
dc.subject.classificationResearch Subject Categories::NATURAL SCIENCES::Physics::Elementary particle physicsen_US
dc.titleFeynman Integral Calculus and Hypergeometric Functions: New Results, Interconnections and Computer Algebraen_US
dc.typeThesisen_US
dc.degree.namePhDen_US
dc.degree.levelDoctoralen_US
dc.degree.grantorIndian Institute of Scienceen_US
dc.degree.disciplineFaculty of Scienceen_US


Files in this item

This item appears in the following Collection(s)

Show simple item record