Analytic and Algebraic Studies of Feynman Integrals

Abstract

Feynman integrals are central to the calculation of scattering amplitudes both in particle and gravitationalwave physics. This thesis presents advancements in both the analytical and algebraicstructure of these integrals and shows how this can be used for efficient evaluation of these integrals.Paper I. In this paper the focus is on one-loop integrals. The singularities of these integrals arefully described and used to derive the full symbol alphabet and canonical differential equation forany number of external particles. It is proven that a large family of one-loop integrals satisfy theCohen-Macaulay property.Paper II. Two infinite families of Feynman integrals satisfying the Cohen-Macaulay propertyare classified. This property implies that both the singularities and the number of master integralsis independent of space-time dimension and propagator powers.Paper III. In this paper the singular locus of a Feynman integral is defined as the critical pointsof a Whitney stratified map. Explicit code and calculations are provided which show that thismethod captures singularities otherwise hard to detect.Paper IV-V. The algebraic properties of the integrand, especially that of its Newton polytopebeing a generalized permutohedron, is leverage together with tropical sampling to provide efficientnumerical evaluation of Feynman integrals with physical kinematics. The connection betweenthe generalized permutohedron property and the Cohen-Macaulay property is also discussed