Prisco, Renato Maria (2025) On the numerical evaluation of Feynman integrals through differential equations. [Tesi di dottorato]

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Abstract

Particle colliders are able to probe the structure of matter at extremely short distances. The increasing precision of present and future experiments demands equally accurate theoretical predictions to test and advance our understanding of particle physics. These predictions are based on the computation of higher-order perturbative corrections to scattering cross sections and, ultimately, on the evaluation of Feynman integrals over virtual loop momenta. Analytical calculations rapidly grow in complexity as the number of loops, particles and energy scales increases, so that numerical approaches become essential and foster the development of automated tools. Within this context, the differential equation method has proved particularly effective. Feynman integrals can indeed be reduced, via integration-by-parts identities, to finite sets of master integrals that satisfy systems of first-order differential equations in the Lorentz-invariant kinematic variables. This thesis focuses on the development of algorithmic procedures for the numerical evaluation of Feynman integrals as series expansion solutions to this kind of differential equations. Within the framework presented here, Feynman integrals are propagated in the phase space of the kinematic invariants by solving the differential equations on a line connecting a boundary point to any given target point. The radius of convergence for series solutions is finite due to the presence of singularities and, therefore, the propagation is split into several steps that iteratively move the solution closer to the target. Boundary conditions are obtained around singular kinematic configurations by imposing that the leading power behaviors of the solution agree with those extracted through techniques based on expansion by regions. To do so, the system is transformed to Fuchsian form and the equations are solved by using the considered singularity as the expansion point. This last approach also serves as a strategy for crossing singular points that lie on the propagation path. Since Feynman integrals exhibit branch point singularities, algorithms are provided to perform the analytic continuation of the solutions in agreement with the Feynman prescription. The methods presented in this thesis have been implemented in a novel open-source computer program called LINE, which stands for Loop Integrals Numerical Evaluation. Aiming to make large-scale cluster computations more feasible, the code has been written in C/C++ to enable good performance while avoiding the limitations imposed by proprietary software. LINE allows both the computation of boundary values and their propagation across the phase space within a unified framework. In particular, for integrals up to two loops, the software includes an in-house implementation of the auxiliary mass flow method. The latter relies on the introduction of an auxiliary mass to obtain boundary conditions in the infinite-mass limit. Alternatively, around a singularity the program is able to receive and analyze expansion-by-regions information in order to minimize the number of leading coefficients to be provided as boundary values. LINE leverages open-source libraries for arbitrary-precision arithmetic, and the numerical accuracy of the results can be assessed without relying on external tools for the evaluation of the integrals. The program has been tested on a variety of one- and two-loop examples, planar and non-planar, with and without masses.

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