# Regularization with Numerical Extrapolation for Finite and UV-Divergent Multi-loop Integrals

## TL;DR

This paper presents advanced numerical techniques for evaluating multi-loop Feynman integrals, including UV divergences, using adaptive multivariate integration and extrapolation methods, with implementations that leverage parallel computing.

## Contribution

It introduces a comprehensive numerical framework combining adaptive multivariate integration and extrapolation for regularizing UV-divergent multi-loop integrals, implemented with parallel computing.

## Key findings

- Successful numerical evaluation of 2-loop vertex and box diagrams.
- Effective regularization of UV divergent integrals using extrapolation.
- Demonstrated scalability with parallel computing techniques.

## Abstract

We give numerical integration results for Feynman loop diagrams such as those covered by Laporta [1] and by Baikov and Chetyrkin [2], and which may give rise to loop integrals with UV singularities. We explore automatic adaptive integration using multivariate techniques from the PARINT package for multivariate integration, as well as iterated in- tegration with programs from the QUADPACK package, and a trapezoidal method based on a double exponential trans- formation. PARINT is layered over MPI (Message Passing Interface), and incorporates advanced parallel/distributed techniques including load balancing among processes that may be distributed over a cluster or a network/grid of nodes. Results are included for 2-loop vertex and box diagrams and for sets of 2-, 3- and 4-loop self-energy diagrams with or without UV terms. Numerical regularization of integrals with singular terms is achieved by linear and non-linear extrapolation methods.