3-loop Feynman Integral Extrapolations for the Baseball Diagram

TL;DR

This paper develops numerical extrapolation techniques for 3-loop Feynman integrals, specifically improving the accuracy of epsilon-expansion coefficients for the baseball diagram by handling internal singularities.

Contribution

It introduces a novel extrapolation method with respect to an additional parameter to enhance epsilon-expansion accuracy for complex Feynman integrals.

Findings

Effective numerical extrapolation for 3-loop integrals demonstrated

Improved accuracy in epsilon-expansion coefficients for baseball diagram

Handling internal singularities enhances computational reliability

Abstract

We focus on numerical techniques for expanding 3-loop Feynman integrals with respect to the dimensional regularization parameter $\varepsilon,$ which is related to the space-time dimension as $\nu = 4-2\varepsilon,$ and describes underlying UV singularities located at the boundaries of the integration domain. As a function of the squared momentum $s,$ the expansion coefficients exhibit thresholds that generally delineate regions for their computational techniques. For the problem at hand, a sequence of integrations with a linear extrapolation as $\varepsilon\rightarrow 0$ may be performed to determine leading coefficients of the $\varepsilon$-expansion numerically. For the "baseball" Feynman diagram, we have used extrapolation with respect to an additional parameter to improve the accuracy of the $\varepsilon$-expansion coefficients in case of singularities internal to the domain.