Solving recurrence relations for multiloop integrals in the limit of large values of the dimensional regularization parameter

TL;DR

This paper introduces a recursive method using linear substitutions to compute the $1/d$ expansion coefficients of Feynman integrals, demonstrated on complex vacuum integrals up to 7 loops.

Contribution

It presents a novel recursive approach for solving integration by parts relations in multiloop Feynman integrals using linear substitutions.

Findings

Applicable to massless vacuum integrals with one massive line

Successfully applied to integrals up to 7 loops

Provides explicit recursive formulas for $1/d$ expansion coefficients

Abstract

A method for calculating the $1/d$ expansion coefficients for solutions of integration by parts relations for Feynman integrals is presented. The idea is to use linear substitutions to transform these relations to an explicitly recursive form. A possible type of such substitutions is proposed for the case of vacuum integrals. Its applicability is shown for several families of massless (with one massive line) vacuum integrals up to the 7-loop level.