Evaluation of the general 3-loop vacuum Feynman integral

TL;DR

This paper presents a systematic method for evaluating 3-loop vacuum Feynman integrals with arbitrary masses, introducing a numerical approach based on differential equations and a new software package for efficient computation.

Contribution

It develops a reduction technique to express complex integrals in terms of basis integrals and provides a public software tool for their numerical evaluation.

Findings

Provides a set of basis integrals for 3-loop vacuum calculations.

Introduces a numerical method using differential equations for basis integral evaluation.

Offers a publicly available software package, 3VIL, for practical computations.

Abstract

We discuss the systematic evaluation of 3-loop vacuum integrals with arbitrary masses. Using integration by parts, the general integral of this type can be reduced algebraically to a few basis integrals. We define a set of modified finite basis integrals that are particularly convenient for expressing renormalized quantities. The basis integrals can be computed numerically by solving coupled first-order differential equations, using as boundary conditions the analytically known special cases that depend on only one mass scale. We provide the results necessary to carry this out, and introduce an implementation in the form of a public software package called 3VIL (3-loop Vacuum Integral Library), which efficiently computes the numerical values of the basis integrals for any specified masses. 3VIL is written in C, and can be linked from C, C++, or FORTRAN code.