Evaluation of three-loop self-energy master integrals with four or five propagators

TL;DR

This paper develops a method to evaluate complex three-loop self-energy master integrals with four or five propagators, using identities and differential equations for efficient numerical computation.

Contribution

It introduces a novel approach to derive identities for master integrals without traditional IBP reduction, facilitating their numerical evaluation and expansion in external momentum.

Findings

Derived identities for master integrals with generic masses.

Enabled straightforward numerical evaluation via differential equations.

Provided a simple formula for arbitrary order expansion in external momentum.

Abstract

I obtain identities satisfied by the 3-loop self-energy master integrals with four or five propagators with generic masses, including the derivatives with respect to each of the squared masses and the external momentum invariant. These identities are then recast in terms of the corresponding renormalized master integrals, enabling straightforward numerical evaluation of them by the differential equations approach. Some benchmark examples are provided. The method used to obtain the derivative identities relies only on the general form implied by integration by parts relations, without actually following the usual integration by parts reduction procedure. As a byproduct, I find a simple formula giving the expansion of the master integrals to arbitrary order in the external momentum invariant, in terms of known derivatives of the corresponding vacuum integrals.