Three-loop vertex integrals at symmetric point

TL;DR

This paper computes three-loop massless three-point integrals at the symmetric point, extending two-loop results and enabling precise QCD calculations within the RI/SMOM scheme.

Contribution

It introduces a method to evaluate three-loop integrals using differential equations in epsilon-form and expresses results with uniform transcendental weight.

Findings

Derived explicit expressions for three-loop integrals up to weight six.

Applied differential equations to transform integrals into epsilon-form.

Provided basis functions in harmonic polylogarithms with six-root of unity.

Abstract

This paper provides details of the massless three-loop three-point integrals calculation at the symmetric point. Our work aimed to extend known two-loop results for such integrals to the three-loop level. Obtained results can find their application in regularization-invariant symmetric point momentum-subtraction (RI/SMOM) scheme QCD calculations of renormalization group functions and various composite operator matrix elements. To calculate integrals, we solve differential equations for auxiliary integrals by transforming the system to the $\varepsilon$-form. Calculated integrals are expressed through the basis of functions with uniform transcendental weight. We provide expansion up to the transcendental weight six for the basis functions in terms of harmonic polylogarithms with six-root of unity argument.