Master Integrals for Fermionic Contributions to Massless Three-Loop Form Factors

TL;DR

This paper advances the calculation of master integrals for massless three-loop form factors, focusing on fermionic contributions, and provides analytical results crucial for precise quantum chromodynamics predictions.

Contribution

It presents new analytical results for specific fermionic master integrals at three loops, including an all-orders epsilon expression and Mellin-Barnes representations.

Findings

Exact epsilon-expansion of one integral with Gamma and hypergeometric functions

Analytical coefficients for Laurent expansion up to transcendentality six

Results enable precise computation of three-loop quark and gluon form factors

Abstract

In this letter we continue the calculation of master integrals for massless three-loop form factors by giving analytical results for those integrals which are relevant for the fermionic contributions proportional to N_F^2, N_F*N, and N_F/N. Working in dimensional regularisation, we express one of the integrals in a closed form which is exact to all orders in epsilon, containing Gamma-functions and hypergeometric functions of unit argument. In all other cases we derive multiple Mellin-Barnes representations from which the coefficients of the Laurent expansion in epsilon are extracted in an analytical form. To obtain the finite part of the three-loop quark and gluon form factors, all coefficients through transcendentality six in the Riemann zeta-function have to be included.