Three loop master integrals for ${\mathcal{O}} (\alpha \alpha_s^2)$ corrections to quark form factor

TL;DR

This paper computes three-loop mixed electroweak and QCD corrections to the quark form factor by calculating 303 master integrals using differential equations, revealing complex algebraic structures in the integrals.

Contribution

It provides the first complete calculation of the master integrals for three-loop mixed electroweak-QCD corrections involving a single massive boson using advanced differential equation techniques.

Findings

Computed 303 master integrals for three-loop corrections.

Expressed results in terms of generalized polylogarithms with complex algebraic structures.

Applied concurrent transformations to handle multiple square roots in integrals.

Abstract

We consider the three-loop mixed strong-electroweak (${\mathcal{O}}(\alpha \alpha_s^2)$) corrections to the quark form factor. We compute the master integrals which are appearing in the Feynman diagrams containing a single massive boson in the loop. We use the state-of-the-art method of differential equations to compute all 303 of them, expressing the results in terms of generalized polylogarithms. We encounter multiple square roots that cannot be simultaneously rationalized using a single transformation. Applying concurrent transformations allows us to express the results through generalized polylogarithms with a simple alphabet, but with multiple interdependent arguments.