One-loop amplitudes for $t\bar{t}j$ and $t\bar{t}\gamma$ productions at the LHC through $\mathcal{O}(\epsilon^2)$

TL;DR

This paper provides detailed analytic one-loop QCD helicity amplitudes for top-quark pair production with a photon or jet at the LHC, expanded through  in , essential for NNLO QCD calculations.

Contribution

It introduces explicit expressions for one-loop amplitudes involving top-quark pairs and photons or jets, expanded to , facilitating higher-order precision in collider predictions.

Findings

Amplitudes expressed as linear combinations of master integrals with rational coefficients.

Derived and solved differential equations for pentagon functions numerically.

Amplitudes are evaluated through  in the dimensional regularisation parameter.

Abstract

We present analytic expressions for the one-loop QCD helicity amplitudes contributing to top-quark pair production in association with a photon or a jet at the Large Hadron Collider (LHC), evaluated through $\mathcal{O}(\epsilon^2)$ in the dimensional regularisation parameter, $\epsilon$. These amplitudes are required to construct the two-loop hard functions that enter the NNLO QCD computation. The helicity amplitudes are expressed as linear combinations of algebraically independent components of the $\epsilon$-expanded master integrals--known as pentagon function--with the corresponding rational coefficients written in terms of momentum-twistor variables. We derive differential equations for the pentagon functions and solve them numerically using the generalised power series expansion method.