A note on quark and gluon energy-momentum tensors

TL;DR

This paper analyzes the constraints on quark and gluon energy-momentum tensors in QCD imposed by Renormalisation-Group invariance, deriving equations that define these tensors and presenting four-loop renormalized results.

Contribution

It introduces a general framework for the renormalization of quark and gluon energy-momentum tensors, extending previous definitions and providing explicit four-loop results.

Findings

Derived Renormalisation-Group equations for energy-momentum tensors.

Showed previous definitions are special cases of the new framework.

Presented four-loop $ar{MS}$-renormalized tensors.

Abstract

We discuss the constraints on quark and gluon energy-momentum tensors in QCD that follow from the requirement of Renormalisation-Group invariance of the traces of these operators. Our study covers the most general form of the latter traces, while the energy-momentum tensors themselves are only subjected to very mild constraints. We derive Renormalisation-Group equations for the two finite independent functions of the strong coupling constant and renormalisation scale of minimal subtraction which completely define the energy-momentum tensors. We demonstrate that previously proposed definitions of the renormalized quark and gluon energy-momentum tensors are special cases of our results assuming no explicit dependence on the renormalisation scale. Finally, we present $\overline{\mathrm{MS}}$-renormalised quark and gluon energy-momentum tensors at four-loop order.