Three-loop corrections to the quark and gluon decomposition of the QCD trace anomaly and their applications

TL;DR

This paper calculates three-loop corrections to the quark and gluon contributions to the QCD trace anomaly, providing explicit formulas and applications to nucleon and pion mass structures and gravitational form factors.

Contribution

It introduces a systematic procedure to derive three-loop order anomalous trace contributions for quark and gluon parts in the QCD energy-momentum tensor.

Findings

Explicit three-loop formulas for trace anomaly decomposition

Application to nucleon and pion mass structures

Constraints on nucleon gravitational form factors

Abstract

In the QCD energy-momentum tensor $T^{\mu\nu}$, the terms that contribute to physical matrix elements are expressed as the sum of the gauge-invariant quark part and gluon part. Each part undergoes the renormalization due to the interactions among quarks and gluons, although the total tensor $T^{\mu\nu}$ is not renormalized thanks to conservation of energy and momentum. We show that, through the renormalization, each of the quark and gluon parts of $T^{\mu\nu}$ receives a definite amount of anomalous trace contribution, such that their sum reproduces the well-known QCD trace anomaly. We provide a procedure to derive such anomalous trace contribution for each quark/gluon part to all orders in perturbation theory, and obtain the corresponding explicit formulas up to three-loop order in the $\overline{\rm MS}$ scheme in the dimensional regularization. We apply our three-loop formulas of the quark/gluon decomposition of the trace anomaly to calculate the anomaly-induced mass structure of nucleons as well as pions. Another application of our three-loop formulas is a quantitative analysis for the constraints on the twist-four gravitational form factors of the nucleon, $\bar{C}_{q,g}$.