Three-loop formula for quark and gluon contributions to the QCD trace anomaly

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 the conservation of energy and momentum. Recently it has been shown 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, $T_{\mu }^{\mu }=(\beta /2g)F^{\mu \nu }{F}_{\mu \nu }+m(1+{\gamma }_m)\bar{\psi }\psi$, and the corresponding formulas have been derived up to two-loop order. We extend this result to the three-loop order, working out all the relevant three-loop renormalization structure for the quark and gluon energy-momentum tensors in the (modified) minimal subtraction scheme in the dimensional regularization. We apply our three-loop formula of the quark/gluon decomposition of the trace anomaly to calculate the anomaly-induced mass structure of nucleons as well as pions.

arXiv comment26 pages, text improved and references added

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