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

In the QCD energy-momentum tensor $T^{μν}$, 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 $T^{μν}$ is not renormalized thanks to conservation of energy and momentum. We show that, through the renormalization, each of the quark and gluon parts of $T^{μν}$ 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, $\bar{C}_{q,g}$.

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