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

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 the conservation of energy and momentum. Recently it has been shown 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, {T}_{μ}^{μ }=(β /2g){F}^{μ ν }{F}_{μ ν }+m(1+{γ}_m)\overline{ψ}ψ , 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.