Proton mass decompositions in the NNLO QCD

Proton matrix elements of the QCD energy-momentum tensor (EMT) are expressed by the gravitational form factors. The forward values of the gravitational form factors allow for a decomposition of the proton mass into contributions from quarks and gluons, and further subdivisions into contributions from quark masses and from the QCD trace anomaly may be considered. We present the most recent evaluations of these mass decompositions, using a recent quantitative evaluation of the forward values of relevant gravitational form factors at the next-to-next-to-leading order (NNLO) QCD. We also calculate the renormalization scale dependence of each component within these decompositions. Furthermore, similar calculations are performed with another decomposition of the proton mass, organized strictly according to the separation into the traceless part and trace part for each of the gauge-invariant quark/gluon parts of the EMT, such that the former (twist-two) quark/gluon contributions of the EMT embody the effects of the partonic motions inside the proton, while the latter (twist-four) contributions are induced as parton correlations by non-perturbative QCD interactions. We demonstrate the advantages of this new decomposition. We also present the results for the pion, which exhibit quite different parton-correlation behaviors from the proton.