Lattice-QCD Validation of Hadron Mass and Trace-Anomaly Decomposition Sum Rules.

We present the first lattice-QCD validation of multiple sum rules associated with quark-gluon decomposition of hadron mass by computing all relevant tensor components of the quark and gluon energy-momentum tensor matrix elements from first principles. We achieve this through nonperturbative renormalization of the QCD energy-momentum tensor, including its trace, in a gradient-flow scheme, followed by continuum extrapolations, two-loop matching to the MS[over ¯] scheme, and zero-flow-time extrapolations. These ingredients enable a direct and simultaneous verification, in a common renormalization scheme and scale, of multiple energy-density-based and trace-based mass decomposition sum rules proposed in the literature. We demonstrate the framework for the η_{c} and J/ψ charmonia using three fine lattice spacings with a physical strange-quark and near-physical up- and down-quark masses. We present the first lattice-QCD results for the gravitational form factor C[over ¯]. We find sizable gluonic contributions to charmonia masses at the hadronic scale, ∼15% in the Lorcé and Metz-Pasquini-Rodini decompositions. The trace-anomaly contribution in the Ji sum rule is ∼6%, while the gluonic component of the trace anomaly in the Hatta-Rajan-Tanaka sum rule is ∼35%. The method is general and can be straightforwardly adopted for lattice-QCD calculations of mass and spin decompositions as well as gravitational form factors of other hadrons and nuclei.