Lattice-QCD validation of hadron mass and trace-anomaly decomposition sum rules

TL;DR

The paper addresses how QCD binds quarks and gluons to produce hadron masses by validating multiple mass-decomposition sum rules from first principles. It introduces a gradient-flow based, nonperturbative renormalization of the energy–momentum tensor, followed by continuum extrapolation and two-loop MSbar matching to ensure a common scheme at μ = 2 GeV. Applying this framework to η_c and J/ψ, it directly verifies HRT, Lorcé, MPR, and Ji decompositions, finds substantial gluon-energy contributions and non-negligible trace-anomaly effects, and reports the first lattice determination of the gravitational form factor C. The method is general and transferable to other hadrons, providing essential nonperturbative insight into mass generation that complements experimental programs exploring hadron structure. The results quantify how much of the mass arises from gluon energy and trace anomalies, offering a concrete, lattice-backed view of confinement dynamics at the hadronic scale.

Abstract

We present the first lattice-QCD validation of multiple sum rules associated with quark-gluon decomposition of hadron mass by computing all components 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 $\overline{\mathrm{MS}}$ 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 $\bar{C}$. We find sizable gluonic contributions to charmonia masses at the hadronic scale, $\sim 15\%$ in the Lorcé and Metz-Pasquini-Rodini decompositions. The trace-anomaly contribution in the Ji sum rule is $\sim 6\%$, while the gluonic component of the trace anomaly in the Hatta-Rajan-Tanaka sum rule is $\sim 35\%$. 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.

Figures (15)

• Figure 1: Continuum extrapolations of hadronic matrix elements of gradient-flow-renormalized EMT components.
• Figure 2: Flow-time dependence and zero-flow-time extrapolations of hadronic matrix elements of $\overline{\mathrm{MS}}$-renormalized EMT components at $\mu=2$ GeV using one-loop (blue) and two-loop (red) perturbative matching.
• Figure 3: Hatta-Rajan-Tanaka (HRT, \ref{['eq:decomp-HRT']}), Lorcé (Lor, \ref{['eq:decomp-Lor']}) , Metz-Pasquini-Rodini (MPR, \ref{['eq:decomp-MPR']}), and Ji (Ji, \ref{['eq:decomp-Ji']}) decompositions of $\eta_c$ (left panel) and $J/\psi$ (right panel) masses. All contributions are presented at 2 GeV $\overline{\mathrm{MS}}$ scale using 2-loop gradient-flow to $\overline{\mathrm{MS}}$ matching. The black error bars indicate combined systematic and statistical uncertainty of each individual components. The blue error bands on top indicate combined systematic and statistical uncertainty of the sum over all components in each cases.
• Figure 4: $E_0/E_0^{\rm PDG}$ParticleDataGroup:2024cfk of $\eta_c$ and $J/\psi$ determined from 2-state fit of the 2pt functions are shown for three gauge ensembles, as a function of $t_{\rm min}$. The fit range is [$t_{\rm min},32a$].
• Figure 5: Left and middle panels: ratios of quark three-point to two-point correlation functions for the $\eta_c$ meson of $a=0.04$ fm lattice at flow time $t_{\rm f}=5\epsilon_{\rm f}$, plotted as functions of $t - \tau/2$. The curved bands represent reconstructions from the two-state fits $\rm Fit_{str2}$ and $\rm Fit_{str4}$, while the horizontal bands indicate the extracted ground-state matrix elements. Right panels: Summed ratios as functions of the source-sink separation $t$, with curved bands representing results from the summation fit SUM. The upper and lower rows show results for temporal components $R^{44}_{q,\gamma_5}$ and spatial components $\sum_{k=1}^3R^{kk}_{q,\gamma_5}/3$ respectively.