This paper reports a first-principles calculation of how the proton’s momentum and angular momentum are split among quarks and gluons. The authors use lattice QCD, a numerical method that puts quantum chromodynamics (QCD) on a space-time grid, to compute matrix elements of the energy–momentum tensor. From those matrix elements they extract gravitational form factors whose values at zero momentum transfer give the fraction of momentum carried by each parton and, via Ji’s sum rule, the total angular momentum carried by quarks and gluons.

The calculation uses four gauge ensembles with Nf = 2+1+1 dynamical quarks (up, down, strange, charm) tuned close to their physical masses. The ensembles have four lattice spacings: a = 0.080, 0.068, 0.057 and 0.049 fm. Having several spacings at the physical pion mass lets the authors take the continuum limit, meaning they extrapolate to zero lattice spacing to remove that source of systematic error. The ensembles also have similar physical volumes and pion masses around 136–141 MeV (with mπL around 3.6–3.9), which avoids the need for a separate chiral extrapolation.

The work includes both quark “connected” and “disconnected” contributions and the separate gluon contribution to the energy–momentum tensor. Practically this means computing two- and three-point correlation functions on the lattice, using standard techniques such as Gaussian smearing for the nucleon fields and sequential inversions through the sink for connected three-point functions. For the gluon operator the authors apply stout smearing (with parameter ρ = 0.129) and test several levels of stout-smearing steps to study how the lattice definition affects the result; such differences are expected to vanish in the continuum limit.

A key technical point is renormalization: lattice operators must be converted to the same continuum scheme used in QCD. The study determines all renormalization functions non-perturbatively, including the mixing between the quark singlet operator and the gluon operator. This operator mixing is important because it couples the quark and gluon contributions under renormalization and must be handled carefully to obtain reliable momentum and angular-momentum fractions. The authors also combine their results for total quark angular momentum with independent determinations of the intrinsic quark spin from the same ensembles to obtain the orbital angular momentum of each quark flavor.