This paper reports a lattice quantum chromodynamics (QCD) calculation of a higher Mellin moment of the unpolarized gluon parton distribution in the proton. In plain terms, the authors extract how strongly gluons carry momentum in the proton beyond the average fraction. They focus on the ratio of the third Mellin moment, ⟨x^3⟩_g, to the gluon momentum fraction, ⟨x⟩_g, and present a determination at a conventional scale of 2 GeV with errors that include both statistical and leading theoretical uncertainties.

The researchers computed matrix elements of nonlocal gluon operators between proton states that were given a momentum boost. The numerical work used one lattice ensemble with Nf = 2+1+1 dynamical quark flavors, employing maximally twisted mass fermions with clover improvement and the Iwasaki-improved gauge action. The pion mass in this ensemble was about 260 MeV, heavier than the physical pion. From these boosted-proton matrix elements they built a reduced gluon Ioffe-time distribution and applied a short-distance operator product expansion (OPE) to relate that distribution to Mellin moments of the gluon parton distribution function (PDF).

At a high level the method relies on short spatial separations between the gluon fields in the nonlocal operator. When the separation is small, the nonlocal operator can be expanded into a series of local, so-called twist-two, operators. The coefficients in that expansion are calculable in perturbation theory and link the lattice-measured quantity to integrals of the gluon PDF, the Mellin moments. The authors use a double-ratio construction that cancels overall renormalization factors and reduces ultraviolet effects tied to the Wilson line. They include perturbative matching kernels and study the small mixing with the quark-singlet contribution, and they test the stability of the extracted moments by evolving them with the DGLAP equations and by varying the matching scale.