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Description
We present a lattice QCD determination of the Mellin moments for the unpolarized gluon PDF in the proton across three ensembles of $N_f = 2 {+} 1 {+} 1$ maximally twisted mass fermions with Iwasaki-improved gluons all at a pion mass of approximately 260 MeV and lattice spacings of $a=0.091,\,0.079,\,0.057$ fm, which allow us to explore the continuum limit. We use the operator product expansion (OPE) for nonlocal gluon operators coupled to boosted proton states and apply the short-distance expansion matching formalism, with perturbatively calculated Wilson coefficients, to relate matrix elements to Mellin moments. We extract the ratio of $\langle x^3 \rangle _g$ to the gluon momentum fraction, $\langle x \rangle _g$ at each lattice spacing. Our analysis investigates the systematic effects caused by truncation in the OPE as well as minimum and maximum spatial separations. Additionally, we assess residual uncertainties associated with the perturbative expansion of the Wilson coefficients by varying the initial matching scale and performing DGLAP evolution to 2 GeV.