Higher order moments of the unpolarized gluon PDF for the proton from lattice QCD at the continuum limit

The x-dependent behavior of partons provides crucial insight for understanding hadron structure. In this talk, we present the continuum limit results for the ratio of gluon moments $\langle x^3 \rangle / \langle x \rangle$ extracted from renormalization group invariant ratios of nonlocal matrix elements using the OPE formalism. The data are produced on four $N_f = 2+1+1$ lattice ensembles of maximally twisted mass fermions with clover improvement with pion mass $m_\pi \approx 250 \rm{ MeV}$ and lattice spacings ranging from $a \approx 0.093-0.057 \rm{ fm}$. We examine the effects of evolution at next-to-leading order and the mixing with the quark singlet, and we compare our results with those from global fitting. Additionally, we present the updated results for the unpolarized gluon PDF for the proton at the continuum limit extracted from the same lattice data.