Nonperturbatively Renormalized Nucleon Gluon Momentum Fraction in the Continuum Limit of $N_f=2+1+1$ Lattice QCD

We present the nonperturbatively renormalized nucleon gluon momentum fraction using ensembles with $2+1+1$ flavors of highly improved staggered quarks (HISQ), generated by MILC Collaboration. The calculation is done using clover fermions for the valence action with three pion masses, $220$, $310$ and $690$ MeV and three lattice spacings, 0.09, 0.12, and 0.15 fm. The renormalization is done using RI/MOM nonperturbative renormalization and using cluster-decomposition error reduction (CDER) to enhance the signal-to-noise ratio of the renormalization constant. We find the CDER technique is particularly important to improve the signal at the finer lattice ensembles where the lattice volume is larger. We extrapolate the gluon momentum fraction to the continuum-physical limit and obtain $\langle x \rangle_g = 0.492(52)_\text{stat.+NPR}(49)_\text{mixing}$ in the $\bar{\text{MS}}$ scheme at 2 GeV, where first error includes the statistical error and uncertainties in nonperturbative renormalization, while the latter systematic accounts for ignoring quark mixing. Our gluon momentum fraction is consistent with other recent lattice-QCD results at physical pion mass.