Accessing the gluon momentum fraction of nucleons through the gradient flow

We calculate the gluon momentum fraction of the nucleon using lattice QCD, with a nonperturbative renormalization technique based on the gradient flow. The gluon momentum fraction is determined on a single Wilson-clover ensemble using <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:msub> <a:mi>N</a:mi> <a:mi>f</a:mi> </a:msub> <a:mo>=</a:mo> <a:mn>2</a:mn> <a:mo>+</a:mo> <a:mn>1</a:mn> </a:math> flavors with pion mass 358 MeV and lattice spacing 0.094 fm. We employ the variational method to reduce excited-state contamination and apply the distillation framework to ensure a large operator basis. To reduce systematic uncertainties, we apply Bayesian model averaging to all fit procedures. We apply matching coefficients to the flow-time dependent lattice results to recover the gluon momentum fraction in the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:mover accent="true"> <c:mi>MS</c:mi> <c:mo stretchy="true">¯</c:mo> </c:mover> </c:math> -scheme at 2 GeV. Our final result is <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mo stretchy="false">⟨</g:mo> <g:mi>x</g:mi> <g:msub> <g:mo stretchy="false">⟩</g:mo> <g:mi>g</g:mi> </g:msub> <g:mo stretchy="false">(</g:mo> <g:mi>μ</g:mi> <g:mo>=</g:mo> <g:mn>2</g:mn> <g:mtext> </g:mtext> <g:mtext> </g:mtext> <g:mi>GeV</g:mi> <g:mo stretchy="false">)</g:mo> <g:mo>=</g:mo> <g:mn>0.482</g:mn> <g:mo stretchy="false">(</g:mo> <g:mn>35</g:mn> <g:mo stretchy="false">)</g:mo> </g:math> , where we quote only statistical uncertainties.