Refined methods for measuring large-distance correlations

Abstract

We present a calculation of the pion and kaon Mellin moment $⟨{x}^{3}⟩$ extracted directly in lattice QCD using a three-derivative local operator. We use one ensemble of gauge configurations with two degenerate light, a strange and a charm quark (${N}_{f}=2+1+1$) of maximally twisted mass fermions with clover improvement. The ensemble reproduces a pion mass $\ensuremath{\sim}260\text{ }\text{ }\mathrm{MeV}$, and a kaon mass $\ensuremath{\sim}530\text{ }\text{ }\mathrm{MeV}$. Excited-states contamination is evaluated using four values of the source-sink time separation within the range of 1.12--1.67 fm. We use an operator that is free of mixing, and apply a multiplicative renormalization function calculated nonperturbatively. Our results are converted to the $\overline{\mathrm{MS}}$ scheme and evolved at a scale of 2 GeV, using three-loop expressions in perturbation theory. The final values are $⟨{x}^{3}{⟩}_{\ensuremath{\pi}}^{{u}^{+}}=0.024(18{)}_{\mathrm{stat}}(2{)}_{\mathrm{syst}}$, $⟨{x}^{3}{⟩}_{K}^{{u}^{+}}=0.035(6{)}_{\mathrm{stat}}(3{)}_{\mathrm{syst}}$, and $⟨{x}^{3}{⟩}_{K}^{{s}^{+}}=0.075(5{)}_{\mathrm{stat}}(1{)}_{\mathrm{syst}}$, where the systematic error is the uncertainty due to excited state contamination. We combine $⟨{x}^{3}⟩$ with the two lower moments, namely $⟨x⟩$ and $⟨{x}^{2}⟩$, to obtain the ratios $⟨{x}^{3}⟩/⟨x⟩$ and $⟨{x}^{3}⟩/⟨{x}^{2}⟩$, as well as $⟨{x}^{3}{⟩}_{K}^{{u}^{+}}/⟨{x}^{3}{⟩}_{\ensuremath{\pi}}^{{u}^{+}}$ and $⟨{x}^{3}{⟩}_{K}^{{u}^{+}}/⟨{x}^{3}{⟩}_{\ensuremath{\pi}}^{{u}^{+}}$. In addition, we reconstruct the $x$-dependence of the pion and kaon PDFs via 2- and 3-parameter fits to our results. We find that the reconstruction is feasible and that our lattice data favor a large $x$-dependence that falls as $(1\ensuremath{-}x{)}^{2}$ for both the pion and kaon PDFs. We integrate the reconstructed PDFs to extract the higher moments with $4\ensuremath{\le}n\ensuremath{\le}6$. Finally, we compare the pion and kaon PDFs, as well as the ratios of their moments, to address the effect of SU(3) flavor symmetry breaking.