The pion and kaon $⟨x^3⟩$ from lattice QCD and PDF reconstruction from Mellin moments

We present a calculation of the pion and kaon Mellin moment $\langle x^3 \rangle$ 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 $\sim 260$ MeV, and a kaon mass $\sim 530$ 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 non-perturbatively. Our results are converted to the $\overline{\mathrm{M}\mathrm{S}}$ scheme and evolved at a scale of 2 GeV, using three-loop expressions in perturbation theory. The final values are $⟨x^3{⟩}_{\pi }^{u^{+}}=0.024(18{)}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}(2{)}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}}$, $⟨x^3{⟩}_K^{u^{+}}=0.035(6{)}_{\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}}(3{)}_{\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}}$, and $\langle x^3 \rangle_K^{s^+}=0.075(5){\rm stat}(1){\rm syst}$, where the systematic error is the uncertainty due to excited state contamination. We combine $⟨x^3⟩$ with the two lower moments to obtain the ratios $⟨x^3⟩\mathrm{/}⟨x⟩$ and $\langle x^3 \rangle/\langle x^2 \rangle$, as well as ratios between the pion and kaon moments. 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 $(1-x{)}^2$ for both the pion and kaon PDFs. We integrate the reconstructed PDFs to extract the higher moments with $4\le n\le 6$. 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.