The pion and kaon \(\langle x^3 \rangle\) 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 \(\sim260\) MeV, and a kaon mass \(\sim530\) 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 \(\overline{\rm MS}\) scheme and evolved at a scale of 2 GeV, using three-loop expressions in perturbation theory. The final values are \(\langle x^3 \rangle_\pi^{u^+}=0.024(18)_{\rm stat}(2)_{\rm syst}\), \(\langle x^3 \rangle_K^{u^+}=0.035(6)_{\rm stat}(3)_{\rm syst}\), 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 \(\langle x^3 \rangle\) with the two lower moments to obtain the ratios \(\langle x^3 \rangle/\langle x \rangle\) 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 \(4\leq n\leq 6\). 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.