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

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 $\sim 260$ 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 $\overline{\mathrm{MS}}$ scheme and evolved at a scale of 2 GeV, using three-loop expressions in perturbation theory. The final values are $⟨x^3{⟩}_{\pi }^{u^{+}}=0.024(18{)}_{\mathrm{stat}}(2{)}_{\mathrm{syst}}$, $⟨x^3{⟩}_K^{u^{+}}=0.035(6{)}_{\mathrm{stat}}(3{)}_{\mathrm{syst}}$, 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 to obtain the ratios $⟨x^3⟩/⟨x⟩$ and $⟨x^3⟩/⟨x^2⟩$, 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.

Comments: 24 pages, 12 figures