Pion and kaon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"><mml:mo stretchy="false">⟨</mml:mo><mml:msup><mml:mi>x</mml:mi><mml:mn>3</mml:mn></mml:msup><mml:mo stretchy="false">⟩</mml:mo></mml:math> 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 three-derivative local operator. use one ensemble gauge configurations with two degenerate light, strange charm quark ($N_f=2+1+1$) maximally twisted mass fermions clover improvement. The reproduces $\sim260$ MeV, $\sim530$ MeV. Excited-states contamination is evaluated four values source-sink time separation within range $1.12-1.67$ fm. an operator that free mixing, apply multiplicative renormalization function calculated non-perturbatively. Our results are converted to $\overline{\rm MS}$ scheme evolved at scale 2 GeV, three-loop expressions perturbation theory. final \rangle_\pi^{u^+}=0.024(18)_{\rm stat}(2)_{\rm syst}$, \rangle_K^{u^+}=0.035(6)_{\rm stat}(3)_{\rm \rangle_K^{s^+}=0.075(5)_{\rm stat}(1)_{\rm where systematic error uncertainty due excited state contamination. combine lower moments obtain ratios \rangle/\langle x x^2 \rangle$, as well between moments. In addition, we reconstruct $x$-dependence PDFs via 2- 3-parameter fits our results. find reconstruction feasible data favor large falls $(1-x)^2$ for both PDFs. integrate reconstructed extract higher $4\leq n\leq 6$. Finally, compare PDFs, their moments, address effect SU(3) flavor symmetry breaking.