LAUSR

We present a calculation of the pion quark momentum fraction, $⟨x⟩$, and its third Mellin moment $⟨{x}^{2}⟩$. We also obtain directly, for the first time, $⟨x⟩$ and $⟨{x}^{2}⟩$ for the kaon using local operators. We use an ensemble of two degenerate light, a strange, and a charm quark (${N}_{f}=2+1+1$) of maximally twisted mass fermions with clover improvement. The quark masses are chosen so that they reproduce a pion mass of about 260 MeV and a kaon mass of 530 MeV. The lattice spacing of the ensemble is 0.093 fm and the lattice has a spatial extent of 3 fm. We analyze several values of the source-sink time separation within the range of 1.12--2.23 fm to study and eliminate excited-state contributions. The necessary renormalization functions are calculated nonperturbatively in the ${\mathrm{RI}}^{\ensuremath{'}}$ scheme and are converted to the modified minimal subtraction scheme at a scale of 2 GeV. The final values for the momentum fraction are $⟨x{⟩}_{{u}^{+}}^{\ensuremath{\pi}}=\phantom{\rule{0ex}{0ex}}0.261(3{)}_{\mathrm{stat}}(6{)}_{\mathrm{syst}}$, $⟨x{⟩}_{{u}^{+}}^{K}=0.246(2{)}_{\mathrm{stat}}(2{)}_{\mathrm{syst}}$, and $⟨x{⟩}_{{s}^{+}}^{K}=0.317(2{)}_{\mathrm{stat}}(1{)}_{\mathrm{syst}}$. For the third Mellin moments, we find $⟨{x}^{2}{⟩}_{{u}^{+}}^{\ensuremath{\pi}}=0.082(21{)}_{\mathrm{stat}}(17{)}_{\mathrm{syst}}$, $⟨{x}^{2}{⟩}_{{u}^{+}}^{K}=0.093(5{)}_{\mathrm{stat}}(3{)}_{\mathrm{syst}}$, and $⟨{x}^{2}{⟩}_{{s}^{+}}^{K}=0.134(5{)}_{\mathrm{stat}}(2{)}_{\mathrm{syst}}$. The reported systematic uncertainties are due to excited-state contamination. We also give the ratio $⟨{x}^{2}⟩/⟨x⟩$ which is an indication of how quickly the parton distribution functions lose support at large $x$.