The Mellin moments \(\langle x \rangle\) and \(\langle x^2 \rangle\) for the pion and kaon from lattice QCD

We present a calculation of the pion quark momentum fraction, \(\langle x \rangle\), and its third Mellin moment \(\langle x^2 \rangle\). We also obtain directly, for the first time, \(\langle x \rangle\) and \(\langle x^2 \rangle\) 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-states contributions. The necessary renormalization functions are calculated non-perturbatively in the RI\('\) scheme, and are converted to the \(\overline{\rm MS}\) scheme at a scale of 2 GeV. The final values for the momentum fraction are \(\langle x \rangle^\pi_{u^+}=0.261(3)_{\rm stat}(6)_{\rm syst}\), \(\langle x \rangle^K_{u^+}=0.246(2)_{\rm stat}(2)_{\rm syst}\), and \(\langle x \rangle^K_{s^+}=0.317(2)_{\rm stat}(1)_{\rm syst}\). For the third Mellin moments we find \(\langle x^2 \rangle^\pi_{u^+}=0.082(21)_{\rm stat}(17)_{\rm syst}\), \(\langle x^2 \rangle^K_{u^+}=0.093(5)_{\rm stat}(3)_{\rm syst}\), and \(\langle x^2 \rangle^K_{s^+}=0.134(5)_{\rm stat}(2)_{\rm syst}\). The reported systematic uncertainties are due to excited-state contamination. We also give the ratio \(\langle x^2 \rangle/\langle x \rangle\) which is an indication of how quickly the PDFs lose support at large \(x\).