The Mellin moments $⟨x⟩$ and $⟨x^2⟩$ for the pion and kaon from lattice QCD

Abstract

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-states contributions. The necessary renormalization functions are calculated non-perturbatively in the RI${}^{\prime }$ scheme, and are converted to the $\overline{\mathrm{MS}}$ scheme at a scale of 2 GeV. The final values for the momentum fraction are $⟨x{⟩}_{u^{+}}^{\pi }=0.261(3{)}_{\mathrm{stat}}(6{)}_{\mathrm{syst}}$, $⟨x{⟩}_{u^{+}}^K=0.246(2{)}_{\mathrm{stat}}(2{)}_{\mathrm{syst}}$, and $⟨x{⟩}_{s^{+}}^K=0.317(2{)}_{\mathrm{stat}}(1{)}_{\mathrm{syst}}$. For the third Mellin moments we find $⟨x^2{⟩}_{u^{+}}^{\pi }=0.082(21{)}_{\mathrm{stat}}(17{)}_{\mathrm{syst}}$, $⟨x^2{⟩}_{u^{+}}^K=0.093(5{)}_{\mathrm{stat}}(3{)}_{\mathrm{syst}}$, 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 PDFs lose support at large $x$.

Comments: 21 pages, 10 figures