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

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 ${\mathrm{RI}}^{\prime }$ scheme and are converted to the modified minimal subtraction scheme at a scale of 2 GeV. The final values for the momentum fraction are $⟨x{⟩}_{u^{+}}^{\pi }=0.261(3{)}_{\mathrm{stat}}(6{)}_{\mathrm{syst}}$, $⟨x{⟩}_{u^{+}}^K=0.246(2{)}_{\mathrm{stat}}(2{)}_{\mathrm{syst}}$, 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 parton distribution functions lose support at large $x$.

Figure 1

Connected diagram for the three-point function entering the calculation of $⟨x⟩$ and $⟨x^2⟩$. The wavy line corresponds to the operator insertion.

Figure 2

Chirally extrapolated results for $Z_{\text{vD}}$ (top) and $Z_{\text{vDD}}$ (bottom). Blue triangles correspond to ${\mathrm{RI}}^{\prime }$ scheme, black circles to $\overline{\mathrm{MS}}$ scheme, and magenta diamonds to the subtracted results in the $\overline{\mathrm{MS}}$ scheme. The data are plotted as a function of the initial ${\mathrm{RI}}^{\prime }$ scale $(a{\mu }_0{)}^2$. The dashed line corresponds to the fit of Eq. (19), and the filled magenta diamonds represent our final values for ${\mathcal{Z}}_{\text{vD}}$ and ${\mathcal{Z}}_{\text{vDD}}$.

Figure 3

Pion (top) and kaon (bottom) mass in the rest frame as a function of the lowest value of $t/a$, $t_{\mathrm{low}}/a$ entering the fit. Results using the plateau method are shown with red squares and results from the two-state fit with green circles. The selected values extracted using the plateau and two-state fits are shown with the purple square and blue circle, respectively.

Figure 4

Pion (top) and kaon (bottom) mass in the boosted frame as a function of the lowest value of $t/a$ entering the fit. The notation is the same as in Fig. 3.

Figure 5

Pion (left) and kaon (right) $m_{\mathrm{eff}}$ in the rest frame. The fitted value from the plateau $m_{\mathrm{plat}}$ is shown with a red band and from the two-state fit applied on $m_{\mathrm{eff}}$ with a green band.

Figure 6

Results for the ratio leading to $⟨x⟩$. We show the up part of the pion, the up and strange parts of the kaon, in the top, center, and bottom panels, respectively. In the left column, the points are the ratios in Eq. (32) multiplied by the renormalization and all kinematic factors. We plot values for $t_s/a=12$ up to $t_s/a=24$. The blue, red, green, magenta, cyan, and orange constant bands are the results obtained from Eq. (34). The purple band is the two-state fit value obtained from Eq. (37). In the right column, we plot the plateau values, together with two-state fit results. The gray band is the function obtained from a two-state fit using $t_s\in [12a-24a]$ and taking $t=t_s/2$.

Figure 7

Comparison of $⟨x⟩$ in the boosted (filled symbols) and rest (open symbols) frame. From top to bottom, we show $⟨x⟩$ for the pion and kaon up and strange contribution. Results at $t_s/a=14$, 16, 18 are shown in the left, center, and right columns, respectively. For this comparison, we use 16 source positions for each momentum frame so that the statistics are consistent.

Figure 8

Ratio for $⟨x{⟩}_u^{\pi }$ (top), $⟨x{⟩}_u^K$ (center), and $⟨x{⟩}_s^K$ (bottom) for $t_s/a=14$, 16, 18, shown with blue circles, red squares, and green triangles, respectively. The corresponding plateau values are shown with bands of the same color. The purple band corresponds to the value extracted using the two-state fit.

Figure 9

Ratio for $⟨x^2{⟩}_u^{\pi }$ (top), $⟨x^2{⟩}_u^K$ (center), and $⟨x^2{⟩}_s^K$ (bottom) for $t_s/a=14$, 16, 18, shown with blue circles, red squares, and green triangles, respectively. The corresponding plateau values are shown with bands of the same color. The purple band corresponds to the value extracted using the two-state fit.