Nucleon and pion structure with lattice QCD simulations at physical value of the pion mass

Abstract

We present results on the nucleon scalar, axial and tensor charges as well as on the momentum fraction, and the helicity and transversity moments. The pion momentum fraction is also presented. The computation of these key observables is carried out using lattice QCD simulations at a physical value of the pion mass. The evaluation is based on gauge configurations generated with two degenerate sea quarks of twisted mass fermions with a clover term. We investigate excited states contributions with the nucleon quantum numbers by analyzing three sink-source time separations. We find that, for the scalar charge, excited states contribute significantly and to a less degree to the nucleon momentum fraction and helicity moment. Our analysis yields a value for the nucleon axial charge agrees with the experimental value and we predict a value of 1.027(62) in the $\overline{\text{MS}}$ scheme at 2 GeV for the isovector nucleon tensor charge directly at the physical point. The pion momentum fraction is found to be $\langle x\rangle_{u-d}^{π^\pm}=0.214(15)(^{+12}_{-9})$ in the $\overline{\rm MS}$ at 2 GeV.

Figures (19)

• Figure 1: Connected (upper) and disconnected (lower) contributions to nucleon three-point functions.
• Figure 2: The ratio of the nucleon mass to the pion mass as a function of the pion mass squared. For determining the pion mass squared the scale is set using the nucleon mass at the physical point as described in the text. The fit only used the $N_f=2+1+1$ ensembles without a clover term (filled circles, diamonds and squares). The plot also shows the $N_f=2$ TMF results (open circles, diamonds and squares) and the $N_f=2$ ensemble with a clover term at the physical point (filled triangle). For the latter action ($N_f=2$ with a clover term) we restrict our analysis of nucleon observables to the ensemble simulated at a physical value of the pion mass only.
• Figure 3: Results for the axial (upper) and tensor(lower) renormalization functions as a function of the momentum square in lattice units. The open (black) circles are the unsubtracted results while the filled triangles (magenta) show the data after ${\cal O}(g^2\,a^\infty)$-terms are subtracted. The filled (red) circle at $(ap)^2=0$ is the value extracted by fitting to the plateau region [2-7] the subtracted data.
• Figure 4: Results for the isovector and isoscalar nucleon scalar charge: Upper two panels is the ratio from which $g_S$ is extracted as a function of $t_{\rm ins}-t_s/2$ for the isoscalar (upper) and the isovector (lower). The blue bands spanning from $(t_{\rm ins}-t_s/2)/a$=-4 to 4 are fits to the ratio for $t_s/a=14$. The dashed lines show the result of the two-state fit method. The dashed (solid) line spanning the entire x-range show the value obtained via the two-state (summation) method, with the band indicating the corresponding statistical error. In the third panel, the summed ratio is shown for the isovector (filled symbols) and isoscalar (open symbols) case. The line shows the result of a linear fit, while the bands show the statistical error based on the jackknife error of the fitted parameters. In the bottom panel, we show the result for $g_S$ when using the plateau method with $t_s/a=10, 12$ and $14$ (squares, circles, and rhombuses respectively), as well as when using the summation method denoted by "sm" (asterisks) and the two-state fit "2-st." (triangles).
• Figure 5: Results for the axial charge. The notation is the same as that in Fig. \ref{['fig:gS']}.