Moments of nucleon unpolarized, polarized, and transversity parton distribution functions from lattice QCD at the physical point

The second Mellin moments $⟨x⟩$ of the nucleon’s unpolarized, polarized, and transversity parton distribution functions are computed. Two lattice QCD ensembles at the physical pion mass are used: these were generated using a tree-level Symanzik-improved gauge action and $2+1$ flavor tree-level improved Wilson Clover fermions coupling via 2-level HEX-smearing. The moments are extracted from forward matrix elements of local leading twist operators. We determine renomalization factors in RI-(S)MOM and match to $\overline{\mathrm{MS}}$ at scale 2 GeV. Our findings show that operators that exhibit vanishing kinematics at zero momentum can have significantly reduced excited-state contamination. The resulting polarized moment is used to quantify the longitudinal contribution to the quark spin-orbit correlation. All our results agree within two sigma with previous lattice results.

Figure 1

Graphical representation of ${\mathrm{C}}_{3\mathrm{pt}}^{{\mathcal{O}}^X}(T,\tau )$: a source nucleon inserted at time $t=0$ and a sink nucleon removed at time $t=T$. A local leading twist operator (1) is inserted on a given time slice $\tau$. The nucleons on the lattice are represented by interpolating operators $\chi$ (4) while ${\mathcal{O}}^X$ is determined by finite differences connected with gauge links.

Figure 2

Ratios, cf. Eq. (7), for the coarse ensemble. Various source-sink separations $T$ are represented by different colors, while the two momenta are distinguished using filled circles and unfilled squares. Each subplot corresponds to a different operator from the different channels organized by column. For the unpolarized (V) case we display operators 2. and 3. for the polarized (A) case we display operators 1. and 2. and for the transversity (T) case we display operators 3. and 1. for the upper and lower panel, respectively.

Figure 3

Ratio sums $\overline{S}(T,{\tau }_{\mathrm{skip}})$ on the coarse and fine lattice, employing the same operators as in 2. Each $\overline{S}(T,{\tau }_{\mathrm{skip}})$ is plotted at fixed ${\tau }_{\mathrm{skip}}=1$. As in Fig. 2, different momenta are displayed with hollow and filled markers.

Figure 4

Extraction result for the matrix elements, cf. Eq. (13), plotted on top of the original ratio data. The same operators and layout as in Fig. 2 are used. Coarse and fine ensemble results are displayed in the first two and second two rows, respectively. Dot-dashed and dotted horizontal lines represent the average slope of the summed ratios divided by the kinematic factor. Solid and dashed lines represent the simultaneous and fully correlated central value fit to the ratios using the 2-state form (10). Surrounding colored areas represent bootstrap uncertainties.

Figure 5

Renormalized moments calculated from the summed ratio (1-state) (gray) (11), and 2-state fit (red) (10). The final average is displayed as a blue solid line while its statistical uncertainty is indicated via the blue dot-dashed line. The blue band represents the statistical and systematic uncertainty, cf. Eq. (15), added in quadrature. The light gray points are not included in the average as per the $T_{\mathrm{plat}}^j$ constraint, compare (14), to reduce excited-state effects. The ordinate limit is cut at $4\sigma$ around the final average to increase resolution of the relevant points.

Figure 6

Continuum extrapolation using a Bayesian fit with the model described in Eq. (19). The limited amount of data makes this extrapolation strongly dependent on the chosen priors for the coefficients $m_i$. Coming from a power counting these are expected to be of $\mathcal{O}(1)$, reflected in Gaussian priors of mean zero and variance 2. Resulting estimates are listed in Table 2.

Figure 7

Analysis results of the ratios (points), slope of summed ratios (horizontal lines), and 2-state fit results (curves) for the operators 1., 2., and 3. corresponding to the unpolarized (vector) PDF.

Figure 8

Analysis results of the ratios (points), slope of summed ratios (horizontal lines), and 2-state fit results (curves) for the operators 1. and 2. corresponding to the polarized (axial) PDF.

Figure 9

Analysis results of the ratios (points), slope of summed ratios (horizontal lines), and 2-state fit results (curves) for the operators 1. and 2. corresponding to the transversity (tensor) PDF.

Figure 10

Analysis results of the ratios (points), slope of summed ratios (horizontal lines), and 2-state fit results (curves) for the operators 3. and 4. corresponding to the transversity (tensor) PDF.

Figure 11

Summary plot of the renormalized moment of PDF of the coarse ensemble resolved for each operator ${\mathcal{O}}^X$, represented by the corresponding ID and irrep, and momentum $p_x$. The purple points correspond to results obtained by the sum-ratio method evaluated at a source-sink separation $T^{\prime }$. The red points are obtained using the two-state fit. The blue solid line corresponds to the overall average as described by Eq. (14) with the dashed line indicating statistical uncertainty and the blue area indicating statistical and systematic uncertainty added in quadrature.

Figure 12

Summary plot of the renormalized moment of PDF of the fine ensemble, similar to Fig. 11.

Figure 13

Ratios of renormalization factors $Z_{DV}^{{\tau }_1^{(3)}}/Z_V$, $Z_{DA}^{{\tau }_4^{(3)}}/Z_V$, and $Z_{DT}^{{\tau }_1^{(8)}}/Z_V$ on the coarse (left) and fine (right) ensembles, determined using the ${\mathrm{RI}}^{\prime }$-MOM (green circles) and RI-SMOM (orange squares) intermediate schemes together with momenta along the four-dimensional diagonal (filled symbols) and two-dimensional diagonal (open symbols) and then matched to $\overline{\mathrm{MS}}$ at scale 2 GeV. For most points, the statistical uncertainty is smaller than the plotted symbol. The solid curves are fits to the 4D data in the ${\mu }^2$ range from 4 to $20{\mathrm{GeV}}^2$, the dashed curves in the range from 10 to $30{\mathrm{GeV}}^2$, and the dotted curves are fits to the 2D data in the range from 4 to $15{\mathrm{GeV}}^2$. The fit curves without the pole term are also plotted in the range $0<{\mu }^2<6{\mathrm{GeV}}^2$. To reduce clutter, uncertainties on the fits are not shown. The symbols filled with black near ${\mu }^2=0$ provide the final estimate for each intermediate scheme; their outer (without end cap) and inner (with end cap) error bars show the total and statistical uncertainties. The filled dark gray diamonds are the final estimates that combine both schemes.

Figure 14

Scale and scheme-invariant ratios of renormalization factors $Z_{DV}^{{\tau }_3^{(6)}}/Z_{DV}^{{\tau }_1^{(3)}}$, $Z_{DA}^{{\tau }_4^{(6)}}/Z_{DA}^{{\tau }_4^{(3)}}$, and $Z_{DT}^{{\tau }_2^{(8)}}/Z_{DT}^{{\tau }_1^{(8)}}$, determined using the ${\mathrm{RI}}^{\prime }$-MOM (green circles) and RI-SMOM (orange squares) intermediate schemes together with momenta along the four-dimensional diagonal (filled symbols) and two-dimensional diagonal (open symbols) and then matched to $\overline{\mathrm{MS}}$ at scale 2 GeV. For most points, the statistical uncertainty is smaller than the plotted symbol. The solid curves are fits to the 4D data in the ${\mu }^2$ range from 1 to $8{\mathrm{GeV}}^2$ and the dotted curves are fits to the 2D data in the range from 1 to $5{\mathrm{GeV}}^2$. To reduce clutter, uncertainties on the fits are not shown. The symbols filled with black near ${\mu }^2=0$ provide the final estimate for each intermediate scheme; their outer (without end cap) and inner (with end cap) error bars show the total and statistical uncertainties. The filled dark gray diamonds are the final estimates that combine both schemes.