Nucleon isovector charges and twist-2 matrix elements with $N_f=2+1$ dynamical Wilson quarks

We present results from a lattice QCD study of nucleon matrix elements at vanishing momentum transfer for local and twist-2 isovector operator insertions. Computations are performed on gauge ensembles with nonperturbatively improved $N_f=2+1$ Wilson fermions, covering four values of the lattice spacing and pion masses down to $M_{\pi }\approx 200\mathrm{MeV}$. Several source-sink separations (typically $\sim 1.0$ to $\sim 1.5\mathrm{fm}$) allow us to assess excited-state contamination. Results on individual ensembles are obtained from simultaneous two-state fits across all observables and all available source-sink separations with the energy gap as a common fit parameter. Renormalization has been performed nonperturbatively using the Rome-Southampton method for all but the finest lattice spacing for which an extrapolation has been used. Physical results are quoted in the $\overline{\mathrm{MS}}$ scheme at a scale of $\mu =2\mathrm{GeV}$ and are obtained from a combined chiral, continuum, and finite-size extrapolation. For the nucleon isovector axial, scalar, and tensor charges we find physical values of $g_A^{u-d}=1.242(25{)}_{\mathrm{stat}}(\genfrac{}{}{0ex}{}{+00}{-31}{)}_{\mathrm{sys}}$, $g_S^{u-d}=1.13(11{)}_{\mathrm{stat}}(\genfrac{}{}{0ex}{}{+07}{-06}{)}_{\mathrm{sys}}$ and $g_T^{u-d}=0.965(38{)}_{\mathrm{stat}}(\genfrac{}{}{0ex}{}{+13}{-41}{)}_{\mathrm{sys}}$, respectively, where individual systematic errors in each direction from the chiral, continuum, and finite-size extrapolation have been added in quadrature. Our final results for the isovector average quark momentum fraction and the isovector helicity and transversity moments are given by $⟨x{⟩}_{u-d}=0.180(25{)}_{\mathrm{stat}}(\genfrac{}{}{0ex}{}{+14}{-06}{)}_{\mathrm{sys}}$, $⟨x{⟩}_{\mathrm{\Delta }u-\mathrm{\Delta }d}=0.221(25{)}_{\mathrm{stat}}(\genfrac{}{}{0ex}{}{+10}{-00}{)}_{\mathrm{sys}}$, and $⟨x{⟩}_{\delta u-\delta d}=0.212(32{)}_{\mathrm{stat}}(\genfrac{}{}{0ex}{}{+20}{-10}{)}_{\mathrm{sys}}$, respectively.

Figure 1

Left panel: Nucleon two-point function. Right panel: Quark-connected nucleon (isovector) three-point function.

Figure 2

Behavior of the fitted gap $\mathrm{\Delta }$ in Eq. (27) as a function of the variable $M_{\pi }{t}_{\mathrm{fit}}$ representing the lower bound on the fit range. Left panel: Ensemble C101; right panel: ensemble N203.

Figure 3

Overview of results for $g_A^{u-d}$, $g_T^{u-d}$ and $⟨x{⟩}_{u-d}$ from the summation method and simultaneous fits on ensemble N203. Left column: Linear fits with error bands for the summation method as given in Eq. (30). Right column: Renormalized lattice data for all values of $t_{\mathrm{sep}}/a$ together with results from the summation method and simultaneous fits. Note that the simultaneous fits use data for all six observables, while in the case of the summation method separate fits were performed for each observable.

Figure 4

Chiral behavior (upper row) and continuum behavior (lower row) for $g_A^{u-d}$. Left column: Results from CCF fit model ABD, i.e., not including finite-size corrections. Right column: Results from model ABDE including finite-size corrections. Lattice data in each panel have been corrected using parameters from the corresponding fits for all extrapolations apart from the one given by the blue band.

Figure 5

Finite-size extrapolation for $g_A^{u-d}$. Lattice data have been corrected to the physical value of the pion mass and the continuum limit using parameters from the fit. Therefore, the corrected data points are highly correlated.

Figure 6

Overview for results for $g_A^{u-d}$ from different variations of the CCF fit model, as detailed in Table 8. Red symbols denote results obtained using $Z_A$ from the Rome-Southampton method. Blue and violet symbols represent data obtained using $Z_A$ from the Schrödinger functional with mass-dependent counterterms included and excluded, respectively. Filled symbols are used for results obtained by fitting data at all four lattice spacings, while open symbols are used for results when excluding data at $\beta =3.7$. Circles and boxes refer to fitting a lattice artifact $\mathcal{O}(a^2)$ and $\mathcal{O}(a)$, respectively.

Figure 7

Results from chiral and continuum model $ABDE$ for local charges. Data on individual ensembles have been obtained from the multistate fit model in Eq. (27). In the left column we show the chiral extrapolation together with the original data from Table 6, while in the right column the lattice data have been corrected for the continuum limit and finite-size extrapolation using the corresponding fit parameters. Therefore, the corrected data points in the right column are highly correlated within the same plot.

Figure 8

Same as Fig. 7 but for twist-2 operator insertions.

Figure 9

Comparison of our results for the twist-2 matrix elements with other recent determinations (ETMC 17 [59], ETMC 15 [13], RQCD 14 [60], LPHC 12 [17], and RBC/UKQCD 10 [58]).

Figure 10

The subtraction functions $D_X(\mu ,a)$ for the operators ${\mathcal{O}}^X$ considered in this study at a lattice spacing of $a=0.086\mathrm{fm}$.

Figure 11

Comparison of the unsubtracted and subtracted values of the RGI tensor renormalization constant $Z_T^{\mathrm{RGI}}$, using the bare coupling $g_0$, the boosted coupling $g_{\mathrm{b}}$, or the BLM coupling $g_{\mathrm{BLM}}$ for the perturbative subtraction. It can be seen that the BLM coupling is most efficient in removing the lattice artifacts, which are otherwise very large.

Figure 12

The final fit used to extract $Z_T^{\mathrm{RGI}}$ for our three value of $\beta$. The solid lines denote the fit ranges, whereas the dashed lines indicate how the fit form of Eq. (a24) extrapolates beyond the fit range, while the different colors correspond to the different lattice spacings. Final fit results for $Z_T^{\mathrm{RGI}}(\beta )$ are shown by the horizontal bands.

Figure 13

Extrapolation of the renormalization constants used in the final analysis to $\beta =3.7$. Shown are the values of $Z_X$ for the local operators $X\in \{A,S,T\}$ (left column) and the one-link operators in irreps $X\in \{v2b,r2a,h1a\}$ (right column) as measured at $\beta \in \{3.4,3.46,3.55\}$, the linear fit in $\beta$ with its error band, and the extrapolated value at $\beta =3.7$ with its tenfold inflated final error.