Improved analysis of isovector nucleon matrix elements with $N_f=2+1$ flavors of $\mathcal{O}(a)$ improved Wilson fermions

We present an update of our determination of the isovector charges $g_A^{u-d}$, $g_S^{u-d}$ and $g_T^{u-d}$, and the isovector twist-2 forward matrix elements $⟨x{⟩}_{u-d}$, $⟨x{⟩}_{\mathrm{\Delta }u-\mathrm{\Delta }d}$ and $⟨x{⟩}_{\delta u-\delta d}$ on the $N_{\mathrm{f}}=2+1$ gauge ensembles generated by the Coordinated Lattice Simulations (CLS) effort. We have significantly extended our coverage of the parameter space by adding ensembles at the physical pion mass and fine lattice spacing, at nearly physical pion masses and very fine lattice spacings, and at very large physical lattice volumes, enabling a well-controlled extrapolation to the physical point. Another major improvement is achieved owing to the extended range of source-sink separations, which allows us to perform two-state fits to summed correlator ratios, leading to a much higher level of control over excited-state effects. Systematic uncertainties from the chiral, continuum and infinite-volume extrapolations are incorporated via model averages based on the Akaike information criterion. Our final results at the physical point are $g_A^{u-d}=1.254(19{)}_{\mathrm{stat}}(15{)}_{\mathrm{sys}}[24{]}_{\text{total}}$, $g_S^{u-d}=1.203(77{)}_{\mathrm{stat}}(81{)}_{\mathrm{sys}}[112{]}_{\text{total}}$, $g_T^{u-d}=0.993(15{)}_{\mathrm{stat}}(05{)}_{\mathrm{sys}}[16{]}_{\text{total}}$, $⟨x{⟩}_{u-d}=0.153(15{)}_{\mathrm{stat}}(10{)}_{\mathrm{sys}}[17{]}_{\text{total}}$, $⟨x{⟩}_{\mathrm{\Delta }u-\mathrm{\Delta }d}=0.207(15{)}_{\mathrm{stat}}(06{)}_{\mathrm{sys}}[16{]}_{\text{total}}$, and $⟨x{⟩}_{\delta u-\delta d}=0.195(17{)}_{\mathrm{stat}}(15{)}_{\mathrm{sys}}[23{]}_{\text{total}}$. While our results for the isovector charges are in excellent agreement with the FLAG 21 averages, we note that our error for the tensor charge $g_T^{u-d}$ is considerably smaller.

Figure 1

Example data for effective form factors of the six observables on the two most chiral ensembles including all available values of $t_{\mathrm{sep}}$. Data for local (twist-2) operator insertions are shown for E250 (E300), respectively. Results with statistical errors for the respective ground state NMEs are indicated by the solid blue line and band. They are obtained from the two-state truncated summation method fit ansatz in Eq. (28) for a choice of $t_{\mathrm{sep}}^{\min }\approx 0.4\mathrm{fm}$. Data with open symbols do not contribute to the sums that enter the fit due to the choice of $t_{\mathrm{sep}}^{\min }$ or the constraint $t_{\mathrm{ex}}/a=1$. The fits to the summed plateau data are carried out simultaneously on any given ensemble with a common parameter for the energy gap $\mathrm{\Delta }$ within each of the two sets of local and twist-2 NMEs.

Figure 2

Examples of simultaneous fits of the two-state truncated summation method fit model in Eq. (28). Top row: Simultaneous fits of the isovector nucleon charges on the E250 ensemble (left panel) and simultaneous fits of twist-2 NMEs on E300 (right panel) corresponding to the data shown in the two columns in Fig. 1. Bottom row: local and twist-2 NMEs on the C101 ensemble. The respective fit ranges in $t_{\mathrm{sep}}$ are indicated by the solid lines and dark shaded parts of the error bands, whereas the dashed lines and light-shaded error bands represent an extrapolation without actual support of the lattice data.

Figure 3

Comparison of the $t_{\mathrm{sep}}^{\min }$ dependence for the plain summation method (open, blue symbols) and two-state summation method (filled, red symbols) fit models on the C101 ensemble. Left: $g_A^{u-d}$; right: $⟨x{⟩}_{u-d}$. The filled symbols at $t_{\mathrm{sep}}^{\min }\approx 0.4\mathrm{fm}$ in both figures are obtained from the corresponding fits shown in the lower two panels of Fig. 2. Data for the two-state fit model are displaced horizontally for clarity.

Figure 4

Examples for the physical extrapolation of $g_A^{u-d}$ using lattice data from the two-state summation fit model ansatz in Eq. (28) with $t_{\mathrm{sep}}^{\min }\approx 0.3\mathrm{fm}$. Upper row: chiral extrapolation as a function of $M_{\pi }^2$ fitting the full set of data to the NNLO model in Eq. (30) (left panel) and with a cut in $M_{\pi }<300\mathrm{MeV}$ using the simplified model in Eq. (37) (right panel). Lower row: continuum (left panel) and infinite volume (right panel) extrapolations fitting the full set of data to the NNLO fit model in Eq. (30). The red data points are obtained by correcting the original lattice data for the extrapolations in all variables but the one on the x-axis, using the parameters from the fit. Therefore, the resulting point errors are highly correlated. Errors are statistical only.

Figure 5

Same as Fig. 4 but for $g_S^{u-d}$.

Figure 6

Same as Fig. 4 but for $g_T^{u-d}$.

Figure 7

Same as Fig. 4 but for $⟨x{⟩}_{u-d}$.

Figure 8

Same as Fig. 4 but for $⟨x{⟩}_{\mathrm{\Delta }u-\mathrm{\Delta }d}$.

Figure 9

Same as Fig. 4 but for $⟨x{⟩}_{\delta u-\delta d}$.

Figure 10

Cumulative distribution functions (CDFs) of the fit models for all six isovector NMEs. Results of the fit models with statistical errors are represented by the individual data points. The color of these data points is referring to the p-value weight which is different from the Akaike weight used in the actual CDF to allow for a visual assessment of the quality of the individual fits. The final result from the model average is given by the solid blue line together with its symmetrized statistical and full error bands as indicated in the plots. The dashed lines represent the (generally nonsymmetric) $1\sigma$-quantiles of the CDF.

Figure 11

Comparison of our results (red diamonds) to our 2019 paper (Mainz 19 [9]), to recent other studies (blue circles: QCDSF/UKQCD/CSSM 23 [13], RQCD 23 [12], PNDME 23 [64], ETMC 23 [65], PNDME 20 [15], NME 20 [16], $\chi \mathrm{QCD}$ 18 [14]) and to the FLAG 2021 [4] and PDG [1] averages (black triangles, where available) for all six isovector NMEs. Studies that entered the FLAG average are not shown separately. Inner error bars are statistical errors only, outer error bars include systematic errors added in quadrature.