Flavor diagonal nucleon charges using clover fermions on MILC HISQ ensembles

Abstract

We present lattice results for the flavor diagonal charges of the proton from the analysis of eight ensembles generated using 2+1+1-flavors of highly improved staggered quarks (HISQ) by the MILC collaboration. The calculation includes all the needed connected and disconnected contributions to nucleon three-point function. For extracting matrix elements using fits to the spectral decomposition of these correlation functions, two strategies to remove excited state contributions are employed and compared. To renormalize these charges, the 2+1-flavor mixing matrix is calculated in the RI-sMOM intermediate scheme on the lattice. The final results are presented in the $\overline{\text{MS}}$ scheme at scale 2GeV. The axial charges for the proton are $g_A^u = 0.781(25)$, $g_A^d = -0.440(39)$, and $g_A^s = -0.055(9)$; the tensor charges are $g_T^u = 0.782(28)$, $g_T^d = -0.195(16)$, and $g_T^s = -0.0016(12)$; and the scalar charges are $g_S^u = 9.39(88)$, $g_S^d = 8.84(93)$, and $g_S^s = 0.37(14)$. Results for the neutron are given by the $u \leftrightarrow d$ interchange. Results for the sigma terms are $σ_{πN}|_{\rm standard} = 42(6)~{\rm MeV}$ from a "standard" analysis and $σ_{πN}|_{N π} = 61(6)~{\rm MeV}$ from a "$Nπ$" analysis that includes the contributions of multihadron $Nπ$ excited states as motivated by chiral perturbation theory. Our preferred value $σ_{πN}|_{N π}$ is consistent with the phenomenological extraction from $π- N$ scattering data. The strangeness content of the proton, for which the "standard" analysis is appropriate, is $σ_{s}|_{\rm standard} = 35(13)~{\rm MeV}$.

Figures (25)

• Figure 1: The top row shows the connected (left) and disconnected (right) diagrams that contribute to the 3-point functions from which the matrix element of flavor-diagonal operators are extracted. The black and gray blobs denote the nucleon source and sink separated by Euclidean time $\tau$. The operator, shown by the symbol $\otimes$, is inserted at all intermediate times $t$ between the nucleon source and sink points. The bottom diagram (redraw of the top right) illustrates why the disconnected contribution for the scalar operator with $u$ and $d$ flavors can have an enhanced contribution due to the $N\pi$-intermediate state.
• Figure 2: Data for the bare charges $g_{A,T,S}^{u,d}$ (sum of the connected and disconnected contributions) obtained at various separations $\{\tau,t\}$ are plotted versus $(t-\tau/2)/a$ for the physical $M_\pi$ ensemble $a09m130$. We show both the "standard" and "$N \pi$" fits, defined in the text, when ESC are manifest. In each case, the result of the fit is shown by lines of the same color as the data for the various $\tau/a$ listed in the label, and the $\tau\to\infty$ value is given by the gray band. We also show, for comparison, the light quark disconnected contributions, $g_{A,S,T}^{l}$, in the panels on the right and the ES fits to them. The $\chi^2/dof$ of the fit with the largest AIC score is given within square parenthesis in the top right corner.
• Figure 3: Data for the bare strange charges, $g_{A,T,S}^{s}$, obtained at various separations $\{\tau,t\}$, are plotted versus $(t-\tau/2)/a$ for the physical $M_\pi$ ensemble $a09m130$. These have only disconnected contributions. Only the "standard" fits to remove ESC are shown and considered since the expected lowest multihadron state, $\Sigma K$, is heavier than the nucleon's radial excitation. Rest is the same as in Fig. \ref{['fig:gAgTgS_ESC']}.
• Figure 4: The two panels shows the 2-state (top) and 3-state (bottom) "$N\pi$" fit to the same data for $g_A^d$ from the $a09m130$ ensemble. Points with $\tau=\{10,12,14,16\}$ and $t_\text{skip}=2$ are included in the fits and are shown using filled circles. These two fits have the largest and similar AIC scores and dominate the model average, but differ qualitatively---the 2-state fit shows a large extrapolation, while the 3-state fit has a large error---and give different central values.
• Figure 5: The ratio plot for $g_S^{u-d,\text{bare}}$ on the a09m130 ensemble. The $\tau=\infty$ value is taken from the fit to $\tau=8, 10$ and 12 data, which show no significant ESC at $t = \tau/2$. The $\tau=14$ and 16 data are neglected as, with current statistics, they have large errors and deviate from the expected behavior of monotonic convergence.