Nucleon Charges and Electromagnetic Form Factors from 2+1+1-Flavor Lattice QCD

TL;DR

This study computes nucleon structure observables from lattice QCD using MILC 2+1+1 HISQ ensembles at $a\approx 0.12$ fm and $M_\pi\approx 310$ and 220 MeV, focusing on the isovector axial, scalar, and tensor charges $g_A$, $g_S$, $g_T$, the connected part of isoscalar charges, and the isovector electromagnetic form factors. It systematically addresses excited-state contamination by analyzing multiple source-sink separations and employing several multi-state fit strategies, establishing that careful treatment of excited states is essential for reliable ground-state matrix elements. Renormalization constants in the RI-sMOM scheme are computed nonperturbatively and converted to $\overline{\text{MS}}$ at 2 GeV, achieving approximately 5% accuracy, with the vector current used to constrain the results. The renormalized isovector charges are $g_A\approx 1.193(68)$, $g_S\approx 0.72(32)$, and $g_T\approx 1.047(61)$ (at or extrapolated to the physical pion mass), while the connected isoscalar charges and the isovector EM form factors are reported with systematic uncertainties. Overall, the work demonstrates control over excited-state and renormalization systematics and highlights the need for additional lattice spacings and lighter quark masses to enable reliable continuum and chiral extrapolations.

Abstract

We present lattice-QCD results on the nucleon isovector axial, scalar and tensor charges, the isovector electromagnetic Dirac and Pauli form factors, and the connected parts of the isoscalar charges. The calculations have been done using two ensembles of HISQ lattices generated by the MILC Collaboration with 2+1+1 dynamical flavors at a lattice spacing of 0.12 fm and with light-quark masses corresponding to pions with masses 310 and 220 MeV. We perform a systematic study including excited-state degrees of freedom and examine the dependence of the extracted nucleon matrix elements on source-sink separation. This study demonstrates with high-statistics data that including excited-state contributions and generating data at multiple separations is necessary to remove contamination that would otherwise lead to systematic error. We also determine the renormalization constants of the associated quark bilinear operators in the RI-sMOM scheme and make comparisons of our renormalized results with previous dynamical-lattice calculations.

Figures (11)

• Figure 1: (top) Nucleon effective-mass plot with one- and two-state fits for the 310-MeV ensemble described in Table \ref{['tab:param1']}. The data are for momenta $\vec{p}=0$ (squares) and $|\vec{p}|^2=5 \left(\frac{2\pi}{La}\right)^2$ (diamonds). Quark propagators with the clover action were calculated using Gaussian-smearing sources with parameters $\{\sigma,n_\text{KG}\}=\{5.5,70\}$ as described in the text. (bottom) A comparison of fitted values of nucleon mass with one- and two-state fits as functions of $t_\text{min}$, the starting value of $t$ used in the fits. The data are for the $M_\pi \approx 220$ MeV ensemble described in Table \ref{['tab:param1']}. The two fits agree for $t_\text{min} \ge 6$, and the two-state fit yields a consistent ground-state mass for all $t_\text{min}>0$.
• Figure 2: The two-simRR (upper) two-sim (lower) methods fit as a function of time to the unrenormalized $g_S$ data from the 220-MeV ensemble with insertion on the $d$ quark. The fits shown are with and without the $\langle 1 | O_\Gamma | 1 \rangle$ term in Eq. \ref{['eq:three-pt']}.
• Figure 3: Estimates of the unrenormalized isovector charges $g_{A,S,T}$ as functions of source-sink separation ($t_\text{sep}$) with 310-MeV (left) and $220$ MeV (right) ensembles at $a \approx 0.12$ fm. Estimates are shown for the different fit types described in the text. The band shows the results of the two-sim fit to data for all $t_\text{sep}$.
• Figure 4: The data for $Z_{A,S,T}/Z_V$ in the $\overline{\text{MS}}$ scheme at $2$ GeV and fits using the ansatz $c/q^2 + Z + d_1 q$. The data for the $M_\pi=310$ ($220$) ensemble are shown by black (brown) symbols. In each case the straight line is the plot of $Z + d_1 q$ where $Z$ is the listed value of $Z_\Gamma/Z_V$.
• Figure 5: (Left) Collected experimental values used in PDG 2012 average (the band) and the latest UCNA (2012) measurements on $g_A$; there has been a slow increase in $g_A/g_V$ over the past 15 years. The lower panel shows $g_A$ values after extrapolating to the physical pion mass collected from dynamical 2+1-flavor and 2-flavor lattice calculations using $O(a)$-improved fermions Khan:2006deLin:2008uzAlexandrou:2010hfCapitani:2012gjEdwards:2005ymLin:2007apYamazaki:2008pyBratt:2010jnGreen:2012ud. Note the change in scale between the experimental and theory plots. Most of the errorbars here are statistical only. In data from the few calculations that also quote systematic errors, we add these to the statistical ones as outer errorbar bands, marked with dashed lines. (right) Calculations of $g_A$ using at least 2+1 flavors $O(a)$-improved dynamical fermions, plotted as a function of $M_\pi^2$, with $m_\pi L > 4$ to avoid systematics due to small spatial extent.