Nucleon Electromagnetic Form Factors from Lattice QCD using 2+1 Flavor Domain Wall Fermions on Fine Lattices and Chiral Perturbation Theory

TL;DR

The authors perform a high-statistics lattice QCD study of nucleon electromagnetic form factors with $N_f=2+1$ domain-wall fermions on fine lattices to determine $F_1(Q^2)$, $F_2(Q^2)$, $G_E(Q^2)$, and $G_M(Q^2)$ up to $Q^2\approx1\ \mathrm{GeV}^2$, including isovector and (connected) isoscalar channels. They employ an overdetermined analysis combining many matrix elements at fixed $Q^2$, assess systematic uncertainties (excited-state contamination, finite volume, discretization), and set the lattice scale by cross-lattice matching and PCAC relations. Chiral extrapolations are performed with ${\mathcal{O}}(\epsilon^3)$ SSE and ${\mathcal{O}}(p^4)$ CBChPT, but the three pion-mass points near $m_\pi\sim300$ MeV do not agree with either EFT framework at these orders, suggesting the need for lighter pions or higher-order terms to connect lattice results with experiment. The study provides precise isovector radii and anomalous magnetic moments, while highlighting limitations in current chiral EFT descriptions and indicating future directions to refine the understanding of nucleon structure from first principles.

Abstract

We present a high-statistics calculation of nucleon electromagnetic form factors in $N_f=2+1$ lattice QCD using domain wall quarks on fine lattices, to attain a new level of precision in systematic and statistical errors. Our calculations use $32^3 \times 64$ lattices with lattice spacing a=0.084 fm for pion masses of 297, 355, and 403 MeV, and we perform an overdetermined analysis using on the order of 3600 to 7000 measurements to calculate nucleon electric and magnetic form factors up to $Q^2 \approx$ 1.05 GeV$^2$. Results are shown to be consistent with those obtained using valence domain wall quarks with improved staggered sea quarks, and using coarse domain wall lattices. We determine the isovector Dirac radius $r_1^v$, Pauli radius $r_2^v$ and anomalous magnetic moment $κ_v$. We also determine connected contributions to the corresponding isoscalar observables. We extrapolate these observables to the physical pion mass using two different formulations of two-flavor chiral effective field theory at one loop: the heavy baryon Small Scale Expansion (SSE) and covariant baryon chiral perturbation theory. The isovector results and the connected contributions to the isoscalar results are compared with experiment, and the need for calculations at smaller pion masses is discussed.

Figures (24)

• Figure 1: One-loop $SU(2)$ ChPT interpolation of the fine lattice values of $F_\pi$ and $m_\pi$. The point with abscissa $9.71 = (m_\pi/F_\pi)^2_\text{coarse}$ was obtained by interpolating $(aF_\pi)$ linearly in $(m_\pi/F_\pi)^2$.
• Figure 2: Comparison of the nucleon form factors extracted from the full overdetermined system, only non-zero equations, uncorrelated fit and averaged equivalence classes for $m_\pi=297\text{ MeV}$. Increased binning of data (eight successive configurations instead of two) shows no increase in estimation of statistical errors. Each form factor value is divided by the central value of the dipole fit. Tab. \ref{['tab:mom_list']} lists the momentum combinations corresponding to each index on the horizontal axis.
• Figure 3: Comparison of the vector and axial vector current renormalization constants. In the chiral limit, these two renormalization constants agree within errors. The errors on all the $Z_A$ points given in Tab. \ref{['tab:ZVZA']} are too small to appear on the figure.
• Figure 4: The top panel shows the lattice results for $F_{1,2}^{u-d}(Q^2)$ at $m_\pi = 297$ MeV along with the dipole fits with three different $Q^2$ cutoffs. The bottom left three panels show the ratios of the lattice results for $F_1^{u-d}$ to the dipole fits using Eq. (\ref{['eq:one-par-dipole']}), and the bottom right three panels show the ratios of the lattice results for $F_2^{u-d}$ to the dipole fits using Eq. (\ref{['eq:two-par-dipole']}). Only the solid data points are included in the fits with cutoff $0.5\text{ GeV}^2$, and the grey bands show the errors for these fits. The dashed and dotted lines show the ratios of dipole fits at cutoffs $0.7\text{ GeV}^2$ and $1.1\text{ GeV}^2$ relative to the fit at $0.5\text{ GeV}^2$.
• Figure 5: The top panel shows the lattice results for $G_{E,M}^{u-d}(Q^2)$ at $m_\pi = 297$ MeV along with the dipole fits with three different $Q^2$ cutoffs. The bottom left three panels show the ratios of the lattice results for $G_E^{u-d}$ to the dipole fits using Eq. (\ref{['eq:one-par-dipole']}), and the bottom right three panels show the ratios of the lattice results for $G_M^{u-d}$ to the dipole fits using Eq. (\ref{['eq:two-par-dipole']}). Only the solid data points are included in the fits with cutoff $0.5\text{ GeV}^2$, and the grey bands show the errors for these fits. The dashed and dotted lines show the ratios of dipole fits at cutoffs $0.7\text{ GeV}^2$ and $1.1\text{ GeV}^2$ relative to the fit at $0.5\text{ GeV}^2$.