[Paper Review] Electromagnetic form factors of the nucleon from $N_f = 2 + 1$ lattice QCD

Dalibor Djukanovic, Georg von Hippel+4 more|arXiv (Cornell University)|Sep 12, 2023

Quantum Chromodynamics and Particle InteractionsPhysics and Astronomy120 references3 citations

TL;DR

This lattice QCD study computes the electromagnetic form factors of the proton and neutron using $N_f = 2+1$ dynamical quarks, including both connected and disconnected quark contractions. It presents the first complete lattice determination of the proton's electric and magnetic radii and magnetic moments with a full error budget, finding the proton's charge radius consistent with muonic hydrogen and PRad, and its magnetic radius compatible with A1 data, supporting emerging experimental consensus.

ABSTRACT

There is a long-standing discrepancy between different measurements of the electric and magnetic radii of the proton. Lattice QCD calculations are a well-suited tool for theoretical investigations of the structure of the nucleon from first principles. However, all previous lattice studies of the proton's electromagnetic radii have either neglected quark-disconnected contributions or were not extrapolated to the continuum and infinite-volume limit. Here, we present results for the electromagnetic form factors of the proton and neutron computed on the $(2 + 1)$-flavor Coordinated Lattice Simulations (CLS) ensembles including both quark-connected and -disconnected contributions. From simultaneous fits to the $Q^2$-, pion-mass, lattice-spacing, and finite-volume dependence of the form factors, we determine the electric and magnetic radii and the magnetic moments of the proton and neutron. For the proton, we obtain as our final values $\langle r_E^2 angle^p = (0.672 \pm 0.014$ (stat)${} \pm 0.018$ (syst)$)$ fm${}^2$, $\langle r_M^2 angle^p = (0.658 \pm 0.012$ (stat)${} \pm 0.008$ (syst)$)$ fm${}^2$, and $μ_M^p = 2.739 \pm 0.063$ (stat)${} \pm 0.018$ (syst). The magnetic moment is in good agreement with the experimental value, as is the one of the neutron. On the one hand, our result for the electric (charge) radius of the proton clearly points towards a small value, as favored by muonic hydrogen spectroscopy and the recent $ep$-scattering experiment by PRad. Our estimate for the magnetic radius, on the other hand, is well compatible with that inferred from the A1 $ep$-scattering experiment.

Motivation & Objective

• To resolve the proton radius puzzle by providing a first-principles lattice QCD prediction of the proton's electromagnetic radii.
• To include both quark-connected and -disconnected contractions in the computation of nucleon form factors, addressing a key systematic omission in prior studies.
• To perform simultaneous fits to $Q^2$, pion mass, lattice spacing, and finite-volume dependence to achieve continuum and infinite-volume extrapolations.
• To extract the electric and magnetic radii and magnetic moments of the proton and neutron with a comprehensive error analysis.
• To test the consistency of lattice QCD predictions with recent experimental results from $ep$ scattering, muonic hydrogen, and dispersive analyses.

Proposed method

• Simulations are performed on $(2+1)$-flavor CLS gauge ensembles with physical pion mass, multiple lattice spacings, and varying volumes.
• The one-end trick (split-even estimator) is employed to efficiently compute quark-disconnected contractions.
• The summation method is applied to suppress excited-state contamination, with conservative source-sink separation windows.
• Covariant baryon chiral perturbation theory is used to match lattice data to physical observables and perform simultaneous fits.
• Simultaneous fits to $Q^2$, pion mass, lattice spacing, and finite-volume dependence are performed in the isospin basis to extract radii and magnetic moments.
• Systematic uncertainties from chiral extrapolation, continuum limit, finite volume, and excited states are fully accounted for in the error budget.

Experimental results

Research questions

• RQ1 What is the electromagnetic charge radius of the proton in lattice QCD when both connected and disconnected quark contractions are included?
• RQ2 How does the lattice QCD prediction for the proton's magnetic radius compare with experimental data from $ep$ scattering and muonic hydrogen?
• RQ3 Can a simultaneous fit to $Q^2$, pion mass, lattice spacing, and finite-volume dependence yield a reliable continuum and infinite-volume extrapolation of nucleon form factors?
• RQ4 Is the magnetic moment of the proton and neutron reproduced accurately in lattice QCD with full control of systematic uncertainties?
• RQ5 Does the lattice result support the emerging consensus that the proton's charge radius is smaller than the A1 $ep$-scattering value?

Key findings

• The proton's electric radius is determined as $ raket{r_E^2}^p = 0.672(14)(18)~\text{fm}^2 $, corresponding to a root-mean-square radius of $ 0.819(9)(11)~\text{fm} $, consistent with muonic hydrogen and the PRad experiment.
• The proton's magnetic radius is $ raket{r_M^2}^p = 0.658(12)(8)~\text{fm}^2 $, compatible with the A1 $ep$-scattering analysis but in tension with other world data.
• The magnetic moment of the proton is $ u_M^p = 2.739(63)(18) $, in excellent agreement with the experimental value, validating the method and error control.
• The neutron magnetic moment is also reproduced within uncertainties, confirming the reliability of the calculation.
• The study presents the first complete lattice QCD result for nucleon electromagnetic radii with a full error budget, including all major systematic effects.
• The results support the emerging consensus that the proton's charge radius is smaller than the A1 $ep$-scattering value, favoring the muonic hydrogen and PRad determinations.