[Paper Review] Nucleon and Pion Form Factors from $N_f=2+1$ Anisotropic Lattices

Huey-Wen Lin, Saul D. Cohen|arXiv (Cornell University)|Apr 21, 2011

Quantum Chromodynamics and Particle Interactions2 references3 citations

TL;DR

This paper presents a lattice QCD calculation of nucleon and pion electromagnetic and axial form factors using $N_f=2+1$ anisotropic lattices, extending the accessible momentum transfer $Q^2$ up to 7 GeV$^2$, significantly beyond the typical 2.5 GeV$^2$ limit. By incorporating excited-state operators and explicit excited-state analysis, the method suppresses noise and enables high-$Q^2$ form factor extraction with improved signal-to-noise, yielding consistent results with experiment and lattice benchmarks at low $Q^2$ while accessing new physics in the large-$Q^2$ regime.

ABSTRACT

We report a recent lattice-QCD calculation of nucleon and pion electromagnetic form factors and nucleon axial form factors, with special emphasis on large $Q^2$. Conventional lattice form-factor calculations can only reach about 2.5 GeV$^2$, but in this work the transfer momentum is pushed as large as 6 GeV$^2$. Here, we demonstrate the results on 2+1-flavor anisotropic clover lattices for the nucleon and pion, comparing with low-$Q^2$ quantities, such as Dirac and Pauli radii, anomalous magnetic moments, $g_A$ and $M_A$. Our approach can be applied to isotropic lattices and lattices with smaller lattice spacing to achieve even larger-$Q^2$ form factors. The form factors are processed to obtain transverse charge and magnetization densities across 2-dimensional impact-parameter space. These measurements could give important theoretical input to experiments, such as those of JLab's 12-GeV program, and provide insight into hadronic structure.

Motivation & Objective

• To extend lattice QCD calculations of hadron form factors to higher momentum transfers ($Q^2$) beyond the conventional 2.5 GeV² limit.
• To address the poor signal-to-noise ratio that plagues standard lattice form factor calculations at large $Q^2$.
• To extract transverse charge and magnetization densities in impact-parameter space for nucleons and pions using high-$Q^2$ form factors.
• To provide theoretical input for Jefferson Lab's 12-GeV program and test the onset of perturbative QCD behavior.
• To develop a method applicable to isotropic lattices and finer lattice spacings for future high-precision studies.

Proposed method

• Utilizes $N_f=2+1$ anisotropic clover lattices with pion masses of 580, 875, and 1350 MeV to compute form factors at high $Q^2$.
• Employs a modified three-point function analysis that explicitly includes excited-state operators to improve ground-state signal extraction at large $Q^2$.
• Applies smearing techniques to interpolating fields to enhance overlap with the ground state at earlier Euclidean times.
• Uses a two-pole fit to lattice data to model the pion form factor and extract transverse densities via Fourier-Bessel transform: $\rho_{\pi}(b) = \int_0^\infty \frac{Q\,dQ}{2\pi} J_0(bQ) F_\pi(Q^2)$.
• Performs extrapolation to the physical pion mass using multiple mass ensembles and compares with experimental data.
• Proposes a step-scaling approach on small, fine-latticed volumes to reduce discretization and finite-volume errors at high $Q^2$.

Experimental results

Research questions

• RQ1 Can lattice QCD calculations achieve reliable form factors at $Q^2 > 2.5$ GeV$^2$, where conventional methods fail due to noise?
• RQ2 How do transverse charge and magnetization densities of the nucleon and pion evolve at high momentum transfer?
• RQ3 To what extent does the pion form factor data support the asymptotic pQCD prediction $F_\pi(Q^2) \sim 1/Q^2$ at large $Q^2$?
• RQ4 Can the inclusion of excited-state operators in the three-point function analysis significantly improve signal-to-noise at high $Q^2$?
• RQ5 What is the behavior of the central charge density $\rho_\pi(b)$ near $b=0$, and does it exhibit a singularity as predicted by pQCD?

Key findings

• The study achieves $Q^2$ values up to nearly 7 GeV$^2$ for the pion form factor, extending the reach of lattice QCD beyond the typical 2.5 GeV$^2$ limit.
• The extrapolated pion form factor at the physical pion mass shows reasonable agreement with Jefferson Lab precision measurements at low $Q^2$.
• Transverse charge densities for the nucleon and pion are computed using high-$Q^2$ form factors, revealing structure in impact-parameter space.
• The right-hand side of Figure 7 shows $b\rho_\pi(b)$ from lattice data, indicating a central singularity whose nature depends on the $Q^2$-dependence of the form factor.
• The method yields consistent results for low-$Q^2$ observables such as Dirac and Pauli radii, anomalous magnetic moments, $g_A$, and $M_A$ when compared to RBC/UKQCD and LHPC results.
• The proposed step-scaling method on fine, small lattices offers a path to reduce systematic errors at high $Q^2$ in future studies.