[Paper Review] Volume Dependence of the Axial Charge of the Nucleon

Nathan L. Hall, A. W. Thomas+2 more|arXiv (Cornell University)|May 8, 2012

Quantum Chromodynamics and Particle Interactions1 references3 citations

TL;DR

This paper explains the strong volume dependence of the nucleon's axial charge $g_A$ in lattice QCD as arising from pion exchange interactions between neighboring nucleons on periodic lattices. Wave function renormalization from pion emission and absorption suppresses $g_A$, with the suppression growing as $\exp(-m_\pi L)$, accounting for up to 20% reduction on small lattices and 10% near physical pion mass, resolving a long-standing systematic error in lattice QCD calculations.

ABSTRACT

It is shown that the strong volume-dependence of the axial charge of the nucleon seen in lattice QCD calculations can be understood quantitatively in terms of the pion-induced interactions between neighbouring nucleons. The associated wave function renormalization leads to an increased suppression of the axial charge as the strength of the interaction increases, either because of a decrease in lattice size or in pion mass.

Motivation & Objective

• To resolve the unexplained strong volume dependence of the nucleon axial charge $g_A$ in lattice QCD simulations.
• To identify the origin of the systematic error in $g_A$ calculations at finite lattice volumes, especially at low pion masses.
• To provide a quantitative, model-independent explanation for the suppression of $g_A$ due to pion-induced interactions between neighboring nucleons.
• To enable reliable extraction of quark spin and orbital angular momentum from lattice data by quantifying finite-volume corrections.

Proposed method

• Model the axial charge suppression using pion exchange between nucleons on a periodic lattice, treating it analogously to self-energy corrections in a single nucleon.
• Apply wave function and vertex renormalization formalism to the nucleon-pion system, with the $N\to\Delta\pi$ transition amplitude suppressed by $\exp[-(m_\Delta - m_N)L]$.
• Compute finite-volume corrections to $g_A$ using a perturbative expansion in the pion-nucleon coupling, truncated at distance $\sqrt{3}L$.
• Include contributions from pion emission and absorption on neighboring nucleons, accounting for periodic boundary conditions.
• Use the $\exp(-m_\pi L)$ dependence as a key scaling variable to describe the volume and pion mass dependence of $g_A$.
• Compare the model predictions with lattice data from QCDSF and RBC collaborations, fitting to $g_A^0(1 - \delta g_A) + B m_\pi^2$ without non-analytic terms.

Experimental results

Research questions

• RQ1 Why does the axial charge $g_A$ in lattice QCD simulations exhibit a strong dependence on lattice size, especially at low pion masses?
• RQ2 What is the physical origin of the finite-volume suppression of $g_A$, beyond standard chiral loop corrections?
• RQ3 How do pion exchange interactions between neighboring nucleons on a periodic lattice affect the axial charge?
• RQ4 To what extent do wave function and vertex renormalization from pion exchange contribute to the observed $g_A$ suppression?
• RQ5 Can the observed volume dependence be quantitatively explained by the $\exp(-m_\pi L)$ scaling of pion exchange effects?

Key findings

• The dominant contribution to $g_A$ suppression in finite-volume lattice QCD arises from pion exchange between neighboring nucleons, not from standard chiral loops or surface terms.
• Wave function renormalization from pion emission and absorption suppresses $g_A$, with a suppression factor scaling as $\exp(-m_\pi L)$.
• Vertex renormalization from $N\to\Delta\pi$ transitions is suppressed by $\exp[-(m_\Delta - m_N)L]$, making it ineffective at counteracting the wave function suppression at typical lattice sizes.
• On a 2 fm lattice, the correction reaches nearly 20% at $m_\pi \approx 0.3$ GeV, and remains significant at 10% near the physical pion mass on a 4 fm lattice.
• The model's predictions show remarkable agreement with lattice data from QCDSF and RBC collaborations, even without including non-analytic $m_\pi^2$ terms.
• The $\exp(-m_\pi L)$ scaling explains the convergence of corrections down to $m_\pi L \simeq 2.5$, validating the truncation at $\sqrt{3}L$.