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Inferring the breakdown scales of the chiral expansions for $g_A$ and $m_N$

Andreas Ekström, Daniel R. Phillips, Lucas Platter, Matthias R. Schindler

TL;DR

The paper addresses how the breakdown scale of single-nucleon chiral perturbation theory depends on the observable by inferring $\Lambda_B$ from order-by-order expansions of $g_A$ and $m_N$ using a pointwise Bayesian approach. It expresses the expansions with a natural-coefficient prior and a likelihood based on $Q = M/\Lambda_B$, then infers $\Lambda_B$ from data at two pion masses across different low-energy-constant (LEC) sets. The main findings are that $\Lambda_B$ for $g_A$ is about $251^{+20}_{-50}$ MeV (Set 1) or $211^{+20}_{-30}$ MeV (Set 2), while for $m_N$ it is about $491^{+60}_{-90}$ MeV, indicating substantially slower convergence for $g_A$ than for $m_N$. This demonstrates observable-dependent convergence in baryon $\chi$PT and emphasizes the influence of resonances like the $\Delta(1232)$, with implications for lattice extrapolations and the interpretation of EFT truncation errors.

Abstract

We apply Bayesian inference to the order-by-order chiral perturbation theory ($χ$PT) expansions for the axial-vector coupling constant $g_A$ and the nucleon mass m_N, and thereby infer the scales at which $χ$PT breaks down for these two observables. Using a pointwise Bayesian analysis, we find that the inferred breakdown scales are notably different for the two observables. For the chiral expansion of $g_A$, we obtain $251^{+20}_{-50}$ MeV and $211^{+20}_{-30}$ MeV using two distinct sets of low-energy constants, while for the chiral expansion of $m_N$ we infer a significantly larger breakdown scale of $491^{+60}_{-90}$ MeV.

Inferring the breakdown scales of the chiral expansions for $g_A$ and $m_N$

TL;DR

The paper addresses how the breakdown scale of single-nucleon chiral perturbation theory depends on the observable by inferring from order-by-order expansions of and using a pointwise Bayesian approach. It expresses the expansions with a natural-coefficient prior and a likelihood based on , then infers from data at two pion masses across different low-energy-constant (LEC) sets. The main findings are that for is about MeV (Set 1) or MeV (Set 2), while for it is about MeV, indicating substantially slower convergence for than for . This demonstrates observable-dependent convergence in baryon PT and emphasizes the influence of resonances like the , with implications for lattice extrapolations and the interpretation of EFT truncation errors.

Abstract

We apply Bayesian inference to the order-by-order chiral perturbation theory (PT) expansions for the axial-vector coupling constant and the nucleon mass m_N, and thereby infer the scales at which PT breaks down for these two observables. Using a pointwise Bayesian analysis, we find that the inferred breakdown scales are notably different for the two observables. For the chiral expansion of , we obtain MeV and MeV using two distinct sets of low-energy constants, while for the chiral expansion of we infer a significantly larger breakdown scale of MeV.

Paper Structure

This paper contains 4 sections, 13 equations, 4 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Chiral expansions of $m_N$ and $g_A$
  3. Analysis and results
  4. Summary and discussion

Figures (4)

  • Figure 1: Pion mass dependence of the nucleon mass $m_N$ using "Set $m_N$" given in Table \ref{['tbl:sets']}.
  • Figure 2: Pion mass dependence of the axial-vector coupling constant $g_A$ for "Set 1" given in Ref. Bernard:2025gto of LECs and $\mathring{g}=1$. For the unknown LEC, we use $C=0$.
  • Figure 3: Posteriors for the breakdown scale $\Lambda_B$ obtained from the analysis of the order-by-order expansion of the axial-vector coupling constant $g_A$ and using the values from Set 1 (top panel) and Set 2 (bottom panel) in Table \ref{['tbl:sets']} for the LECs.
  • Figure 4: Posteriors for the breakdown scale $\Lambda_B$ obtained from the analysis of the order-by-order expansion of nucleon mass $m_N$ and using the values from "Set $m_N$" in Table \ref{['tbl:sets']} for the LECs.