Method for high-precision determination of the nucleon axial structure using lattice QCD: Removing $πN$-state contamination

TL;DR

The paper tackles the challenge of accurately determining the nucleon axial structure in lattice QCD by addressing strong πN-state contamination in the induced pseudoscalar and pseudoscalar form factors. It introduces a leading πN subtraction method that leverages time-derivative combinations of axial-vector correlator ratios to remove πN contributions, yielding FP and GP in agreement with pion-pole dominance and experimental data. Using 2+1 flavor PACS10 ensembles at two lattice spacings and large volumes, the study obtains g_P^* and g_{π NN} with percent-level precision and demonstrates consistency across volumes and discretizations. The results reinforce the validity of the generalized GT relation at the ground state and provide a practical, model-independent procedure for extracting low-energy constants from lattice data, with implications for neutrino-nucleon and muon capture phenomenology.

Abstract

We performed a precise calculation of physical quantities related to the axial structure of the nucleon using 2+1 flavor lattice QCD gauge configuration (PACS10 configuration) generated at the physical point with lattice volume larger than $(10\;{\mathrm{fm}})^4$ by the PACS Collaboration. The nucleon matrix element of the axial-vector current has two types of the nucleon form factors, the axial-vector ($F_A$) form factor and the induced pseudoscalar ($F_P$) form factor. Recently lattice QCD simulations have succeeded in reproducing the experimental value of the axial-vector coupling, $g_A$, determined from $F_A(q^2)$ at zero momentum transfer $q^2=0$, at a percent level of statistical accuracy. However, the $F_P$ form factor so far has not reproduced the experimental values well due to strong $πN$ excited-state contamination. Therefore, we proposed a simple subtraction method for removing the so-called leading $πN$-state contribution, and succeeded in reproducing the values obtained by two experiments of muon capture on the proton and pion electro-production for $F_P(q^2)$. The novel approach can also be applied to the nucleon pseudoscalar matrix element to determine the pseudoscalar ($G_P$) form factor with the help of the axial Ward-Takahashi identity. The resulting form factors, $F_P(q^2)$ and $G_P(q^2)$, are in good agreement with the prediction of the pion-pole dominance model. In the new analysis, the induced pseudoscalar coupling $g_P^\ast$ and the pion-nucleon coupling $g_{πNN}$ can be evaluated with a few percent accuracy including systematic uncertainties using existing data calculated at two lattice spacings.

Figures (17)

• Figure 1: Using the standard ratio method, the values of $F_A$ (upper-left panel), $2M_N F_P$ (upper-right panel) and $\widetilde{G}_P$ (lower panel) calculated from PACS10/L160 with $t_{\mathrm{sep}}/a=13$ (diamonds), $16$ (squares) and $19$ (circles) for seven nonzero momentum transfers (labeled from Q1 to Q7) as functions of the current insertion time slice $t$.
• Figure 2: Results of the ratio $\mathcal{R}^{5z}_{A_4}(t;\bm{q})/(iq_3 K^{-1})$ calculated from PACS10/L160 with $t_{\mathrm{sep}}/a=13$ (diamonds), $16$ (squares) and $19$ (circles) for seven nonzero momentum transfers (labeled from Q1 to Q7) as functions of the current insertion time slice $t$. The nucleon matrix element of $A_4$ exposes the presence of $\pi N$-state contribution described by $\Delta_-(t)$ defined in Eq. (\ref{['Eq:Delta_ex']}).
• Figure 3: Schematic view of the ground-state contribution (A) and two types of the leading $\pi N$ contributions (B) and (C) for the axial-vector matrix element.
• Figure 4: Using the leading $\pi N$ subtraction method as defined in Eq. (\ref{['eq:new_FP']}), the values of the $F_P$ form factor multiplied by $2M_N$ calculated from PACS10/L160 with $t_{\mathrm{sep}}/a=13$ (diamonds), $16$ (squares) and $19$ (circles) are shown for seven nonzero momentum transfers (labeled from Q1 to Q7) as functions of the current insertion time slice $t$. The horizontal dashed line together with yellow band in each panel is calculated from the PPD model ($2M_N F_P^{\rm PPD}(q^2)$) given in Eq.(\ref{['eq:PPD_FP']})
• Figure 5: $t_{\mathrm{sep}}$ dependence on the $F_P$ form factor is shown for seven lowest momentum transfers, along with a comparison of the summation method. Results are obtained from the coarse $128^4$ lattice (left) and fine $160^4$ lattice (right).