Extraction of the nucleon axial form factor from Lattice QCD using NNLO chiral perturbation theory

Technical Summary: Extraction of the Nucleon Axial Form Factor from Lattice QCD using NNLO Chiral Perturbation Theory

Problem Statement
The nucleon axial form factor, $F_A(Q^2)$, is a fundamental quantity governing weak interactions involving nucleons, particularly neutrino-nucleon scattering cross-sections essential for oscillation experiments. Extracting $F_A(Q^2)$ from experimental data is challenging due to model dependencies (e.g., dipole vs. $z$-expansion parametrizations) and experimental biases. While Lattice QCD (LQCD) offers a first-principles approach, simulations are affected by systematic artifacts, including finite volume, discretization spacing, and unphysical light quark masses (pion masses $M_\pi$). Reaching the physical point requires a robust chiral extrapolation. Previous analyses often relied on non-relativistic approximations or leading-order loop corrections ($O(p^3)$), which may be insufficient to describe the $M_\pi$ dependence of the axial charge ($g_A$) and radius ($\langle r_A^2 \rangle$) across the range of available LQCD data.

Methodology
The authors perform a global analysis of recent LQCD data for the axial form factor using Relativistic Baryon Chiral Perturbation Theory (B$\chi$PT) up to Next-to-Next-to-Leading Order (NNLO), corresponding to $O(p^4)$.

• Theoretical Framework: The calculation employs the Extended On-Mass-Shell (EOMS) renormalization scheme to preserve power counting and analytic properties in the presence of baryons. The $\Delta(1232)$ resonance is treated as an explicit degree of freedom using the small-scale expansion.
• Lagrangian and Diagrams: The effective Lagrangian includes terms up to $O(p^4)$ involving nucleon ($N$) and $\Delta$ fields. The calculation incorporates Feynman diagrams up to $O(p^4)$, including contributions from $O(p^2)$ vertices and baryon mass insertions.
• Data Set: The fit utilizes a combined set of LQCD results from RQCD, NME, Mainz, PNDME, PACS, and ETMC collaborations. The analysis focuses on ensembles with $M_\pi L \geq 3.5$ to neglect finite-volume effects.
• Fit Strategy:
  • The fit range is restricted to $M_\pi \leq 402$ MeV and $Q^2 \leq 0.36$ GeV$^2$ (determined via $\chi^2$/dof plateaus).
  • Lattice-spacing ($a$) effects are corrected phenomenologically using specific dependence terms for different lattice actions.
  • Truncation errors of the chiral expansion are estimated and added in quadrature to LQCD statistical errors. This is done iteratively to account for the correlation between the truncation error and the fitted Low-Energy Constants (LECs).
  • Naturalness Gaussian priors are applied to free LECs to reduce correlations, except for the axial charge in the chiral limit ($\mathring{g}_A$).

Key Contributions

1. NNLO Relativistic Calculation: The study provides the first extraction of $F_A(Q^2)$ using NNLO ($O(p^4)$) relativistic ChPT with explicit $\Delta$ degrees of freedom. This improves upon previous $O(p^3)$ or non-relativistic analyses.
2. Systematic Uncertainty Quantification: The authors explicitly model the truncation error of the chiral expansion, assigning larger uncertainties to data points at higher $M_\pi$ and $Q^2$ where convergence is poorer.
3. Role of the $\Delta$ Resonance: The analysis demonstrates that the explicit inclusion of the $\Delta$ resonance is necessary to reproduce the lattice data's $M_\pi$ dependence for both $g_A$ and $\langle r_A^2 \rangle$, as well as the momentum dependence of the form factor.
4. Model-Independent Parametrization: The work offers a parametrization of $F_A$ rooted in QCD symmetries rather than ad-hoc functional forms, facilitating the extrapolation of LQCD results to the physical point.

Results

• Axial Charge ($g_A$): At the physical point, the analysis yields $g_A = 1.257 \pm 0.011$. This value is in good agreement with the FLAG review ($1.263 \pm 0.010$) but shows mild tension ($\sim 1.5\sigma$) with the experimental value from $\beta$-decay ($1.2754 \pm 0.0013$). The authors note that unaccounted radiative corrections could affect this comparison.
• Axial Radius ($\langle r_A^2 \rangle$): The extracted axial radius squared is $\langle r_A^2 \rangle = 0.312 \pm 0.037$ fm$^2$. This result favors lower values compared to some dipole parametrizations of neutrino scattering data but is consistent with recent $z$-expansion analyses and individual LQCD determinations.
• Low-Energy Constants (LECs):
  • $d_{16} = -0.82 \pm 0.72$ GeV$^{-2}$ (governs $M_\pi$ dependence of $g_A$).
  • $d_{22} = 1.1 \pm 1.4$ GeV$^{-2}$ (enters the axial radius).
  • $g_1 = 0.40 \pm 0.46$ (correlated with $d_{22}$).
  • The large uncertainties in $d_{16}$ and $d_{22}$ reflect the slow convergence of the chiral series for $g_A(M_\pi)$ and correlations between LECs.
• Convergence: The $O(p^4)$ calculation with explicit $\Delta$ provides a realistic description of the data up to $M_\pi \approx 400$ MeV and $Q \approx 600$ MeV. However, the authors note that the $O(p^4)$ contribution to $g_A$ at $M_\pi \gtrsim 200$ MeV is sizable, suggesting that full convergence may require $O(p^5)$ (two-loop) calculations.

Significance
The paper claims to provide a "model-independent and systematically improvable parametrization" of the pion-mass and momentum dependence of the axial form factor. By successfully describing LQCD data up to NNLO with explicit $\Delta$ contributions, the work validates the necessity of including the $\Delta$ resonance for accurate chiral extrapolations of nucleon properties. The resulting values for $g_A$ and $\langle r_A^2 \rangle$ offer a benchmark for improving predictions of low-energy weak interactions involving nucleons, particularly for neutrino oscillation experiments. The authors emphasize that their approach avoids the model dependence of traditional dipole fits while providing a rigorous estimate of theoretical uncertainties.