The Nucleon Axial Form Factor from Averaging Lattice QCD Results

TL;DR

The paper addresses constraining the nucleon isovector axial form factor $F_A(Q^{2})$ for neutrino scattering by averaging five independent lattice QCD results with complete error budgets. It develops two averaging strategies: (i) fitting derivatives of $F_A$ at selected $Q^{2}$ points within the $z$-expansion framework, and (ii) stochastic sampling of the form factor over a $Q^{2}$ grid, both with careful treatment of unknown covariances via covariance derating and Schmelling-style correlations. The averaged result yields improved precision (about 1.2% at $Q^{2}=0.5$ GeV$^{2}$ and 2.2% at $Q^{2}=1$ GeV$^{2}$) and is broadly consistent with the most precise lattice determinations, while indicating tension with neutrino-deuterium data and compatibility with MINERvA antineutrino-hydrogen data. The findings provide a tighter, lattice-grounded constraint on neutrino-nucleon cross sections and inform future analyses in neutrino oscillation experiments.

Abstract

Flagship neutrino oscillation experiments depend on precise and accurate theoretical knowledge of neutrino-nucleon cross sections across a variety of energies and interaction mechanisms. Key ingredients to the amplitudes that make up these cross sections are parameterized form factors. The axial form factor describing a weak interaction with a nucleon is part of one of the primary neutrino-nucleon interaction mechanisms, quasielastic scattering, yet this form factor is uncertain and its precision is limited by the availability of data for a neutrino scattering with nucleons or small nuclear targets. Lattice Quantum Chromodynamics (LQCD) now offers another approach for obtaining mathematically rigorous constraints of the axial form factor from theoretical calculations with complete systematic error budgets. In this work, strategies for averaging LQCD results are explored, including both a random sampling of form factor values across momentum transfers as well as an averaging strategy based on analytic calculations of form factor derivatives. Fits to z expansion parameterizations are reported and compared against neutrino-hydrogen and neutrino-deuterium scattering data.

Figures (13)

• Figure 1: Plot of the axial form factor fits from this work normalized by the $z$ expansion deuterium result from Ref. Meyer:2016oeg. The uncertainty band for the fit with $k_{\rm max}=5$ and up to first-order derivatives only is given by the teal shaded region bordered by dashed lines. A similar fit with $k_{\rm max}=6$ ($k_{\rm max}=7$) is shown as the orange band with a dot-dashed border (blue-violet band with a dotted border). For reference, the normalized range $0\pm1$ of the $z$ expansion result is shown as black lines.
• Figure 2: Plot of the axial form factor fits from this work normalized by the $z$ expansion deuterium result from Ref. Meyer:2016oeg. The uncertainty band for the nominal result with $k_{\rm max}=6$ and up to first-order derivatives only is given by the teal shaded region bordered by dashed lines. Fits with up to second derivatives for $k_{\rm max}=6$, 7, and 8 are shown as an orange band with a dot-dashed border, a blue-violet band with a dotted border, and a pink band with a triple-dash border, respectively. For reference, the normalized range $0\pm1$ of the $z$ expansion result is shown as black lines.
• Figure 3: Plot of the axial form factor fits from this work normalized by the $z$ expansion deuterium result from Ref. Meyer:2016oeg. The uncertainty band for the nominal result, evaluating the form factor and its derivatives for each separate LQCD result at its own value of $Q^{2}=-t_{0}$, is given as the teal shaded region bordered by dashed lines. When $t_{0}=0$, the minimum value $Q^{2}_{0}$ is used in place of $t_{0}$, as described in the text. Fits evaluating the form factor and its derivatives at $Q^{2}=0.25~\text{GeV}^{2}$ and $0.50~\text{GeV}^{2}$ are shown as the orange band with a dot-dashed border and the blue-violet band with a dotted border, respectively. For reference, the normalized range $0\pm1$ of the $z$ expansion result is shown as black lines.
• Figure 4: Plot of the axial form factor fits from this work normalized by the $z$ expansion deuterium result from Ref. Meyer:2016oeg. The teal dashed, orange dot-dashed, and blue-violet dotted regions correspond to setting the minimum $Q^{2}$ evaluation point, denoted by $Q^{2}_{\rm min}$, equal to $0$, $0.05~\text{GeV}^{2}$, and $0.10~\text{GeV}^{2}$, respectively. This value is used for evaluation of the form factors and derivatives for results that set $t_{0}=0$ (ETM and Djukanovic et al.). The nominal fit corresponds to the choice $0.05~\text{GeV}^{2}$, given by the orange dot-dashed region. For reference, the normalized range $0\pm1$ of the $z$ expansion result is shown as black lines.
• Figure 5: Plot of the axial form factor fits from this work normalized by the $z$ expansion deuterium result from Ref. Meyer:2016oeg. The teal dashed region corresponds to the nominal fit with no anchor point. The orange dot-dashed, blue-violet dotted, and pink triple-dashed ranges, which all lie on top of each other, correspond to $k_{\rm max}=7$ fits with an anchor point at $Q^{2}_{\rm anchor}=0.10$, $0.75$, and $1.00~\text{GeV}^{2}$, respectively. For reference, the normalized range $0\pm1$ of the $z$ expansion result is shown as black lines.