Nucleon 3D intrinsic spin structure from the weak-neutral axial-vector form factors

TL;DR

This paper investigates the relativistic 3D weak-neutral axial-vector spin distributions inside nucleons, clarifies the physical meaning of related form factors and radii, and proposes neutrino scattering measurements to explore nucleon spin structure.

Contribution

It introduces a new understanding of the relativistic 3D axial charge and spin distributions, clarifies the non-existence of a genuine axial radius, and derives neutrino scattering cross sections for experimental insights.

Findings

The axial charge distribution is described by the induced pseudotensor form factor $G_T^Z(Q^2)$.

The root-mean-square axial radius $R_A$ is not physically meaningful.

The relativistic 3D weak-neutral spin radius $r_ ext{spin}$ is physically meaningful and approximately 2.1 fm.

Abstract

Relativistic 3D weak-neutral axial-vector four-current and spin distributions inside a nucleon (or a general spin-$\frac{1}{2}$ hadron) including three weak-neutral axial-vector form factors are investigated for the first time. We clarify that the relativistic 3D axial charge distribution in the Breit frame is completely described by the induced pseudotensor form factor $G_T^Z(Q^2)$ rather than by the axial form factor $G_A^Z(Q^2)$. We demonstrate that $R_A \equiv \sqrt{ \frac{-6}{G_A^Z(0) }\frac{\text{d} G_A^Z(Q^2) }{\text{d} Q^2} \Big|_{Q^2=0} }$ can not be interpreted as the physically meaningful 3D root-mean-square axial radius of a spin-$\frac{1}{2}$ hadron. The genuine axial radius for any spin-$\frac{1}{2}$ hadron in fact does not exist. We also show that the relativistic 3D weak-neutral spin radius $r_\text{spin} = \sqrt{\langle r_\text{spin}^2 \rangle}$, with $\langle r_\text{spin}^2 \rangle \equiv R_A^2 + \frac{ 1 }{ 4M^2 }\left[ 1 + \frac{ 2 G_P^Z(0) }{ G_A^Z(0) } \right]$ based on the relativistic and intrinsic 3D weak-neutral spin distribution in the Breit frame, is a physically meaningful radius that can be unambiguously defined for the nucleon, which provides an additional key motivation for the further determination of the induced pseudoscalar form factor $G_P^Z(Q^2)$. Numerically, we find that $R_A \approx 0.6510~\text{fm}$, $r_\text{spin} \approx 2.1054~\text{fm}$ and $\overline r_\text{spin} \equiv r_\text{spin}/3 \approx 0.7018~\text{fm}$. For future experimental measurements of $G_A^Z(Q^2)$ and $G_T^Z(Q^2)$, we also derive the full tree-level unpolarized differential cross sections for neutrino-proton and antineutrino-proton elastic scattering in the lab frame, in hoping to provide a complementary and new perspective to unveil the nucleon spin structure by using (anti)neutrino-based facilities.