Nucleon 3D intrinsic spin structure from the weak-neutral axial-vector form factors
Yi Chen
TL;DR
The paper addresses how to characterize the nucleon spin structure in a fully relativistic 3D framework using weak-neutral axial-vector form factors. It develops relativistic 3D axial-vector densities in the Breit frame from the three FFs $G_A^Z$, $G_P^Z$, and $G_T^Z$, showing that the axial charge density is controlled by $G_T^Z$ rather than $G_A^Z$ and establishing a physically meaningful spin radius $r_\text{spin}$ with $\langle r_\text{spin}^2 \rangle = R_A^2 + \frac{1}{4M^2}\left[ 1 + \frac{2 G_P^Z(0)}{ G_A^Z(0) } \right]$. The work argues that the traditional 3D axial radius is not well-defined for spin-$\tfrac{1}{2}$ hadrons and provides numerical values $R_A \approx 0.6510$ fm and $r_\text{spin} \approx 2.1054$ fm (with $\overline r_\text{spin} \approx 0.7018$ fm). It also derives full tree-level unpolarized differential cross sections for neutrino–proton elastic scattering in the lab frame, offering a practical avenue to probe the spin structure with (anti)neutrino facilities and motivating measurements of $G_A^Z(Q^2)$, $G_T^Z(Q^2)$, and especially $G_P^Z(Q^2)$ to constrain the spin radii.
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.