Energy-momentum tensor form factors and spin density distribution in the nucleon calculated in a quantized Skyrme model with vector mesons

Abstract

We investigate energy-momentum tensor (EMT) form factors and the spatial spin density distribution in the nucleon within a framework of the quantized Skyrme model with vector mesons. We construct both the canonical and Belinfante improved EMTs and analyze how pseudogauge uncertainty influences local spin and momentum densities while leaving the global nucleon properties unchanged. Using the inversion formulas from nucleon matrix elements in the forward limit, we extract the form factors, $A(t)$, $D(t)$, and $J(t)$, in both pseudogauges and the additional antisymmetric form factor associated with the canonical EMT. We find that the pseudogauge choice leads to sizable differences in the local spin and momentum densities. In particular, the canonical EMT naturally encodes spin density through the antisymmetric tensor structure, while the Belinfante EMT is sensitive to the total angular momentum only. Our results illustrate explicitly how different pseudogauges correspond to different spatial interpretations of nucleon spin structure within the same underlying dynamics. These findings provide a concrete model realization of the pseudogauge ambiguity in QCD-inspired nucleon structure and offer useful intuition for interpreting spatial distributions.

Figures (10)

• Figure 1: Numerical solutions of $F(r)$ (blue solid curve), $G(r)$ (red dashed curve), $\omega(r)$ (green dot-dashed curve), $\xi_1(r)$ (cyan solid line), $\xi_2(r)$ (magenta dotted curve), and $\phi(r)/r$ (yellow dot-dashed line). Only $\omega(r)$ is rescaled with $f_\pi$ to be dimensionless.
• Figure 2: Momentum form factor, $A(t)$, extracted from $T^{\mu\nu}_\text{can}$ and $T^{\mu\nu}_\text{Bel}$. In both cases, the normalization condition, $A(0)=1$, is satisfied within numerical accuracy.
• Figure 3: Mechanical form factor, $D(t)$, extracted from $T^{ij}_\text{can}$ and $T^{ij}_\text{Bel}$ using the static stress decomposition and Fourier-Bessel relations in Sec. \ref{['sec:inversion']}.
• Figure 4: Angular-momentum form factor, $J(t)$, from the mixed components of the EMT. For the canonical EMT, we also quantify the antisymmetric contribution through an additional form factor, $\mathcal{G}(t)$, which vanishes identically for the Belinfante EMT.
• Figure 5: Canonical total angular-momentum density, $J^z_\text{can}$, for a nucleon polarized along $\hat{\bm z}$. Left: transverse plane ($z=0.05fm$). Right: longitudinal plane ($y=0$). These two slices visualize the characteristic geometry implied by rotational symmetry along the polarization axis.