On pseudo-gauge ambiguity in the distributions of energy density, pressure, and shear force inside the nucleon

TL;DR

This study demonstrates that, within a two-flavor Skyrme model augmented by vector mesons, the local distributions of energy density, pressure, and shear inside the nucleon depend on the chosen energy-momentum tensor (EMT) representation. Canonical and Belinfante EMTs yield distinct spatial profiles due to total-derivative surface terms tied to vector-meson spin currents, though global quantities like the nucleon mass and the von Laue condition remain invariant. The pressure distribution exhibits strong pseudo-gauge sensitivity, with qualitative changes in the small-radius behavior between forms, while the integrated shear-related surface energy also varies significantly. Despite these ambiguities, the work clarifies that mapping EMT-derived local mechanical properties to physical observables (e.g., confinement strength or single-nucleon EoS) requires careful consideration of the EMT choice, especially in the presence of dynamical vector fields, and highlights the need for pseudo-gauge–invariant characterizations for robust interpretation in GPD/DVCS analyses.

Abstract

We study the spatial distributions of pressure, energy density, and shear forces inside the nucleon within the two-flavor Skyrme model including vector mesons. This framework has the advantage that nucleon configurations can be stabilized without the Skyrme term. In contrast to the model without vector mesons, however, we realize that the energy-momentum tensor (EMT) becomes pseudo-gauge dependent. We explicitly demonstrate that all these distributions differ between the canonical and Belinfante forms of the EMTs. We identify the pseudo-gauge ambiguity as originating from nonvanishing surface terms associated with spin currents generated by the vector-meson field strength tensors. Furthermore, we show that the pressure and shear-force distributions in the canonical EMT develop singularities at the nucleon center, whereas the corresponding Belinfante distributions remain finite. Finally, we discuss the implications of pseudo-gauge dependence for extracting the confining force and for constructing the equation of state inside the nucleon.

Figures (8)

• Figure 1: Schematic illustration for $p_r(r)$ and $p_{\theta,\phi}(r)$.
• Figure 2: Distributions of the vector-meson magnetic profile in the flavor-$j$ component. The $\rho$-meson Ansatz leads to the axial symmetric profile with respect to the $x_j$ axis.
• Figure 3: Distributions of the spin current in the $k$ direction. The corresponding spin charge density is $S^{0ij}$, which vanishes in our vector-meson Ansätze.
• Figure 4: Distributions of the energy density in two different forms multiplied by $4\pi r^2$ as functions of $r$.
• Figure 5: Distributions of the pressure in two different forms multiplied by $4\pi r^2$ as functions of $r$.