Quantum stress and torsion distributions in the deuteron

Abstract

Stress distributions in the deuteron are related to form factors of the asymmetric energy-momentum tensor through three-dimensional Fourier transforms. There are eleven such form factors, which we calculate in an impulse approximation. We compare the obtained form factors to prior results for the six form factors that have been previously calculated. We then elaborate on the formalism for relating the form factors to internal distributions of mass, mass flux, momentum, stresses, and forces, and obtain results for all of these distributions. We obtain the principal stresses for the symmetric part of the stress tensor, and show that the antisymmetric part describes reorientation of fermion spin by torsion stress when the nucleon moves between the S- and D-waves. Force distributions in the nucleons depend on the so-called non-conserved form factors through the Cauchy momentum equation, and are non-radial owing to the presence of tensor forces and spin-orbit coupling.

Figures (13)

• Figure 1: Numerical results for the six symmetric and conserved EMT-FFs of the deuteron. For our result, we used the AV18 deuteron wave function Wiringa:1994wb and the same nucleon EMT form factors as He and Zahed He:2023ogg. The results we compare to are from Refs. Freese:2022yurHe:2024vzzPanteleeva:2024abz, though the results of Ref. Freese:2022yur have been modified to use He and Zahed's nucleon form factors.
• Figure 2: Comparisons between the deuteron $D$-like form factors for different deuteron wave functions and different nucleon EMT-FFs. Blue curves use the AV18 wave function Wiringa:1994wb while orange curves use CD-Bonn Machleidt:2000ge. Solid curves use dipole forms for the nucleon EMT-FFs, while dashed curves assume pointlike nucleons.
• Figure 3: Comparisons between the deuteron $\bar{c}$-like form factors for different deuteron wave functions and different nucleon EMT-FFs. Blue curves use the AV18 wave function Wiringa:1994wb while orange curves use CD-Bonn Machleidt:2000ge. Solid curves use dipole forms for the nucleon EMT-FFs, while dashed curves assume pointlike nucleons.
• Figure 4: Comparisons between the EMT-FFs appearing in the antisymmetric part of the stress tensor, using either a dipole form for $S_N(\bm{\varDelta}^2)$ (solid blue curve) or assuming pointlike nucleons (dashed orange curve).
• Figure 5: Mass density of a deuteron in an $m_j=0$ state (left panel) and an $m_j=\pm1$ state (right panel). The images on the walls of the plots are slices of the mass density at $x=0$ (left wall), $y=0$ (back wall) and $z=0$ (floor). This calculation uses the AV18 deuteron wave function Wiringa:1994wb and the meson dominance nucleon form factors of Broniowski and Ruiz Arriola Broniowski:2025ctl.