Entanglement and accidental symmetries in the nucleon-nucleon system

TL;DR

This work investigates how accidental spin-isospin symmetries, notably $SU(4)$ Wigner and Serber symmetries, constrain spin entanglement in two-nucleon scattering within chiral EFT. By embedding symmetry-enhanced LO potentials and computing the full $S$-matrix, it quantifies entanglement via the two-entanglement measure $\mathcal{E}_{\text{2E}}$ and the entanglement power $\mathcal{E}_{\text{EP}}$, and extends the analysis to $N^2LO$ for both $np$ and $nn$, with comparison to NijmI. Key findings show entanglement suppression in $np$ at low energies when $SU(4)$ is approximately preserved, while tensor forces drive entanglement at higher momenta and nonforward angles; $nn$ scattering exhibits no suppression due to Pauli constraints, yielding stronger entanglement. Forward-scattering results match the analytic $S$-wave formulas at low energy, highlighting distinct behavior between forward and nonforward directions and underscoring entanglement as a diagnostic tool for symmetry content and EFT power counting in nuclear interactions.

Abstract

We study the connection between accidental symmetries in the nuclear interaction and spin entanglement in two-nucleon scattering. Specifically, we incorporate different levels of Wigner $SU(4)$ and Serber symmetries into leading-order potentials derived from chiral effective field theory. We conduct a quantitative analysis by computing the full $S$ matrix, demonstrating that the neutron-proton spin entanglement can be related to the symmetry properties of the interaction and the presence of certain operators and partial waves. Furthermore, we study the order-by-order evolution of the spin entanglement, up to next-to-next-to-leading order in Weinberg power counting, for both neutron-proton and neutron-neutron scattering. Entanglement suppression is not observed in neutron-neutron scattering, which can be attributed to the Pauli principle and the absence of accidental symmetries in this system. We conclude that entanglement is a useful guide for studying the power counting and symmetries in nuclear interactions derived from effective field theories.

Figures (5)

• Figure 1: NijmI Stoks:1994wp$S$-wave phase shifts. The red stars are the ${^1\!S_0}$ and ${^3\!S_1}$ phase shifts at $p = 22$ MeV ($T_{\text{lab}}=1$ MeV). The blue diamond is the average $S$-wave phase shift at $p = 119$ MeV ($T_{\text{lab}}=30$ MeV).
• Figure 2: Entanglement power of the $M$ matrix $\mathcal{E}_{\text{EP}}(\boldsymbol{M})$ for the $np$ system using the LO WPC and the modified LO WPC potentials listed in \ref{['tab:modified_LO_potentials_np']}. Each LEC is tuned to a single $S$-wave phase shift, see Section \ref{['subsubsec:LECs']}, in all potentials (even the LO WPC potential). $\mathcal{E}_{\text{EP}}(\boldsymbol{M})$ is evaluated for scattering angles $\theta_{\text{c.m}}\in[1^\circ, 180^\circ]$ and relative momenta $p\in [0.01,300]$ MeV.
• Figure 3: Entanglement power of the $M$ matrix $\mathcal{E}_{\text{EP}}(\boldsymbol{M})$ for the $np$ system using the WPC potential at different chiral orders and using the phenomenological NijmI potential Stoks:1994wp. The LECs in the WPC potentials are from Ref. Carlsson:2015vda. $\mathcal{E}_{\text{EP}}(\boldsymbol{M})$ is evaluated for scattering angles $\theta_{\text{c.m.}}\in[1^\circ, 180^\circ]$ and relative momenta $p\in[7,350]$ MeV.
• Figure 4: Entanglement power of the $M$ matrix $\mathcal{E}_{\text{EP}}(\boldsymbol{M})$ for the $nn$ system using the WPC potential at different chiral orders. The lower right panel is the $pp$ result using the NijmI potential Stoks:1994wp without the electromagnetic contribution. Due to charge symmetry, the nuclear $pp$ and $nn$ M matrices should be very similar. The LECs in the WPC potentials are from Ref. Carlsson:2015vda. $\mathcal{E}_{\text{EP}}(\boldsymbol{M})$ is evaluated on the same angle-energy grid as in \ref{['fig:EP_LO_NLO_N2LO_Nijm_np']}.
• Figure 5: Entanglement power of the $S$ matrix in the forward scattering direction using the N$^2$LO WPC potential for $np$ and $nn$ scattering. The solid lines are the result for the full $S$ matrix while the dashed lines are the $S$-wave result using the analytical formula for the entanglement power as derived in Ref. Beane:2018oxh, see Eqs. \ref{['eq:S_wave_EP_np']} and \ref{['eq:S_wave_EP_nn']}. Note that the exact solution and the $S$-wave solution align for momenta $p\lesssim 50$ MeV for both $np$ and $nn$ scattering.