[Paper Review] Low scale saturation of Effective NN Interactions and their Symmetries

E. Ruiz Arriola|arXiv (Cornell University)|Sep 21, 2010

Quantum, superfluid, helium dynamics4 citations

TL;DR

This paper demonstrates that effective nucleon-nucleon (NN) interactions, specifically $V_{\rm lowk}$ potentials, reach saturation in their low-energy parameters at momentum scales of $\Lambda \approx 250$ MeV for S-waves and $\Lambda \approx 100$ MeV for P-waves, uniquely determining Skyrme force parameters from physical scattering threshold data. The saturation of Wigner and Serber symmetries at these scales confirms their universality and links them to underlying quark-gluon dynamics via large-$N_c$ relations.

ABSTRACT

The Skyrme force parameters can be uniquely determined by coarse graining the NN interactions at a characteristic momentum scale. We show how exact Vlowk potentials to second order in momenta are accurately and universally saturated with physical NN scattering threshold parameters at CM momentum scales of about Lambda=250 MeV for the S-waves and Lambda=100 MeV for the P-waves. The pattern of Wigner and Serber symmetries unveiled previously is also saturated at these scales.

Motivation & Objective

• To determine the unique momentum scale $\Lambda$ at which effective NN interactions saturate in their low-energy parameters.
• To establish a direct link between physical NN scattering threshold parameters and the parameters of Skyrme-type effective interactions.
• To investigate the saturation of Wigner and Serber symmetries in $V_{\rm lowk}$ potentials at low momentum scales.
• To validate the universality and accuracy of $V_{\rm lowk}$-based effective interactions by comparing with ab initio calculations and phenomenological fits.

Proposed method

• Use of the $V_{\rm lowk}$ method to coarse-grain NN interactions by integrating out momentum modes above a scale $\Lambda$.
• Matching of $V_{\rm lowk}$ potentials to second-order momentum expansions in the form of Skyrme-like pseudo-potentials.
• Calculation of effective parameters $t_0, t_1, t_2, x_0, x_1, x_2, t_T, t_U, W_0$ from on-shell NN scattering threshold parameters (scattering length, effective range).
• Comparison of derived parameters with exact $V_{\rm lowk}$ results from Argonne-V18 interactions to validate accuracy.
• Analysis of symmetry saturation via $x_0 = 0$ (Wigner) and $x_2 = -1$ (Serber) conditions in the effective interaction.
• Use of large-$N_c$ QCD relations to interpret symmetry saturation as a fingerprint of underlying quark-gluon dynamics.

Experimental results

Research questions

• RQ1 At what momentum scale $\Lambda$ do $V_{\rm lowk}$ potentials for S- and P-waves saturate in their low-energy parameters?
• RQ2 Can Skyrme force parameters be uniquely determined from physical NN scattering threshold parameters via $V_{\rm lowk}$ coarse graining?
• RQ3 To what extent are Wigner and Serber symmetries saturated at low momentum scales in effective NN interactions?
• RQ4 How accurately do the derived effective parameters reproduce exact $V_{\rm lowk}$ values from Argonne-V18 potentials?
• RQ5 What is the role of higher-order terms and three-body contributions in the saturation of low-energy parameters?

Key findings

• The $V_{\rm lowk}$ potential for S-waves saturates at $\Lambda \approx 250$ MeV, with effective parameters matching exact $V_{\rm lowk}$ values to within 10%.
• For P-waves, saturation occurs at $\Lambda \approx 100$ MeV, with similar accuracy in reproducing $V_{\rm lowk}$ parameters.
• The spin-orbit coupling strength $W_0$ is predicted as $72$ MeV·fm⁴, matching the exact $V_{\rm lowk}$ value within 10%.
• Wigner symmetry ($x_0 = 0$) and Serber symmetry ($x_2 = -1$) are saturated with high accuracy at the respective saturation scales.
• The saturation of symmetries at $\Lambda \sim 250$ MeV supports their role as fingerprints of large-$N_c$ QCD dynamics.
• Higher-order terms beyond second order in momenta are found to be negligible for saturation, provided on-shell scattering parameters are correctly reproduced.