Chiral Symmetry and the Nucleon Nucleon Interaction

TL;DR

This paper explores the application of Effective Field Theory to nucleon-nucleon interactions, emphasizing chiral symmetry and renormalization group methods, and assesses the convergence of the small momentum expansion in this context.

Contribution

It introduces a systematic EFT framework for NN interactions incorporating chiral symmetry and analyzes the convergence issues of the small momentum expansion.

Findings

Expansion around fixed points reproduces effective range expansion.

Two-pion exchange effects can be isolated in certain partial waves.

Slow convergence of the small momentum expansion is observed at third order.

Abstract

Various aspects of the application of Effective Field Theory (EFT) to the Nucleon-Nucleon (NN) interaction are considered. We look for contributions beyond One Pion Exchange which are predicted by Chiral Symmetry. Using the formalism of the Wilson Renormalisation Group (RG) we review power counting in a simplified EFT containing only nucleons. For weak scattering at low energy, we find a natural expansion of the scattering matrix around the unique trivial fixed point of the RG. For strong scattering at low energy, the calculation can be organised in a useful and systematic way by expanding the potential around a non-trivial fixed point corresponding to a bound state of two nucleons at threshold. The resulting expansion of the inverse of the scattering matrix reproduces the effective range expansion order by order. The extension of this EFT to include pions in a manner consistent with chiral symmetry is discussed. By considering a modified effective range expansion, we find that the small momentum expansion in S-wave scattering converges slowly, if at all. The NN potential is written down to third order in small momenta. With a cut-off in coordinate space, we calculate phase shifts in peripheral partial waves for which the EFT predictions are parameter-free, and for which we may use the expansion around the trivial fixed point. We find several partial waves in which the effects of two-pion exchange can be isolated, but no strong evidence is found for the convergence of the small momentum expansion at this order.