[Paper Review] Chiral Symmetry and the Nucleon Nucleon Interaction

Keith G. Richardson|ArXiv.org|Aug 11, 2000

Quantum Chromodynamics and Particle Interactions5 references3 citations

TL;DR

This paper applies chiral effective field theory (EFT) to the nucleon-nucleon interaction, using Wilson renormalization group methods to establish two distinct power counting schemes: one for weak scattering around a trivial fixed point, and another for strong scattering around a non-trivial fixed point corresponding to a two-nucleon threshold bound state. The key result is that expanding the inverse scattering matrix around the non-trivial fixed point reproduces the effective range expansion order by order, and the $ν=3$ potential captures leading two-pion exchange contributions, showing improved agreement with phase shifts in peripheral partial waves like ${}^{1}F_{3}$ and ${}^{3}G_{5}$, though convergence of the small-momentum expansion remains uncertain.

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.

Motivation & Objective

• To develop a systematic power counting framework for nucleon-nucleon scattering using effective field theory (EFT) that respects chiral symmetry.
• To address the limitations of phenomenological meson exchange potentials by constructing a model-independent EFT approach with controlled systematic uncertainties.
• To investigate whether the small-momentum expansion in S-wave scattering converges, especially in the presence of two-pion exchange contributions.
• To isolate and test the effects of two-pion exchange in peripheral partial waves using a parameter-free EFT approach.

Proposed method

• Employing the Wilson renormalization group (RG) formalism to identify fixed points in a nucleon-only EFT at low energies.
• Using expansion around the trivial RG fixed point for weak scattering, and around a non-trivial fixed point (corresponding to a two-nucleon threshold bound state) for strong scattering.
• Deriving a power counting scheme for the inverse scattering matrix $1/K$ that reproduces the effective range expansion order by order.
• Constructing the NN potential up to order $\nu=3$ in small momenta, including leading and next-to-leading two-pion exchange contributions.
• Applying a coordinate-space cutoff to the potential and calculating phase shifts in peripheral partial waves ($\ell > 1$) where scattering is weak and power counting is valid.
• Comparing results with Nijmegen partial-wave analysis (PWA) data to assess convergence and predictive power.

Experimental results

Research questions

• RQ1 Can a systematic power counting scheme be derived for nucleon-nucleon scattering using EFT and RG methods in both weak and strong scattering regimes?
• RQ2 Does the small-momentum expansion in S-wave scattering converge, or are long-range or short-distance physics causing slow convergence?
• RQ3 Can two-pion exchange contributions be isolated and tested in peripheral partial waves without free parameters?
• RQ4 How do the $\nu=3$ potential and its components compare to OPE and leading-order two-pion exchange in reproducing phase shifts?
• RQ5 Is there evidence for convergence of the small-momentum expansion at $\nu=3$, or do higher-order terms dominate?

Key findings

• The expansion of the inverse scattering matrix around the non-trivial fixed point reproduces the effective range expansion order by order, validating the method for strong scattering.
• In peripheral partial waves such as ${}^{1}F_{3}$, ${}^{3}G_{5}$, ${}^{1}G_{4}$, and ${}^{3}F_{2}$, the EFT predictions at $\nu=3$ are parameter-free and show improved agreement with Nijmegen PWA data compared to OPE or leading-order two-pion exchange.
• The $\nu=3$ potential captures the first-order $\sigma$-like contributions, and in the ${}^{1}G_{4}$ wave, a single adjustable parameter $\alpha$ allows a satisfactory fit to data up to 300 MeV.
• The $\nu=3$ potential is an improvement over lower-order potentials, but the $\nu=3$ term is often larger than the $\nu=2$ correction, indicating no clear convergence of the small-momentum expansion at this order.
• In isoscalar waves, the full two-pion exchange potential tends to provide too little attraction, while in isovector waves (especially for $R<1.4$ fm), it provides too much; next-to-leading-order isovector contributions may correct this imbalance.
• The calculation to $\nu=4$ involves no new counterterms for $F$-waves and above, offering a cleaner test of convergence, though this order is not computed in the current work.