The multiple-scattering series in pion-deuteron scattering and the nucleon-nucleon potential: perspectives from effective field theory

TL;DR

This paper uses effective field theory to analyze the multiple-scattering series in pion-deuteron scattering and nucleon-nucleon potentials, revealing divergence issues, resummation effects, and limitations of chiral perturbation theory at short distances.

Contribution

It provides EFT-based insights into the divergence and resummation of the multiple-scattering series, clarifying the applicability limits of chiral perturbation theory in nucleon-nucleon interactions.

Findings

Resummation cancels divergences in the scattering series.

Chiral perturbation theory breaks down below ~1 fm without explicit Delta degrees of freedom.

Poles in the resummed series indicate the theory's breakdown scale.

Abstract

Important contributions to meson-nucleus scattering are produced by terms in the multiple-scattering series, which is defined as the sum of all diagrams where the meson scatters back and forth between a pair of static nucleons before leaving the nucleus. In particular, the sum of this series is needed for an accurate description of kaon-deuteron scattering, and appears as part of the nucleon-nucleon potential. In this article we present some effective-field-theory (EFT)-based insights into this series in the case of two-nucleon systems. In particular, we discuss the fact that, if meson-nucleon scattering is approximated by the scattering-length term, individual terms of the series are divergent, and enhanced with respect to the straightforward expectation from chiral perturbation theory ($\chi$PT). This apparently indicates the presence of similarly enhanced counterterms. However, we show that when the series is resummed the divergences cancel, such that no additional information on short-range interactions is needed to obtain predictions for observables after resummation. We discuss the conditions under which this resummation is justified. We show that the same issues arise in the $NN$ potential, where the resummed series produces poles whose appearance indicates the breakdown scale of the $\chi$PT expansion for that quantity. This demonstrates unequivocally that $\chi$PT cannot be applied to compute $V(r)$ for distances smaller than $r \sim 1$ fm at least in the theory without explicit Delta(1232) degrees of freedom. We briefly discuss whether this bound can be lowered if the Delta resonance is included in the EFT as an explicit degree of freedom.