Reconciling threshold and subthreshold expansions for pion–nucleon scattering

D Siemens,J Ruiz De Elvira,E Epelbaum,M Hoferichter,H Krebs,B Kubis,U.-G Meißner

doi:10.1016/j.physletb.2017.04.039

Abstract

Heavy-baryon chiral perturbation theory (ChPT) at one loop fails in relating the pion–nucleon amplitude in the physical region and for subthreshold kinematics due to loop effects enhanced by large low-energy constants. Studying the chiral convergence of threshold and subthreshold parameters up to fourth order in the small-scale expansion, we address the question to what extent this tension can be mitigated by including the Δ(1232) as an explicit degree of freedom and/or using a covariant formulation of baryon ChPT. We find that the inclusion of the Δ indeed reduces the low-energy constants to more natural values and thereby improves consistency between threshold and subthreshold kinematics. In addition, even in the Δ-less theory the resummation of 1/mN corrections in the covariant scheme improves the results markedly over the heavy-baryon formulation, in line with previous observations in the single-baryon sector of ChPT that so far have evaded a profound theoretical explanation.

Highlights

The approximate chiral symmetry of QCD imposes strong constraints on low-energy hadron dynamics, which can be explored systematically in the framework of chiral perturbation theory (ChPT) [1,2,3]
In this paper we address the question to what extent consistency between subthreshold and physical region can be restored by introducing the ∆ as an explicit degree of freedom, and/or by using a covariant formulation of baryon ChPT
We start off emphasizing that based on the employed power counting, there is no a priori argument why a manifestly covariant scheme should give improved results compared to the HB approach

Summary

Introduction

The approximate chiral symmetry of QCD imposes strong constraints on low-energy hadron dynamics, which can be explored systematically in the framework of chiral perturbation theory (ChPT) [1,2,3]. The matching to HBChPT revealed that, in contrast, the chiral representation is not accurate enough to relate these two regions [25] These findings can be best illustrated considering the parameters in the expansion around threshold and the subthreshold point: with LECs determined in the subthreshold region, where due to the absence of unitarity cuts ChPT is expected to converge best [34], the chiral series fails to reproduce some of the threshold parameters. The reason for this behavior can be traced back to loop diagrams producing terms that scale as g2A(c3 − c4) ∼ −16 GeV−1, an enhancement that is, at least partially, generated by saturation of the LECs ci with the ∆(1232) resonance. Details on large-Nc constraints and correlation coefficients of the extracted LECs are summarized in the appendices

Formalism

Results in a covariant formulation

Conclusions