Positivity constraints on the low-energy constants of the chiral pion–nucleon Lagrangian

Abstract

Positivity constraints on the pion-nucleon scattering amplitude are derived in this article with the help of general S-matrix arguments, such as analyticity, crossing symmetry and unitarity, in the upper part of Mandelstam triangle, R. Scanning inside the region R, the most stringent bounds on the chiral low energy constants of the pion-nucleon Lagrangian are determined. When just considering the central values of the fit results from covariant baryon chiral perturbation theory using extended-on-mass-shell scheme, it is found that these bounds are well respected numerically both at O(p^3) and O(p^4) level. Nevertheless, when taking the errors into account, only the O(p^4) bounds are obeyed in the full error interval, while the bounds on O(p^3) fits are slightly violated. If one disregards loop contributions, the bounds always fail in certain regions of R. Thus, at a given chiral order these terms are not numerically negligible and one needs to consider all possible contributions, i.e., both tree-level and loop diagrams. We have provided the constraints for special points in R where the bounds are nearly optimal in terms of just a few chiral couplings, which can be easily implemented and employed to constrain future analyses. Some issues about calculations with an explicit Delta(1232) resonance are also discussed.