Dynamical fit to low-energyπ−Nphase shifts and determination of the threshold parameters

Abstract

For the description of low-energy $\pi N$ scattering, [1/1] Padé approximants have had limited success starting from Lagrangian-induced power series. We have shown elsewhere that, from a formal power series whose generating kernel can in principle be approximated by a kernel of finite rank $N$, we can construct a democratic approximant $A^N$ with $N$ perturbative terms which provides as good an approximation to the true solution as a Padé approximant [$\frac{N}{N}$] with $2N$ perturbative terms. Here we use the two available orders of perturbative terms $g^2$ and $g^4$ of the Lagrangian $g\tilde{\psi }{\gamma }_5\psi \phi$ to construct a democratic approximant $A^{N=2}$. We apply it to the low-energy $\pi N$ phase-shift analysis of Carter, Bugg, and Carter and show empirically that a reasonably good fit can be obtained in the low-energy region with the two available orders of perturbative terms. Extrapolating this fit to threshold we determine scattering lengths and effective ranges for $S$ and $P$ waves which are in reasonably good agreement with more conventional dispersion-relation determinations. The method indicates how the concept of Lagrangian can be made dynamically relevant in a strong-interaction context.