Singularities and Accumulation of Singularities of $\pi$N Scattering amplitudes

TL;DR

This paper proves that specific points in the complex s-plane are accumulation points of poles for the $ o$N scattering amplitudes, revealing intricate singularity structures relevant to theoretical physics.

Contribution

It demonstrates, to all orders of chiral expansions, the accumulation of singularities at certain points for $ o$N scattering amplitudes, advancing understanding of their complex analytic structure.

Findings

Identifies accumulation points of poles in $ o$N amplitudes.

Shows these points are on the second sheet of the complex s-plane.

Validates the proof to all orders of chiral expansions.

Abstract

It is demonstrated that for the isospin $I=1/2$ $\pi$N scattering amplitude, $T^{I=1/2}(s,t)$, $s={(m_N^2-m_\pi^2)^2}/{m_N^2}$ and $s=m_N^2+2m_\pi^2$ are two accumulation points of poles on the second sheet of complex $s$ plane, and are hence accumulation of singularities of $T^{I=1/2}(s,t)$. For $T^{I=3/2}(s,t)$, $s={(m_N^2-m_\pi^2)^2}/{m_N^2}$ is the accumulation point of poles on the second sheet of complex $s$ plane. The proof is valid up to all orders of chiral expansions.