Singularities and accumulation of singularities of πN scattering amplitudes

It is demonstrated that for the isospin I = 1/2 πN scattering amplitude, T^I=1/2(s, t), $s={\left({m}_{N}^{2}-{m}_{\pi }^{2}\right)}^{2}/{m}_{N}^{2}$ and $s={m}_{N}^{2}+2{m}_{\pi }^{2}$ 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), $s={\left({m}_{N}^{2}-{m}_{\pi }^{2}\right)}^{2}/{m}_{N}^{2}$ is the accumulation point of poles on the second sheet of the complex s plane. The proof is valid up to all orders of chiral expansions.