Breakdown of the Pomeranchuk Theorem and the Behavior of the LeadingJ-Plane Singularity

We prove that the leading $J$-plane singularity in the symmetric partial-wave amplitude $F_J^{}{}_{}{}^{+}\mathrm{}\left(t\right)\mathrm{}$ near $t=0\mathrm{}$ should behave like ${\alpha }_{\pm }\mathrm{}\left(t\right)=1\mathrm{}\pm A{t}^{\frac{1}{2}}+$ terms of higher order in $t$; namely, the Pomeranchuk pole (or cut) must be a pair of complex-conjugate poles (cuts) if the total cross sections $\sigma _T^{}{}_{}{}^{p,a}\mathrm{}\left(s\right)\mathrm{}\underset{s\to \infty }{\to }\mathrm{const}$ and $\sigma _T^{}{}_{}{}^P\left(\infty \right)\ne \sigma _T^{}{}_{}{}^a\left(\infty \right)$, where $p$ and $a$ denote particle-particle and antiparticle-particle scattering, respectively. We use only unitarity and analyticity to prove this.