Self-Consistent Pomeranchukon Singularities. I

We formulate a functional self-consistency equation for the Pomeranchukon pole trajectory. The only ingredients are unitarity in the $t$ channel (the channel of the Pomeranchukon) and the specification of $\alpha \mathrm{}\left(0\right)\mathrm{}$. The three solutions studied here are: $\alpha \mathrm{}\left(0\right)=1\mathrm{}$, cut dominant; $\alpha \mathrm{}\left(0\right)=1-\epsilon$, cuts dominant; and $\alpha \mathrm{}\left(0\right)=1\mathrm{}$, pole dominant. We further provide expressions for the $t$-channel partial-wave amplitudes in terms of a small number of free parameters; these expressions are adequate for the calculation of the high-$s$ behavior of ${\sigma }_{\mathrm{tot}}$, $\frac{\mathrm{Re}F(s,0\mathrm{})\mathrm{}}{\mathrm{Im}F(s,0\mathrm{})\mathrm{}}$, and the diffraction peak in the domain $\left|\frac{t\left[\ln \right(\frac{s}{s_0}\left)\right]}{\left[\ln \ln \right(\frac{s}{s_0}\left)\right]}\right|\lesssim \mathrm{constant}$. Three prominent conclusions are: (1) multi-Pomeranchukon phase space plays a leading role in determining the relative pole-cut strength; (2) there are fixed cuts in the vacuum partial-wave amplitude that accumulate at $j=1\mathrm{}$ even when $\alpha \mathrm{}\left(0\right)=1-\delta$; (3) Schwarz trajectories $\alpha \mathrm{}\left(t\right)=1+\gamma t^{\frac{1}{2}}+\cdots$ have a special status.