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 α(0). The three solutions studied here are: α(0)=1, cut dominant; α(0)=1-ε, cuts dominant; and α(0)=1, 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 σₜₒₜ, ReF(s,0)/ImF(s,0), and the diffraction peak in the domain |t[ln(ss₀)][lnln(ss₀)]|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 $j=1$ even when α(0)=1-δ; (3) Schwarz trajectories α(t)=1+γt1/2+⋯ have a special status.