A scaling property of shrinking diffraction peaks

Abstract

We prove that b(s)≡d/dtln(dσ^A/dt(s,t)|_{t=0 s→∞}≤1/4t₀[ln(s²/dσ^A/dt(s,0))]², where (dσ/dt)^A(s,t) is the absorptive contribution to the elastic unpolarized differential cross section for particles with arbitrary spins at c.m. energy and momentum transfer √s, √−t, and t₀ is the right extremity of its Lehmann-Martin ellipse for s→∞. If this inequality is saturated apart from a constant factor, then there must exist sequences of s_n→∞ such that lim_sn→∞[dσ^A/dt(s_n,t=−τ/b(s_n))/dσ^A/dt(s_n,0)]=ƒ(τ), where ƒ(τ) is an entire function of order ½.