A scaling law for s-channel partial wave amplitudes from t-channel unitarity

Abstract

A solution is given for the following problem: What are the properties of s-channel partial-wave amplitudes in order to produce t-channel unitarity at threshold t = 4 μ 2? It turns out that the absorptive parts of the s-channel partial wave amplitudes Im ≈ l( s) have to obey the following scaling law: Im ≈( s)=√2 π exp (−4 sx/ μ 2) ( μ 6/ s 3) β( x)/ √ x, fors→∞ where x= l ( μ/√ s) 3 = fixed. The structure function β( x) is simply related to important physical quantities: i) The behaviour at x → 0 is determined by the leading Regge singularity (Pomeron) J = α( t=4 μ 2. ii) The asymptotic behaviour x → ∞, is completely determined by the s-wave scattering length. iii) The moments ∝ ∞ 0 dx x J+ 1 2 β (x) are determined by the scattering length of all the higher partial waves J = 2, 3, ….