Residues of Regge Poles and the Diffraction Peak
Abstract
It is shown, on the basis of potential theory, that for a Regge pole with position α(t)>12 the reduced residue β(t)ν^{α(t)} behaves essentially as [α'(t)R]R^{2α(t)} near threshold (ν<~0), where R is the effective radius interaction. The quantity γ(t)=(m_{0}^{2}ν)^{α}β(t) which occurs in the asymptotic term γ(t)(s2m_{0}^{2})^{α} can therefore be considered as a slowly varying function of t only if one takes m_{0}=R^{1}. Assuming that our threshold expression gives a qualitative description in the relativistic case, we note that if Mr>1, where M is the nucleon mass, then the normalization m_{0}=M used in highenergy phenomenology will give rise to an exponential falloff of γ(t) with (width)^{1}~2α'(0) ln (MR). The values of R in the t channel of ππ, πN, and NN, estimated from the knowledge of the nearest lefthand branch points or factorization, occur in increasing order, and for each of them MR>1. Our results for the Pomeranchuk trajectory roughly reproduce the exponential diffraction shape and indicate that a larger total cross section implies a sharper falloff of the amplitude in the diffraction region. A connection between f^{0} and the diffraction width is discussed, as well as the question of zeros in βν^{α}.
 Publication:

Physical Review
 Pub Date:
 June 1965
 DOI:
 10.1103/PhysRev.138.B1174
 Bibcode:
 1965PhRv..138.1174D