The Riemann Surface of a Static Dispersion Model and Regge Trajectories

The S-matrix in the static limit of a dispersion relation is a matrix of a finite order N of meromorphic functions of energy ω in the plane with cuts (−∞,−1], [+1,+∞). In the elastic case it reduces to N functions Si(ω) connected by the crossing symmetry matrix A. The scattering of a neutral pseodoscalar meson with an arbitrary angular momentum l at a source with spin 1/2 is considered (N=2). The Regge trajectories of this model are explicitly found. The analytic structure of physical amplitudes in gauge theories with confinement was investigated in ref.[1]. It was shown that the analytic structure of hadron physical amplitudes established in old proofs of dispersion relations remains valid in QCD. It is well known[2]that the static limit of a dispersion relation is equivalent to the system of nonlinear integral eguations[3].Below,we will study this type of equations reducing them to a nonlinear boundary value problem [4]. It consists of the following series of conditions on Si, S–matrix elements: a) Si(z)−meromorphic functions in the complex plane z with cuts (−∞,−1], [+1,+∞), b) S i (z) = Si(z ), c) | Si(ω + i0) |2= 1 at ω ≥ 1 Si(ω + i0) = lim ǫ→+0 Si(ω + iǫ), d) Si(−z) = ∑N j=1 AijSj(z). (1)