Unified treatment: analyticity, Regge trajectories, Veneziano amplitude, fundamental regions and Moebius transformationsAbdur Rahim Choudhary
Bell Laboratories, 6426 Grendel Place, Bowie, MD 20720, USA
(Received: 22 October 2002 / Revised version: 19 February 2003 Published online: 23 May 2003)
In this paper we present a unified treatment that combines the analyticity properties of the scattering amplitudes, the threshold and asymptotic behaviors, the invariance group of Moebius transformations, the automorphic functions defined over this invariance group, the fundamental region in (Poincaré) geometry, and the generators of the invariance group as they relate to the fundamental region. Using these concepts and techniques, we provide a theoretical basis for Veneziano type amplitudes with the ghost elimination condition built in, related the Regge trajectory functions to the generators of the invariance group, constrained the values of the Regge trajectories to take only inverse integer values at the threshold, used the threshold behavior in the forward direction to deduce the Pomeranchuk trajectory as well as other relations. The enabling tool for this unified treatment came from the multi-sheet conformal mapping techniques that map the physical sheet to a fundamental region which in turn defines a Riemann surface on which a global uniformization variable for the scattering amplitude is calculated via an automorphic function, which in turn can be constructed as a quotient of two automorphic forms of the same dimension.
© Società Italiana di Fisica, Springer-Verlag 2003