DOI: 10.1140/epjc/s2003-01179
Unified treatment: analyticity, Regge trajectories, Veneziano amplitude, fundamental regions and Moebius transformations
Abdur Rahim ChoudharyBell Laboratories, 6426 Grendel Place, Bowie, MD 20720, USA
(Received: 22 October 2002 / Revised version: 19 February 2003 Published online: 23 May 2003)
Abstract
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
