Scattering amplitude expansion in the transverse-momentum plane

Abstract

1. -The relativistic generalizations of the eikonal approximation provide a useful to01 for a phenomenological description of high-energy scattering, though the extension to large angles contains a good deal of arbitrariness (1). The only condition that is generally fulfilled in these models fixes the l imit of the impact parameter representation at high energy and small angles to coincide with the earlier eikonal formulation. In this context, the important role played by the Bessel functions of first kind can be justified, for example, either by means of an integral representation for the Legendre polynomials in the partial-wave expansion or simply by taking the l imit of iP~(cos0) for large values of I. I n the two cases, however, one gets a different form for the angular dependence of the Bessel function. In recent years the impact parameter representation has been obtained from a grouptheoretic analysis of the S-matrix element for the two-body elastic scattering P l Y P 2 + P a k P4 (3,3). The derivation is carried out by noticing that a rotation about the axis orthogonal to the scattering plane is not the only Lorentz transformation that brings the vector q~ = (Pl--P2)~ into q~ = ( P a P 4 ) ~ . In the c.m. frame, for example, the same task is accomplished by a boost along P l followed by a rotat ion and a subsequent boost along a direction lying in the transverse plane. The generators of these transformations form the Lie algebra of the similitude group in two dimensions $2 (2.3), that is closely related to the Euclidian group of motion in a two-dimensional plane E~. In fact, if we take the z-axis along the direction of the incident momentum P l , $2 can also be generated by adding the dilation operator tiYa to the Lie algebra of E 2. Moreover, utilizing elements in the enveloping algebra of the Poincar4-Lie algebra, i t is possible to (~ project ~> the similitude group onto the direct product of Ea and a one-dimensional Abelian group. In ref. (a) this procedure led to the expansion of the scattering amplitude in terms of the un i ta ry principal-series representation of the Euclidian