Operator product expansions at large distances and Regge asymptotic behavior

Abstract

The class of theories is considered in which the simplest high-energy scattering processes are described by the exchange of a Regge pole or cut and in which light-cone behavior is described by operator product expansions. The scattering processes are those in which an initial particle interacts with an initial cluster of other particles to produce a final particle and final cluster. The high energy limit is that in which the energy of the initial particle relative to the energy of any particle in the clusters becomes large with fixed masses, momentum transfers and cluster subenergies. It is shown that these limits can be completely characterized by an operator statement on the product of the interpolating fields of the initial and final particles, independently of the initial and final clusters. Given then that these limits correspond to large distance limits in space-time, operator product expansions are obtained which describe in an operator manner the behavior of a product A( x) B(0) of local fields in the large distance limit x 0 → ∞ with x 2 fixed. This provides an exact operator picture of a Regge pole being built out of an infinite set of elementary spin exchanges. The crucial ingredient in the derivation is the factorization property of Regge poles or Regge cut discontinuities. The leading light-cone contribution to the scattering amplitude will then have the same factorization properties. The operator expansion is derived first for single particle matrix elements, first using commutativity of the Regge and scaling limits and second using diagonalized Bethe-Salpeter equations for the matrix elements of the local fields in the light-cone expansion. It is then deduced for single-double particle matrix elements using the second method where now helicity sums play an important role. It is finally derived for arbitrary matrix elements. Non-leading Regge poles and Regge cuts can be similarly described by using also non-leading light-cone singularities.