Operator product expansions and recoil

TL;DR

This paper investigates recoil effects in AdS/CFT using OPE expansions for generalized free fields, revealing how conformal symmetry encodes collision dynamics and when recoil can be neglected.

Contribution

It introduces a detailed analysis of recoil effects via conformal group structures and Clebsch-Gordan coefficients, providing criteria for when recoil is negligible in AdS collisions.

Findings

Recoil effects are encoded in conformal group structures.

Maximum probability for heavy particle to remain in ground state.

Reduced mass concept emerges in slow-moving particle collisions.

Abstract

Some issues of recoil effects in AdS/CFT are studied from the point of view of OPE expansions for generalized free fields. We show that the conformal group structure encodes the relative energies and momenta at a collision center. This is done by being careful with the analysis of Clebsch-Gordan coefficients for an $SL(2)$ subalgebra of the conformal group. The collision fraction of kinetic energy carried by the particles is derived from a probability distribution that arises from these coefficients. We specifically identify a precise statement of when recoil of a heavy particle in AdS can be ignored: the maximum probability is for the heavy particle to be in its ground state. We also argue how a notion of reduced mass appears in these collisions, in the limit where the particles are moving slowly with respect to each other. This controls the notion of the impact parameter of the collision.