Operator product expansions and recoil

David Berenstein,Ruwanmali Bernadette De Zoysa

doi:10.1103/physrevd.105.026019

David Berenstein, Ruwanmali Bernadette De Zoysa

Open Access

https://doi.org/10.1103/physrevd.105.026019

Copy DOI

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.

Highlights

In the AdS=CFT correspondence [1,2,3], the gravity intuition is supposed to be equivalent to conformal field theory intuition, once we have a dictionary that properly identifies how these two are supposed to match
From certain bound state problems, is that if we have a light particle bound to a heavy particle, we can assume that the center of mass is located at the position of the heavy particle
In this paper we have studied the kinematics of tensor products of representations of the conformal group

Summary

TWO INTUITIONS COLLIDE

In the AdS=CFT correspondence [1,2,3], the gravity intuition is supposed to be equivalent to conformal field theory intuition, once we have a dictionary that properly identifies how these two are supposed to match. Goldstone boson should appear that restores the symmetry This does happen, but because the commutation relations of the spontaneously broken generators with the Hamiltonian do not vanish, the modes are massive (this is important in the discussion of the difference between mass and dimension for particles in AdS [4]). These modes generate the descendants of the representation of the conformal group associated to the heavy object. One uses the same symbol O3 ∼ O1ð∂↔ÞnO2 to indicate that for very large n (the dimension of the primaries) the right-hand side will generically be very similar to the result of generalized free fields between the two operators.

SLð2Þ CLEBSCH-GORDAN COEFFICIENTS

RECOIL EFFECTS

ANGULAR MOMENTUM AND AdS GEOMETRY

CONCLUSION