Abstract
The Operator Product Expansion is a useful tool to represent correlation functions. In this note we extend Conformal Regge theory to provide an exact OPE representation of Lorenzian four-point correlators in conformal field theory, valid even away from Regge limit. The representation extends convergence of the OPE by rewriting it as a double integral over continuous spins and dimensions, and features a novel “Regge block”. We test the formula in the conformal fishnet theory, where exact results involving nontrivial Regge trajectories are available.
Highlights
Theories [1,2,3]
In this note we extend Conformal Regge theory to provide an exact OPE representation of Lorenzian four-point correlators in conformal field theory, valid even away from Regge limit
We test the formula in the conformal fishnet theory, where exact results involving nontrivial Regge trajectories are available
Summary
2.1 Review of conformal Regge kinematics A conformal four-point correlator in Minkowski space Md−1,1 can be expressed as. To evaluate the Regge limit, the Lorentzian correlator must be obtained from the Euclidean theory described above. It is calculated by analytically continuing the theory from the region where z = z∗, namely by rotating z around the branch point at z = 1 while keeping z fixed [3]. To understand the continuation path a little more explicitly, we recall that for Lorentzian correlators, time-like distances acquire a small imaginary part x223 → −|x23|2 ± i0 which is positive if the operators are in time-ordering and negative otherwise. The different phases originate from the prefactor in eq (2.1) These two discontinuities contain effectively the same information, and the fourth independent operator ordering, G(z, z ), can be reached by complex conjugation
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