Conformal perturbation theory for n-point functions: structure constant deformation

Abstract

We consider conformal perturbation theory for n-point functions on the sphere in general 2D CFTs to first order in coupling constant. We regulate perturbation integrals using canonical hard disk excisions of size ϵ around the fixed operator insertions, and identify the full set of counter terms which are sufficient to regulate all such integrated n-point functions. We further explore the integrated 4-point function which computes changes to the structure constants of the theory. Using an sl(2) map, the three fixed locations of operators are mapped to 0, 1, and ∞. We show that approximating the mapped excised regions to leading order in ϵ does not lead to the same perturbative shift to the structure constant as the exact in ϵ region. We explicitly compute the correction back to the exact in ϵ region of integration in terms of the CFT data. We consider the compact boson, and show that one must use the exact in ϵ region to obtain agreement with the exact results for structure constants in this theory.