Conformal Perturbation Theory for $n$-Point Functions: Structure Constant Deformation

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 $\epsilon$ 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 $\infty$. We show that approximating the mapped excised regions to leading order in $\epsilon$ does not lead to the same perturbative shift to the structure constant as the exact in $\epsilon$ region. We explicitly compute the correction back to the exact in $\epsilon$ region of integration in terms of the CFT data. We consider the compact boson, and show that one must use the exact in $\epsilon$ region to obtain agreement with the exact results for structure constants in this theory.