First-order conformal perturbation theory by marginal operators

TL;DR

This work investigates first-order conformal perturbation theory by a marginal operator ${\mathcal O}$ with dimension $\Delta = d$ in a $d$-dimensional CFT. By combining the embedding-space formalism with dimensional regularization, the authors regulate the deformation and construct a renormalization scheme that preserves conformal invariance of the deformed correlators to linear order, explicitly computing the deformed two- and three-point functions. They show that the deformed two-point functions yield a shift in scaling dimensions, $\Delta_i \to \Delta_i - \lambda \mathcal{C}_{\mathcal{O}ii}$, with wavefunction renormalization $Z_i^{-1} = 1 + \lambda \mathcal{C}_{ii\mathcal{O}}/\epsilon$, and that the deformed three-point functions retain the conformal form with $\mathfrak{F}_{ijk}$ coefficients. The results extend to STT operators and provide a concrete framework for verifying conformal invariance under marginal perturbations, laying groundwork for general representations and relaxing genericity assumptions. The work also raises questions about the inverse problem and the completeness of marginal deformations in generating CFT families, with potential implications for the structure of conformal manifolds.

Abstract

We perform conformal perturbation theory by marginal operators to first order. A suitable renormalization method is needed that makes the conformal invariance of the deformed correlation functions manifest. Combining the embedding space formalism with the dimensional regularization, we explicitly check that the deformed and renormalized two and three point functions are conformally invariant.