Systematic approach to correlators in $T\bar{T}$ deformed CFTs

We investigate higher-order corrections to correlators in a general CFT (conformal field theory) with the double-trace $T\bar{T}$ deformation. Standard perturbation theory proves inadequate for this problem due to the intricate stress-tensor flow induced by the deformation. To tackle this challenge, we introduce a novel technique termed the conservation equation method. This method leverages the stress tensor's trace relation and conservation property to establish relationships between higher and lower-order corrections, and subsequently determine the correlators by enforcing symmetry properties. As an illustration, we compute first- and higher-order corrections to various types of correlators. Our results align with existing calculations in the literature and demonstrate the viability of our method for general CFTs.