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

We investigate higher-order corrections to correlators in a general CFT with the double trace $T\bar{T}$ deformation. Traditional perturbation theory proves inadequate for addressing this issue, 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 trace relation and conservation property of the stress tensor to establish relationships between higher and lower-order corrections and subsequently determine the correlators by enforcing symmetry properties. As an illustration, we compute both first and higher-order corrections, demonstrating the impact of stress tensor deformation on correlators in a general deformed CFT. Our results align with existing calculations in the literature.