Closing the loop on $Φ^4$ in AdS$_3$

Abstract

We compute the one-loop correction to the CFT data of all double-trace operators $[ϕϕ]_{n,\ell}$ for a $Φ^4$ theory in AdS$_3$, for arbitrary values of $n$, $\ell$, and of the scaling dimension $Δ_ϕ>1$. Working in the spectral representation, the $t$-channel one-loop bubble diagram is reduced to a product of spectral integrals dressed by the conformal $6j$ symbol. Both the spectral integrals and the subsequent sums over residues are performed analytically, yielding finite closed-form expressions for the anomalous dimensions in terms of higher hypergeometric functions. We discuss the structure of the results, including their large-spin and high-energy behaviors, and show that the anomalous dimensions are completely monotonic in spin.