Stability analysis of a non-unitary CFT

Abstract

A bstract We study instability of lowest dimension operator (i.e., the imaginary part of its operator dimension) in the rank- Q traceless symmetric representation of the O ( N ) Wilson-Fisher fixed point in D = 4 + ϵ . We find a new semi-classical bounce solution, which gives an imaginary part to the operator dimension of order $$ O\left({\epsilon}^{-1/2}\exp \left[-\frac{N+8}{3\epsilon }F\left(\epsilon Q\right)\right]\right) $$ O ϵ − 1 / 2 exp − N + 8 3 ϵ F ϵQ in the double-scaling limit where $$ \epsilon Q\le \frac{N+8}{6\sqrt{3}} $$ ϵQ ≤ N + 8 6 3 is fixed. The form of F ( ϵQ ), normalised as F (0) = 1, is also computed. This non-perturbative correction continues to give the leading effect even when Q is finite, indicating the instability of operators for any values of Q . We also observe a phase transition at $$ \epsilon Q=\frac{N+8}{6\sqrt{3}} $$ ϵQ = N + 8 6 3 associated with the condensation of bounces, similar to the Gross-Witten-Wadia transition.