We study instability of the lowest dimension operator (\it i.e., \rm the imaginary part of its operator dimension) in the rank-$Q$ traceless symmetric representation of the $O(N)$ Wilson-Fisher fixed point in $D=4+\epsilon$. We find a new semi-classical bounce solution, which gives an imaginary part to the operator dimension of order $O\left({ { { \epsilon^{ -1/2}}}}\exp\left[-\frac{ N+8}{ 3\epsilon}F(\epsilon Q)\right]\right)$ in the double-scaling limit where $\epsilon Q \leq \frac{ N+8}{ 6\sqrt{ 3}}$ is fixed. The form of $F(\epsilon Q)$, 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 $\epsilon Q=\frac{ N+8}{ 6\sqrt{ 3}}$ associated with the condensation of bounces, similar to the Gross-Witten-Wadia transition.
Categories: hep-th cond-mat.str-el