  Recently, Ai, Bender and Sarkar gave a prescription on how to obtain
$\mathcal{PT}$-symmetric field theory results from an analytic continuation of
Hermitian field theories. I perform this analytic continuation for the massless
(critical) O(N) model with quartic interaction in 3+1 dimensions. In the large
N limit, this theory is exactly solvable, and has negative $\beta$-function in
the ultraviolet, and a stable bound state in the infrared. The coupling
diverges at a scale $\Lambda_c$, but can be continued into the far infrared. At
finite temperature, the theory exhibits two phases separated by a second-order
phase transition near $T_c\simeq \Lambda_c/\sqrt{e}$.
