A solvable quantum field theory with asymptotic freedom in 3+1 dimensions

Recently, Ai, Bender and Sarkar gave a prescription on how to obtain $\mathcal{P}\mathcal{T}$-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 ${\mathrm{\Lambda }}_c$, but can be continued into the far infrared. At finite temperature, the theory exhibits two phases separated by a second-order phase transition near $T_c\simeq {\mathrm{\Lambda }}_c\mathrm{/}\sqrt{e}$.