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

TL;DR

This paper constructs an exactly solvable, asymptotically free quantum field theory in 3+1 dimensions using a PT-symmetric analytic continuation of the massless O(N) model, revealing phase transitions and bound states.

Contribution

It introduces a novel solvable quantum field theory in four dimensions via PT-symmetry, extending the understanding of asymptotic freedom and phase structure in such models.

Findings

The theory is exactly solvable in the large N limit.

It exhibits asymptotic freedom with a negative beta function.

A second-order phase transition occurs at finite temperature.

Abstract

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 $T_c\simeq \Lambda_c/\sqrt{e}$.