Infinite Heat Order in 3+1 Dimensions

TL;DR

This paper demonstrates the existence of a four-dimensional quantum field theory where spontaneous symmetry breaking persists at arbitrarily high temperatures, challenging the usual expectation of symmetry restoration.

Contribution

It extends previous large-N analyses to finite N, providing explicit finite-N models exhibiting infinite heat order with symmetry non-restoration at high temperatures.

Findings

Finite-N fixed-flow equations derived and solved numerically.

Explicit models found with negative scalar thermal mass at high temperature.

Gauge sector remains asymptotically free and potential bounded from below.

Abstract

We investigate whether spontaneous symmetry breaking can persist up to arbitrarily high temperature in ultraviolet-complete quantum field theories in four spacetime dimensions. We focus on completely asymptotically free models with gauge group $\mathrm{SU}(N_{c1})\times \mathrm{SU}(N_{c2})$ and two complex scalar fields, each transforming in the fundamental representation of one gauge factor and singlet under the other. The scalar potential contains quartic self-interactions together with a negative portal coupling between the two sectors. In the Veneziano limit, this class of theories was previously shown to admit fixed-flow trajectories for which one scalar acquires a negative thermal mass at asymptotically large temperature, leading to symmetry non-restoration. Here we extend that analysis to finite numbers of colours and flavours. We derive the finite-$N$ fixed-flow equations, compute the leading $1/N$ corrections to the large-$N$ solutions, and solve the full finite-$N$ system numerically. We find explicit finite-$N$ benchmark theories for which the scalar potential remains bounded from below, the gauge sector is asymptotically free, and one scalar thermal mass stays negative at arbitrarily high temperature. This provides an explicit perturbative example of infinite heat order in a four-dimensional ultraviolet-complete quantum field theory with a finite field content.