(In)stability of the Higgs vacuum from the $\textrm{O}(N)$ model at large $N$

TL;DR

This paper reexamines the large-$N$ $ extrm{O}(N)$ model for the Higgs field, demonstrating it is stable and well-defined, with the true vacuum lacking spontaneous symmetry breaking, contrary to earlier dismissals.

Contribution

It clarifies the stability and phase structure of the large-$N$ $ extrm{O}(N)$ model, resolving historical misconceptions about its 'sickness' and identifying the true vacuum state.

Findings

The model is stable and not 'sick' at large $N$.

The true vacuum has no spontaneous symmetry breaking.

Spontaneous symmetry breaking occurs only at high temperatures.

Abstract

The theory of an independent Higgs field is given by an $\textrm{O}(N)$ model with an $N$-component scalar $\vec{\phi}$ and a quartic $\lambda(\vec{\phi}\cdot\vec{\phi})^2$ potential when $N=4$. The phase structure of the theory can be studied analytically for all values of the coupling $\lambda$ using the large-$N$ limit, both at zero and finite temperature. However, authors in the 70s and 80s argued the theory at large $N$ was "sick" and "futile", and dismissed the theory. This was based on two points: (1) a failure to identify the stable phases and vacuum of the theory and (2) the issue of a negative bare coupling $\lambda<0$ in the UV. We provide evidence that the theory is not, in fact, "sick". Issue (2) is dealt with through the modern understanding of $\mathcal{P}\mathcal{T}$-symmetric non-Hermitian theories with "wrong-sign" couplings. Issue (1) is resolved by realizing that the true vacuum has no spontaneous symmetry breaking (SSB) and that the SSB phase only becomes preferred at high temperatures.