The non-topological $Z^\prime$ string in the 331 model and its classical stability

TL;DR

This paper investigates the classical stability of a non-topological $Z^\prime$ string in the 331 model, finding it stable only near a specific parameter limit, which questions its existence in larger unified theories.

Contribution

It provides a detailed stability analysis of a non-topological string in the 331 model, revealing stability conditions and implications for theories based on larger Lie algebras.

Findings

The string is stable only near the semilocal limit of $ heta_S o rac{ ext{ extpi}}{2}$.

Higgs self-couplings do not significantly extend the stability region.

Non-topological strings are unlikely in $SU(N>5)$ unified theories.

Abstract

We study the classical stability of a non-topological $Z^\prime$ string in the minimal 331 model, which arises from the maximal symmetry breaking pattern of an ${{\mathfrak s}{\mathfrak u}}(6)$ toy model. Two Higgs triplets are introduced according to the emergent global symmetries in the fermionic sector of the ${{\mathfrak s}{\mathfrak u}}(6)$ toy model, which will achieve the sequential symmetry breaking of ${{\mathfrak s}{\mathfrak u}}(3)_c\oplus {{\mathfrak s}{\mathfrak u}}(3)_W \oplus {\mathfrak u}(1)_X\to {{\mathfrak s}{\mathfrak u}}(3)_c\oplus {{\mathfrak s}{\mathfrak u}}(2)_W \oplus {\mathfrak u}(1)_Y$. By analyzing small perturbations around the string background and solving the coupled Helmholtz equations numerically, we find that the string is stable only near the semilocal limit of $\vartheta_S \approx \frac{\pi}{2}$, even when Higgs self-couplings are tuned to minimize instabilities. This suggests that such non-topological strings are unlikely to exist in unified theories based on ${{\mathfrak s}{\mathfrak u}}(N>5)$ Lie algebras.