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

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 $\mathrm{SU}(6)$ toy model. Two Higgs triplets are introduced according to the emergent global symmetries in the fermionic sector of the $\mathrm{SU}(6)$ toy model, which will achieve the sequential symmetry breaking of $\mathrm{SU}(3{)}_c\otimes \mathrm{SU}(3{)}_W\otimes \mathrm{U}(1{)}_X\to \mathrm{SU}(3{)}_c\otimes \mathrm{SU}(2{)}_W\otimes \mathrm{U}(1{)}_Y$. 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 $\mathrm{SU}(N>5)$ Lie algebras.

Comments: 23 pages with references, 1 table, and 4 figures, matches the published version