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

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.