The non-topological Z string in the 331 model and its classical stability

We study the classical stability of a non-topological Z^ Z ′ string in the minimal 331 model, which arises from the maximal symmetry breaking pattern of an SU(6) SU ( 6 ) toy model. Two Higgs triplets are introduced according to the emergent global symmetries in the fermionic sector of the SU(6) SU ( 6 ) toy model, which will achieve the sequential symmetry breaking of SU(3)c⊗ SU(3)W ⊗ U(1)X→ SU(3)c⊗ SU(2)W ⊗ U(1)Y SU ( 3 ) c ⊗ SU ( 3 ) W ⊗ U ( 1 ) X → SU ( 3 ) c ⊗ SU ( 2 ) W ⊗ 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 ϑ S ≈ π /2 ϑ S ≈ π 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 SU(N>5) SU ( N > 5 ) Lie algebras.