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Axion is a promising candidate of dark matter. After the Peccei-Quinnsymmetry breaking, axion strings are formed and attached by domain walls whenthe temperature of the universe becomes comparable to the QCD scale. Suchobjects can cause cosmological disasters if they are long-lived. As a solutionfor it, the Lazarides-Shafi mechanism is often discussed through introductionof a new non-Abelian (gauge) symmetry. We study this mechanism in detail andshow configuration of strings and walls. Even if Abelian axion strings with adomain wall number greater than one are formed in the early universe, each ofthem is split into multiple Alice axion strings due to a repulsive forcebetween the Alice strings even without domain wall. When domain walls areformed as the universe cools down, a single Alice string can be attached by asingle wall because a vacuum is connected by a non-Abelian rotation withoutchanging energy. Even if an Abelian axion string attached by domain walls arecreated due to the Kibble Zurek mechanism at the chiral phase transition, suchstrings are also similarly split into multiple Alice strings attached by wallsin the presence of the domain wall tension. Such walls do not form stablenetworks since they collapse by the tension of the walls, emitting axions.

Note on a solution to domain wall problem with the Lazarides-Shafi mechanism in axion dark matter models