Note on a solution to domain wall problem with the Lazarides-Shafi
  mechanism in axion dark matter models
  Axion is a promising candidate of dark matter. After the Peccei-Quinn
symmetry breaking, axion strings are formed and attached by domain walls when
the temperature of the universe becomes comparable to the QCD scale. Such
objects can cause cosmological disasters if they are long-lived. As a solution
for it, the Lazarides-Shafi mechanism is often discussed through introduction
of a new non-Abelian (gauge) symmetry. We study this mechanism in detail and
show configuration of strings and walls. Even if Abelian axion strings with a
domain wall number greater than one are formed in the early universe, each of
them is split into multiple Alice axion strings due to a repulsive force
between the Alice strings even without domain wall. When domain walls are
formed as the universe cools down, a single Alice string can be attached by a
single wall because a vacuum is connected by a non-Abelian rotation without
changing energy. Even if an Abelian axion string attached by domain walls are
created due to the Kibble Zurek mechanism at the chiral phase transition, such
strings are also similarly split into multiple Alice strings attached by walls
in the presence of the domain wall tension. Such walls do not form stable
networks since they collapse by the tension of the walls, emitting axions.
