  We investigate the decay of condensates of scalars in a field theory defined
by $V({\cal A})=m{}^2\,f{}^2\,[1-\cos({\cal A}\,/\,f)]$, where $m$ and $f$ are
the mass and decay constant of the scalar field. An example of such a theory is
that of the axion, in which case the condensates are called axion stars. The
axion field, $\cal A$, is self adjoint. As a result the axion number is not an
absolutely conserved quantity. Therefore, axion stars are not stable and have
finite lifetimes. Bound axions, localized on the volume of the star, have a
coordinate uncertainty $\Delta x \sim R \sim 1/(m_a \Delta)$, where $R$ is the
radius of the star and $\Delta = \sqrt{1-E_0{}^2\,/\,m_a^2}$. Here $m_a$ and
$E_0$ are the mass, and the ground state energy of the bound axion. Then the
momentum distribution of axions has a width of $\Delta p\sim m_a\,\Delta$. At
strong binding, $\Delta={\cal O}(1)$, bound axions can easily transfer a
sufficient amount of momentum to create and emit a free axion, leading to fast
decay of the star with a transition rate $\Gamma \sim m_a$. However, when
$\Delta\ll 1$, the momentum distribution is more restricted, and as shown in
this paper, the transition rate for creating a free axion decreases as $\exp
(-p \, \Delta x) \sim \exp (-\Delta^{-1})$. Then sufficiently large, weakly
bound axion stars, produced after the big bang, survive until the present time.
We plot the region of their stability, limited by decay through axion loss and
by gravitational instability, as a function of the mass of the axion and the
mass of the star.
