  We compute the quantum tunneling rate of dilute axion stars close to the
maximum mass [P.H. Chavanis, Phys. Rev. D {\bf 84}, 043531 (2011)] using the
theory of instantons. We confirm that the lifetime of metastable states is
extremely long, scaling as $t_{\rm life}\sim e^N\, t_D$ (except close to the
critical point), where $N$ is the number of axions in the system and $t_D$ is
the dynamical time ($N\sim 10^{57}$ and $t_D\sim 10\, {\rm hrs}$ for typical
QCD axion stars; $N\sim 10^{96}$ and $t_D\sim 100\, {\rm Myrs}$ for the quantum
core of a dark matter halo made of ultralight axions). Therefore, metastable
equilibrium states can be considered as stable equilibrium states in practice.
We develop a finite size scaling theory close to the maximum mass and predict
that the collapse time at criticality scales as $t_{\rm coll}\sim N^{1/5}t_D$
instead of being infinite as when fluctuations are neglected. The collapse time
is smaller than the age of the universe for QCD axion stars and larger than the
age of the universe for dark matter cores made of ultralight axions. We also
consider the thermal tunneling rate and reach the same conclusions. We compare
our results with similar results obtained for Bose-Einstein condensates in
laboratory, globular clusters in astrophysics, and quantum field theory in the
early Universe.
