  The mass of ultralight axions is determined in order to get the explicit
U(1)$_A$ symmetry breaking scale $ \Lambda $ at a Peccei-Quinn scale of the
magnitude of the Planck mass. It is assumed that the dominant contribution to
the mass of a galaxy with low surface brightness is only determined by one
axionic species in the sense of fuzzy dark matter (lumps). For rotation curve
fits to galactic rotation curves, therefore the Soliton-Navarro-Frenk-White
model is used, which assumes a condensate core plus correlated axions in the
halo according to the solution of the Poisson-Schr\"odinger system. In
addition, three commonly used mass density profiles are considered:
Navarro-Frenk-White, pseudo-isothermal and the Burkert model. An axion mass
$m_a$ of $0.675\times10^{-23}\,$eV is extracted, which reproduces previous
results in the literature. This implies an effective Yang-Mills scale of
$\Lambda \sim 287\,$ eV, which is only a factor of $15$ smaller than the
Yang-Mills scale of an SU(2) theory that is used to describe the first lepton
family. The cosmological model SU(2)$_{\rm CMB} $ suggests that three SU(2)
Yang-Mills theories, each for the formation of the lepton doublets (e,$\nu_e$),
($\mu$,$\nu_\mu$), and ($\tau$,$\nu_\tau$) are equally responsible for
contributing to the current density of dark matter. Parameters of an isolated
lump, such as the gravitational Bohr radius or the virial mass, are determined
solely by the Planck mass and the corresponding lepton mass. If the dominant
constituent of the dark mass contained in a galaxy is represented by e-lumps, a
mixture of $ \tau- $ and $ \mu$-lumps could possibly explain the presence of
massive compact objects in galactic centers, and $ \tau$-lumps could be related
to globular clusters and the halo mass. This might provide a theoretical
explanation for the mass gap between stellar and super massive black holes.
