  We have long ago derived a theoretical relation between the mass of the
electron and the fine structure constant \cite{Nottale1994}, which writes to
lowest order $\alpha \ln (m_{\mathbb{P}}/m_e) = 3/8$ (where $m_{\mathbb{P}}$ is
the Planck mass). We suggest the existence of a similar relation valid for
neutrinos, $\alpha \ln ({m_{\mathbb{P}}}/{m_\nu} ) =1/2$. From this relation,
we theoretically predict a lightest neutrino mass $m_{\nu} = m_{\mathbb{P}}\exp
(-\alpha^{-1}/2 )=0.0214$ eV. The masses of the two heavier neutrinos, $0.0231$
eV and $0.0552$ eV, can then be obtained from experimental results of neutrino
oscillations.
