  A proposal for the neutrino mass, based on neutrino-scalar field interaction,
is introduced. The scalar field is also non-minimally coupled to the Ricci
scalar and hence relates the neutrino mass to the matter density. In a dense
region, the scalar field obeys the $Z_2$ symmetry, and the neutrino is
massless. In a dilute region, the $Z_2$ symmetry breaks and neutrino acquires
mass from the non-vanishing expectation value of the scalar field. We consider
this scenario in the framework of a spherical dense object whose outside is a
dilute region. In this background, we study the neutrino flavors oscillation,
along with the consequences of the theory on oscillation length and MSW effect.
This preliminary model may shed some lights on the existing anomalies within
the neutrino data, concerning the different oscillating behavior of the
neutrinos in regions with different densities.
