  In analogy with spontaneous magnetization of ferromagnets below the Curie
temperature, a neutron star (NS), with a compactness above a certain critical
value, may undergo spontaneous scalarization and exhibit an interior nontrivial
scalar configuration. Consequently, the exterior space-time is changed, and an
external scalar field appears, which subsequently triggers a scalarization of
its companion. The dynamical interplay produces a gravitational scalar
counterpart of tensor gravitational waves. In this paper, we resort to
scalar-tensor theory and demonstrate that the gravitational scalar counterpart
from double neutron star (DNS) and neutron star-white dwarf (NS-WD) become
massive. We report that (i) a gravitational scalar background field, arising
from convergence of external scalar fields, plays the role of gravitational
scalar counterpart in scalarized DNS binary, and the appearance of a
mass-dimensional constant in Higgs-like gravitational scalar potential is
responsible for a massive gravitational scalar counterpart with mass of order
of Planck scale; (ii) a dipolar gravitational scalar radiated field, resulting
from different binding energy of NS and WD, plays the role of gravitational
scalar counterpart in scalarized orbital shrinking NS-WDs, which oscillates
around a local and scalar-energy-density dependent minimum of the gravitational
scalar potential and gains a mass of the order of about $10^{-21} ev/c^2$.
