Massive scalar counterpart of gravitational waves in scalarized neutron star binaries

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 $10^{-21} ev/c^2$.