Unitarity relation and unitarity bounds for scalars with different sound speeds

Motivated by scalar-tensor gravities, we consider a theory which contains massless scalar fields with different sound speeds. We derive unitarity relations for partial wave amplitudes of scattering, with explicit formulas for contributions of two-particle intermediate states. Making use of these relations, we obtain unitarity bounds both in the most general case and in the case considered in the literature for the speed of sound, equal to unity. These bounds can be used to estimate the strong coupling scale of a pertinent effective field theory (EFT). We illustrate our unitarity relations by explicit calculation to the first non-trivial order in couplings in a simple model of two scalar fields with different sound speeds.