Unitarity relation and unitarity bounds for scalars with different sound speeds

We consider a theory which contains massless scalar fields with different sound speeds. For these theories we derive unitarity relations for partial wave amplitudes of $ 2 \to 2$ scattering, with explicit formulas for contributions of two-particle intermediate states. We also 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. We illustrate our unitarity relations by explicit one-loop calculation to the first nontrivial order in couplings in a simple model of two scalar fields with different sound speeds. Obtained unitarity bounds can be used to estimating the strong coupling scale of a pertinent effective field theory (EFT).