Tail effects of self-interacting scalar fields

We consider the effects of quartic self-interactions on the dynamics of a binary system due to a (nearly) massless scalar field conformally coupled to matter. We investigate the deviations from General Relativity at the conservative level and put a bound on the self-coupling $\lambda \lesssim (\beta^2 G_N M_\odot^2)^{-1}$ where $\beta$ is the conformal coupling of the scalar to matter. We also consider the radiative sector where we use the Schwinger-Keldysh formalism to find the tail interactions which couple the multipoles of the binary system and induce a small advance of the periastron.