Newtonian self-gravitation in the neutral meson system

We derive the effect of the Schrödinger-Newton equation, which can be considered as a nonrelativistic limit of classical gravity, for a composite quantum system in the regime of high energies. Such meson-antimeson systems exhibit very unique properties, e.g., distinct masses due to strong and electroweak interactions. This raises an immediate question: what does one mean by mass in gravity for a state that is a superposition of mass eigenstates due to strong and electroweak interactions? We find conceptually different physical scenarios due to lacking of a clear physical guiding principle to explain which mass is the relevant one and due to the fact that it is not clear how the flavor wave function relates to the spatial wave function. There seems to be no principal contradiction. However, a nonlinear extension of the Schrödinger equation in this manner strongly depends on the relation between the flavor wave function and spatial wave function and its particular shape. In opposition to the continuous spontaneous localization collapse models we find a change in the oscillating behavior and not in the damping of the flavor oscillation.