Classical and quantum decay of oscillations: Oscillating self-gravitating real scalar field solitons

The oscillating gravitational field of an oscillaton of finite mass M causes it to lose energy by emitting classical scalar field waves, but at a rate that is nonperturbatively tiny for small μ≡GMm/c, where m is the scalar field mass: dM/dt≈-3797437.776(c³/G)μ^-2e^-39.433795197/μ[1+O(μ)]. Oscillatons also decay by the quantum process of the annihilation of scalarons into gravitons, which is only perturbatively small in μ, giving by itself dM/dt≈-0.008513223935(m²c²/)μ⁵[1+O(μ²)]. Thus the quantum decay is faster than the classical one for μ39.4338/[ln(c/Gm²)+7ln(1/μ)+19.9160]. The time for an oscillaton to decay away completely into free scalarons and gravitons is tdecay~2⁶c³/G⁵m¹¹~10³²⁴yr(1meV/mc²)¹¹. Oscillatons of more than one real scalar field of the same mass generically asymptotically approach a static-geometry $U(1)$ boson star configuration with μ=μ₀, at the rate ${d(GM/c}³)/dt{≈}[(C/{{μ}}⁴{)e}^{{-}{α}/{μ}}{+Q(m/m}Pl{)}²{{μ}}³]({{μ}}²{-}{{μ}}₀²),$ with ${{μ}}₀$ depending on the magnitudes and relative phases of the oscillating fields, and with the same constants C, ${α},$ and Q given numerically above for the single-field case that is equivalent to ${{μ}}₀=0.$