  One of the most interesting candidates for dark matter are massive real
scalar particles. A well-motivated example is from a pure Yang-Mills hidden
sector, which locks up into glueballs in the early universe. The lightest
glueball states are scalar particles and can act as a form of bosonic dark
matter. If self-interactions are repulsive this can potentially lead to very
massive boson stars, where the inward gravitational force is balanced by the
repulsive self-interaction. This can also arise from elementary real scalars
with a regular potential. In the literature it has been claimed that this
allows for astrophysically significant boson stars with high compactness, which
could undergo binary mergers and generate detectable gravitational waves. Here
we show that previous analyses did not take into proper account $3 \to 2$ and
$4 \to 2$ quantum mechanical annihilation processes in the core of the star,
while other work misinterpreted the classical $3 \to 1$ process. In this work,
we compute the annihilation rates, finding that massive stars will rapidly
decay from the $3 \to 2$ or $4 \to 2$ processes (while the $3 \to 1$ process is
typically small). Using the Einstein-Klein-Gordon equations, we also estimate
the binding energy of these stars, showing that even the densest stars do not
have quite enough binding energy to prevent annihilations. For such boson stars
to live for the current age of the universe and to be consistent with bounds on
dark matter scattering in galaxies, we find the following upper bound on their
mass for $O(1)$ self-interaction couplings: $M_*<10^{-18}M_{sun}$ when $3 \to
2$ processes are allowed and $M_*<10^{-11}M_{sun}$ when only $4 \to 2$
processes are allowed. We also estimate destabilization from parametric
resonance which can considerably constrain the phase space further.
Furthermore, such stars are required to have very small compactness to be long
lived.
