  We study the formation of a stable self-trapped spherical quantum ball in a
binary Bose-Einstein condensate (BEC) with two-body inter-species attraction
and intra-species repulsion employing the beyond-mean-field Lee-Huang-Yang and
the three-body interactions. We find that either of these interactions or a
combination of these can stabilize the binary BEC quantum ball with very
similar stationary results, and for a complete description of the problem both
the terms should be considered. These interactions lead to higher-order
nonlinearities, e.g. quartic and quintic, respectively, in a nonlinear
dynamical equation compared to the cubic nonlinearity of the two-body contact
interaction in the mean-field Gross-Pitaevskii equation. The higher-order
nonlinearity makes the energy infinitely large at the center of the binary ball
and thus avoids its collapse. In addition to the formation of stationary binary
balls, we also study a collision between two such balls. At large velocities,
the collision is found to be elastic, which turns out to be inelastic as the
velocity is lowered. We consider the numerical solution of a beyond-mean-field
model for the binary ball as well as a single-mode variational approximation to
it in this study.
