  The dynamics of the 4-body problem allows for two binary collisions to occur
simultaneously. It is known that in the collinear 4-body problem this
simultaneous binary collision (SBC) can be block-regularised, but that the
resulting block map is only $C^{8/3}$ differentiable. In this paper, it is
proved that the $C^{8/3}$ differentiability persists for the SBC in the planar
4-body problem. The proof uses several geometric tools, namely, blow-up, normal
forms, dynamics near normally hyperbolic manifolds of equilibrium points, and
Dulac maps.
