  We investigate the possibility of self-tuning of the effective 4D
cosmological constant in 6D supergravity, to see whether it could naturally be
of order 1/r^4 when compactified on two dimensions having Kaluza-Klein masses
of order 1/r. In the models we examine supersymmetry is broken by the presence
of non-supersymmetric 3-branes (on one of which we live). If r were
sub-millimeter in size, such a cosmological constant could describe the
recently-discovered dark energy. A successful self-tuning mechanism would
therefore predict a connection between the observed size of the cosmological
constant, and potentially observable effects in sub-millimeter tests of gravity
and at the Large Hadron Collider. We do find self tuning inasmuch as 3-branes
can quite generically remain classically flat regardless of the size of their
tensions, due to an automatic cancellation with the curvature and dilaton of
the transverse two dimensions. We argue that in some circumstances
six-dimensional supersymmetry might help suppress quantum corrections to this
cancellation down to the bulk supersymmetry-breaking scale, which is of order
1/r. We finally examine an explicit realization of the mechanism, in which
3-branes are inserted into an anomaly-free version of Salam-Sezgin gauged 6D
supergravity compactified on a 2-sphere with nonzero magnetic flux. This
realization is only partially successful due to a topological constraint which
relates bulk couplings to the brane tension, although we give arguments why
these relations may be stable against quantum corrections.
