  We compute how one-loop bulk effects renormalize both bulk and brane
effective interactions for geometries sourced by codimension-two branes. We do
so by explicitly integrating out spin-zero, -half and -one particles in
6-dimensional Einstein-Maxwell-Scalar theories compactified to 4 dimensions on
a flux-stabilized 2D geometry. (Our methods apply equally well for D dimensions
compactified to D-2 dimensions, although our explicit formulae do not capture
all divergences when D>6.) The renormalization of bulk interactions are
independent of the boundary conditions assumed at the brane locations, and
reproduce standard heat-kernel calculations. Boundary conditions at any
particular brane do affect how bulk loops renormalize this brane's effective
action, but not the renormalization of other distant branes. Although we
explicitly compute our loops using a rugby ball geometry, because we follow
only UV effects our results apply more generally to any geometry containing
codimension-two sources with conical singularities. Our results have a variety
of uses, including calculating the UV sensitivity of one-loop vacuum energy
seen by observers localized on the brane. We show how these one-loop effects
combine in a surprising way with bulk back-reaction to give the complete
low-energy effective cosmological constant, and comment on the relevance of
this calculation to proposed applications of codimension-two 6D models to
solutions of the hierarchy and cosmological constant problems.
