  It is generic for the bulk fields sourced by branes having codimension two
and higher to diverge at the brane position, much as does the Coulomb potential
at the position of its source charge. This complicates finding the relation
between brane properties and the bulk geometries they source. (These
complications do not arise for codimension-1 sources, such as in RS geometries,
because of the special properties unique to codimension one.) Understanding
these relations is a prerequisite for phenomenological applications involving
higher-codimension branes. Using codimension-2 branes in extra-dimensional
scalar-tensor theories as an example, we identify the classical matching
conditions that relate the near-brane asymptotic behaviour of bulk fields to
the low-energy effective actions describing how space-filling codimension-2
branes interact with the surrounding extra-dimensional bulk. We do so by
carefully regulating the near-brane divergences, and show how these may be
renormalized in a general way. Among the interesting consequences is a
constraint relating the on-brane curvature to its action, that is the
codimension-2 generalization of the well-known modification of the Friedmann
equation for codimension-1 branes. We argue that its interpretation within an
effective field theory framework in this case is as a relation $4\pi U \simeq
\kappa^2 (T')^2$ between the codimension-2 brane tension, T, and its
contribution to the low-energy on-brane effective potential, U. This relation
implies that any dynamics that minimizes a brane contribution to the on-brane
curvature automatically also minimizes its couplings to the extra-dimensional
scalar.
