We compute how threshold effects obtained by integrating out a heavy particle localized on a codimension-2 brane influence the properties of the brane and the bulk fields it sources in D=d+2 dimensions. We do so using a recently developed formalism for matching the characteristics of higher codimension branes to the properties of the bulk fields they source. We show that although the dominant dependence on a heavy mass M that is induced in the low-energy codimension-2 tension has the generic size expected, T_2 propto M^d, the very-low-energy effective potential governing the on-brane curvature once bulk KK modes are integrated out can be additionally suppressed, by factors of order k^2 M^d, where k is the bulk gravitational coupling. In the special case of a codimension-2 brane in a 6D supersymmetric bulk we also estimate the size of the contributions of short-wavelength bulk loops near the brane, and find these can be similarly suppressed.
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