  Extra-dimensional models can involve bulk pseudo-Goldstone bosons (pGBs)
whose shift symmetry is explicitly broken only by physics localized on branes.
Reliable calculation of their low-energy potential is often difficult because
it requires details of the stabilization of the extra dimensions. In rugby ball
solutions, for which two compact extra dimensions are stabilized in the
presence of only positive-tension brane sources, the effects of brane
back-reaction can be computed explicitly. This allows the calculation of the
shape of the low-energy pGB potential and response of the extra dimensional
geometry as a function of the perturbing brane properties. If the
pGB-dependence is a small part of the total brane tension a very general
analysis is possible, permitting an exploration of how the system responds to
frustration when the two branes disagree on what the proper scalar vacuum
should be. We show how the low-energy potential is given by the sum of brane
tensions (in agreement with common lore) when only the brane tensions couple to
the pGB. We also show how a direct brane coupling to the flux stabilizing the
extra dimensions corrects this result in a way that does not simply amount to
the contribution of the flux to the brane tensions. We calculate the mass of
the would-be zero mode, and briefly describe several potential applications,
including a brane realization of `natural inflation,' and a dynamical mechanism
for suppressing the couplings of the pGB to matter localized on the branes.
Since the scalar can be light enough to be relevant to precision tests of
gravity (in a technically natural way) this mechanism can be relevant to
evading phenomenological bounds.
