Bulk Casimir densities and vacuum interaction forces in higher dimensional brane models

Abstract

Vacuum expectation value of the energy-momentum tensor and the vacuum interaction forces are evaluated for a massive scalar field with general curvature coupling parameter satisfying Robin boundary conditions on two codimension one parallel branes embedded in ($D+1$)-dimensional background spacetime ${\mathrm{AdS}}_{{D}_{1}+1}\ifmmode\times\else\texttimes\fi{}\ensuremath{\Sigma}$ with a warped internal space $\ensuremath{\Sigma}$. The vacuum energy-momentum tensor is presented as a sum of boundary-free, single brane-induced, and interference parts. The latter is finite everywhere including the points on the branes and is exponentially small for large interbrane distances. Unlike to the purely anti--de Sitter (AdS) bulk, the part induced by a single brane, in addition to the distance from the brane, depends also on the position of the brane in the bulk. The asymptotic behavior of this part is investigated for the points near the brane and for the position of the brane close to the AdS horizon and AdS boundary. The contribution of Kaluza-Klein modes along $\ensuremath{\Sigma}$ is discussed in various limiting cases. The vacuum forces acting on the branes are presented as a sum of the self-action and interaction terms. The first one contains well-known surface divergences and needs a further renormalization. The interaction forces between the branes are finite for all nonzero interbrane distances and are investigated as functions of the brane positions and the length scale of the internal space. We show that there is a region in the space of parameters in which these forces are repulsive for small distances and attractive for large distances. As an example, the case $\ensuremath{\Sigma}={S}^{{D}_{2}}$ is considered. An application to the higher dimensional generalization of the Randall-Sundrum brane model with arbitrary mass terms on the branes is discussed. Taking the limit with infinite curvature radius for the AdS bulk, from the general formulas we derive the results for two parallel Robin plates on background of ${R}^{({D}_{1},1)}\ifmmode\times\else\texttimes\fi{}\ensuremath{\Sigma}$ spacetime.