Vacuum polarization and classical self-action near higher-dimensional defects

We analyze the gravity-induced effects associated with a massless scalar field in a higher-dimensional spacetime being the tensor product of $(d-n)$-dimensional Minkowski space and $n$-dimensional spherically/cylindrically-symmetric space with a solid/planar angle deficit. These spacetimes are considered as simple models for a multidimensional global monopole (if \mbox{$n\geqslant 3$}) or cosmic string (if $n=2$) with $(d-n-1)$ flat extra dimensions. Thus, we refer to them as conical backgrounds. In terms of the angular deficit value, we derive the perturbative expression for the scalar Green's function, valid for any $d\geqslant 3$ and $2\leqslant n\leqslant d-1$, and compute it to the leading order. With the use of this Green's function we compute the renormalized vacuum expectation value of the field square $\langle \phi^{2}(x)\rangle_{\mathrm{ren}}$ and the renormalized vacuum averaged of the scalar-field's energy-momentum tensor $\langle T_{M N}(x)\rangle_{\mathrm{ren}}$ for arbitrary $d$ and $n$ from the interval mentioned above and arbitrary coupling constant to the curvature $\xi$. In particular, we revisit the computation of the vacuum polarization effects for a non-minimally coupled massless scalar field in the spacetime of a straight cosmic string. The same Green's function enables to consider the old purely classical problem of the gravity-induced self-action of a classical pointlike scalar or electric charge, placed at rest at some fixed point of the space under consideration. To deal with divergences, which appear in consideration of the both problems, we apply the dimensional-regularization technique, widely used in quantum field theory (QFT). The explicit dependence of the results upon the dimensionalities of both the bulk and conical submanifold, is discussed.