The goal of this paper is to extend to higher dimensionality the methods and computations of vacuum polarization effects in black hole spacetimes. We focus our attention on the case of five dimensional Schwarzschild-Tangherlini black holes, for which we adapt the general method initially developed by Candelas and later refined by Anderson and others. We make use of point splitting regularization and of the WKB approximation to extract the divergences occurring in the coincidence limit of the Green function and, after calculating the counterterms using the Schwinger-De Witt expansion, we explicitly prove the cancellation of the divergences and the regularity of the vacuum polarization once counterterms are added up. We finally handle numerically the renormalized expression of the vacuum polarization. As a check on the method we also prove the regularity of the vacuum polarization in the six dimensional case in the large mass limit.
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