In a recent work, it has been pointed out that certain observables of the massless scalar field theory in a static spherically symmetric background exhibit a universal behavior at large distances. More precisely, it was shown that, unlike what happens in the case the coupling to the curvature xi is generic, for the special cases xi=0 and xi = 1/6 the large distance behavior of the expectation value <T^{mu}_{ u}> turns out to be independent of the internal structure of the gravitational source. Here, we address a higher dimensional generalization of this result: We first compute the difference between a black hole and a static spherically symmetric star for the observables <phi^2> and <T^{mu}_{ u}> in the far field limit. Thus, we show that the conformally invariant massless scalar field theory in a static spherically symmetric background exhibits such universality phenomenon in Dgeq 4 dimensions. Also, using the one-loop effective action, we compute <T^{mu}_{ u}> for a weakly gravitating object. These results lead to the explicit expression of the expectation value <T^{mu}_{ u}> for a Schwarzschild-Tangherlini black hole in the far field limit. As an application, we obtain quantum corrections to the gravitational potential in D dimensions, which for D=4 are shown to agree with the one-loop correction to the graviton propagator previously found in the literature.
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