Heat Kernel and Scaling of Gravitational Constants

We consider the non-local energy-momentum tensor of quantum scalar and spinor fields in $2 w$-dimensional curved spaces. Working to lowest order in the curvature we show that, while the non-local terms proportional to $\Box {\cal R}$, $\Box \Box{\cal R}$, $\ldots, \Box^{w-2} {\cal R}$ are fully determined by the early-time behaviour of the heat kernel, the terms proportional to ${\cal R}$ depend on the asymptotic late-time behaviour. This fact explains a discrepancy between the running of the Newton constant dictated by the RG equations and the quantum corrections to the Newtonian potential.