Deviations from the $1/r^2$ Newton law due to extra dimensions and higher-derivative terms

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Abstract
We systematically examine corrections to the gravitational inverse square law, which are due to compactified extra dimensions, and to higher-derivative R^2-terms in the effective gravitational action. We find that both induce Yukawa-type potentials for which we calculate the strength \alpha and range. In general the range of the Yukawa correction is given by the wavelength of the lightest Kaluza-Klein state and its strength, relative to the standard gravitational potential, by the corresponding degeneracy. In particular, when n extra dimensions are compactified on an n-torus, we find that the strength of the potential is \alpha=2n, whereas the compactification on an n-sphere gives \alpha=n+1. For Calabi--Yau compactifications the strength can be at most \alpha=20. Finally, when higher-derivative terms in four space-time dimensions are considered, we find a repulsive Yukawa potential with \alpha=-1.

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