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

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 α 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 α = 2n, whereas the compactification on an n-sphere gives α = n + 1. For Calabi–Yau compactifications the strength can be at most α = 20. Finally, when higher-derivative terms in four space-time dimensions are considered, we find a repulsive Yukawa potential with α = −1.