We consider the long range forces between two BPS particles on the Coulomb branch of N=2 and N=4 supersymmetric gauge theories. The 1/r potential is unambiguously fixed, even at strong coupling, by the moduli dependence of central charges supported by the BPS states. The effective Coulombic coupling vanishes on marginal stability curves, while sign changes on crossing these curves explain the restructuring of the spectrum of composite BPS states. This restructuring proceeds via the delocalization of the composite state on approach to the curve of marginal stability. Therefore, the spectrum of BPS states can be inferred by analyzing the submanifolds of the moduli space where the long range potential is attractive. This method also allows us to find certain non-BPS bound states and their stability domains. As examples, we consider the dissociation of the W boson and higher charge dyons at strong coupling in N=2 SU(2) SYM, quark-monopole bound states in N=2 SYM with one flavor, and composite dyons in N=2 SU(3) SYM.
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We are grateful to Philip Argyres, Oren Bergman, Gregory Gabadadze, Mikhail Shifman, Mikhail Voloshin, Mikhail Olshanetsky, Ed Witten, and Alexei Yung for helpful discussions, and we thank Philip Argyres and K. Narayan for helpful comments on a preliminary draft of this paper. This work was supported by the DOE under grant number DE-FG02-94ER408.