Long-Range Electromagnetic Forces on Neutral Particles

We consider the long-range forces, i.e., those falling off as a power of the distance, which may act between pairs of particles, one of which is neutral and spinless. It is shown that these forces may easily be calculated from the discontinuity function in the momentum transfer of the scattering amplitude for the two particles. In particular, we have investigated the two-photon exchange force between two neutral, spinless systems, and the three-photon exchange force between a charged and a neutral, spinless system. In the former case, we find that the potential behaves as $r^{-7\mathrm{}}$ for large $r$, in contradiction to the London expression for the Van der Waals force, and in agreement with the result of Casimir and Polder. For the latter case, the potential is odd under charge conjugation and hence can convert a $K_2$ meson to a $K_1$ meson. We find again that the potential behaves as $r^{-7\mathrm{}}$. It is found that such long-range electromagnetic interactions are presently unobservable in particle physics.