Higher-Order Poles in Electron-Hydrogen Scattering: Who Ordered That?

J. Sucher

Department of Physics

University of Maryland

College Park, MD 20742

It has long been known but little noted that the amplitude f for the elastic scattering of an electron by a hydrogen atom, as calculated in nonrelativistic quantum mechanics (NRQM) and in Born approximation, has peculiar properties when considered as an analytic function of the energy and momentum transfer. In particular, the exchange part fex of f Born has double and triple poles in the energy. Such singularities are unexpected in the context of particle physics, where only simple poles are encountered. Equally strange is the fact that the direct part fdir of f has a double and a single pole in the momentum transfer. I will show that these singularities are a manifestation of the long-range character of the interaction of the bound electron with the proton core, i.e. the Coulomb binding potential. They are not present in the analogous case of neutron-deuteron scattering.

At first sight it seems that one can easily understand (without carrying out any integrations) the existence and location of the singularities in fex and fdir on the basis of quantum field theory (QFT) and Feynman diagrams: One simply treats the H-atom as an elementary particle coupled trilinearly with a constant ge,p;H to the electron and the proton, or, more generally, as a composite particle with a vertex function GH (pe, pp) which describes the virtual transition H <--> e- + p of the H-atom into an electron and proton. However, there is no free lunch: This approach appears to yield only a simple pole for fex and only a branch point for fdir , as in the analogous case of neutron-deuteron scattering. Further analysis shows that the situation is much more complicated: In order for the NR limit of the QFT approach to agree with NRQM it seems to be necessary that GH is singular when the constituents go on the mass shell: pe2 = me2, pp2 = mp2. I will provide direct evidence that GH indeed has this property. As a consequence, the would-be (e.p;H) coupling constant ge,p;H, defined in direct analogy with the (n.p;d) coupling constant is infinite and the failures of the naïve model are explained. It seems that in this ancient problem there are lessons to be learned for both atomic and particle physics. We have here another manifestation of the sometimes hidden consequences of long-range forces, this time involving a charged and neutral particle.