Abstract

The Dirac equation in the field of an extended SU(2) dyon is studied for a massless fermion isodoublet with nonzero angular momentum J. The scattering amplitudes for all partial waves are obtained, and the dependence on the energy and the dyon's charge are explicitly given. Because the Coulomb potential of a dyon in the time independent gauge does not vanish at large distances, the asymptotic kinetic energy of a positively and negatively charged massless fermion E± does not coincide with the energy E, the eigenvalue of the Dirac hamiltonian. In the limit of small dyon radius and electric charge, the charge exchange scattering amplitude can be parametrized by a suppression factor ∣4E+E−r02∣^{√J√(J+1)} = ∣(e q/(4π))^2 − (2 E r_0)^2∣√J(J+1) where e is the SU(2) gauge coupling constant, q is the electric charge of the dyon, and r0 is the “radius” of the dyon. For sufficiently low energies E, this amplitude is only suppressed by powers of . However, a dyon radiates away electric charge through fermion pairs. When the self-consistency requirement is imposed that the overall charge conservation holds for the combined dyon-fermion system, the dyon can only have a very small charge and E+ ∼ E− ∼ E. Consequently, the charge exchange scattering found in the external field problem becomes completely negligible for low-energy fermions with J ⩾ 1. Therefore, the only physically significant charge exchange scattering is that of the s-wave fermions. The details are given for a specific model (which we call the asymptotic model) in which the dyon has a sharp edge: the exterior potentials are approximated by their asymptotic forms, while the interior potentials are replaced by their values at the origin. In an appendix we show that our result is quite general, and the dependence on a particular dyon potential used is characterized by a numerical factor of order unity.