  Dyonium induced fermion number violation is studied in a SU(2) x U(1) gauge
theory with a doublet Higgs field. Dyonium is a generalization of Nambu's
monopolium, having finite sized pair of dyon and anti-dyon, connected with a
thin string under the linear plus Coulomb force. This is a follow-up of the
author's paper on monopolium induced neutrino mass (including the lepton and
baryon number violations), studied with a SO(10) Grand Unified model in 1983.
To fulfill the requirement from the chiral anomaly or its index theorem, an
electric field should be excited in parallel to the dipole magnetic field. This
crucial dynamical problem, not fully answered then, has been solved, by
considering not monopolium but dyonium. The fermionic zero modes in the dyonium
background fields are necessary in evaluating the transition rate of the
fermion number violation processes. Their Dirac equation can be reduced to a
single component partial differential equation, similar to the renormalization
group equation, which may be useful to estimate the reaction rates.
