Collision Induced Decays of Electroweak Solitons: Fermion Number Violation with Two and Few Initial Particles

We consider a variant of the standard electroweak theory in which the Higgs sector has been modified so that there is a classically stable weak scale soliton. We explore fermion number violating processes which involve soliton decay. A soliton can decay by tunnelling under the sphaleron barrier, or the decay can be collision induced if the energy is sufficient for the barrier to be traversed. We present a classical solution to the Minkowski space equations of motion in which a soliton is kicked over the barrier by an incoming pulse. This pulse corresponds to a quantum coherent state with mean number of $W$ quanta $\sim 2.5/g^2$ where $g$ is the $SU(2)$ gauge coupling constant. We also give a self-contained treatment of the relationship between classical solutions, including those in which solitons are destroyed, and tree-level quantum amplitudes. Furthermore, we consider a limit in which we can reliably estimate the amplitude for soliton decay induced by collision with a single $W$-boson. This amplitude depends on $g$ like $\exp (-cg^{-1/3})$, and is larger than that for spontaneous decay via tunnelling in the same limit. Finally we show that in soliton decays, light $SU(2)_L$ doublet fermions are anomalously produced. Thus we have a calculation of a two body process with energy above the sphaleron barrier in which fermion number is violated.