In the standard electroweak theory that describes nature, the unstable sphaleron solutions play a crucial role in the baryon number violating processes. For the periodic sphaleron potential, we find the one-dimensional time-independent Schrödinger equation with the Chern-Simons number as the coordinate. We recall why the Chern-Simons number should be generalized from a set of discrete values to a dynamical (quantum) variable. Via the construction of an appropriate Hopf invariant and the winding number, we discuss how the geometric information in the gauge fields is also captured in the Higgs field. We then discuss the choice of the Hopf variable in relation to the Chern-Simons variable. For the effective Schrödinger equation, we construct the Bloch wave function and determine t...[ Read more ]

In the standard electroweak theory that describes nature, the unstable sphaleron solutions play a crucial role in the baryon number violating processes. For the periodic sphaleron potential, we find the one-dimensional time-independent Schrödinger equation with the Chern-Simons number as the coordinate. We recall why the Chern-Simons number should be generalized from a set of discrete values to a dynamical (quantum) variable. Via the construction of an appropriate Hopf invariant and the winding number, we discuss how the geometric information in the gauge fields is also captured in the Higgs field. We then discuss the choice of the Hopf variable in relation to the Chern-Simons variable. For the effective Schrödinger equation, we construct the Bloch wave function and determine the corresponding conducting (pass) band structure. We discuss the impact of such periodic potential and band structure on the understanding of anomalous baryon/lepton number violation. It is suggested that such processes at zero temperature may not be very suppressed at energies higher than the sphaleron barrier height at 9 TeV.