Bloch wave function for the periodic sphaleron potential and unsuppressed baryon and lepton number violating processes

Abstract

For the periodic sphaleron potential in the electroweak theory, we find the onedimensional time-independent Schrödinger equation with the Chern-Simons number as the coordinate, construct the Bloch wave function and determine the corresponding conducting (pass) band structure. We show that the baryon-lepton number violating processes can take place without the exponential tunneling suppression (at zero temperature) at energies around and above the barrier height (sphaleron energy) at 9.0 TeV. Phenomenologically, probable detection of such processes at LHC is discussed.

Band structure in Yang-Mills theories

“…[8] which challenged the earlier consensus that B + L-violating electroweak processes should be invisible at LHC. 7 The standard wisdom is that the collision energy, initially carried by two hard quanta, has no chance of being streamlined to a coherent excitation of one (or a few) degrees of freedom in a weakly-coupled theory like the electroweak Standard Model.…”

“…The resulting potential is non-analytic at the identification point 0 = q = 1, but it remains twice-continuously differentiable there. The above action has the same gross features as that in [8], namely a harmonic potential around q = 0 and a barrier height of order ∼ 1/g 2 at weak coupling. Since q is a compact variable, the energy spectrum is discrete with no bands.…”

“…A quantum-mechanical action for the Chern-Simons variable, similar to the one proposed by Tye and Wong [8], can be derived by putting the SU(2) theory on the 3-sphere and then projecting onto the SO(4)-invariant sector. This is an amusing exercise, but since the details of this action play no role we relegate it to the appendix.…”

“…Let us now assume, following Tye and Wong [8,9], that it makes sense to write down an effective action for the variable n CS . We will comment on the validity of this assumption in the end.…”

“…While revisiting recently the question of electroweak B + L violation [1] in collider experiments (for reviews and more references see [2][3][4][5][6][7]) Tye and Wong made a bold proposal [8,9]. They argued that the relevant degree of freedom in the space-time interaction region for such processes is the Chern-Simons number of the electroweak gauge fields, whose effective quantum mechanics exhibits band structure similar to that of an electron in a solid.…”