ℤ 2 as the topological origin of B + L violation in the hot electroweak theory

Abstract

The space of static finite-energy configurations in the electroweak theory admits a ℤ 2 2 topological structure. More precisely, we show that this space contains two disconnected sectors of unstable gauge-Higgs fields odd under a properly defined generalized parity. This classification extends the description of baryon- and lepton-number violating electroweak processes to the symmetric phase of the theory. Configurations with odd pure-gauge behaviour at spatial infinity, such as the sphaleron, multisphalerons, as well as an infinite surface of their equivalents, have half-integer Chern—Simons number and mediate B + L violating processes in the early universe. Finite-energy configurations with even pure-gauge behaviour, such as the S* new sphaleron, are topologically equivalent to the vacuum and are irrelevant for B + L violation. Electroweak string configurations comprise continuous families of sphaleron equivalents as well as configurations equivalent to the vacuum. We discuss the possible formation of B + L violating quark-lepton condensates in the symmetric high-temperature phase of the electroweak theory.