$Z_2$ 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 $Z_2$ 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, electroweak strings 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 and electroweak strings, are topologically equivalent to the vacuum and are irrelevant for B+L violation. We discuss the possible formation of B+L violating quark-lepton condensates in the symmetric high-temperature phase of the electroweak theory.