Sphaleron

A sphaleron is a static, unstable finite-energy field configuration in the electroweak sector of the Standard Model. It is a saddle point of the energy functional for the SU(2) gauge fields coupled to the Higgs field and marks the top of the energy barrier separating vacua that differ by one unit of Chern-Simons number. Transitions between these vacua are nonperturbative and, due to the chiral anomaly, change baryon and lepton numbers.

The sphaleron is an unstable solution with a single negative mode, and its energy defines the barrier height, typically of order 10 TeV in the electroweak theory (often quoted as about 9–12 TeV depending on parameters). The rate of transitions over the barrier is temperature dependent: at zero temperature the process is exponentially suppressed, while at temperatures near or above the electroweak scale it can proceed at a significant rate, enabling anomalous B and L violation. In the broken electroweak phase, the rate is further suppressed by a Boltzmann factor exp(-E_sph/T). Sphaleron processes are thus central to discussions of electroweak baryogenesis and the fate of any pre-existing baryon asymmetry, since they violate B+L while conserving B−L.

The concept was introduced in 1984 by Manton and by Klinkhamer and has since become a standard description of saddle-point configurations in non-Abelian gauge theories.