sphalerons

A sphaleron is a static, unstable solution to the electroweak sector of the Standard Model, representing a saddle point of the energy in the SU(2) gauge–Higgs theory. It sits atop the energy barrier that separates vacua with different Chern-Simons numbers. In the Standard Model these vacua differ by one unit, and transitions between them change baryon number B and lepton number L through the chiral anomaly, while conserving B−L. The term “sphaleron” was introduced by Klinkhamer and Manton in 1984, reflecting its unstable, barrier-crossing character.

The sphaleron has an energy Esph of order 8–10 TeV for physical values of the Higgs and gauge couplings. Its field configuration involves a nontrivial SU(2) gauge field intertwined with the Higgs field. At zero temperature, transitions between vacua are exponentially suppressed, but at high temperature the barrier can be overcome by thermal fluctuations.

In the early universe, sphaleron transitions were rapid in the electroweak symmetric phase (temperatures above the electroweak crossover, around 100 GeV). The rate per unit volume is temperature dependent and, up to logarithmic factors, scales with powers of the weak coupling and temperature. Because sphalerons violate B+L but conserve B−L, they can erase any preexisting B+L asymmetry unless a nonzero B−L asymmetry is present. This makes sphalerons a central element in scenarios of baryogenesis and leptogenesis, where a lepton asymmetry can be partially converted into a baryon asymmetry through these processes.

Direct observation of sphalerons in colliders has not been achieved; their effects remain primarily of cosmological and theoretical significance.