High-temperature sphalerons

The SU(2) part of electroweak gauge theory has sphaleronlike configurations even in its symmetric phase (temperature $T>\mathrm{}{T}_c$) which mediate baryon-number-violating processes. These sphalerons sit on top of a potential barrier (which we construct explicitly) whose height rises linearly with $T$, and always exceeds $T$ by a substantial factor. Such symmetric sphalerons are entirely nonperturbative, and we can at present only give lower bounds to the potential height, that is, the sphaleron mass $M_s$. In terms of the Boltzmann factor $\exp (-\beta M_s)\equiv e^{-A}$, when $T\gg T_c$, $A$ is a pure number independent of both $T$ and $g$, the electroweak coupling constant. We estimate $13<A\lesssim 40$, corresponding to a Boltzmann factor between 2×${10}^{-6\mathrm{}}$ and 4×${10}^{-18\mathrm{}}$. We do not discuss the full problem of small fluctuations around the sphaleron (necessary to find sphaleron-induced rates from the Boltzmann factor) but our explicitly constructed potential barrier gives a reasonable estimate of the single imaginary eigenvalue of small fluctuations. We also investigate high-temperature sphalerons in the presence of a finite baryon-number density, or equivalently a tachyonic Chern-Simons mass term. Such a term tends to reduce the sphaleron mass and increase the Boltzmann factor; the sphaleron never becomes tachyonic, no matter how large the expectation value of the Chern-Simons density.