High-temperature sphalerons.

Abstract

The SU(2) part of electroweak gauge theory has sphaleronlike configurations even in its symmetric phase (temperature $T>{T}_{c}$) 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 $\mathrm{exp}(\ensuremath{-}\ensuremath{\beta}{M}_{s})\ensuremath{\equiv}{e}^{\ensuremath{-}A}$, when $T\ensuremath{\gg}{T}_{c}$, $A$ is a pure number independent of both $T$ and $g$, the electroweak coupling constant. We estimate $13<A\ensuremath{\lesssim}40$, corresponding to a Boltzmann factor between 2\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}6}$ and 4\ifmmode\times\else\texttimes\fi{}${10}^{\ensuremath{-}18}$. 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.