Free energy of the sphaleron in the Weinberg-Salam model

We show that the eigenvalue equation for excitation modes around a sphaleron background is divided into an infinite number of coupled equations, each of which is composed of only a finite number (2-7) of channels and hence solvable numerically. For spherically symmetric excitation modes we solve the eigenvalue equations and calculate the free energy of the sphaleron. We find no extra entropy suppression other than the damping factors observed in the previous analyses on sphaleron-induced processes. We also find that there is only one unstable mode whose frequency ${\omega }_{-}$ is estimated as ${\omega }_{-}^2\simeq -2.3\mathrm{}{m}_W^2$. We also study contributions of higher partial waves to the free energy in the Born approximation and find even an entropy enhancement for the channels with angular momenta greater than two ($L\ge 3$). Our result puts the previous calculation of the rate of the baryon-number-violating processes via the sphaleron on a much firmer ground. However, there remain many issues that need to be examined.