Relaxation of Baryon Number in the Standard Electroweak Model

Abstract

The presence of many topologically distinct vacua is an essential ingredient of the baryon-number nonconservation in the standard electroweak model. t'Hoofe) showed that the baryon-number violation takes place at zero temperature via quantum tunneling (instanton) transitions between the topologically inequivalent vacua and the amplitude is exponentially suppressed by the WKB factor exp (-27[/(12). Since the SU(2) gauge coupling constant (12 is very small, the instanton processes are completely negligible. On the other hand, it has been suggested) that diffusion over potential barriers interpolating between neighbouring vacua may occur at much larger rate. Recently, several authors3) have calculated the rate for thermal transitions using the sphaleron solution of Klinkhamer and Manton) which corresponds to a saddle-point field configuration connecting the neighbouring vacua. They found that the baryonnumber violation is unsuppressed around temperature T ~ O(lOO)Ge V. This being the case, the baryon-number asymmetry of the universe, produced, e.g., at GUT scale, might be substantially affected by the sphaleron-induced processes at the electroweak scale. In the above picture, the difference of fermion free energies LlF between the neighbouring vacua is crucial for producing net changes id the baryon number of the universe (see Eq. (20)). Cohen, Dugan and Manohar) have claimed that the rate of the net baryon-number change is exponentially small although there are unsuppressed fluctuations in the baryon number at high temperatures. In their analysis, they have argued that there is no change in the fermion free energy when the system goes over the potential barrier. The purpose of this paper is to calculate carefully effects of the fermion fields on the potential of Higgs and gauge field-and obtain the fermion free energy in the static background fields of arbitrary topological charges. We discuss the relaxation of the net baryon number, using the Langer-Afflech theory.6) We assume that the system of quarks and leptons is in thermal equilibrium, while a collective excitation mode of the Higgs and gauge fields relevant to the baryon number violation produces inequilibrium transitions along the way developed in Ref. 6). These physical assumptions lead to a conclusion different from Cohen et al.)