Fermion back reaction and the sphaleron.

Using a simple model, a new sphaleron solution which incorporates finite fermionic density effects is obtained. The main result is that the height of the potential barrier (sphaleron energy) decreases as the fermion density increases. This suggests that the rate of sphaleron-induced transitions increases when the fermionic density increases. However the rate increase is not expected to change significantly the predictions from the standard sphaleron-induced baryogenesis scenarios. 11.15.Kc,11.10.Kk Typeset using REVTEX 1 Nonconservation of baryon number in the standard model through quantum effects (anomalies) is well known [1]. While instanton mediated baryon decays are negligible, the same cannot be said for transitions occuring because of monopole catalysis [2], high temperature [3,4], high densities [5,6], or in the presence of a heavy particle [7] (for an excellent review of these four mechanisms, see ref. [8]). The basis for all these transitions is the level crossing phenomenon [9] which is usually illustrated by looking at adiabatic changes in the gauge field configuration and at the accompanying variation in the energy levels for the fermions resulting in a change in the fermion number. This description neglects the effect of the fermion back-reaction: a change in the fermion density can introduce a change in the gauge field configuration, just as a change of gauge field configuration can change the fermion density. This is most easily seen in the Schwinger model where this back-reaction of the fermions is responsible for oscillation in the fermion number [10]. The fermion back-reaction is a purely quantum mechanical effect being a direct consequence of the anomaly equation. Since the focus of fermion number violation has been in the study of solutions to the classical equation of motion (e.g. instantons and sphalerons), it is, therefore, not surprising that little attention has been paid to this back-reaction. Further, it might be very difficult to properly take into account the fermion back-reaction in realistic 3+1 dimensional theories since the resolution of even seemingly straightforward related issues, like the gauge invariance of the free energy at finite temperature and fermionic density [11], require a careful treatment of non-perturbative effects to be properly resolved [12]. Fortunately, the situation is simpler in 1+1 dimensional models where one can, through bosonization [13], take into account the fermion back-reaction at the classical level. The present work illustrates some of the non-trivial effects of this fermion back-reaction using a simple model which has been extensively studied in the past, namely the Abelian Higgs model axially coupled to fermions (see for example [6,14–18]) . The Lagrangian density describing this model is