Fermions destabilize electroweak strings.

Z-strings in the Weinberg-Salam model including fermions are unstable for all values of the parameters. The cause of this instability is the fermion vacuum energy in the Z-string background. Z-strings with non-zero fermion densities, however, may still be stable. ⋆ e-mail: [email protected] The recent discovery of cosmic string solutions in the Weinberg-Salam model of electroweak interactions [1, 2] has fueled a burst of activity in the study of defects in the standard model and its extensions, and of their possible consequences for astrophysics and cosmology [3, 4]. The existence of these “electroweak strings,” which are essentially Nielsen-Olesen vortices embedded in the Weinberg-Salam model, was previously neglected because they do not possess topological stability. They may nonetheless be stable if they sit at a local minimum of the energy. Because they owe their existence to energetic rather than topological reasons, the stability of electroweak strings is sensitively dependent on the field content and the values of the parameters in the theory. For example, in a simplified version of the Weinberg-Salam model containing only bosonic fields, Z-strings [2] are stable only for light Higgs masses (≤ mZ), and for sin θW fairly close to unity [5], a region that obviously does not include the physical world. Attempts have been made to increase the range of stability of the Z-string by extending the model to include other fields [3, 6, 7]. One idea, familiar from the study of nontopological solitons, is to include particles whose mass arises from the Higgs mechanism. Such particles remain massless at the center of the string where the Higgs field vanishes, and the presence of such particles at the core would resist the string’s dissolution, because that would increase their energy. Indeed, the presence of charged scalar bound states was shown to lower the value of sin θW for which the string is stable [6]. It has been suggested that a similar enhancement of stability could be achieved by using fermion bound states on the Z-string [6]. The existence of Z-string zero modes, fermion states localized on the string with zero energy, lends support to this idea [8–10]. Another advantage of this suggestion is that fermions are already contained in the standard electroweak model! In this letter, we show that, on the contrary, the presence of fermions in the electroweak theory destabilizes Z-strings. More precisely, the lowest-energy (or ground) state of the Z-string is always a local maximum of the energy functional 2 with respect to (at least) one of the modes of instability. The Z-string is therefore unstable for all values of the parameters of the Weinberg-Salam model. (It is possible, however, that a higher-energy state of the Z-string, with a finite quark density, could be locally stable.) This instability results from the fermion vacuum energy, which also has an important effect on other types of solitons [11]. One cannot consistently consider the effects of positive-energy fermion states without also taking account of the (filled) negative-energy states, particularly because, with the existence of zero modes, there is no gap between them. We will show that the contribution to the energy functional of the filled Dirac sea, i.e., the fermion vacuum energy, is a local maximum for the Z-string. First, we will describe the fermion spectrum in the presence of the Z-string; then we will show how the fermion vacuum energy changes under certain small perturbations away from the Z-string. The electroweak Lagrangian is L = Lboson + ∑