Bound States Can Stabilize Electroweak Strings

We show that the electroweak Z−string can be stabilized by the presence of bound states of a complex scalar field. We argue that fermions coupled to the scalar field of the string can also make the string stable and discuss the physical case where the string is coupled to quarks and leptons. This stabilization mechanism is expected to work for other embedded defects and also for unstable solutions such as the sphaleron. It is now known that vortex solutions1 may be embedded in almost any field theoretic model that exhibits spontaneous symmetry breaking2. In particular, two distinct vortex solutions are known to be embedded in the standard electroweak model2,3,4,5. These are called the τ−string and the Z−string in the literature. The τ−string is conjectured to be unstable for all values of the parameters while the Z−string has been shown to be stable only for sinθW ≈ 1. The Z−string in the standard electroweak model with sinθW = 0.23 is unstable6 and remains so even at high temperatures such as would be present in the early universe7. If the string solutions are indeed unstable under all circumstances, their relevance to physical processes would probably be negligible. However, in this letter we shall show that the strings can be stabilized by the presence of other scalar or fermionic fields in the theory. The idea behind this result is quite simple to understand, especially if one is aware of the reason that permits the existence of non-topological solitons8. Suppose that we have a theory in which the Higgs mechanism is responsible for generating the mass of a certain scalar. Then, after the symmetry breaking, the Higgs field gives the scalar a mass but the back-reaction of the scalar field on the Higgs field is to try and prevent the Higgs field from acquiring its vacuum-expectation-value (vev). In other words, the scalar would rather live in a region where the Higgs field vanishes since the mass of the scalar field is zero wherever the Higgs field is zero. But the center of the string is precisely a region where the Higgs field vanishes. Therefore the scalar likes to accumulate on the string and tends to maintain the string configuration with its region of vanishing Higgs field that is, the scalar adds to the stability of the configuration. Yet another way of stating this idea is that the string is a “bag” in which the scalar prefers to sit and, hence, hold together. In what follows, we shall only consider the case of a scalar field interacting with the electroweak Z−string. To start with, we shall describe the effect of scalar bound states on semilocal strings9 where it is fairly clear that the stability improves due to the bound state. 2 This in itself shows that the electroweak Z−string will become more stable when it has scalar bound states since, after all, the Z−string is nothing but the semilocal string when sinθW = 1. However, we go further and explicitly examine the case of the Z−string with a scalar bound state. Our results suggest that it may be possible to get stable Z−strings even when sinθW = 0.23. This does not immediately imply that stable Z−strings occur in the standard electroweak model since there is no extra scalar field in this model. However, the standard model does contain leptons and quarks which will also have bound states on the string. We expect the arguments of the previous paragraphs to apply in this case too since, once again, it is favorable for the fermion to sit in the string “bag” and to prevent the bag from decaying. Whether the lepton and quark bound states are sufficient to stabilize the Z-string is another story that needs detailed investigation. We hope to undertake this task in the near future. The Lagrangian that yields semilocal string solutions with an additional complex scalar field is: Lsl = (D μφ)†(Dμφ) + (∂μχ)∗(∂μχ)− 1 4 F μνF μν Z − V (φ, χ) (1) where, V (φ, χ) = λ1 ( φ†φ− η 2 2 )2 + λ2|χ| + 2λ3(φ†φ±m2)χ∗χ . (2) The field φ is a global SU(2) doublet carrying a gauged U(1) charge, while χ is a single complex field. The covariant derivative is defined by, Dμ = ∂μ + i 2 αZμ . (3) There are two approaches to finding solutions that describe a string with a non-trivial χ configuration. The first is that, for the negative sign in (2) and for some values of the parameters, the string configuration together with χ = 0 is unstable, and the stable 3 ground state solution is one that has a non-trivial χ condensate on the string10. It may be speculated that the presence of a condensate11 might improve the stability of the string. Indeed we have checked that there is an improvement in string stability due to a condensate but the improvement is only marginal and is certainly not enough to stabilize the string when sinθW = 0.23. The second approach is to consider the string in the presence of χ particles that is, the string with χ bound states. The χ particles carry a conserved U(1) global charge which is derived from the conserved current j = i 2 (χ∗∂μχ− χ∂μχ∗) . (4) Hence, we consider a string in the presence of a certain amount of global U(1) charge. This, together with cylindrical symmetry, leads to the following ansatz for χ: