Abstract
The vacuum manifold of the standard electroweak model is a three-sphere when one considers homogeneous Higgs field configurations. For inhomogeneous configurations we argue that the vacuum manifold is the Hopf fibered three sphere and that this viewpoint leads to general criteria to detect electroweak monopoles and Z-strings. We extend the Kibble mechanism to study the formation of electroweak monopoles and strings during electroweak symmetry breaking. The distribution of magnetic monopoles produces magnetic fields that have a spectrum Bλ ∝ λ−2, where λ is a smearing length scale. Even as the magnetic monopoles annihilate due to the confining Z-strings, the magnetic field evolves with the turbulent plasma and may be relevant for cosmological observations.
Highlights
For inhomogeneous configurations we argue that the vacuum manifold is the Hopf fibered three sphere and that this viewpoint leads to general criteria to detect electroweak monopoles and Z-strings
If we restrict attention to asymptotic Higgs fields configurations that have vanishing potential and gradient energy, the Higgs VEV would have to lie on an S2 and this has the right topology for magnetic monopoles
In gauge theories, inhomogeneous configurations can have minimum energy provided they lie on gauge orbits on the vacuum manifold
Summary
The vacuum manifold of the electroweak manifold is the set of all spatially homogeneous and static Higgs fields for which the energy function vanishes. We have learned from semilocal strings that the vacuum manifold does not give the complete picture when one considers inhomogeneous Higgs fields configuration for the gradient energy terms can be important. Let us clarify this further by discussing the semilocal limit of the electroweak model. One can consider Higgs configurations that lie entirely on the vacuum manifold but whose energy cannot vanish This is because the gauged U(1)Y symmetry defines S1 gauge orbits on the vacuum manifold. Because of the non-trivial global structure of the Hopf fibration, the Z-string is attached to the electroweak monopole
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