Abstract
We argue that strings (vortices) and monopoles are confined, when fields receiving nontrivial Aharonov-Bohm (AB) phases around a string develop vacuum expectation values (VEVs). We illustrate this in an SU(2)×U(1) gauge theory with charged triplet complex scalar fields admitting Alice strings and monopoles, by introducing charged doublet scalar fields receiving nontrivial AB phases around the Alice string. The Alice string carries a half U(1) magnetic flux and 1/4 SU(2) magnetic flux taking a value in two of the SU(2) generators characterizing the U(1) modulus. This string is not confined in the absence of a doublet VEV in the sense that the SU(2) magnetic flux can be detected at large distance by an AB phase around the string. When the doublet field develops VEVs, there appear two kinds of phases that we call deconfined and confined phases. When a single Alice string is present in the deconfined phase, the U(1) modulus of the string and the vacuum moduli are locked (the bulk-soliton moduli locking). In the confined phase, the Alice string is inevitably attached by a domain wall that we call an AB defect and is confined with an anti-Alice string or another Alice string with the same SU(2) flux. Depending on the partner, the pair annihilates or forms a stable doubly-wound Alice string having an SU(2) magnetic flux inside the core, whose color cannot be detected at large distance by AB phases, implying the “color” confinement. The theory also admits stable Abrikosov-Nielsen-Olesen string and a ℤ2 string in the absence of the doublet VEVs, and each decays into two Alice strings in the presence of the doublet VEVs. A monopole in this theory can be constructed as a closed Alice string with the U(1) modulus twisted once, and we show that with the doublet VEVs, monopoles are also confined to monopole mesons of the monopole charge two.
Highlights
Alice strings are an example of AB strings and exhibit a peculiar electromagnetic property
In order to discuss what happens when a field receiving a non-trivial AB phase develops a vacuum expectation value (VEV), we further introduced doublet scalar fields receiving non-trivial AB phases in the presence of an Alice string, and found that a soliton or domain wall which we call an AB defect3 is attached to the Alice string
We argue that the Alice string is not confined in the absence of a doublet VEV because the SU(2) magnetic flux can be detected at large distance by an AB phase of the doublet fields around the string, while the confinement occurs once the doublet field develops VEVs
Summary
We consider an G = SU(2) × U(1) gauge theory coupled with one charged complex triplet (adjoint) scalar field Φ =. Since the matter content is the same with the gauge and Higgs sectors of the triplet Higgs model beyond the SM, we label the gauge group as G = SU(2)W × U(1)Y .6. The U(1)Y charge of the triplet is. 1 2 is the choice of the triplet Higgs model). E and g are the coupling constants of the U(1)Y and SU(2)W gauge fields, respectively. The charge conjugation of the doublet field is defined as Ψc = iσ2Ψ∗. Here let us summarize the symmetry breaking patterns in figure 1 and vacuum manifolds and corresponding lower dimensional homotopy groups in table 1 Before discussing details here let us summarize the symmetry breaking patterns in figure 1 and vacuum manifolds and corresponding lower dimensional homotopy groups in table 1
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