Confinement and moduli locking of Alice strings and monopoles

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.

Topological confinement of vortices in two-flavor dense QCD

“…On the other hand, this 2SC+ dd phase also admits magnetic monopoles. Probably these monopoles are also confined as discussed in the SU(2) toy model [21] for which monopoles are twisted Alice strings and confined by AB defects. If monopoles are confined, this may show a duality between quark matter and hadronic matter; monopoles are confined where quarks are condensed in quark matter while quarks are confined where monopoles are condensed in hadron matter, analogous to the CFL phase [93].…”

“…In this paper, we show that in the confined phase, these non-Abelian vortices are confined to either a baryonic or mesonic bound state in which constituent vortices are connected by AB defects [19][20][21] that appear to compensate nontrivial color non-singlet (generalized) AB phases of the 2SC condensates around vortices. The baryonic bound state consists of three non-Abelian Alice strings with different color magnetic fluxes with the total flux canceled out, which are connected by a domain wall junction.…”

“…Here h(ϕ) is a kink profile inserted to compensate the AB phase, satisfying the boundary conditions h(ϕ = 0) = 1 and h(ϕ = 2π) = −1. The domain walls or solitons appearing to compensate an AB phase are called as the AB defects [19][20][21].…”

“…Another mechanism responsible for vortex confinement is given in the so-called modified XY model [16][17][18] in which the modified gradient term with half periodicity compared with the canonical gradient term plays an essential role. More recently, a novel vortex confinement mechanism based on topology was proposed; when some fields receiving nontrivial Aharonov-Bohm (AB) phases around a vortex develop vacuum expectation values (VEVs), there must appear kinks or domain walls attached to the vortex in order to compensate the AB phases to maintain singlevaluedness of the VEVs [19][20][21]. We referred such defects AB defects.…”

Vortex-bound solitons in topological superfluid 3He

“…Besides stabilizing novel superfluid phases, confinement by nafen features even more complicated symmetry breaking patterns not possible in bulk samples. Such symmetry-breaking phase transitions can be used to study cosmological and dark matter models with SSB [21][22][23][24][25][26][27][28] and, in particular, topological defects that appear in such theories. In nafen samples, the following symmetry breaking pattern can be realized:…”