Confinement and moduli locking of Alice strings and monopoles

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 $SU(2) \times U(1)$ 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 ${\mathbb Z}_2$ 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.


Introduction
Color confinement is one of the most challenging problems in modern high energy physics.
Particles transforming under color gauge group cannot be observed and are confined. Here, we discuss Aharonov-Bohm (AB) effects [1] may be one of key ingredients to understand color confinement. When a charged particle scatters from a solenoid with non-zero magnetic flux inside, a gauge potential rather than a field strength (a magnetic or electric field) yields an AB phase to the charged particle [1], resulting in a non-trivial differential scattering cross section. Such an AB effect was experimentally observed [2,3]. In the theory side, cosmic strings (vortices) exhibiting AB effects were proposed in cosmology [4][5][6] as well as in string theory [7][8][9][10][11]. Non-Abelian vortices in supersymmetric gauge theories [12][13][14][15][16][17][18][19][20] exhibit AB effects once a part of flavor symmetry is gauged [21][22][23]. In dense QCD, a color magnetic flux tube in the 2SC phase exhibits AB effects for quarks [24], and a non-Abelian vortex (color magnetic flux tube) in the color-flavor locked phase [25][26][27][28][29][30] exhibits (electromagnetic) AB effects for charged particles [31] as well as Z 3 (color) AB effects for quarks [32][33][34][35][36]. Non-Abelian Alice strings in two-flavor dense QCD exhibit nontrivial AB phases [37]. Alice strings are not only an example of AB strings but also exhibit a peculiar electromagnetic property. When electrically charged particles encircle around an Alice string, the signs of their electric charges are flipped because of non-single valuedness of an electromagnetic generator [38,39]. The simplest Alice string is present in an SO(3) gauge theory with fiveplet scalar fields (real traceless symmetric tensor), where the SO(3) gauge symmetry is spontaneously broken down to O(2) identified with the electromagnetic U (1) group, which is not singly defined around the string [38,[40][41][42][43]. In quantum field theory, Alice strings were extensively studied for their exotic properties such as topological obstruction, non-local (Cheshire) charge, and non-Abelian statistics [40,[43][44][45][46][47][48][49][50]. In condensed matter physics, a global analogue of Alice strings was found in spinor Bose-Einstein condensates of ultracold atomic gases [51][52][53][54]. Recently, a U (1) × SU (2) (supersymmetric) gauge theory coupled with complex triplet scalar fields was found to admit a Bogomol'nyi-Prasad-Sommerfield (BPS) [55,56] Alice string [57][58][59]. 1 The theory also admits a stable (BPS) Abrikosov-Nielsen-Olesen (ANO) string carrying a unit U (1) magnetic flux [63,64], and a Z 2 string connecting the two elements of the center of SU (2) and consequently carrying one half SU (2) magnetic flux [64]. 2 In this case, the Alice string is a hybrid of these two kinds of vortices and carries one half U (1) magnetic flux of an ANO vortex and one quarter SU (2) magnetic flux taking a value in two of the SU (2) generators characterizing the U (1) modulus. On the other hand, a monopole in this theory can be constructed as a closed Alice string with the U (1) modulus twisted once, which was found in the global analogue [52]. Such a construction of monopole was also known in the conventional Alice theory, the SO(3) gauge theory with fiveplet scalars [40][41][42][43].
In the previous paper, 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 defect 3 is attached to the Alice string. 4 This theory admits multiple BPS Alice strings at arbitrary positions, when we turn off the doublet field. Since a doublet encircling two strings receives no AB phase, the two Alice strings should be connected by one AB defect when the doublet develops a VEV. One natural question was how each Alice string find a partner, when there are many Alice strings.
In this paper, we define the "color" (de)confinement as a situation that a color cannot (can) be detected at large distance by AB phases of some fields belonging to a representation of the color group. We then argue that the Alice string is not confined in the absence of a doublet VEV because the SU (2) (color) 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 (in the certain phase defined below). 5 When the doublet field develops VEVs, there appear deconfined and confined phases with unbroken Z 4 and Z 2 symmetries, respectively. In the confined phase, the Alice string is inevitably attached by 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, where the "color" cannot be seen at large distance by AB phases, implying the "color" confinement. We also show that with the doublet VEVs, monopoles are also confined to pair annihilate or to form monopole mesons of the monopole charge two. We thus call the unbroken phase without the doublet VEVs and the confined phase with the doublet VEVs as deconfined and confined phases, respectively.
Another interesting phenomenon is that in the presence of a single Alice string, the phase is enforced to be in the deconfined phase to annihilate an AB defect, and consequently the U (1) modulus of the string and the vacuum moduli of the doublet fields are locked. We call this phenomenon as the bulk-soliton moduli locking. We also show that while an ANO string and a Z 2 string are stable in the absence of the doublet VEVs, each decays into two Alice strings to each of which an AB defect is attached, once the doublet field develops the VEVs. We discuss that whether a set of two strings exhibits confinement or decay can be determined from AB phases of the doublet field encircling the two strings. This paper is organized as follows. In Sec. 2, we introduce our model admitting BPS Alice strings and discuss in detail symmetry breaking patterns when the triplet field develops VEVs at high energy and the doublet field also develops VEVs at low energy. We also discuss the opposite hierarchy in which the doublet field develops VEVs at high energy and the triplet field develops VEVs at low energy. In Sec. 3, we introduce an ANO string, a Z 2 string, an Alice string, and an Alice monopole. In Sec. 4, we first discuss AB phases the conventional case, an explicit symmetry breaking term of the U (1) symmetry around which a string winds induces a domain wall. String-wall composites in axion models [69], two Higgs doublet models [70,71], and domain wall skyrmions [72,73] belong to this class. We may classify the two cases as follows; The conventional case is induced by "explicit symmetry breakings," while the AB defect is induced by "spontaneous symmetry breakings." 5 In this definition, although a non-Abelian vortex in the CFL phase of dense QCD carries a color magnetic flux [25][26][27][28][29][30], it exhibits only a Z3 AB phase within SU (3) color symmetry, which is color singlet. Namely, a color flux of the non-Abelian vortex cannot be seen from large distance, and consequently it is already confined as it is. The same holds for non-Abelian vortices in supersymmetric gauge theories [12][13][14][15][16][17][18][19][20] unless a part of flavor symmetry is gauged [21][22][23].
of the doublet fields around a single Alice string and around two strings. We then discuss, in the presence of doublet VEVs, the bulk-soliton moduli locking, confinement of Alice strings and monopoles, and decay of ANO and Z 2 strings. Sec. 5 is devoted to a summary and discussion. In Appendix A, we show that complex adjoint scalar fields are equivalent to complex 2 × 2 symmetric tensor scalar fields. In Appendix B, we give a brief review of previous results for BPS Alice string solution [57,58] and an AB defect [59].
2 The Alice theory

The model
We consider an G = SU (2) × U (1) gauge theory coupled with one charged complex triplet (adjoint) scalar field Φ = 3 α=1 Φ α σ α and one (charged) doublet scalar field Ψ. 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 Y = 1, while the doublet fields q (q = 1 2 is the choice of the triplet Higgs model). In the previous paper [59], we considered the cases q = 1 2 and q = 0, but here we mostly consider the former.
The Lagrangian is given by Here e and g are the coupling constants of the U (1) Y and SU (2) W gauge fields, respectively. The potential term is given by where λ g and λ e are two couplings of the triplet, ξ is a parameter giving a VEV of the triplet, M is the bare mass of the doublet, λ ψ is the quartic coupling of the doublet field, and µ, λ 1 , λ 2 are couplings between the doublet and triplet scalar fields. The charge conjugation of the doublet field is defined as Ψ c = iσ 2 Ψ * . Before discussing details in the next section, here let us summarize the symmetry breaking patterns in Fig. 1 and vacuum manifolds and corresponding lower dimensional homotopy groups in Table 1. Vacuum manifolds

Symmetry breaking by the triplet field
First, let us turn off the doublet field and consider the triplet field. We choose the vacuum expectation value of the triplet field Φ without loss of generality as This triplet VEV breaks the gauge symmetry group spontaneously as 7 Here, U (1) W 1 is generated by σ 1 , and denotes a semi-direct product implying that the U (1) W 1 acts on the (Z 4 ) Y +W 2,3 symmetry. The (Z 4 ) Y +W 2,3 symmetry is defined by a 7 When we do not have a doublet field, the actual gauge group is SU (2)/Z2 = SO(3) since the center Z2 does not act on the triplet. Even in this case, it is useful to consider the universal covering SU (2) for the symmetry breaking. simultaneous action of a π rotation of the U (1) Y group and an SU (2) W element generated by a linear combination of σ 2 and σ 3 , e i π 2 n· σ = i cos β sin β − sin β −i cos β , n = (0, sin β, cos β), (2.8) i. e., the minimum element of (Z 4 ) Y +W 2,3 is given by Because of (e iπ ) 2 = 1 and (e i π 2 n· σ ) 2 = −1 2 , the double action in Eq. (2.9), is an element of the center of SU (2) W . 8 We thus have a relation By noting SU (2) W /(Z 2 ) Y +W = SO(3) W with the footnote 8, the vacuum manifold is found to be In the rightmost expression, the (Z 2 ) Y +W 2,3 symmetry is generated by the first factor of the right hand side of Eq. (2.11). The lower dimensional homotopy groups for this manifold can be calculated as indicating the absence of domain walls, the existence of stable strings (vortices), and monopoles. 9

Symmetry breaking by the doublet field
Next, let us turn on the doublet field supposing that the triplet field develops VEVs at high energy and the doublet develops VEVs at low energy. The U (1) W 1 group in H in Eq. (2.7) must be spontaneously broken by the doublet VEVs, but the breaking pattern of the (Z 4 ) Y +W 2,3 symmetry depends on the VEVs. The (Z 4 ) Y +W 2,3 symmetry in Eq. (2.9) acts on the doublet field Ψ = (a, b) (2.14) 8 One has to be careful about the fact that this action behaves as a center only on the triplet field. If there is another field with a non-integer U (1)Y charge, a non-trivial action of the U (1)Y action remains on such a field. Our doublet field is such a field, see Eq. (2.18), below. Thus, we write it as (Z2)Y +W with "Y +." 9 From the nontrivial third homotopy group π3, Hopfions are possible at least for global analogues [75], but it is unclear whether gauged Hopfions stably exist for the local case.
From now on, we consider the case of q = 1/2. We call the case in which the (Z 4 ) Y +W 2,3 symmetry in Eq. (2.9) is unbroken as the deconfined phase, and otherwise the confined phase. Two typical solutions of the deconfined phase are where (Z 4 ) Y +W 2,3 in Eqs. (2.8) and (2.9) is generated by a π rotation of U (1) and +σ 3 or −σ 3 in the SU (2) group, respectively, and so we have denoted it by (Z 4 ) Y ±W 3 . More general solution of the deconfined phase is found to be with (Z 4 ) Y +W 2,3 in Eq. (2.9) and U defined by (2.17) Here, β = 0 and β = π correspond to the two cases in Eq. (2.15) (up to the constant factor i), respectively. The parameter β corresponds to the U (1) W 1 (would-be) Nambu-Goldstone mode associated with the U (1) W 1 symmetry spontaneously broken by the doublet. From the (Z 4 ) Y +W 2,3 action on Ψ in Eq. (2.14) with q = 1/2, the double action, always leaves arbtrary Ψ invariant, thanks to the particular choice of q = 1/2. As noted in footnote 8, this (Z 2 ) Y +W action behaves as a center of SU (2) W when acting only on the triplet field Φ as in Eq. (2.10) but it is not so on the doublet field Ψ because of the −1 element of U (1) Y . This fact becomes important in the vacuum manifolds. Thus, in the confined phase, the (Z 4 ) Y +W 2,3 is spontaneously broken to a (Z 2 ) Y +W subgroup. 10 One can easily verify that the configurations in Eq. (2.16) satisfy the gauge invariant (covariant) condition Thus, the phases and unbroken symmetries can be summarized as for the deconfined and confined phases, respectively. 10 For different choice of q, this (Z2)Y +W is broken in general.
The vacuum manifolds for the second symmetry breaking are deconfined phase: For the both cases, the S 1 U (1) W 1 part of the vacuum manifolds is parameterized by β in Eq. (2.16). The associated lower dimensional homotopy groups for the second symmetry breaking are These show that both the phases allow another type of vortices, and that the confined phase allows Z 2 domain walls while the deconfined phase does not. Finally, the vacuum manifolds of the overall breaking by the both triplet and doublet fields are deconfined phase: for the deconfined and confined phases, respectively. 11 The lower dimensional homotopy groups are (2.24) The both phases have nontrivial first homotopy groups. However, the minimal elements are different as it will become important later.
For the case of q = 0, the condition Ψ = Ψ = (a, b) T with Eq. (2.14) cannot be satisfied, implying that the Z 4 symmetry is broken completely by the VEV of the doublet (where no Z 2 remians).

The opposite hierarchy
Although we mainly consider the case that the triplet VEVs are much greater than the doublet VEVs, let us briefly mention the opposite case (with the doublet U (1) Y charge q = 1/2), in which the doublet VEVs are much greater than the triplet VEVs. As the case of the SM, when the doublet field develops VEVs, we can take without loss of generality. The symmetry is broken as 12 which is the electromagnetic U (1) em in the context of the SM. The vacuum manifold is with lower dimensional homotopy groups which are topologically almost trivial as is well known. Then, when the triplet develops VEVs at low energy, there are two phases as summarized in Eq. (2.20): for Eq. (2.25). The unbroken symmetryH = U (1) Y +W 3 is further broken to either Z 4 or Z 2 (unlike the triplet Higgs model beyond the SM in whichH should be unbroken). In the deconfined phase, Φ is invariant under the Z 4 symmetry whose minimum element is given by In the confined phase, Φ = ξσ 3 is invariant under the double action of Eq. (2.31) generating (Z 2 ) Y +W in Eq. (2.18). Thus, we have unbroken symmetries consistent with the previous discussion as it should be so. One can easily check that these discrete groups are subgroups of the U (1) Y +W 3 group in Eq. (2.27). Thus, the vacuum manifolds for the second symmetry breaking are found to be deconfined phase: 33) and the lower dimensional homotopy groups are There are only nontrivial first homotopy groups admitting strings, which look the same for the deconfined and confined phases. However, the vacuum manifolds in Eq. (2.33) show that the deconfined and confined phases allow 1/4 quantized and 1/2 quantized strings, respectively. When both the triplet and doublet fields develop VEVs, the unbroken symmetries and physics are the same between the two cases that the triplet (doublet) develops VEVs at high energy and the doublet (triplet) develops VEVs at low energy. In the above discussion, these two cases look different at first glance; for instance, in the former, the doublet field Ψ contains a continuous parameter β in Eq. (2.16), and in the lattter, the triplet field Φ contains a continuous parameter γ in Eq. (2.30). This is just because of a difference of the gauge fixings: for the former (latter) the gauge fixing in Eq. (2.6) [(2.25)] was convenient, and these two gauges can be of course transformed to each other by a gauge transformation. In fact, the classification of the deconfined and confined phase is the identical in a gauge invariant description, as can be seen in Eq. (2.20),

Strings and monopoles
In this section, we discuss string and monopole configurations associated with homotopy groups in Eq. (2.13) in the absence of the doublet field.

Abrikosov-Nielsen-Olesen string and Z 2 string
The ANO string [63,64] is a vortex having a phase winding only in the U (1) gauge group. In our case, it has the form of This has a unit U (1) magnetic flux and no SU (2) flux. This is BPS when we take the BPS limit λ e = e and λ g = g.
Next, we consider a Z 2 string which exists in an SU (2) gauge theory with a complex adjoint field [63]. 13 It has the form of The SU (2) group element e i θ 2 σ 3 varies from 1 to −1 when θ changes from θ = 0 to θ = 2π. Thus, the Z 2 string connects the center elements ±1 of SU (2) and carries a half amount of the SU (2) magnetic flux compared with one corresponding to a closed loop in SU (2). 14 This is non-BPS. However, it is stable.

Alice strings
Let us introduce an Alice string discussed in Refs. [57][58][59]. The configuration of an Alice string can be expressed at a large distance as with an angular coordinate θ. It carries a half amount of the U (1) magnetic flux and 1/4 amount of the SU (2) color magnetic flux, taking a value on ±σ 3 in this case. It is important to note that the 1/4 quantization of the SU (2) color magnetic flux of the Alice string is a consequence of (Z 4 ) Y +W 2,3 in Eq. (2.12). The Alice string also can be regarded as a half ANO string as well as a half Z 2 string. A full configuration including finite r can be obtained by complementing profile functions in the asymptotic form in Eq. (3.3). See Appendix B. 13 In the original case [63], U (1) is a global symmetry. 14 Since the center does not act on the triplet in the adjoint action, the actual gauge group is SO(3) = SU (2)/Z2 unless we introduce a doublet field. In this case, Eq. (3.2) gives a closed loop in SO(3). While it carries a half amount of flux in terms of SU (2), it carries a unit flux in terms of SO(3) because it is given by a closed loop in SO(3). In this paper, however, we also consider the doublet, and so our gauge group is SU (2) rather than SO(3).
These two configurations are merely typical configurations belonging to a continuous family of more general configurations, given by Here, β is a U (1) modulus characterizing an SU (2) color magnetic flux between σ 2 and σ 3 , and β = 0 (β = π) corresponds to the configuration in Eq.
On the other hand, anti-strings carrying a −1/2 U (1) magnetic flux and 1/4 SU (2) magnetic flux can be constructed as Again, more generally, they belong to a continuous family of configuration: with U defined in Eq. (2.17) andσ defined in Eq. (3.6). Let us briefly show the Alice property. The triplet field approaches Φ(r → ∞, θ = 0) = ξσ 1 asymptotically along the x 1 -axis (θ = 0), and the θ-dependence is is given by where the holonomies are defined as These can be understood by using the condition of topological vortex where the order parameter Φ is covariantly constant at large distances (D i Φ → 0 as r → ∞). When we encircle the string by an angle θ, the unbroken U (1) generator Q transforms as with Q 0 = σ 1 . We thus have a sign flip of the U (1) generator after one complete encirclement as implying that the unbroken generator is not singly defined around the string, which is nothing but the Alice property and sometimes called a topological obstruction. For later convenience, let us give a doubly-wound Alice string configuration, which is the most important among multiple Alice string configurations. It has the form of for a specific orientation and with U defined in Eq. (2.17) andσ defined in Eq. (3.6), for more general configuration.
Here, β is again a U (1) modulus characterizing an SU (2) color magnetic flux between σ 2 and σ 3 . Each carries a unit U (1) magnetic flux and a half SU (2) magnetic flux, which are just twice of a single Alice string. This can also be regarded as a composite of one ANO vortex and one Z 2 vortex. This should be BPS, since it is just twice of one Alice string (although a Z 2 string is not BPS). The unbroken generator Q is singly defined around the doubly-wound Alice string, unlike a single Alice string.

Alice monopoles
An Alice string carries a U (1) modulus β in Eq. (3.5), corresponding to the internal direction of the flux [58]. Let us consider a closed Alice string as in Fig. 2 (a). The black arrows represent the phase winding of the string. The antipodal points of the ring (cut by a plane in the figure) are a pair of a vortex and anti-vortex, and therefore the ring is usually unstable to shrink. When a U (1) modulus β (represented by red arrows) is twisted along the closed Alice string as in Fig. 2 (a), it can be stable depending on a parameter region and is called a vorton, as discussed below. This twisted Alice ring carries a magnetic monopole charge.
(a) (b) In Ref. [52], Ruostekoski and Anglin found such a stable magnetic monopole solution in the theory without gauge fields (in the context of spinor Bose-Einstein condensates), which is therefore a global monopole.
A similar case for local monopoles were studied in the SO(3) gauge theory with a fiveplet real scalar field (real symmetric traceless tensor) [41,42,74] admitting a conventional Alice string, in which the vacuum manifold is RP 2 S 2 /Z 2 rather than Eq. (2.12). In this case, it was observed that a usual spherical 't Hooft-Polyakov monopole decays into a twisted Alice ring (in a certain parameter region).
Let us discuss the issue of the stability. On the plane cutting the ring, the vortex and anti-vortex are given by Eqs. (3.3) and (3.7), respectively. Note that because of the U (1) modulus twist, the anti-vortex is not the one in Eq. (3.8). The SU (2) fluxes are the same, while the U (1) fluxes are opposite to each other. In general, the U (1) gauge field mediates an attraction between a vortex and anti-vortex, while the SU (2) gauge fields mediates a repulsion between the same magnetic fluxes. Thus, at least when the SU (2) gauge field is lighter than the U (1) gauge field (g < e) and in the type-II regime (implying that Higgs fields are heavier than gauge fields), they can repel and contribute to the stability.
By cutting the Alice ring and stretch it to a straight string (along the z-direction), we obtain an Alice string along which the U (1) modulus is twisted once, as in Fig. 2 (3.17) In this case, the monopole charge density is broadly present where the U (1) modulus is twisted. Namely, the U (1) modulus β of the string winds once along it from β = 0 at z → −∞ to β = 0 at z → +∞. However, this charge cannot be localized and is spread over the string in order to reduce the gradient energy. 15

Confinement of Alice strings and monopoles
In this section, we first discuss the AB phase around an Alice string. We then introduce AB defects appearing when the doublet field develops VEVs, and discuss the bulk-soliton moduli locking in the deconfined phase. After classifying configurations of two strings by AB phases of the doublet field, the confinement of Alice strings and decays of an ANO string and Z 2 string are discussed in the confined phase. Finally, we discuss the confinement of monopoles.

Aharonov-Bohm phases around an Alice string
When the doublet field Ψ = (a, b) T encircles around an Alice string, it receives an AB phase as This phase depends on the U (1) modulus of the Alice string. For instance, it is given by As it can be noticed that after a complete encirclement, the doublet field gets an AB phase as (4. 3) The important observation is that the doublet field can feel the flux (modulus) of the Alice string from infinitely far away from it by encircling around it. From this fact, we say that color is not confined for a single Alice string.

Bulk-soliton moduli locking
When the doublet field Ψ develops VEVs in the absence of an Alice string, the VEV can be arbitrary: parametrizing S 3 (if we turn off the interaction between the triplet and doublet fields).
On the other hand, in the presence of a single Alice string, the VEVs of Ψ is singlevalued for (4.5) More generally, the single-valuedness condition is given by with U defined in Eq. Here, β = 0 and β = π correspond to the first and second lines in Eq. (4.5), respectively (up to the constant factor i). Otherwise, the doublet field cannot be single-valued. Thus, we have found that the U (1) modulus of the Alice string and the vacuum moduli of the doublet fields are locked. We call this the bulk-soliton moduli locking mechanism. We should point out that Ψ is in the deconfined phase satisfies Ψ † ΦΨ = 0 as can be seen in Eq. (2.20). We have two schemes to interpret this phenomenon. When the triplet VEVs are much larger than the doublet VEVs, the symmetry breaking by the triplet field occurs at high energy and subsequently the symmetry breaking by the doublet field occurs at low energy. This is the scheme that we are mainly considering in this paper. In this case, we can fix a triplet configuration (with a string) as a background. Then, if we turn on the doublet VEVs gradually, they develop as in Eq. (4.5) or more generally as in Eq. (4.6). Thus, the doublet choses the deconfined phase. It is interesting that the phase of matter is determined by the presence of a soliton.
In the opposite scheme discussed in Sec. 2.4, in which the doublet VEVs are much larger than the triplet VEVs, the symmetry breaking by the doublet field occurs at high energy and subsequently the symmetry breaking by the triplet field occurs at low energy. In this case, if one creates an Alice string, its U (1) modulus β should be aligned to the doublet field, as Eq. (4.5) or more generally as Eq. (4.6). Namely, the flux is determined by the doublet moduli in the bulk. This can be understood from the vacuum manifold in the first line in Eq. (2.33) allowing strings. In this case, one can observe that strings are purely ANO-like, in contrast to Eq. (2.12) in which the U (1) group is present in the denominator of the coset space, endowing the U (1) modulus to the string. When one creates multiple Alice strings, all of their U (1) moduli must be aligned to the doublet field. In other words, the U (1) moduli of strings are killed by the doublet field.

Aharonov-Bohm defects
Next, let us consider the confined phase. 16 Then, it is inevitable for the doublet fields to receive an AB phase which must be canceled for the single-valuedness as where the factors e ±i θ 2 come from the AB-phase, and α corresponds to the direction of the AB defect. Here, for the single-valuedness of Ψ, we have introduced a multi-valued function f (θ) on S 1 which changes the sign under a 2π shift: f (θ + 2π) = −f (θ). This creates a soliton attached to the Alice string as schematically shown in Fig. 3(a). The  existence of the soliton is topologically supported by the nontrivial zeroth homotopy group in Eq. (2.22) in the confined phase. We called such a soliton as an AB defect [59]. There are two ways to create an AB defect.
1. An ansatz for the doublet for the first case is to take f (θ) = e iφ(θ)/2 [59]: which is nothing but the sine-Gordon model.
2. The second case is to consider a kink configuration by a real function f varying from +1 to −1 at θ = 0 to θ = 2π.
In either case, the AB defect is a linearly extended to infinity, with having infinite energy in infinite space. So it should decay and the configurations in Eq. (4.7) go to those in Eq. (4.5) by changing the orientation of Ψ in the internal space, and the phase is in the deconfined phase. We conclude that the bulk-soliton moduli locking mechanism is energetically realized.
On the other hand, there appear two AB defects around an ANO string and a Z 2 string as in Fig. 3(b), since the both components of the doublet field Ψ receive AB phases around those strings. The doublet fields behave around an ANO string as where f 1 and f 2 are multi-valued functions, α controls the relative directions of the AB defect (we have taken a constant in the upper component to be zero without loss of generality). Energetically, it may be natural to consider α = π as in Fig. 3(b). The two different AB defects are present in the two different components of the doublet field Ψ, and so they have different tensions in general, depending on a and b. Similarly, the doublet fields behave around a Z 2 string as for Φ(r, θ) ∼ ξ 0 e iθ e −iθ 0 (4.11) with multi-valued functions f 1 and f 2 . In these cases, it is impossible to eliminate the two AB defects simultaneously in any choice of the doublet VEVs, although one can eliminate at least one AB defect by taking either a = 0 or b = 0.
In the case of q = 0, one cannot eliminate an AB defect.

Classification of configurations of two strings
When the double VEVs are much larger than the triplet VEVs, we can fix a doublet configuration as a background. Let us consider the confined phase in which the both components of the doublet are nonzero and the Z 4 symmetry is broken to Z 2 . In this case, Alice strings are inevitably attached by AB defects. Here we discuss that configurations  of two strings can be classified by the AB phases of the doublet field encircling around the two strings, as in Fig. 4.
The configurations of two strings can be classified into either the confinement [ Fig. 4(a)] or the decay [ Fig. 4(b)], depending on AB phases of the doublet field encircling around the strings. Without loss of generality, let us consider the case that when the doublet field Ψ = (a, b) T encircles the right string along a smaller loop, it becomes Ψ = (−a, b) T which is not single-valued if there is no AB defect, implying the existence of an AB defect attached to it. Next, when the doublet field Ψ = (a, b) T encircles the left string, let us suppose that it becomes either (a) Ψ = (−a, b) T or (b) Ψ = (a, −b) T . In either case, an AB defect of a different type is attached to the left string.
In the case of (a), the two AB defects attached to the two strings are of the same type, and so they can be connected. In fact, if the doublet field Ψ = (a, b) T encircles the both strings together along the larger loop in Fig. 4(a), it comes back to Ψ = (a, b) T which is single-valued. This shows the absence of AB defect crossing the large loop, implying that the two strings must be connected by a single AB defect. This case implies the confinement of the two strings.
On the other hand, in the case of (b), the two AB defects attached to the two strings are of the different type, and so they cannot be connected. In fact, if the doublet field Ψ = (a, b) T encircles the both strings together along the larger loop in Fig. 4(b), it becomes Ψ = (−a, −b) T which is not yet single-valued, implying the existence of the two kinds of AB defects crossing the large loop. Thus, no AB defect connects the two strings, implying that the two strings are pulled to the opposite (or different) directions. In the following subsections, we discuss the confinement in Fig. 4(a) and the decay in Fig. 4(b), respectively.

Confinement of Alice strings
First, let us discuss the case of confinement corresponding to Fig. 4(a). There are two types of the confinement: 1) a pair of an Alice string and anti-Alice string (colorless confinement) and 2) a pair of the same type of Alice strings (colorful confinement). Let us discuss them one by one. and with constants α 1 and α 2 describing the directions of the AB defects. Here (r, θ) are polar coordinates from the origin, and (r 1 , θ 1 ) [(r 2 , θ 2 )] are polar coordinates from (a, 0) [(−a, 0)]. The configurations (Φ 1 , A 1 , a 1 , Ψ 1 ) and (Φ 2 , A 2 , a 2 , Ψ 2 ) denote a vortex placed at (a, 0) and anti-vortex at (−a, 0). These two vortices are connected by an AB defect as in Fig. 4(a), if one considers the total configuration constructed by an Abrikosov-like ansatz, (4.14) This configuration corresponds to take α 1 = π, α 2 = 0 in the individual configurations in order to connect them. In this case, they annihilate in pair as indicated by arrows in the above equation, and the final state is the vacuum Φ = σ 1 and Ψ = constant as expected.
2) A pair of the same type of Alice strings (colorful confinement). Next, let us consider a pair of configurations and Φ 2 (r, θ) ∼ ξ 0 e iθ 2 1 0 = ξe i θ 2 2 e +i θ 2 4 σ 3 σ 1 e −i θ 2 4 σ 3 , The coordinates are the same with Fig. 5, and the configurations (Φ 1 , A 1 , a 1 ) and (Φ 2 , A 2 , a 2 ) denote vortices of the same kind placed at (a, 0) and (−a, 0). Again, these two vortices are also connected by an AB defect as in Fig. 4(a), if the total configuration is constructed by an Abrikosov-like ansatz, Then, the final state indicated after the arrows will be the doubly-wound Alice string configuration with a singly quantized U (1) magnetic flux and half-quantized SU (2) magnetic flux in Eq. (3.14). In this case, the SU (2) color flux remains in the confined string. One may think that color is not confined. However, this color flux is screened and cannot be seen from the large distance by AB phases. If we consider the deconfined phase, either of the two components of the doublet is zero, the Z 4 symmetry is unbroken, and domain walls are not topologically supported. In this case, either of ANO string is free from an AB defect as can be seen in Eq. (4.5), and consequently it is not confined (but the other is confined).
Comparing the vacuum manifolds for the deconfined and confined phases in Eq. (2.23), one can understand that the confined phase does not admit a half U (1) winding (and therefore a half-quantized U (1) magnetic flux), while the deconfined phase does, consistent with the above discussion.
From the vacuum manifolds in Eq. (2.33) for the second symmetry breaking in the opposite hierarchy, one can understand that in the confined phase strings are half-quantized strings corresponding to doubly-wound Alice strings, in contrast to the deconfined phase admitting 1/4-quantized strings corresponding to the Alice strings.

Decay of Abrikosov-Nielsen-Olesen string and Z 2 string
Next, let us discuss the case of decay corresponding to Fig. 4(b).
An ANO vortex is attached by two AB defects in the confined phase as in Eq. (4.10) and Fig. 3 (b). It decays and is split into a pair of Alice strings with the opposite SU (2) fluxes: and Φ 2 (r, θ) ∼ ξ 0 1 e iθ 2 0 = ξe i θ 2 2 e −i θ 2 4 σ 3 σ 1 e +i θ 2 4 σ 3 , . (4.21) Here, the coordinates are the same with Fig. 5, and the configurations (Φ 1 , A 1 , a 1 , Ψ 1 ) and (Φ 2 , A 2 , a 2 , Ψ 2 ) denote vortices placed at (a, 0) and (−a, 0) as in Fig. 4(b). These two vortices cannot be connected by an AB defect. Instead, two different AB defects are attached to each of them and pull the vortices to the opposite (or different) directions.
A Z 2 vortex is also attached by two AB defects in the confined phase as in Eq. (4.11) and Fig. 3 (b). It also decays and is split into a pair of Alice string and anti-Alice string with the same SU (2) magnetic fluxes: and . (4.23) Again, the coordinates are the same with Fig. 5, and the configurations (Φ 1 , A 1 , a 1 , Ψ 1 ) and (Φ 2 , A 2 , a 2 , Ψ 2 ) denote vortices placed at (a, 0) and (−a, 0) as in Fig. 5. These two vortices are attached and pulled by the two different AB defects as in Fig. 4(b). If we consider the deconfined phase in which either of the two components of the doublet is zero, only one AB defect is attached to the ANO (or Z 2 ) string, and either of two Alice strings is free from an AB defect as can be seen in Eq. (4.5). However, the decay of the ANO string and Z 2 string always occurs since either of constituents is attached by an AB defect.

Confinement of Alice monopoles: monopole mesons
Here we turn to discuss that monopoles are also confined. We first discuss the deconfined phase in which one of components of the doublet fields is zero: Ψ = (a, 0) T . Then, we point out that the same story holds for the confined phase as well.
First, let us consider a monopole on an Alice string. Let an Alice string be placed along the z direction as in Eq. (3.17), where and the U (1) modulus β of the string winds once along it from β = 0 at z → −∞ to β = 2π at z → +∞. Then, there is one monopole on the string at β = π as Sec. 3.3. In the absence of the doublet VEV, the monopole charge density is spread over the string. However, it is not the case in the presence of the doublet VEV.
To see this, we consider a Wilson loop for the doublet encircling the string. We consider the doublet fields have a VEV in the second component and an Alice string has a winding in the upper-right component, as the first line Eq. (4.5) with the upper sign. Then, the Wilson loop is trivial for β = 0, 2π at z = ±∞. However, the Wilson loop encircling around the monopole (β = π) is inevitably nontrivial, resulting in an AB defect (string) attached to the monopole as in Fig. 6 (a). In this sense, this can be regarded as a defect of the (a) (b) Wilson loop. The U (1) modulus β is a sine-Gordon-like soliton along the string. If the AB defect is heavy enough compared with the tension of the Alice string, it will pull the string with constituting a Y -shape junction. Next let us consider that the U (1) modulus β of the Alice string winds twice along the string from β = 0 at z → −∞ to β = 2π at z → +∞. Then, there exist two monopoles, and each of them is attached by an AB defect string as in Fig. 6 (b) (left). Energetically, it is naturally to consider that these AB defects are connected, and the two monopoles are confined to be a monopole meson as in Fig. 6 (b) (right). Thus, the monopole confinement occurs on the Alice string. Now, we are ready to discuss an isolated monopole. An isolated monopole can be obtained as a twisted Alice ring as discussed in Sec. 3.3, which can be done by gluing the two ends of an Alice string in Fig. 6. Then, as shown in Fig. 7(a), an AB defect is attached on the monopole point on the Alice ring. The U (1) modulus also becomes a (a) sine-Gordon-like soliton on a ring. This monopole must be confined. As the case of Alice strings, it can be confined with an anti-monopole. In that case, they are unstable and annihilate in pair. On the other hand, a monopole can be confined with another monopole as in Fig. 7(b). This configuration has the monopole charge two in total. There are two more possibilities of the monopole of the charge two. One is a double winding of the U (1) modulus β along the Alice ring in Fig. 7(c), and the other is a singly twisted U (1) modulus β along a ring of a doubly-wound Alice string as in Fig. 7(d). Interestingly, the latter is not associated with any AB defect, since doubly-wound Alice strings do not have AB effects. It is a highly nontrivial question depending on parameter choices which configurations among Fig. 7(b), (c) and (d) is the most stable.
In summary, monopoles of a single (or odd) monopole charge is confined, and monopoles of an even monopole charge are not confined and are allowed to exist.
Let us discuss how monopoles look like in the deconfined phase. In this case, AB defects appear around β = 0 and β = π. Therefore, the AB defects disappear around β = ±π/2. Monopoles on a straight string become like Fig. 8(a). If we twist β from once (a) (b) there appear two monopoles of the half charge, each of which is attached by two different AB defects corresponding to two components of the doublet. Therefore, in principle, only a half-monopole is possible to exist alone on a string. The effective theory for β would be double sine-Gordon-like. If we consider monopoles of charge two on a string, they are confined as in Fig. 8(b). On the other hand, isolated monopoles can be constructed as twisted Alice rings as before. See Fig. 9. A single monopole is attached by two AB defects as in (a). This must be confined as either (b) or (c): two single monopoles are connected by two AB defects as in (b), and a doubly-twisted Alice ring have two AB defects connecting two sets of antipodal points of the ring as in (c).

Summary and discussion
We have studied an SU (2) × U (1) gauge theory with charged triplet complex scalar fields admitting Alice strings and Alice monopoles, and introduced the charged doublet scalar fields receiving nontrivial AB phases around the Alice string. An ANO string and a Z 2 string carry unit U (1) magnetic flux and a half SU (2) magnetic flux, respectively. First, we have studied the vacua in the presence of the doublet VEVs. The unbroken symmetry is Z 4 if one of VEVs is zero (the deconfined phase), and it is Z 2 if both VEVs are nonzero (the confined phase). The Alice string carries a half U (1) magnetic flux and 1/4 SU (2) magnetic flux taking a value in a linear combination of σ 2 and σ 3 (when the VEV of the triplet is σ 1 ). In the absence of a doublet VEV, the Alice string is deconfined in the sense that the color flux can be seen by AB phases of some fields (like our doublet fields) encircling the string. When the doublet field develops VEVs in the presence of a single Alice string, the bulk-soliton moduli locking occurs; the vacuum moduli are locked with the U (1) modulus of the Alice string. The deconfined phase is realized to eliminate a domain wall called an AB defect responsible for single-valuedness of the doublet fields. On the other hand, in the confined phase, the Alice string is inevitably attached by an AB defect. Consequently, it is confined with either an anti-Alice string or the same type of an Alice string. In the former case, the pair annihilates. In the latter case, the pair forms a stable doubly-wound Alice string having a color magnetic flux inside the core, but the color cannot be seen at large distance by AB phases and so the color confinement occurs. We have referred these phases without and with doublet VEVs as the deconfined and confined phases, respectively. The results are also consistent with the vacuum manifold in Eq. (2.12) admitting 1/4 quantized SU (2) fluxes for the deconfined phase, and the vacuum manifold in the confined phase in Eq. (2.23) admitting only 1/2 quantized SU (2) fluxes for the confined phase. In the scheme that the doublet field develops VEVs at high energy, the vacuum manifolds in Eq. (2.33) for the second symmetry breakings by the triplet field at low energy imply that the deconfined and confined phases allow 1/4 and 1/2 quantized W 3 magnetic fluxes, which are a single Alice string and doubly wound Alice string, respectively, implying the confinement in the confined phase. We also have shown that the ANO string and Z 2 string decay into two Alice strings once the doublet field develops VEVs. Furthermore, a monopole in this theory can be constructed as a closed Alice string with the U (1) modulus twisted once, and we have found that when the doublet field develops VEVs, monopoles are confined to mesons of the monopole charge two. When a charged particle encircles an Alice string in the absence of the doublet VEVs, its electric charge is flipped. This is an example of topological obstructions. In addition to this, when a monopole encircles an Alice string, a monopole charge is also flipped, which is an example of topological influence (see Ref. [53] for a global analogue of this phenomenon). When the doublet field develops VEVs, Alice strings are confined to doubly-wound Alice strings and monopoles are confined to monopole mesons as discussed in this paper. In this case, it seems that the topological obstructions and topological influence disappear. This suggests that deconfinement and confinement phases may be characterized by the presence and absence, respectively of the topological obstruction and topological influence.
Several discussions are addressed here. If we turn off the U (1) gauge coupling, e = 0, and regard this global U (1) symmetry as the Peccei-Quinn symmetry for axions, our model becomes an axion dark matter model in Refs. [60,61]. After chiral symmetry breaking, an axion string is attached by two axion domain walls and the domain wall number is two in the model. When the domain wall number is more than one, usually there is a domain wall problem (if domain walls cannot decay and dominate Universe) [69]. In our case, one axion string attached by two domain walls can decay into two Alice strings each of which is attached by only one domain wall, thereby free from the domain wall problem. In Ref. [60], the doublet scalar field was also considered. This doublet field receives an AB phase as discussed in the present paper. If the doublet develops a VEV, one axion string (corresponding to ANO string in this paper) can decay into two Alice strings, thereby a domain wall problem does not occur even before chiral symmetry breaking.
The third homotopy group for the first symmetry breaking by the triplet in Eq. (2.13) implies the existence of gauged Hopfions (for a global analogues of the same breaking, see Ref. [75]). The second homotopy group for the second symmetry breaking by the doublet in Eq. (2.22) implies that there are another type of vortices in the both deconfined and confined phases. Studying these topological objects remains as a future problem.
Fermions receiving non-trivial AB phases can contribute to AB defects if they form pair condensations. In fact, such an example can be found in dense QCD [37], in which an with the same gauge field configurations in Eq. (3.14) and (3.15), respectively.

B Alice string configurations
Here, we briefly summarize BPS Alice strings [57,58] in the absence of the doublet field. setting all the scalar couplings to be zero except λ g and λ e . The static Hamiltonian density is given by