Note on a solution to domain wall problem with the Lazarides-Shafi mechanism in axion dark matter models

Axion is a promising candidate of dark matter. After the Peccei-Quinn symmetry breaking, axion strings are formed and attached by domain walls when the temperature of the universe becomes comparable to the QCD scale. Such objects can cause cosmological disasters if they are long-lived. As a solution for it, the Lazarides-Shafi mechanism is often discussed through introduction of a new non-Abelian (gauge) symmetry. We study this mechanism in detail and show configuration of strings and walls. Even if Abelian axion strings with a domain wall number greater than one are formed in the early universe, each of them is split into multiple Alice axion strings due to a repulsive force between the Alice strings even without domain wall. When domain walls are formed as the universe cools down, a single Alice string can be attached by a single wall because a vacuum is connected by a non-Abelian rotation without changing energy. Even if an Abelian axion string attached by domain walls are created due to the Kibble Zurek mechanism at the chiral phase transition, such strings are also similarly split into multiple Alice strings attached by walls in the presence of the domain wall tension. Such walls do not form stable networks since they collapse by the tension of the walls, emitting axions.


I. INTRODUCTION
The standard model successfully explains various phenomena in experiments. However there exist several unsolved problems. One of the problems is the strong CP problem. We may have a CP violating term in QCD: where θ is a constant parameter, g s is the strong coupling constant, G a µν is the gluon field strength, andG a µν is its dual. Measurements of the neutron electric dipole moment gives a constraint that |θ| 10 −10 [1], while a naive expectation is that |θ| = O(1). Without any mechanisms, this would require a fine-tuning. The problem can be naturally solved if one introduces the Peccei-Quinn (PQ) mechanism with a global U (1) symmetry denoted by U (1) PQ [2][3][4]. After the spontaneous U (1) PQ breaking, a pseudo Nambu-Goldstone boson called the QCD axion appears. When the U (1) PQ symmetry is explicitly broken by QCD instanton effect via chiral anomaly between U (1) PQ and QCD, the axion vacuum is at a CP-conserving minimum of the potential and the axion solves the strong CP problem. The axion decay constant f a is of order the U (1) PQ symmetry breaking scale, and the so-called axion window is given by 10 9 GeV f a 10 12 GeV. ( The lower bound comes from the SN 1987A neutrino burst duration [5]. The upper bound comes from the dark matter abundance by the misalignment mechanism without a tuning of the initial misalignment, in which the coherent oscillation of the axion around the vacuum accounts for the abundance [6][7][8]. See e.g. Refs. [9,10] for reviews. Physics of axion is related to the history of the universe. Below temperature of 1 GeV, QCD instantons breaks the U (1) PQ to Z NDW symmetry, where N DW is an integer called the domain wall number, then there appear N DW vacua. Along with the PQ symmetry breaking, domain walls attached to strings are formed [11][12][13][14][15] once one of the vacua is chosen. The cosmological scenario depends on when the breakdown of the PQ symmetry takes place. If the PQ symmetry is broken before or during inflation, strings and walls are inflated away because the axion field value becomes homogeneous over the scale of Hubble horizon after inflation. They cannot affect on cosmological observations, however, there exist a stringent constraint on isocurvature perturbation produced by the axion during the inflation [16]. If the PQ symmetry is broken after inflation, such a constraint does not exist, but walls and strings may survive until late time and can affect on evolution of the universe. We will focus on the latter case in this paper.
A stability of walls attached to strings is known to depend on N DW , which is related to a topological charge. For N DW = 1, the domain wall attached to a string collapses owing to its tension, emitting axions. On top of the misalignment, axions produced by their decays accounts for a fraction of the dark matter abundance [17,18]. For N DW > 1, the domain walls attached to a string constitute complex networks, which are called string-domain wall networks. The walls in the network pull each other with their tensions and they do not shrink to a point, thus networks can be long-lived. They eventually dominate the energy density of the universe beyond those of radiation and matter, and conflict with the standard cosmology. This is called a domain wall problem [14].
For a solution to the domain wall problem, several ideas have been proposed so far [19][20][21][22]. Among them, we will focus on an axion model associated with a continuous non-Abelian gauge symmetry proposed by Lazarides-Shafi [20] 1 . One might think that a topologically stable domain wall is formed when Z NDW symmetry is spontaneously broken by choosing the vacuum. However, when the non-Abelian symmetry are also spontaneously broken at the same time and a combination of Z NDW and the broken non-Abelian rotation can make the vacuum invariant, the vacua are also continuously connected also by the non-Abelian group without changing energy. This is because the non-Abelian rotation is equivalent to a travel in the space of would-be NG modes. Then, a topological property of such a domain wall for any N DW becomes trivial like in a case with N DW = 1. Hence domain wall problem is solved. This is called the Lazarides-Shafi mechanism.
However, behaviors of strings and walls in the mechanism has not been much discussed in literatures, while the authors of Ref. [22] discussed the decay of Abelian axion strings to Alice axion strings to solve the domain wall problem based on the mechanism. In this model, we may have N DW = 2 and a single Axion string is attached by two domain walls, seemingly having a domain wall problem. The Alice string produced by the decay plays a crucial role to realize a situation similar to the N DW = 1 case, in which one Alice string is attached by one domain wall. Hence the network is unstable in the presence of the domain wall tension. (See Refs. [24,25] for Abelian string and also Refs. [26,27] for reviews of cosmic strings.) Alice strings have a peculiar property that when the electric charge of a charged particle encircles around an Alice string, it changes its sign [28,29]. Other peculiar propeties such as a topological obstruction, a non-local charge called the Chesire charge, and non-Abelian statistics, have been studied in the literature [30][31][32][33][34][35][36][37]. While a typical Alice string is present in an SO(3) gauge theory with scalar fields in the fiveplet representation (a traceless symmetric tensor), recently it has been found that a U (1) × SO(3) gauge theory with complex triplet scalar fields also admits an Alice string, which is a Bogomol'nyi-Prasad-Sommerfield state [38,39] and is stable, thereby possible to be embedded into supersymmetric theories [40,41]. A global analog was known in the context of Bose-Einstein condensates in condensed matter physics [42][43][44][45]. The Alice axion string proposed in Ref. [22] is the case that only the U (1) part is global identified with axion, while the SU (2) part is a gauge symmetry.
In this paper, we show why the domain wall problem is solved physically in more detail, focusing on dynamics of domain walls and two types of axion strings. It is found that even if Abelian axion strings are formed in the early universe, the string decays into multiple Alice axion strings owing to a repulsive force between the Alice strings. When domain walls are formed at the chiral phase transition as the universe cools down, a single Alice axion string is attached by a single wall because a vacuum is connected by a non-Abelian rotation without changing energy. Such walls do not form stable networks since they collapse owing to the tension of the walls, emitting axions similarly to the N DW = 1 case. Also, at the chiral phase transition, two types of domain walls may be created by the Kibble-Zurek mechanism [46,47], and can be glued along an Abelian axion string. The Abelian axion string is pulled by these domain walls and is splitted into multiple Alice axion strings, each of which is attached by one domain wall. In either of these cases, the domain wall problem can be solved.
The rest of this paper is organized as follows. In Sec. II, we briefly review the QCD axion and domain wall problem. In Sec. III, we introduce an axion model with heavy quarks and a new gauge symmetry for solving the domain wall problem. In Sec. IV, properties of strings and domain walls are studied. In Sec. V, we study domain walls attached to strings. Sec. VI is devoted to discussion and conclusions.

II. REVIEW OF THE QCD AXION AND DOMAIN WALL PROBLEM
In this section, we review the QCD axion based on the Kim-Shifman-Vainstein-Zakharov (KSVZ) model [48,49] and domain wall problem for simplicity. (The Dine-Fischler-Srednicki-Zhitnitsky (DFSZ) model [50,51] can be also discussed in a similar way.) Let us consider a coupling Here, (Q, Q) are N DW pairs of extra heavy quark in the SU (3) c gauge symmetry, and Φ is a complex scalar singlet under the Standard Model gauge symmetry. This coupling is invariant under the global U (1) PQ symmetry: Here θ is a transformation parameter. Suppose that after inflation Φ develops vacuum expectation value (VEV) η and the U (1) PQ is spontaneously broken down. Thus the scalar field is parametrized as Here σ is supposed to be stabilized and we neglect it throughout this paper, a is the QCD axion. Note that a U (1) PQ rotation of a → a + 2π · (2η) is a symmetry. All pairs of extra quarks obtain heavy mass √ 2yη × e i a 2η . After integrating out extra quarks with the rotation ofQQ → e −i a 2ηQ Q, we obtain via the chiral anomaly between U (1) PQ and SU (3) c . Thus a = 0 means the CP-conserving vacuum. After the chiral symmetry breaking in QCD, gluons and light quarks are integrated out, and the axion potential can be written as Here, m π is the pion mass and f π is the pion decay constant. As desired, a = 0 is obtained in the vacuum and the strong CP problem is then solved. The axion mass is given by m a m π f π f a 6 × 10 −4 eV 10 10 GeV f a .
Note that a → a+2πf a is the Z NDW symmetry against the above potential, in addition to the original larger symmetry of a → a+2πN DW f a . Thus, there exist N DW vacua. Once one of the vacua is chosen, Z NDW symmetry is spontaneously broken and topologically stable domain walls (attached to Abelian strings) appear between vacua for N DW > 1. When one classically travels from a vacuum to the next one in the axion field space, it is necessary to climb the potential energy. The walls pull each other with their tensions and they do not shrink to a point, thus can be long-lived. The presence of stable domain walls conflict with the standard cosmology, because they eventually dominate the energy density of the universe beyond those of radiation and matter. It is verified in simulations that domain walls survive until late time for N DW > 1, while they decay for N DW = 1 or in the presence of bias potential for N DW > 1 [17,18]. For N DW = 1, the axion dark matter abundance produced by decays of walls and strings is estimated as Ω a h 2 ∼ 10 −2 f a 10 10 GeV 1.19 . (9)

III. THE MODEL WITH A NON-ABELIAN GAUGE SYMMETRY
Following the Ref. [22], we explain the model to implement the Lazarides-Shafi mechanism in the KSVZ case. We shall start with the hidden SU (2) H gauge theory on top of the global U (1) PQ symmetry. The Lagrangian is given as where F µν is the SU (2) (2) generators τ a = 1 2 σ a as Φ = φ a τ a and A µ = A a µ τ a , and the covariant derivative is defined as Here g is the SU (2) H gauge coupling. The potential for Φ is given as This is an usual potential for complex adjoint scalar field that breaks U (1) PQ × SU (2) H spontaneously. Later we will consider an explicit violation term for U (1) PQ , which is relevant to axion mass and domain wall construction. The ground state is given by the solution of the equations The vacuum solution we may generally choose as The VEV breaks U (1) PQ and SU (2) H , then there exists the unbroken U (1) H gauge symmetry with a generator of τ 1 . This vacuum is invariant under O(2) rather than SO (2). The elements are given as Here α, c 1,2 are parameters and the conditon of c 2 1 + c 2 2 = 1 is satisfied. The first entry of each element of H is U (1) PQ and the second entry is the element of U (1) ⊂ SU (2) H group, which can act as adjoint representation on Φ . The first element is usual element of U (1) H ⊂ O(2) connected to identity element (1, 1). However, the second element is a non-trivial Z 2 . Note that e iπQPQ = −1 for Q PQ = 1, where Q PQ is the PQ charge of the Φ, and e iπτ a = iσ a for a = 2, 3: The first entry is π rotation of U (1) PQ and the part of second entry i(c 1 σ 2 + c 2 σ 3 ) gives π rotation of broken SU (2) H around an axis orthogonal to τ 1 . So the second element of H is referred to the disconnected elements of O(2).
More specifically we may describe the symmetry breaking procedure as where denotes the semi-direct product because the Z 2 caused by π rotation in the above discussion does not commute with the unbroken U (1) H generated by τ 1 . The vacuum manifold 3 is found to be The fundamental group is π 1 (G/H) = Z, and this shows the existence of strings. It is noted that Φ has six degrees of freedom. In this vacuum, two of three NG modes associated with the broken SU (2) H are eaten by the gauge fields A 2 and A 3 , which get massive. The rest (pseudo) NG mode relevant to U (1) PQ is the QCD axion denoted by a. The remaining three modes of Φ also become massive in the vacuum [22]. By integrating out extra heavy quarks with a mass matrix of we have Here, N DW = 2 and N DW = 3, and F µν is the unbroken U (1) H gauge field strength. After the chiral symmetry breaking in QCD, the axion potential can be written as Here, m a m π f π /f a and f a = 2η NDW . To study domain wall later, we parametrize this potential with Φ as Here µ is assumed to be of O(m 2 a /N 2 DW ) constant 4 . A constant term is added to make the vacuum energy positive definite. This form is motivated by the fact that axion and walls do not appear for Φ = 0. (See also [52].) V DW (Φ) will be added to V (Φ) of Eq. (11) in the following sections about domain walls. Since N DW = 2 vacua are connected by Z 2 embedded both in the U (1) PQ and the broken SU (2) H , which acts as Φ → − Φ , the domain wall problem is solved as seen later. For a general N DW , a model with an SU (N DW ) H (gauge) symmetry is viable to implement the Lazarides-Shafi mechanism, because Z NDW of the center in SU (N DW ) H connects N DW vacua. Now, we have also monopole since π 2 (G/H) = Z. It can be also a candidate of dark matter with mass of 10 10 GeV which is supposed to be created by the first order phase transition of the SU (2) H at a high temperature around 3 × 10 9 GeV [22]. The monopole can become a dyon with an electric U (1) H charge via the Witten effect [53] when the axion cannot cancel the CP phase in the hidden sector θ-term. Then the monopole may have mini electric charge via kinetic mixing between the photon and hidden photon in the U (1) H . We assume that the mini charge is sufficiently small to evade experimental bounds [54,55]. The monopole will not give further effects to solution to the domain wall problem in the KSVZ case. For N DW > 2 with SU (2) H , however, the Witten effect to the axion mass in the early universe [56] can give a significant effect to solve domain wall problem in the presence of monopoles [21,22]. 5

IV. CONSTRUCTION OF STRINGS
Since the fundamental group of the vacuum manifold is an integer so we may expect topological string solutions. In this section we discuss two kinds of string solutions. One is an Abelian axion string originating from U (1) PQ symmetry breaking and this kind of strings can be thought of an usual global Abelian strings. Another is a non-trivial Alice axion strings which contain non-Abelian magnetic flux. This string can be classified by the elements of the unbroken group H = O(2). In the following, we discuss the splitting of one Abelian axion string into two Alice axion strings. We show full numerical results of the splitting and will set V DW (Φ) = 0 in this section.

IV.1. Abelian axion strings
Here we would like to discuss the fully Abelian axion string which results from the U (1) PQ breaking. The axisymmetric ansatz for this string is given by where m is winding number, r and θ are radial coordinate and azimuthal angle respectively. The boundary condition is given by f (0) = 0, f (ρ) = 1, where ρ is the system size in the radial direction. The profile function f (r) can be calculated from the axisymmetric equations of motion with this boundary condition. After inserting the above ansatz for m = 1, we easily find the static Hamiltonian for Φ with Eq. (10). Two dimensional integral of the Hamiltonian density is given by Here prime denotes the derivative with respect to r. Note that because the above Hamiltonian is written in a static case we have H (2) = −L, where L = d 2 xL is two dimensional integral of the Lagrangian Eq. (10) with the ansatz.
With the radial coordinate, one dimensional Euler-Lagrange equation of the profile function reads The numerical result is shown in Fig. 1, while the analytical solution of the above equation is not known. The tension (energy per unit length) of these strings is found from the above H (2) . Approximate value of the tension far from the string core is easy to compute if we insert the ansatz at a large distance into the Hamiltonian: We may notice that energy is logarithmically divergent and energy depends on the square of winding number m 2 .
The energy stored inside of the core is estimated as of order η 2 , to which the scalar potential contributes, and can be neglected at a large distance. It is always energetically favorable to decay higher winding number string into lower winding strings. Since an Alice axion strings have m = 1/2 as seen later, so an Abelian axion string with m = 1 decays into two Alice axion strings.

IV.2. Alice axion strings
The Alice axion string is a kind of topological string which changes the sign of electric charge of a probe particle with an original gauge symmetry after one encirclement around the string. In our case, the generator of the unbroken U (1) H changes its sign with the SU (2) H after one rotation. This is because a particle charged under the U (1) H is affected by the broken SU (2) H flux inside the Alice string. To understand this better, let us first consider the field value rotating around an Alice axion string, which depends on the azimuthal angle at a large distance ρ, Here, x 2 + y 2 = r 2 , Φ(ρ, 0) = 2ητ 1 = Φ and the holonomy Ω(θ) rotating the Φ(ρ, 0) by θ can be defined by the broken SU (2) H gauge field: where we used ∂ i θ = − ij x j /r 2 . It is easy to compute the (non-Abelian) flux in the broken SU (2) H trapped inside the string: Ths SU (2) holonomy can also be Ω(θ) = e i θ 2 τ 2 or more generally Ω(θ) = e i θ 2n withn = sin ατ 2 + cos ατ 3 = e iατ 1 τ 3 e −iατ 1 . Correspondingly, the flux is alongn. The presence of α can be understood as follows. The unbroken symmetry U (1) H of the vacuum is generated by τ 1 . However, this U (1) H acts on the Alice string solution with a parameter α. Namely, the Alice string configuration spontaneously breaks U (1) H symmetry of the vacuum, implying the appearance of a Nambu-Goldstone mode in the vicinity of the Alice string. Therefore, α parametrizes a continuous family of the Alice string solution of the same energy, and is a U (1) modulus of the Alice string [41].
While Φ in Eq. (13) is invariant under the U (1) H with τ 1 , Φ(ρ, θ) in Eq. (25) is not invariant under such unbroken elements in Eq. (14) because the unbroken transformation for Φ(ρ, θ) becomes angle dependent. The U (1) H generator must be changed by the holonomies with the gauge field when it goes around the Alice string as Here where β is a transformation parameter. After encircling a full loop around the Alice string, we find that Ω(2π) = e iπτ 3 ∈ Z 2 ⊂ H, hence the unbroken generator becomes This is nothing but the most characteristic property of the Alice string; the charge of a charged particle flips its sign when it encircles around an Alice string.
To find a solution of the Alice axion string, we shall consider an axisymmetric ansatz as where f 1,2 (r) and a(r) are profile functions of the scalar fields and gauge field with the boundary condition that f 1 (0) = f 2 (0) = 0, f 1 (ρ) = f 2 (ρ) = 1, a(0) = 0, a(ρ) = 1. After inserting the above ansatz, two dimensional integration of the static Hamiltonian density can be expressed as 6 As in the Abelian string case, the equations of motion of the profile functions read −r d dr 1 r d dr a(r) + 4g 2 η 2 f 1 (r) 2 1 − a 2 + 2g 2 η 2 f 2 (r) 2 a(r) = 0.
The profile functions are solved numerically for several values of λ 1 and λ 2 , and plotted in Fig. 2. It is noted that in the equation of motion for f 2 the potential contribution can vanish if λ 1 = 2λ 2 is taken. The tension far from the string core can be approximately computed with the ansatz at a large distance ρ in Eq. (25): It should be noted that the energy of Alice axion string is also logarithmically divergent and the same as that of the Abelian axion string with m = 1/2 in Eq. (24). Here there would be contribution from SU (2) magnetic field 1 2 TrF 2 ij , however that we have neglected due to the logarithmic divergence of the leading term and the energy stored inside the core of the Alice string is estimated as of order η 2 , to which fluxes and the scalar potential contribute. As mentioned in the previous subsection, an Abelian axion string (with m = 1) can decay into two Alice axion strings (with m = 1/2), since the energy of the Abelian strings gets lower by the decay: 1 2 > (1/2) 2 + (1/2) 2 = 1/2 in the m = 1 string tension unit, whereas the winding number is conserved. The remaining 1/2 energy is thought to be converted to axions at the decay, and they will contribute to a fraction of the final abundance of the axion.

IV.3. The decay of an Abelian axion string to two Alice axion strings
In this subsection, we try to understand the decay of the Abelian axion string to the Alice axion strings. As seen above, the tension for the Abelian string is different from the Alice strings by a factor 4. So the Abelian string (with m = 1) is always energetically favorable to decay into two Alice strings (with m = 1/2) with the conserved winding number. These two Alice axion strings produced by the decay must have opposite flux direction, since the parent Abelian axion string contains no flux and a field configuration at a large distance does not change through the decay. The configuration of Φ at an angle θ and a large distance ρ, which is far from the string core, can be in general written by where Ω = Ω(θ) is a path dependent rotation matrix with two entries around the strings in the axisymmetric case.
For an Abelian string, the gauge field is vanishing, whereas for an Alice string with the positive flux the gauge field is given by the ansatz in the previous subsection. For a full loop we have Ω(2π) = (h 0 (2π), h(2π)) ∈ H of O(2), where h(2π) = P e ig A·dl . For a rotation around the Abelian axion strings by θ, Ω 0 (θ) = (e iθ , 1) and Ω 0 (2π) = (1, 1) ∈ H, where zero denotes for one rotation around the Abelian axion string. For a rotation by θ around a single axisymmetric Alice axion string with a positive flux, Ω + (θ) = (e i θ 2 , e i θ 2 τ 3 ) and Ω + (2π) = {−1, iσ 3 } ∈ H. This is similar to Ω − for the Alice string with a negative flux: Ω − (θ) = (e i θ 2 , e −i θ 2 τ 3 ) and Ω − (2π) = {−1, −iσ 3 } ∈ H. Now let us understand what happens at the decay of the Abelian axion string to two Alice axion strings. Just at the time when a single Abelian axion string is split into two Alice strings, the boundary condition remains unchanged because it is understood with Ω. To show it, we draw a loop C around the Abelian axion string before splitting. In this case, the Ω is given by Ω 0 (C) = (1, 1). Just after the splitting, we divide the loop in two parts as C = C 1 + C 2 and we close the loops by connecting the points a and b by the path R as shown in the Fig. 3 In this case we have So we find This shows that field configurations at a large distances do not change, while the splitting takes place.  To estimate the force between two Alice axion strings, we take a large distance approximation, following Ref. [59] for the same problem in the context of non-Abelian strings in dense QCD [60]. Suppose that there exist two heavy Alice strings, which do not move, on the x-axis in the x − y plane at a large distance of 2a apart as shown in Fig. 4. So the fields are approximately written as where A Here we neglected small contribution from the gauge field. It is found also that So the interaction energy can be expressed as = 4η 2 · 2π drr r 2 − a 2 r 4 + a 4 − 2r 2 a 2 cos 2θ Here we used ∂ i θ = − ij x j /r 2 and this is similar to θ 1,2 . The force between two Alice axion strings is found to be repulsive: This repulsive force is mediated by the light QCD axion at a large distance apart. 7 This is analogous to a Coulomb force between particles with the same charge in two spatial dimension. Hence, the distance between two Alice axion strings would increase with time and it is confirmed from numerical calculation. However, note that our simulation is done in a relaxation method but not in a real time dynamics.  For numerical solutions, we used 500 × 500 lattice with lattice spacing 0.2. We relaxed the system in 1000000 time steps with each time step ∆t = 10 −3 . The other parameters are taken as g = 1, η = 0.5, λ 1 = 2.0 and λ 2 = 0.25.

V. DOMAIN WALL-STRING COMPOSITE
So far we discussed two types of axion strings and decay of a Abelian axion string to two Alice axion strings. In this section, we study configuration of domain walls attached to the Abelian axion string or to the Alice string. The former situation involving the Abelian strings may be realized at the chiral phase transition through the Kibble-Zurek mechanism [46,47]: At the chiral phase transition, two kinds of domain walls may be created elsewhere. If these domain walls collide, they can be glued along an Abelian axion string. The latter situation involving the Alice strings will always take place in our case. To find the domain walls attached to an Abelian axion string, we start with µ = 0 in the potential of Eq. (20). The static Hamiltonian density (in A 0 = 0 gauge) is given by We shall first discuss vacua in the presence of Abelian axion strings and the Alice axion strings and focus a parameter region, in which 2η 2 λ 1 > µ is satisfied, for V DW not to affect Φ significantly. This is natural for axion domain wall since µ is expected to be of order m 2 a .

V.1. Abelian axion string-domain wall composite
First let us consider an Abelian axion strings. Below we show that a single Abelian axion string is attatched by two domain walls. To understand a situation in the presence of the walls attached to the Abelian axion string, we shall consider an approximate ansatz of the string at a large distance as Substituting the above ansatz into Eq. (46), we find The potential in the above Hamiltonian is nothing but the potential of Eq. (19) wth α = a/2η. As noted already, α sweeps full circle (0 ≤ α ≤ 2π) around the axion string, however, the potential shows that system have two vacua at α = 0 or π and this would create two domain walls attached to the Abelian axion string: N DW = 2. α would be almost zero or π everywhere however changes at the place where domain walls are created. In other words, there are two different domain walls: One is connecting the vacuum at α = 0 and that at α = π. Another is doing the vacuum at α = π and that at α = 2π ≡ 0. We call them DW1 and DW2 respectively. The whole configuration can be regarded as a junction of these two domain walls (DW1 and DW2), whose junction line is nothing but an Abelian axion string. Fig. 7 shows a full numerical simulation of such a configuration, obtained by the relaxation method. We used larger lattice of 700 × 700 points with lattice spacing 0.2. We have taken g = 1, µ = 1.4, η = 0.5, λ 1 = 2.0 and λ 2 = 0.25 for our computation of domain wall-string composites. Here, we put a large friction around a string to prevent this configuration to decay, as described below. In this subsection we discuss the formation of domain walls in the presence of two Alice strings produced via the decay of the parent Abelian axion string. To understand behavior of the walls, we similarly start with field configurations at a large distance as With this ansatz, the static Hamiltonian density in Eq. (46) reads Here we may check the difference from Eq. (48). In this case, the vacuum is still at α = 0 (or 2π) whereas the field range is given by 0 ≤ α ≤ 2π. As a result, there will exist only one domain wall attached to one Alice axion string. This is a similar to the vacuum with N DW = 1. The model is also identical to the sine-Gordon model in two dimension and a domain wall solution along the x-axis interpolating between the two vacua can be written as α = 4 tan −1 e ± √ 2µx . For µ = 0 in computation, when an Abelian axion string attached by two domain walls is initially created and decays into two Alice axion strings, each domain walls remains attached to Alice strings as shown in the Fig. 8. For numerical calculations, the relaxation method is used and the same parameters are chosen as before. There exist two kinds of the Alice axion strings when one focuses on Eq. (25). One has the non-Abelian flux parallel to the orientation defined by the U (1) PQ : e iθ/2 Ω = e iθ/2 e iθτ 3 /2 , where e iθ/2 ∈ U (1) PQ . Another has the flux opposite to the orientation: e iθ/2 Ω = e iθ/2 e −iθτ 3 /2 (with a modulus parameter of π). We call these strings Alice1 and Alice2 respectively. To flip the sign simultaneously by θ → −θ gives the same configuration because this is to see the same string in different ways: the string seen from a positive z coordinate or from a negative z coordinate. So, Alice2 with e iθ/2 e −iθτ 3 /2 is physically equivalent to that with e −iθ/2 e iθτ 3 /2 . Factors of e ±iθ/2 in U (1) PQ (with the same Ω = e iθτ 3 /2 ) imply the way to approach domain walls in the axion space. After one rotation around the Alice1 with 0 ≤ θ/2 ≤ π, one meets DW1. Further, with Ω(2π) = e iπτ 3 ∈ Z 2 ⊂ H, the vacuum is smoothly connected. At the chiral phase transition, like in N DW = 1 case, a single Alice1 is attached by a single DW1, whereas a single Alice2 is attached by a single DW2. See Fig. 9. In the actual cosmological history, once an Abelian string is created after the PQ symmetry breaking, it will quickly decay into two Alice strings in the presence of the repulsive force between them. Then domain walls, which are created after the chiral symmetry breaking, attach to Alice strings, and they shrink to a point owing to the wall tension like in the N DW = 1 case. Even if an Abelian sting attached by two domain walls is created at the chiral phase transition by the Kibble-Zurek mechanism, the Abelian string can be split into two Alice strings in the presence of the wall tension. Then, one Alice string is attached by one wall and each of the walls similarly shrinks to a point by the tension. See Fig. 10.

VI. CONCLUSION AND DISCUSSIONS
Axion is an attractive candidate of dark matter, while stable networks composed by stable strings and walls may be created after the breakdown of U (1) PQ and Z NDW embedded in U (1) PQ . They can cause cosmological disasters since the energy density of them can finally dominate the that of the universe. The Lazarides-Shafi mechanism is one of solutions to the domain wall problem. We have studied this mechanism in detail based on a recently proposed model [22], showing dynamics of axion strings and walls. Even if Abelian axion strings are formed in the early universe, each of them is split into multiple Alice axion strings due to a repulsive force between the Alice strings even without domain wall. When domain walls are formed as the universe cools down, a single Alice string is attached by a single wall because a vacuum is connected by a non-Abelian rotation without changing energy. Such walls do not form stable networks since they collapse by the tension of the walls, emitting axions. Even if domain walls attached to the Abelian axion strings is created by the Kibble-Zurek mechanism at the chiral phase transition, the Abelian string can be split into Alice strings and one domain wall is attached to one Alice axion strings. Such walls can shrink to a point owing to the wall tension like in N DW = 1 case.
Several discussions are addressed here. The model of Ref. [22] was also proposed as a model for a monopole dark matter. A monopole in the conventional Alice theory (SO(3) gauge theory with scalar fields of the fiveplet) admitting Alice strings was studied in Refs. [61][62][63][64]. In particular, a monopole is not spherical and decays into a twisted Alice ring depending on choice of parameters [62,63]. It would be interesting to study if the same would happen in our case. In fact, it is known that a global analog (global monopole) shows this property [43]. A dyon with an electric U (1) H charge may be realized as a vorton, namely a persistent electric current flows along a ring. Dyons can be dark matter if their (mini) electric charge, which may be obtained via a kinetic mixing between U (1) H and electromagnetism, is below experimental bounds. It is also worth to point out that the conventional monopole charge of π 2 is not well defined in the presence of an Alice string, because a monopole becomes an anti-monopole when it encircles around an Alice string, as a dual of electric charge encircling around the Alice string. Instead of using the usual homotopy group π 2 , a monopole charge must be defined in terms of the Abe homotopy [44].
There can exist infinitely long (Abelian) strings produced after the PQ breaking. In such cases, the scaling solution is found to be violated by a logarithmic growth of the string scaling parameter in time [65,66]. It might be hard for wide walls attached to long strings to shrink to a point, hence simulations for them may also be altered. When two Abelian cosmic strings collide, they reconnect each other, which is important process for cosmic strings to reduce their number. Alice strings have U (1) moduli corresponding to fluxes, and so it is unclear if they can reconnect. As this regards, two non-Abelian strings with non-Abelian fluxes were shown to always reconnect [67], and so it would be true for Alice strings. Further, a nature of reconnection among Alice strings may be different from that among Abelian strings due to a force existing among Alice strings, so the number of long Alice strings could differ from that of long Abelian string. The number of long Alice strings would be significant to a solution to the domain wall problem. In any cases, the axion abundance needs to be correctly estimated and may be modified from Eq. (9). Thus, an allowed region for the axion decay constant may be altered. If there is no allowed region, the PQ symmetry breaking might be required to take place before or during inflation, and constraint on insocurvature may be important then.
In future observations, gravitational waves produced by the decay of strings and walls may be detected, depending on axion model [68,69]. That can be an important signal to verify the presence of axion dark matter produced by the topological objects.
If topological objects appear in dark matter models, it is necessary to study the nature of the objects in detail, for precise estimation of dark matter.