Non-Abelian Global Vortices

We study topologically stable non-Abelian global vortices in the U(N) linear sigma model. The profile functions of the solutions are numerically obtained. We investigate the behaviour of vortices in two limits in which masses of traceless or trace parts of massive bosons are much larger than the others. In the limit that the traceless parts are much heavier, we find a somewhat bizarre vortex solution carrying a non-integer U(1) winding number 1/\sqrt{N} which is irrational in general.


Introduction
Superfluid vortices appear in various condensed matter systems such as helium superfluid. They are global vortices in relativistic field theories [1]. When a global U(1) symmetry is spontaneously broken, for instance by the order parameter φ = v with φ a complex scalar field in the Goldstone model, a U(1) Nambu-Goldstone boson appears. Then in general there appears a global string asymptotically φ ∼ ve iθ winding around the vacuum manifold (or the order parameter space) U (1). The energy of global strings is logarithmically divergent in the infinitely large space, unlike local vortices with gauged U (1). So less attention have been paid to global vortices as a candidate of cosmic strings compared with local counter part. When the net number of strings is zero, for instance, a pair of string and anti-string or string loops, their energy is finite and can be considered as cosmic strings [2,3]. Axion strings are such objects.
Natural non-Abelian extension of U(1) vortices is U(N) vortices in the U(N) linear sigma model for which the order parameter is extended to an N by N complex matrix Φ = v1 N . 1 Such non-Abelian vortices are expected to form during the chiral phase transition in QCD, in which case the field Φ of a 3 by 3 matrix (N = 3) is a condensate of quark-anti-quark Φ ∼ qq .
At the chiral phase transition [4], the chiral symmetry SU(N) L × SU(N) R is spontaneously broken down to its diagonal symmetry SU(N) V . According to this breaking, massless Nambu-Goldstone bosons appear as pions (or more generally mesons). At the same time, the axial symmetry U(1) A is also spontaneously broken and the η ′ meson appears. However U(1) A is explicitly broken by the axial anomaly at zero temperature, giving a mass to the η ′ meson. It has been argued that the axial anomaly might disappear and U(1) A is approximately recovered at high temperature [5]. Although there is still an ambiguity if it occurs below the temperature of the chiral phase transition, let us consider such a situation. Then the breaking pattern is U(1) A × SU(N) L × SU(N) R → SU(N) V apart from the discrete symmetry, and the vacuum manifold is (1.1) maximum when two strings wind around the same component of the vacuum expectation value [for instance diag (e iθ , 1, · · · , 1) and diag (e iθ , 1, · · · , 1)], and it vanishes when the two strings wind around different components of the vacuum expectation value [for instance diag (e iθ , 1, 1, · · · , 1) and diag (1, e iθ , 1, · · · , 1)]. This result implies that a U(1) η ′ -string is marginally decomposed into N pieces of non-Abelian strings: e iθ 1 N → diag (e iθ , 1, 1, · · · , 1)+diag (1, e iθ , 1, · · · , 1)+· · · . Such decomposition necessary occurs at finite temperature where the free energy is minimized instead of the energy. Therefore at low temperature with the axial anomaly, a domain wall junction [10] is unstable, because a U(1) string is pulled by each of N domain walls and is decomposed into N non-Abelian strings to each of which one domain wall is attached. In the end each piece is pulled to infinity in each of N directions. 4 In this paper we study the purely string solution without domain walls in the linear sigma model without the axial anomaly. Numerical solutions themselves were previously obtained in [11]. Here we study profile functions of solutions in much more detail with more accuracy. By using the relaxation method, we numerically determine the shooting parameters of the solutions up to fifth order. We investigate the dependence of the profiles of the vortex to the parameters in the linear sigma model. We also study the two limits in which the masses of traceless or trace parts of the massive bosons in the linear sigma model are much larger than the others. In these limits, the model reduces some nonlinear sigma models. In the limit that the trace parts are much heavier, the equations for the profiles of the U(N) vortex become sine-Gordon-like.
The solution remains to be regular. On the other hand, in the limit that the traceless parts are much heavier, we find somewhat surprising solution; the U(N) vortex solution reduces to a singular U(1) vortex with the U(1) winding number 1/ √ N which is irrational specifically for N = 2, 3, 5, 6, · · · . In general, profiles of U(1) vortices with a non-integer U(1) winding number (≤ 1) are of course singular. Interesting is that such an "irrational vortex" naturally appears in a particular limit of a regular non-Abelian vortex solution. As far as we know such a vortex has not been reported yet in the literature.
This paper is organized as follows. In Sec. 2 we review the U(1) global vortex solution in the Goldstone model. We give numerical solutions and study their asymptotic behaviours. In Sec. 3 we study the non-Abelian vortex solution in the U(N) linear sigma model. We derive profile 4 Stable domain wall junctions or networks exist in gauged U (N ) linear sigma models with appropriate masses [18].
functions of the minimum U(N) vortex with the U(1) winding number 1/N for N = 2, 3, · · · , 10 and determine the shooting parameters up to fifth order. We also discuss various limits by sending some of masses to infinity. We find that in a particular limit the non-Abelian U(N) vortex reduces to an Abelian vortex with the irrational U(1) winding number 1/ √ N . Sec. 4 is devoted to conclusion and discussion. Behaviors of vortex solutions in the large N limit is discussed in Appendix.

The Goldstone Model and Vortex Solutions
Let us begin with giving a review on a global vortex-string solution in the Goldstone model with a complex scalar field φ(x) We choose λ > 0 and v 2 > 0 for stable vacua with broken U(1) symmetry. The scalar potential is like a wine bottle, so we have an S 1 vacuum space with radius |φ| = v. When we choose a vacuum φ = v and consider small fluctuations as φ = v + ϕ 1 +iϕ 2 √ 2 (ϕ 1,2 ∈ R), the Lagrangian in terms of the small fluctuations up to quadratic terms is of the form This shows that ϕ 1 is massive and ϕ 2 is a massless Nambu-Goldstone mode: The equation of motion of φ reads A global vortex-string extending linearly to the x 3 -axis is obtained by solving (2.4) with an axisymmetric vortex ansatz in the cylindrical coordinates x 1 + ix 2 = re iθ , given by Plugging the ansatz (2.5) into Eq. (2.4), we get a second order differential equation Numerical solutions for k = 1, 2, · · · , 10 with the boundary conditions (2.6) are plotted in the left panel of Fig. 1. The energy of the vortex solution can be expressed as where the first two terms come from the derivative of the field φ and the last term is from the scalar potential. Since f → 1 as r → ∞, the kinetic energy logarithmically diverges. So the energy of the vortex-string consists of a finite part and a logarithmically divergent part as where we have defined Here we have introduced an IR cut-off L which is the size of the system and r 0 ≃ m −1 1 is a typical size of the U(1) global vortex such as |f − 1| ≪ 1 as r ≫ r 0 . We can analytically calculate E pot as follows. Let us first introduce a dimensionless coordinate ρ ≡ m 1 r. The equation becomes independent of the coupling constants Then we see that E pot is independent of the scalar mass m 1 : (2.14) To calculate this, let us make a trick By using this equation, we can bring E pot in the following form This formula is called the Derrick-Pohozaev identity in the literature [19]. To derive the rightmost equality, we have used the asymptotic behaviours f ∝ ρ k at ρ ≪ 1 and f = 1 − a 2 /ρ 2 + O(ρ −4 ) for ρ ≫ 1, which will be obtained in the next subsection. We plot the total and potential energy densities E and E pot , respectively of our numerical solutions in the right panel of Fig. 1. We also have numerically checked E pot /(πv 2 k 2 ) = 0.999 + O(10 −4 ). We can show that E der is also finite.
Note that higher winding solutions (k ≥ 2), especially co-axial vortices as we assumed above, are not stable. Since the energy of the co-axial vortices is almost proportional to k 2 , distant vortices are energetically preferred (k 2 > k for k ≥ 2). Therefore, the above static solutions with k > 1 are artifacts of our co-axial ansatz and boundary conditions.

Asymptotic behaviors at r → 0 and r → ∞
Let us investigate asymptotic behavior at m 1 r = ρ ≪ 1 where the profile function is very small |f | ≪ 1. We expand the profile function as a n ρ n . (2.17) By substituting this in Eq. (2.13), we find that the first non-zero coefficient is that of ρ k a k = lim ρ→0 f (ρ) ρ k (2.18) which is sometimes called the shooting parameter and which may be determined by making use of numerical solutions. Such parameters are important since we can uniquely determine solutions with the parameters. In the minimal winding (k = 1) vortex, we got a 1 = 0.4123772 and . (2.19) In principle, one can infinitely increase accuracy of the approximation with the unique parameter a 1 . Although we assumed ρ ≪ 1 in the beginning, we can reach at ρ 0 1 with a good accuracy by increasing the order of expansion.
Let us next discuss asymptotics at r → ∞. We expand solution in the following way and determine b i by comparing terms order by order. Then we get b odd = 0 and We can go on up to order which we desire.

Global U (N ) Vortices
Let us study non-Abelian global vortices. Basic analysis on them have been done in [11]. Here we are going to push forward the analysis of [11] on vortex solutions by investigating the equations of motion in much detail.

The U (N ) Linear Sigma Model and Vortex Solutions
The model that we consider here is a natural extension of the Goldstone model given in Eq. (2.1).
where the last constant term is introduced for the vacuum energy to vanish. For a stability of vacua, we consider the parameter region µ 2 > 0, λ 2 > 0 and Nλ 1 + λ 2 > 0. The chiral symmetry However, unlike the usual case in the absence of U(1) A broken by the axial anomaly, the structure of discrete symmetries becomes somewhat complicated in the presence of U(1) A . Here we explain it in detail. First the group action G on Φ is not Eq. (3.2) itself but is given by where the following Z N × Z N action does not act on Φ and therefore is removed from G: For later use let us redefine these discrete groups as By using the symmetry G, any vacuum can be transformed into the form .
We can consider this vacuum without loss of generality. All other degenerate vacua are obtained from this by the G transformations. The symmetry G is spontaneously broken down to the isotropy group where Z N × Z N is the one of Eq. (3.4) which can be rewritten by Eqs. (3.5) and (3.6), and SU(N) V is given by When the vacuum Φ is transformed by G, the isotropy group H is also transformed by a similarity transformation, but all of them are isomorphic to the original isotropy group (3.8).
Therefore the vacuum manifold (the order parameter space) can be written as a coset space: which is eventually a group manifold (because H is a normal subgroup of G). Since the first homotopy group of the vacuum manifold is non-trivial, it admits topological vortex-string solutions.
In order to find the mass spectrum, let us perturb Φ with small fluctuations as Then it is turned out that imaginary parts of φ and χ a are massless Nambu-Goldstone modes parametrizing the vacuum manifold U(N) A in Eq. (3.10), while their real parts are massive: The original coupling constants are expressed by these masses as (3.14) The scalar potential can be rewritten by the dimensionfull parameters v, m φ and m χ as where we have introduced a notation X ≡ X − Tr[X] N 1 N for N × N matrix X (note 1 N = 0). Let us introduce a ratio of the two masses 5 We will see that τ determines all properties of non-Abelian global vortices. 5 Instead of this, κ = λ 1 /λ 2 was used to parametrize solutions in [11].
Let us construct vortex-string solutions in this model. Firstly, one may consider the following simple boundary condition A natural ansatz with this boundary condition is with boundary conditions f (0) = 0 and f (∞) = 1. The equation of motion for the profile So the solution is just embedding of the global U(1)-vortex of Eq. (2.13). It is important to observe that neither N nor m χ appears. Furthermore, only the overall U(1) phase winds once when we go around the vortex solution. This Abelian vortex solution is called the η ′ string [7].
However, as we will see, this solution is not minimal [11] and is broken into N minimal solutions [13].
A minimal winding vortex configuration can be obtained by taking the following boundary (3.20) The key point here is that the overall U(1)-phase winds only 2π/N around the vortex and the rests 2π(N − 1)/N or −2π/N are inside the non-Abelian group SU(N) A . As a consequence, the tension of the vortex is 1/N of the above Abelian solution (3.18). Because the vacuum space is U(N) A , the above vortex is called a non-Abelian global vortex or more specifically a global with the boundary conditions Since f is the profile function of a winding scalar field, it must go to zero at the origin. On the other hand, g does not have to vanish.
For a fixed θ = θ 0 , the field Φ behaves at the boundary as Φ(θ = θ 0 , r → ∞) = v diag(e iθ 0 , 1, · · · , 1) where the isotropy group is not the same with the one in Eq. (3.8). Instead, it is obtained , and therefore it is isomorphic to in Eq. (3.8). Around the vortex where g(r) differs from f (r) (especially at the center Φ(0) = v diag(0, g(0), · · · , g(0))), the isotropy group This gives rises to further Nambu-Goldstone modes [11] CP In the case of the local U(N) vortices for which U(1) and SU(N) L are gauged [12], the corresponding modes are called the orientational zero modes of a vortex. However, these are nonnormalizable for a global U(N) vortex (in the infinite space), unlike the local U(N) vortex, because the isotropy groups H θ depends on θ = θ 0 and differ from each other, and consequently the wave functions of the Nambu-Goldstone modes CP N −1 spread to infinity.
Let us investigate concrete profile functions of vortex solutions. The Hamiltonian density and the energy densities in terms of the profile functions f (r) and g(r) is given by 24) (3.26) By minimizing the Hamiltonian H, we get the equations of motion for f and g The third term in the left hand side of Eq. (3.27) is typical for global vortices. This leads to logarithmic energy divergence. With respect to a dimensionless coordinate ρ = m φ r, the above equations can be written in the following forms It is clear in this form that solutions depend on τ only. We should solve these ordinary differential equations with boundary conditions (3.22) with replacing r by ρ.
Let us see the non-Abelian vortex in a special case τ = 1. In this case, G N [f, g; 1] = 0 can be solved by g = 1 while the other equation F N [f, g = 1; τ = 1] = 0 is the same as (2.13). Therefore in the case of τ = 1 the Abelian vortex is embedded into Φ as a non-Abelian vortex solution.
Qualitative behaviors of vortex profiles change at τ = 1. As we take τ to larger than 1, the f r Τ  value g(0) gradually goes down toward zero and the energy density becomes sharp (but remains regular for finite τ ), see Fig. 2. On the other hand, g(0) grows and the vortex remains regular and finite when we take τ smaller than 1, see Fig. 3. As we will see below, g(0) is an important value.
In order to see N dependence of the non-Abelian global vortex, we plot two figures in Fig. 4 by changing N form 2 to 10 with τ being fixed to 1/2 and 2.

Asymptotic behaviors
Let us investigate asymptotics of the non-Abelian global vortex. This has been studied in [11] and here we want to extend it with a better accuracy and determine some fundamental parameters (shooting parameters) accompanied by the differential equations (3.29) and (3.30).
We start with study on asymptotics at r ≫ max{m −1 φ , m −1 χ }. As usual, we expand the fields f, g in the following By plugging these into Eqs. (3.29) and (3.30) and equating them with 0 order by order, we can analytically determine the coefficients. It is immediately turned out that a 2i+1 and b 2i+1 vanish.
The leading order terms are of the form Note that f is always less than 1 while g 1 for 1 τ . As observed in [11], the behavior of g(r) depends on the coupling constants λ 1 , λ 2 . With respect to the physical masses m χ and m φ , we now understand that the behavior depends on the ratio τ of the two masses. The higher order terms are determined as 36) . . . The two parameters c 1 = f ′ (0) and d 0 = g(0) cannot be obtained analytically, but we can do it numerically. Note that they depend on τ and N, see Table 1.  has analytic values, given in Eq. (3.52), below.

Let us next consider asymptotics at
Higher order terms are determined by c 1 and d 0 : (3.42)

Heavy Particle Limits
In this subsection, we are going to study how the vortices behave in two regions i) m φ ≫ m χ and ii) m χ ≫ m φ . Before doing it, remember that the linear sigma model (3.1) becomes the chiral Lagrangian in the limit m χ , m φ → ∞. This is because, after the heavy fields are integrated out, only massless NG modes (Φ ∈ U(N)) survive in a low energy theory. Our purpose of this subsection is to clarify effects of lightest massive fields to the chiral Lagrangian. For later convenience, let us rewrite the equations of motion (3.27) and (3.28) as follows A point is that m φ appears only in the first equation while m χ is in the second equation.
Similar limits (with some particles being infinitely heavy) have been studied recently for a local U(N) vortex, for which U(1) and SU(N) L are gauged, at the critical coupling (with a certain relation between gauge and scalar couplings) [20].
Let us first consider the mass of the trace part of Φ is much greater than that of traceless part, (3.45) In other words, we consider a limit τ → 0 by sending m φ → ∞ with m χ being fixed 6 . Then we integrated out the heavy modes with mass m φ . The first term in the scalar potential in Eq. (3.15) becomes very high and sharp, so that the scalar fields are squeezed into the following manifold To solve this condition, let us expand Φ by Φ = ϕ · T , T = (1 N / √ N , T 1 , T 2 , · · · , T N 2 −1 ), (3.47) with N 2 complex vector ϕ = (ϕ 0 , ϕ 1 , · · · , ϕ N 2 −1 ). Then the condition becomes Therefore the linear sigma model (3.1) reduces to S 2N 2 −1 non-linear sigma model. The scalar potential takes the form The vacuum remains as U(N) A . By using the chiral symmetry, we can choose Let us next consider a vortex-string solution in this model by taking the same diagonal ansatz (3.21) as before. However, f and g are no longer independent because of the condition (3.48).
They should satisfy With this at hand, we are aware of an interesting phenomenon that the shooting parameter g(0) = d 0 approaches to an analytic value as Here we have used f (0) = 0. Our numerical result matches with this result, see Table 1. The same thing can be found from the view point of equations of motion (3.43). By dividing its both hands by m 2 φ and taking the limit m φ → ∞, we get Eq. (3.51) again.
To simplify the other equation (3.44), we may rewrite the fields by The reduced model is like the Sine-Gordon model: the potential is The corresponding equation of motion is of the form We solve this with the boundary conditions We numerically solved this and the solution is shown in Fig. 5.

m χ → ∞ limit (τ → ∞)
Next let us investigate the other limit We send m χ infinity with m φ being kept finite. Then N 2 − 1 real scalar fields in Φ become infinitely heavy and we integrated them out from the theory. The scalar potential (3.15) gives us the following condition We can solve this by This decomposition is up to the Z k identification (s, U) ∼ (ω k s, ω −k U) with ω = e 2πi/N (k = 0, 1, 2, · · · , N − 1). Thus the theory in this limit is the SU(N) chiral Lagrangian coupled with a complex scalar field s. The kinetic term is given by From the second expression, we see that the metric of the target space is a cone over U(N). The scalar potential becomes in this limit as Thus the vacuum manifold remains to be U(N) A . The vacuum expectation value v of |s| gives the pion decay constant F 2 π = 16|s| 2 of the U(N) chiral Lagrangian.
We are ready to consider vortex-string solutions in the limit. With respect to the profile functions f, g, the condition (3.58) forces us the following condition f (r) = g(r). (3.62) In terms of s and U, the non-Abelian vortex solution (3.21) can be expressed as (3.63) Note that the overall U(1) winding of the non-Abelian vortex is 1/N as before. The condition (3.62) explains the behavior of g(0) which tends to go down toward zero as τ is sent to ∞, see r ∼ 0 (see Table 1

Conclusion and Discussion
In this article, we have investigated non-Abelian global vortices in SU(N) L × SU(N) R × U(1) A linear sigma model in detail. We push forward the analysis in [11] and determined important numerical parameters c 1 and d 0 which determines all the properties of the solutions. Furthermore, we have obtained expansion formulae for asymptotics at large distance. We have found that interesting two limits i) m φ ≫ m χ , ii) m χ ≫ m φ . The original linear sigma model reduces S 2N 2 −1 non-linear sigma model in the i) limit and we have found a sort of non-Abelian global string solution there. In the second limit ii), we have obtained the SU(N) chiral Lagrangian coupled with a complex scalar field. We have also found a sort of Abelian global vortex solution.
It is a singular solution and can be identified with Abelian global vortex with an irrational U(1) Here we give several discussions. The coupling between a global U(1) string and the U (1) Nambu-Goldstone boson can be constructed by using a duality between a boson φ and a twoform field B µν [21]. In the same way the coupling of a global U(N) string and the U(N) Nambu-Goldstone bosons will be possible by using non-Abelian two-form [22].
In the presence of the U(1) A axial anomaly a term which gives a sine-Gordon potential to the phase. Then a U(N) vortex becomes a boundary of a domain wall [10,9]. In the presence of quark masses, a term with H a mass matrix. It remains as a future problem to study detailed structure of vortex solutions in the presence of these terms because the authors in [10,9] assumed constant profiles.
In Subsec. 3.3.2 we have simply sent the mass m χ to infinity to obtain the SU(N) chiral Lagrangian coupled with a complex scalar field s. However quantum mechanically we should integrate out the massive fields. This procedure generally induces higher derivative terms for remaining massless fields. The Skyrme term is such a term of the fourth order [23]. In our case with massless field s we will obtain the effective Lagrangian of the form where e is a parameter determined by an explicit calculation and V is the potential in Eq. (3.61).
Note that there is no fourth order term for s. Because of the relation Φ † ∂ µ Φ = s * ∂ µ s1 N + In this paper we have studied U(N) vortices. Local and semi-local vortices with different groups [U(1) × G]/Z n 0 , where G is arbitrary group with the center n 0 [25], have been studied recently. In this framework the U(N) vortex corresponds to the case of G = SU(N) with n 0 = N.
Global version of these vortices are also possible, especially the case of G = SO is related to vortices in the B-phase of 3 He superfluids.
Before closing this paper let us compare our global U(N) vortices with other types of U(N) vortices in the related models: 1) semi-superfluid U(N) vortices in high density QCD and 2) local U(N) vortices. In these models, the group structure is completely the same with the global case in this paper. However the energetics/interactions of vortices and the (non-)normalizability of the zero modes are significantly different.  [26].
In the asymptotic behaviour of the scalar field of a U(N) vortex, Φ ∼ v diag(e iθ , 1, · · · , 1) = v e i θ N diag e i N−1 N θ , e −i θ N , · · · , e −i θ N , the latter non-Abelian part can be eliminated by a gauge transformation U(r, θ) = diag e −i N−1 N θF (r) , e i θ N F (r) , · · · , e i θ N F (r) with an arbitrary function F (r) satisfying the boundary conditions F (r = 0) = 0 and F (r → ∞) = 1. 9 2) Next let us to compare the global U(N) vortices studied in this paper with the local U(N) vortices [12]. In this case too the symmetry breaking pattern is the same but a crucial difference 9 This transformation is well-defined because of the triviality of the first homotopy group: π 1 [SU (N )] = 0.
This transformation brings Φ to Φ ∼ v e i θ N 1 N . This property makes the orientational zero modes of CP N −1 to be normalizable and to become the moduli (collective coordinates) of the vortex [27]. In other words, the isotropy groups H θ at the infinities in the presence of a global U (N ) vortex were physically different and depend on θ (although they are isomorphic), whereas, in the SU (N ) gauged case, the isotropy group at the infinities in the presence of a semi-superfluid U (N ) vortex is physically equivalent for any θ. The U (1) B is global and so the energy of a vortex remains logarithmically divergent. The asymptotic interaction between two semi-superfluid U (N ) vortices is also essentially the same with the one between U (1) global vortices because of the above property [27,14]. Therefore it gives the universal repulsion between separated vortices.
is that the U(1) symmetry is also gauged in addition to the color SU(N), and therefore there remain no Nambu-Goldstone bosons. Namely the vacuum is the unique at the infinity even in the presence of a U(N) vortex because the U(1) symmetry is gauged . Nevertheless we cannot gauge transform the asymptotic behavior of the scalar fields from Φ ∼ v e i θ N 1 N to 1 N because we can define no regular gauge transformation well-defined in the entire space because of non-triviality of the first homotopy group: π 1 [U(1)] = 0. Unlike the global U(N) vortices or semi-superfluid U(N) vortices, these local U(N) vortices have finite energy because of the gauged U(1). The CP N −1 zero modes are normalizable. At the critical (BPS) coupling with a particular relation between gauge and scalar couplings, there is no static force among multiple vortices, with allowing the multi-vortex moduli space [28,29] (see [30] for the moduli spaces of local U(N) vortices on a cylinder and a torus). Static interactions exist between vortices at non-critical (non-BPS) couplings. The force between two U(N) vortices was shown to depend on both CP N −1 orientations and positions [31].
If we gauge all the symmetry, the vortices are those in quiver gauge theories [32]. In this case, the diagonal gauge symmetry remains unbroken, and the vortices do not have orientations in the internal space. The final possibility which was not studied so far is the case that only U(1) is gauged.
scale in N. From the observation of loop corrections (perturbation series in λ 1 and λ 2 ) to twobody meson scattering amplitude in a single channel, one can set λ 1 ∼ O(N −2 ) and λ 2 ∼ O(N −1 ) in order to have a tenable perturbative expansion [33]. and the other equation (3.27) becomes This is the equation for k = 1 Abelian global vortex. Note that m χ is shown up in the equation and solutions are independent of m φ unlike the case of Abelian global vortex given in Eq. (3.19).
The solutions themselves happen to be identical to those for m φ = m χ (τ = 1) with any finite N.