Multiple Layer Structure of Non-Abelian Vortex

Bogomol'nyi-Prasad-Sommerfield (BPS) vortices in U(N) gauge theories have two layers corresponding to non-Abelian and Abelian fluxes, whose widths depend nontrivially on the ratio of U(1) and SU(N) gauge couplings. We find numerically and analytically that the widths differ significantly from the Compton lengths of lightest massive particles with the appropriate quantum number.

1. Introduction. Many important properties of Abelian (ANO) vortex were found [1,2,3,4,5] since its discovery [6]. Recently vortices in U(N) gauge theories (called non-Abelian vortices) were found [7,8] and have attracted much attention [9] because they play an important role in a dual picture of quark confinement [8,10] and are a candidate of cosmic strings [11] (see [12] for review report). The moduli space of U(N) non-Abelian vortices was determined in [13] and study on interactions between non-BPS configurations started in [14]. Non-Abelian vortices in other gauge groups have been studied [15].
Although there have been much progress and wide applications, internal structures and dependence on gauge coupling constants have not yet been studied for (color) magnetic flux tubes.
It is particularly important to study physical widths of vortices qualitatively and quantitatively, although it is not easy because no analytic solutions are known. It may be tempting to speculate that the width is determined by the Compton lengths of lightest massive particles with the appropriate quantum number. Purpose of this letter is to clarify intricate multiple layer structures of non-Abelian vortices by investigating numerically and analytically the equations of motion.
Non-Abelian vortices have two distinct widths for SU(N) and U(1) fluxes. We clarify properties of these widths by making use of several approximations. It turns out that non-Abelian vortices are very different from ANO vortices and have much richer internal structures.

Vortex equations and solutions.
Let us consider a U(N) gauge theory with gauge fields W µ for SU(N) and w µ for U (1) and N Higgs fields H (N-by-N matrix) in the fundamental representation. We consider the Lagrangian L = K − V which can be embedded into supersymmetric theory with eight supercharges where X stands for a traceless part of a square matrix X. Our notation is We have three couplings: SU(N) gauge coupling g, U(1) gauge coupling e and Fayet-Iliopoulos parameter c > 0.
The Higgs vacuum H = √ c 1 N is unique and is in a color-flavor SU(N) C+F locking phase.
Mass spectrum is classified according to representations of SU(N) C+F as m g ≡ g √ c for non-Abelian fields φ N = (W, H ) and m e ≡ e √ 2Nc for Abelian fields φ A = (w, Tr(H − √ c1 N )). The non-Hermitian part of H is eaten by the U(N) gauge fields. A special case of m g = m e [7] has been mostly considered so far, which is equivalently but we study general cases in this Letter.
Let us consider static vortex-string solutions along x 3 -axis. The BPS equations for the non-Abelian vortex areD analytic solutions have been known whereas a numerical solution was first found in [8]. For and ψ e and ψ g are real functions of the radius r > 0 of the polar coordinates (r, θ) in x 1 , x 2 plane. Then we get with △f (r) = ∂ r (r∂ r f (r))/r. The boundary conditions are ψ e , ψ g → 0 (r → ∞) and Nψ e , N N −1 ψ g → − log r 2 (r → 0). The fluxes and the Higgs fields are expressed by with . The amount of the Abelian flux is 1/N and the non-Abelian flux is (N − 1)/N of the ANO vortex.
Numerical solutions for N = 2 for a wide range of γ (including γ = 0, ∞) are shown in Fig. 1. Winding field h 1 is not sensitive on γ while unwound field h 2 is. As m g being sent to ∞ (γ → ∞), the non-Abelian flux F 12 becomes very sharp and finally gets to singular. Interestingly, the Abelian flux f 12 is kept finite there. In a region γ < 1 (m e > m g ), on the other hand, the Abelian flux is a bit smaller than the non-Abelian tube. Surprisingly, the fluxes remain finite even in m e → ∞ limit.
3. Asymptotic width. Let us investigate the vortex solution by expanding (5) and (6) in region We keep only the lowest-order term in ψ e while keeping terms up to next to leading order in ψ g in Eq. (5): The solution is given by the second modified Bessel function K 0 (r), and approximated as with c e,g being dimensionless constants, see  [3,4]. The origin of ψ e (ψ g ) is (non-)Abelian fields φ A (φ N ) with mass m e (m g ), and the γ = 1/2 threshold can be interpreted as follows. The expansion of the Lagrangian with respect to small φ A,N contains the triple couplings φ A φ 2 N . For m e ≤ 2m g , asymptotics for φ A,N are given by K 0 (m e,g r) as the two-dimensional Green's function. When m e > 2m g , the particles φ A decay into two particles φ 2 N through these couplings, and thus, φ A exhibits the asymptotic behavior e −2mg r below γ = 1/2 like Eq. (9). On the contrary, even for γ > 2, φ N does not behave as e −2mer since there is no triple coupling φ N φ 2 A due to the traceless condition for φ N .
Let us define asymptotic width of the vortex by an inverse of the decay constant in Eq. (9): Here the factor 2 is put in the numerator to match with another definition in Eq. (14). The asymptotic width of Abelian vortex is bigger than the non-Abelian one when γ > 1 and vice versa for 1/2 ≤ γ < 1. For γ = 1, the two widths are the same. The case γ < 1/2 indicates a significant modification, where the Abelian flux tube is supported by the non-Abelian flux tube.
This answers the question why the Abelian vortex does not collapse even in the m e → ∞ limit.
When γ ≫ 1, the thin non-Abelian flux hidden by fat Abelian flux cannot be correctly measured by L g . We now turn to another definition of vortex width which reflect the size of the vortex core more faithfully.
4. Core widths. Let us consider a region near the vortex core. We expand fields by Dimensionless constants b e,g are related to h 2 (r = 0) by See Table 1. These are important since they are related to the maximum values of the magnetic fluxes at r = 0 Widths of the magnetic fluxes can be estimated by using a step function Θ(x) as F 12 = B g Θ(L g − r)T and f 12 = B e Θ(L e −r) keeping amounts of the fluxes as |B e | ×πL 2 e = 2π/N and |B g | ×πL 2 g = 2(N − 1)π/N:L We callL e andL g as the core widths of the vortex. In the case of γ = 1,L e andL g coincide because of a γ=1 = h 2 = 1. In Fig. 2, we show the core widths numerically in the case of N = 2, which are analytically reinforced as we will discuss. We again observe that the Abelian core does not collapse even when m e ≫ 1 (γ ≪ 1).
Mass dependence of the core widths coincides with one of the asymptotic widths L e,g given in Eq. (10), except forL g (γ > 1), see Fig. 2. The asymptotic width L g is independent of m e whereas the core widthL g depends on m e . This is becauseL e,g more faithfully reflects the multilayer ւ Fig. 2: The core and asymptotic widths v.s log γ.
structure in the large intermediate region of r for the strong coupling regime (γ ≫ 1), to which we now turn.

Two strong coupling limits.
Here we study two limits: i) m g → ∞ with m e fixed, and ii) m e → ∞ with m g fixed. In the former limit (γ → ∞ with m e fixed), all the fields with mass m g become infinitely heavy and integrated out from the theory. As a result, the original U(N) gauge theory becomes the Abelian theory with one SU(N) singlet field B ≡ det H. Note that where suffix ∞ denotes γ = ∞: ψ g,∞ ≡ ψ g | γ→∞ . The boundary condition tells that vorticity is fractional k = 1/N. This way the non-Abelian flux tube collapses and the U(N) non-Abelian vortex reduces to the 1/N fractional Abelian vortex. This solution helps us to understand the non-Abelian vortex for γ ≫ 1 better. Since e −ψe ≈ (m e b e,∞ r) 2/N (r ≪ 1/m e ) for γ ≫ 1, ψ g for r ≪ 1/m e is well approximated by a solution of the following where the parameterm g ≡ m g (b e,∞ γ −1 ) 1 N+1 has a mass dimension. Therefore ψ g has asymptotic behavior in the middle region 1/m g ≪ r ≪ 1/m e and ψ g ≈ − 2(N −1) N log(b gmg r) for r ≪ 1/m g . Hereb g ,c g are determined numerically and independent of γ, for instance,b g = 0.74672,c g = 0.63662 for N = 2. Comparing this with Eq. (11), In the second limit (γ → 0 with m g fixed), all the fields with the mass m e are integrated out.
6. Summary and Discussion. We have proposed two length scales for fluxes of non-Abelian vortices: asymptotic widths L e,g in Eq. (10) and core widthsL e,g in Eq. (14). By using the asymptotics of a γ obtained above, the core width is summarized as where α and β depend only on N and are determined numerically, for instance α = 0.55010, β = 0.97022 for N = 2. The core and asymptotic widths have the same mass dependence except forL g and L g for γ ≫ 1. Interestingly, the Abelian flux does not collapse even when m e → ∞ (γ ≪ 1). For γ ≫ 1, the thin non-Abelian flux is hidden by fat Abelian flux, so that the true width cannot be captured by the asymptotics at r ≫ m −1 e . Instead we should use improved approximation given in Eq. (17) to measure the non-Abelian flux. Indeed, the decay constant in Eq. (17) ism −1 g whose mass dependence is the same as one ofL g for γ ≫ 1.
In the limit m g → 0, the original U(N) gauge theory reduces to U(1) gauge theory coupled to N 2 Higgs fields. Eq. (19) tells us that the vortex is diluted and vanishes in this limit. This is consistent with the fact that there is no (smooth) vortex solution with a winding number 1/N in that U(1) theory. The minimal vortex in the U(1) theory corresponds to N vortices in the original theory. The dilution is expected to be avoided and all the fields with mass of the order of m g decouple, when N vortices are arranged as H = f (r)1 N . In the limit m e → 0, there is no BPS vortex solution since the U(1) gauge field is decoupled from the Higgs fields.
Actually, according to Eq. (19), one can find both of the Abelian and the non-Abelian fluxes are diluted again even in this limit due to the factor γ 1 N+1 . Monopoles/instantons attached by vortices are known to exist [10]. The above observation implies that such configurations reduce to a monopole/instanton configurations in the SU(N) gauge theory in that limit, and strongly supports the correspondence between the moduli spaces of them.