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Dynamics of Non-Abelian Vortices

The scattering is studied using moduli space metric for well-separated vortices of non-Abelian vortices in (2+1)-dimensional U(N) gauge theories with N Higgs fields in the fundamental representation. Unlike vortices in the Abelian-Higgs model, dynamics of non-Abelian vortices has a lot of new features; The kinetic energy in real space can be transfered to that of internal orientational moduli and vice versa, the energy and charge transfer between two vortices, the scattering angle of collisions with a fixed impact parameter depends on the internal orientations, and some resonances appear due to synchronization of the orientations. Scattering of dyonic non-Abelian vortices in a mass deformed theory is also studied. We find a bound state of two vortices moving along coils around a circle, like a loop of a phone code.


Introduction
Topological solitons are localized finite energy solutions of classical equations of motion in field theories. Their stability is protected by topological winding numbers. Various topological solitons are known so far in field theories, for instance, instantons, magnetic monopoles, vortices and kinks (domain walls) [1]. They are not only the classical solutions of the equations of motion but also play important roles in non-perturbative quantum effects through, for example, strong-weak dualities such as the electromagnetic dual in four dimensions. While topological solitons have been observed in condensed matter systems, in cosmology, solitons, especially cosmic strings, may be observed in a future by a direct detection of gravitational waves, the gravitational lens or the cosmic microwave background.
In each dimension, we can consider dynamics of various topological solitons which behave as if they are particle-like objects ; kinks in d = 1 + 1, vortices in d = 2 + 1, magnetic monopoles in d = 3 + 1, and instantons in d = 4 + 1. In principle, the motion of solitons are determined by field equations which are usually highly nonlinear differential equations. Therefore a very complicated analysis is required to understand the soliton dynamics in general. For complicated physical systems, it is important to extract essential degrees of freedom by throwing away other unimportant degrees of freedom. This procedure is a challenging and interesting problem. Among various kinds of solitons, gauge theories often admit an important class of solitons, called local solitons; Yang-Mills instantons [2], 't Hooft-Polyakov monopoles [3], Abrikosov-Nielsen-Olesen (ANO) vortices [4] and CP 1 kinks (domain walls) [5,6]. A nice way of extracting such essential degrees of freedom has been established for a particularly important class of local solitons, called Bogomol'nyi-Prasad-Sommerfield (BPS) solitons [7]. BPS solitons saturate the lower energy bound and are the most stable among configurations with a fixed topology [7]. They further naturally appear in supersymmetric gauge theories, break/preserve a fraction of supersymmetry, and consequently are quantum mechanically stable under perturbative or non-perturbative quantum corrections [8]. Since static BPS solitons exert exactly no forces among them at any distance, multiple solitons can statically coexist at any position, which become parameters of multi-soliton solutions, namely moduli parameters associated with massless modes of solitons. Even though no static forces exist, solitons feel forces depending on their velocity. As long as solitons move slowly, it is sufficient to consider the motion of massless modes to describe the dynamics neglecting all the massive modes. This approximation is called the moduli space (geodesic, or Manton) approximation, which was proposed by Manton to discuss the dynamics of BPS monopoles [9,1].
The moduli space dynamics is a good approximation when the kinetic energy of solitons is much smaller than any mass scales of the theory. According to the moduli space dynamics, solitons move along geodesics of the moduli space of the soliton, where the forces depending on velocities of solitons are represented as geodesic forces.
Unfortunately, it is not very easy to get the moduli space metric in general. Only in few cases, the metrics are known explicitly; the prime example is the Atiyah-Hitchin metric of k = 2 BPS monopoles in the SU(2) gauge theory. For multiple monopoles, only asymptotic metrics were obtained when monopoles are well separated [11]. The moduli spaces of well-separated monopoles were also obtained in gauge theories with arbitrary gauge groups [12]. The moduli space approximation has been successfully applied to many other solitons such as ANO vortices in the BPS limit [13,14,15,16,17,18], lumps [19], and BPS domain walls [6]. It has been applied even to BPS composite solitons of different kinds [20,21], such as domain wall networks (webs) [22,23,24] and vortex-strings stretched between parallel domain walls [25,23,24]. However only the asymptotic metric is explicitly known for well-separated ANO vortices [16,17,18], because the ANO vortex equations are not integrable. 1 These metrics have been used to discuss the scattering problems. In particular, scattering processes of solitons have attracted attention of many mathematicians and physists. It is well known that the magnetic monopoles scatter, surprisingly, with 90 degree, when they collide head-on [10]. The same has been seen for the head-on collision of ANO vortices in the BPS limit [14,15], and vortex-strings stretched between parallel domain walls [21].
In general the moduli space of BPS solitons is a surprisingly big space. Usually, the dimensions of the moduli space is proportional to the topological number (the number of the solitons) k.
For example, the ANO vortices in the Abelian-Higgs model (Ginzburg-Landau model) at the critical coupling have 2k moduli parameters, which correspond to positions of the vortices [28]. This is much larger than the dimension of the symmetry group of the theory. Some topological solitons admit more moduli than the degrees of freedom of their positions. For example, the k BPS monopoles in the SU(2) gauge theory is known to have 4k = (3 + 1)k degrees of freedom.
As in the vortex case, 3k can be identified as the monopole positions. The remaining k degrees of freedom are U(1) phases of the internal space, which can be called as internal orientations. There are other examples of solitons which possess the U(1) orientational moduli; kinks in the CP 1 model [5] and U(1) or U(N) gauge theories coupled to Higgs fields with non-degenerated masses [6,29]. Hence these solitons can be called Abelian solitons. On the other hand, non-Abelian moduli are associated with non-Abelian solitons; Yang-Mills instantons, non-Abelian monopoles [30], non-Abelian vortices [31], and non-Abelian kinks (in U(N) gauge theory coupled to Higgs fields with degenerated masses) [32]. The internal orientations can be regarded as Nambu-Goldstone zero modes corresponding to global symmetries or (global parts of) local symmetries, 1 For ANO vortices in a hyperbolic plane with a particular curvature, the vortex equations become integrable [26] and consequently the moduli space metric can be calculated [27].
which are unbroken in the vacuum but are spontaneously broken in the presence of solitons. In general, the motion of internal U(1) orientations give preserved charges to the solitons, making them BPS dyonic solitons.
Since the discovery of non-Abelian vortices [31], much progress has been made in recent years [33,23,24]. Unlike the ANO vortices in the Abelian-Higgs model, the non-Abelian vortices have non-Abelian internal orientations and associated conserved charges; In the case of U(N) gauge theory with N flavors of Higgs fields in the fundamental representation, the internal orientation is the complex projective space CP N −1 , which corresponds to Nambu-Goldstone modes associated with the SU(N) C+F color-flavor locked global symmetry spontaneously broken in the presence of vortices. Because of non-Abelian internal orientations, we can expect that the dynamics of the non-Abelian vortices is much richer and more interesting compared to the ANO vortices, although the analysis gets much more complicated. The moduli space of multiple vortices with full moduli parameters was completely determined without metric by partially solving BPS vortex equations [25,34,24,35]; The moduli space for k separated vortices is a k-symmetric product of the single vortex moduli space [34] while the whole space M k is regular. General formula for the moduli space metric and its Kähler potential were given in [36]. The metric of the moduli subspace for two coincident vortices [39,38,37], which is supplement to M sep k=2 inside the whole space M k=2 , was found, and it surprisingly shows that two non-Abelian vortices scatter with 90 degree in head-on collision even though they have different internal orientations CP N −1 as the initial conditions [39]. Most recently, we have obtained the asymptotic metric on the moduli space M sep k of k well-separated non-Abelian vortices which is valid when the separation of vortices are much larger than the inverse Compton wave length of massive vector bosons, which is the length scale of the vortices [40].
In this paper, we study the dynamics of non-Abelian vortices by using the recently found asymptotic metric of non-Abelian vortices in the U(N) gauge theory with N Higgs scalar fields in the fundamental representation. In order to solve the dynamics, we will make use of the technique of the moduli space approximation. The asymptotic metric of the moduli space allows us to solve the dynamics of the well-separated and slowly moving non-Abelian vortices. We find that the dynamics of the non-Abelian vortex is quite different from that of the Abelian one. The major reason of the difference can be traced back to the conserved charges which are absent in the Abelian case. The vortices with the same charges repel while those with the opposite charges attract. Since the charges can change during the scattering process, non-Abelian vortices experience rich and subtle forces, which produce sometimes the counter-intuitive or unexpected dynamics. We find several new features of the dynamics of non-Abelian vortices: i) the scattering angle depends on the internal orientation, especially parallel orientations give repulsion while anti-parallel orientations give attraction, ii) the energy of real and internal spaces can be transfered, iii) the energy and charge transfer between two vortices occur, and iv) some resonances appears due to synchronization of the orientations.
We also study the dynamics of the dyonic non-Abelian vortices in the mass deformed theory [41], with the method of the moduli space dynamics. A new feature in this case is that a potential term appears in the low energy effective action. Therefore, the solitons experience two forces: The one is the geometric force and the other is the potential force. In a special situation, the two dyonic vortices drift on a circular orbit. This fact strongly suggests that the non-Abelian dyonic vortices can have a bound state.
The paper is organized as follows. In Section 2 we introduce the model which allows the BPS non-Abelian vortices and review briefly the low energy effective theory of the two non-Abelian vortices, and the moduli space dynamics. We also give the asymptotic metric on the moduli space and the geodesic equations for the well-separated vortices. In Sec. 3, we study the scattering of two non-Abelian vortices. After defining the Noether charges of vortices, we give typical examples of numerical solutions of geodesic motion on the moduli space in Subsec. 3.2.
We then analytically study properties of dynamics; the geodesic force in Subsec. 3.3, scattering of two vortices with a large impact parameter by free motion approximation in Subsec. 3.4, and dynamics with zero impact parameter in Subsec. 3.5. In Sec. 4, we consider a mass deformation of the theory and investigate the dynamics of the dyonic non-Abelian vortices. We find a bound state of two dyonic vortices moving along coils around a circle. Sec. 5 is devoted for conclusion and discussion. In Appendix A, the effective action of vortices is written in the U(2) case in terms of unit three-component vectors.

Asymptotic metric for non-Abelian vortices 2.1 Lagrangian and BPS equations
We consider a U(N) gauge theory in (2 + 1)-dimensional spacetime with gauge fields w µ for U (1) and W a µ (a = 1, . . . , N 2 − 1) for SU(N) C , which couple to N Higgs fields H A (A = 1, . . . , N) in the fundamental representation of the SU(N) C gauge group. The Lagrangian is given by where ξ is the Fayet-Iliopoulos parameter, e and g are gauge coupling constants for U(1) and SU(N) C , respectively. The overall scalar coupling constants in the potential V are chosen to be equal to the square of the gauge coupling constants, so that the model admits the BPS non-Abelian vortices. Thus the model has the three coupling constants e, g, ξ. In the three dimensional spacetime, all of the mass dimensions of e 2 , g 2 , ξ are unity. Our convention is The matrices t 0 and t a are the generators of U(1) and SU(N) C , normalized as As is well known, the Lagrangian Eq. (2.1) can be embedded into a supersymmetric theory with eight supercharges. The Higgs fields can also be expressed as an N-by-N matrix on which the SU(N) C gauge transformations act from the left and the SU(N) F flavor symmetry acts from the right The vacuum of this model (V = 0) is in an SU(N) C+F color-flavor locking phase ( where the vacuum expectation values (VEVs) of the Higgs fields are In this vacuum, we have a mass gap with the mass m e for singlet and m g for adjoint representa- (2.6) The energy density for a static configuration can be rewritten as For configurations with vorticity k (vortex number), the energy of is bounded from below by the following BPS bound where we have assume that the last term in Eq. (2.7) vanishes at infinity. This bound is saturated if the following BPS equations are satisfied: where z = x 1 +ix 2 is a complex coordinate. One can easily verify that all the solutions of the BPS isometry acting on β I , which descends from the SU(N) C+F global symmetry in the vacuum.

Asymptotic metric for non-Abelian vortices
The low-energy dynamics of the vortex system can be described by an effective Lagrangian in which the moduli parameters are promoted to dynamical variables. The effective Lagrangian for these moduli parameters is given in terms of the Kähler metric on the moduli space where φ i are the holomorphic coordinates on the moduli space: The metric g ij of the moduli space consists of the free part and the interaction part which are given in terms of the corresponding Kähler potentials (2.13) The free part describes the dynamics of completely isolated vortices 2  Table 1: Numerical data for k = 1 U(2) vortex. The constants c e and c g are not well-defined for m e < 2m g and m g → ∞, respectively [44].
where M v = 2πc is the tension of the vortex and 4π/g 2 corresponds to the radius of CP N −1 . On the other hand, the interaction part of the Kähler potential describes the leading interactions between well-separated vortices [40] where K 0 stands for the modified Bessel function of the second kind. The interaction term K (I,J) between I-th and J-th vortices is a function of the relative distance and an SU(N) invariant quantity Θ IJ defined, respectively, by The origin of the modified Bessel function K 0 appearing in the interaction term can be traced back to the asymptotic tail of a profile function of the vortex. Since the leading term in the modified Bessel function K 0 (m|z IJ |) is of order e −m|z IJ | , the interactions exponentially vanish for large |z IJ |. Note that the mass scales m e and m g can be interpreted as inverse widths of "Abelian core" and "non-Abelian core" of a vortex, respectively. The strength of the asymptotic coupling is controlled by the constants c e and c g , which depend on the ratio m g /m e and N [44]. The numerical values of c e and c g for U(2) vortices are given in Table 1.
For later convenience, let us factor out 4π/g 2 from the Kähler potential and rewrite it in terms of three coupling constants g, m e and m g instead of e, g, c

Equation of motion along geodesics
The overall coefficient 1/g 2 of the effective Lagrangian plays a role of the Planck constant (loop counting parameter) for fixed mass scales m g and m e . Therefore, the quantum effects are negligible and the classical analysis of the effective Lagrangian is valid for the energy scale E much larger than g 2 Note that the energy scale E should be much smaller than mass gaps to justify the use of the lowenergy effective Lagrangian. In the classical analysis, the dynamics are completely independent of the gauge coupling g.
The equations of motion for the moduli parameters φ i take the following form of the geodesic Since the interactions are sufficiently small (K (I,J) ≈ e −m|z IJ | ) for well-separated vortices, the Christoffel symbol Γ i jk can be approximated as whereΓ i jk ,ĝ ij and∇ i are the free part of the Christoffel symbol, metric and covariant derivative respectively. Then, the equations of motion for z I can be written as where we have defined a differential operator δ IJ by The equations of motion for the orientations arê where∇ t˙ β I is the free part of the covariant derivative along the trajectory on CP N −1 We refer the right-hand sides of the equations of motion (2.22) and (2.24) as forces, more precisely, geodesic forces. In general, they are proportional to squares of velocities

Scattering of U (2) non-Abelian vortices
In this section, we discuss the asymptotic scattering of the non-Abelian vortices in the simplest example of the U(2) case which shows essential differences between the Abelian and the non-Abelian vortices. In this case, the orientational moduli space becomes a sphere S 2 ≃ CP 1 . The inhomogeneous coordinate β I is given by the stereographic projection from the north pole of S 2 and related to the standard spherical coordinates as One can also make use of the following three-dimensional unit vector as coordinates of S 2 n I = (sin θ I cos ϕ I , sin θ I sin ϕ I , cos θ I ).
Since SO(3) ≃ SU(2)/Z 2 symmetry is manifest on n I , the dynamics of the orientational modes can be better understood in terms of n I . The Lagrangian written in terms of n I is given in Appendix A.

Noether charge
One of the sharp contrast of the dynamics of non-Abelian vortex to that of Abelian vortex comes from the internal orientations. As mentioned in the introduction, such internal degrees of freedom leads to conserved Noether charges. So let us begin with describing the conserved charges of non-Abelian vortices.
Corresponding to the SU(2) global symmetry, we have one set of conserved charges Q. In the original theory, they are given by where σ are the Pauli matrices. Although Q = 0 for static non-Abelian vortices, these charges arise from the motion of the orientational modes.
In the effective action of vortices, the charges of vortices are given by where ξ I are the holomorphic Killing vectors associated with SU(2) rotation on the moduli space.
Explicit form of the SU(2) triplet vector is given by For the isolated vortices, the dynamics of the orientation is described by free equation of motion, which is nothing but the conservation law of the charge of each individual vortex This equation is equivalent to the geodesic equation on a sphere, so that the trajectory of the orientation is a great circle, as shown in Fig. 1: where n I0 and q I are unit constant vectors satisfying n I0 · q I = 0. The charge of this solution is given by Note that we can always set ω I ≥ 0 without loss of generality.

Features of the scattering of non-Abelian vortices
As stressed in the previous subsection, the internal orientations and the conserved Noether charges make the dynamics of non-Abelian vortices quite different from that of the well-known 3 The normalization of Q I is chosen so that they obey the half-integer quantization condition in the quantum theory. Although the coupling constant g appears in the classical equations of motion because of this normalization, it can be absorbed by rescaling Q I appropriately.
ANO vortices. Since the internal orientations are continuous parameters, there exist continuously different initial conditions, which can make the dynamics quite complicated. In order to see how much the dynamics of non-Abelian vortices differs from that of the ANO vortices, we show several numerical solutions for the geodesic equations (2.22) and (2.24).
The relative internal orientations between the first and second vortices are denoted as ∆θ = We specify initial conditions at sufficiently past. We can always use the SU(2) symmetry to set the initial values at sufficiently past as This initial condition corresponds to a pair of vortices whose orientations are rotating around great circles at the longitudes ϕ 10 and ϕ 20 = −ϕ 10 . Therefore, we can choose five initial conditions for the orientations: θ I ,θ I (I = 1, 2) and ∆ϕ.

Scattering of vortices without initial Q-charges
As the first example of the scattering of two non-Abelian vortices, we examine the case of m e = m g and choose the following initial conditions for the relative orientation ∆ϕ 0 and for the velocities of the orientationsθ I0 : In this setting, the vortices have no initial charges Q I0 = 0. Because of these initial conditions, the variation of the internal orientation is very small before the vortices passes through the interaction region.
In Fig. 2-(a), we show the scattering orbits for the initial relative orientation ∆θ 0 = π/2 by changing the impact parameter a as a = 2.0, 2.5, 3.0, · · · . One can clearly see that the moving vortices feel the repulsive geodesic force between each other. At a glance, this result appears to be very similar to the scattering of the ANO vortices, in which case the scattering is repulsive and the orbits are uniquely determined if the initial velocity and the impact parameter are fixed. However, it is actually different. See Fig. 2-(b) where the impact parameter is fixed at a = 3 and the initial relative orientation is varied as ∆θ 0 = 0, π 4 , π 2 , 3π 4 , π. When ∆θ 0 = 0 (∆θ 0 = π 4 , π 2 ...), they scatter (almost) in the same way as the Abelian vortices. On the other hand, non-Abelian vortices just pass through each other without feeling any interactions for ∆θ 0 = π (a pair of vortices with anti-parallel orientations). This behavior is markedly different from the scattering of ANO vortices. We thus have found that the scattering angle is sensitive to the relative orientation.
The dependence on the relative orientation can be roughly understood from the fact that the interaction part of the Kähler potential K int given in Eq. (2.18) is proportional to 1 + Θ 12  Otherwise, the interaction is always repulsive with reaching the maximum at ∆θ 0 = 0 (parallel orientations), where the interaction reduces to that of the ANO vortices.
for m g = m e (c g = c e ). Since the relative orientation is almost unchanged from the initial condition until the vortices leave the interaction region, it follows that the scattering angle is also proportional to 1 + Θ 12 . For example, the scattering angle is maximized (vanishes) for ∆θ 0 = 0, (∆θ = π) at which 1 + Θ 12 = 2, (1 + Θ 12 = 0). We emphasize again that, in contrast to the non-Abelian vortex scattering, the scattering of the ANO vortices is uniquely determined with the initial velocity and the impact parameter fixed.
In this example, we have taken the initial condition without initial Q-charges. The numerical calculations show that the vortices are charged Q 1 = − Q 2 = 0 after the scattering, even if orientational moduli is initially static. We will see this phenomenon via an analytic discussion in section 3.4.

Scattering of vortices with maximized non-Abelian effect
As the second example let us consider the case where non-Abelian effects are maximized. If the Abelian vector boson mass is sent to infinity 4 m e → ∞, the Abelian part of the interaction  case, which has not been seen in the Abelian case. As before, the scattering angle is affected by the initial relative angle of the internal orientations, as shown in Fig. 3-(b). In the present case, the interaction is proportional to Θ 12 = {1, 0, −1} for ∆θ 0 = {0, π 2 , π}, respectively. Thus, the scattering angle changes its sign at ∆θ 0 = π 2 . For ∆θ 0 < π 2 , the vortices repel as in the Abelian case, while they attract for ∆θ 0 > π 2 . The parallel orientation (∆θ 0 = 0) gives the maximal repulsion and the anti-parallel orientation (∆θ 0 = π) gives the maximal attraction.

Scattering of vortices with non-zero Q-charges
The last interesting example is the scattering of non-Abelian vortices which have non-zero initial charges Q I Since a repulsive (attractive) force works between the same (opposite) charges, the scattering is very different from that with vanishing charges. The results are shown in Fig. 4, where γ I ≡θ I0 /(m g v) is defined by the initial relative velocity v and the angular velocitiesθ I0 . charges feel attractive force as shown in Fig. 4-(b). We can see that their orbits are slightly wavy. This is due to oscillations of the forces caused by the rotations of the orientations. In the next section, we will discuss an "average" over the rapid motions of the orientations to find out effective forces between non-Abelian vortices with Q-charges.

Forces between non-Abelian vortices
Now let us examine the geodesic forces between two vortices induced by motions of the moduli parameters. To this end, let us first rewrite the equations of motion Eqs. (2.22) and (2.24) in terms of n I as 5 where α I (I = 1, 2) is a complex three-vector defined by Note that the equation of motion for n 2 can be obtained by exchanging n 1 and n 2 in Eq. (3.13).
The geodesic force given in the right-hand side of Eq. (3.12) shows that the motion in orientational moduli space generally induces a force between the vortices even if there is no spatial motion initially. Similarly, Eq. (3.13) shows that a spatial motion induces a motion in orientational moduli space, even if orientational moduli is initially static. This implies that the kinetic energy in spatial motion and in orientational moduli transmute each other.

Average over rapid motion of internal orientations
As we have seen in the numerical example Fig. 4-(b), the sign and magnitude of forces between them oscillate due to the rotations of n I . Since motion in orientational moduli is (almost) periodic, its physical effect is best seen by averaging over the periods of two orientational moduli individually. Denoting this time average for the high-frequency modes by , we can express the averaged equation of motion for the relative position z 12 as This approximation is valid only when the motion in space is slow compared to the velocities of the orientations, ω 1 , ω 2 ≫ m g |ż 12 | and ω 1 ≈ ω 2 . It has been shown that the interaction of non-Abelian vortices are well described by regarding them as point-like sources of the Higgs fields and massive vector fields [40]. The motion of the orientation of the I-th vortex induces the following (color) electric charge distribution 6 and the massive vector field Therefore, the electrostatic potential between I-th and J-th vortices is given by where we have used n I ⊗ n T I = 1 2 1 − q I ⊗ q T I . The equation for Q 2 can be obtained just by exchanging the two vortices (1 ↔ 2). These equations describe precessions of the orientations, that is, motions along great circles with slowly moving axes. We can see from Eq. (3.19) that the sum Q tot ≡ Q 1 + Q 2 and the inner product Q 1 · Q 2 are conserved. By solving averaged equation of motion, we find that each charge Q I slowly rotates around Q tot as (see Fig. 5) 20) where a + b = 1, Q osc · Q tot = 0 and We can see that the angular frequency of the rotation is quite smallθ ≪ ω 1,2 . In contrast to the Abelian vortex, the non-Abelian vortex has the characteristic feature that the interactions depend on the relative internal orientations. Since the inner product Q 1 · Q 2 is almost conserved during the scattering process, the interaction can be, for instance, kept to be attractive by choosing near anti-parallel orientations. In such a situation, one can imagine a possibility for two non-Abelian vortices to be bound together. However, it is not the case. It is because the balanced point for the exponentially decaying attractive force (∼ e mg |z 12 | ) with the centrifugal force (∼ 1/|z 12 | 2 ) is unstable. Of course, this conclusion is valid only for the well-separated vortices at distances |z 12 | > 1/m g . Although one may expect that bound states can exist when |z 12 | < 1/m g , such a region is out of range of validity of our approximation. In Sec. 4, we will see that a bound state of two (dyonic) vortices exists in a mass deformed theory where a certain extra force, which is not a geodesic force, is induced by a mass term.

Limit of slowly moving internal orientations
Let us next consider the opposite limit of Eq. (3.16),  Since the interaction for m e < m g is equivalent to that of the Abelian case, the vortices receive repulsive force in the scattering process. For m e > m g , the forces depend on n 1 · n 2 . When n 1 · n 2 > 0 ( n 1 · n 2 < 0), they repel (attract) each other and the interaction accidentally vanishes at n 1 · n 2 = 0. Although the interaction depends on n 1 · n 2 also in the case of m e = m g , it is qualitatively similar to the Abelian case, that is, the vortices receive the repulsive force except for the case n 1 · n 2 = −1. This explains the behaviors of the scattering given in Figs. 2 and 3.

Internal orientations at a head-on collision
As a special case, let us consider a pair of vortices which are going to collide head-on with each From the equation of motion for the orientation Eq. (3.13), we find that This equation implies that the orientations of the vortices tend to align before a head-on collision.
This result for well-separated vortices naturally extends a similar result obtained by an analysis around a vicinity of coincident vortices [39].

Large impact parameter
If two vortices are completely separated, interactions between them can be neglected, and the solution of the equations of motion is given by the free motion (moving parallel to the x-axes) Here we have rewritten the solution Eq. (3.8) in terms of a complex vector c I which is related to n I0 and q I as This can be carried out explicitly by using the following formulas of Fourier transformations Eventually, we obtain the following deviation of the relative velocity Note that the frequencies ω I appear only through the following ratio The free-motion approximation is valid for orbits with small scattering angles. This means that the kinetic energy of the spatial motion should be sufficiently larger than that of the internal orientations. If vortices with non-zero charges have small relative velocities, the interaction Eq. (3.18) lasts for a long time interval ∆t ≈ 1/(m g v) and the deviation of the trajectory from the free motion becomes large. Therefore γ 1 and γ 2 should be generically small so that In Fig. 6-(a), we compare the scattering angles obtained by numerical calculations and those  with the free-motion approximation. One finds that the free-motion approximation is indeed valid only for the scattering with small scattering angles. We also show the numerical results of the scattering with various initial conditions in Fig. 6-(b). Among them, there are orbits whose scattering angles exceed π/2. For such collisions, the free-motion approximation cannot be applied.
The terms with γ I in the exponents (terms other than the first line) in Eq.

Exchange of Q-charges
In the scattering process, the charge of individual vortex in Eq.
Here, we have used the following integration formula For generic values of the angular velocities of the orientations γ I , the leading contribution is given by Therefore, the vortices are charged after the scattering even if Q I = 0 initially.

Exchange of energy
Let us now show that the energy can be transfered between two non-Abelian vortices through the scattering. This is another characteristic feature of the non-Abelian vortices. Since two Abelian vortices are identical objects without any internal degree of freedom, the energy transfer never occurs in the scattering of two Abelian vortices since only the elastic scattering is possible for two identical vortices.
The energy of an isolated vortex is given by the sum of the kinetic energies of the spatial motion and the orientation Assuming that ∆ż 12 and ∆ Q 1 are sufficiently small, we can calculate the total change in energy, where E 0 ≡ πm 2 g v 2 /(2g 2 ) is the kinetic energy associated to the initial relative velocity. Note thatż 1 = 1 2ż 12 in the center of mass frame. This shows that the energy is exchanged between two vortices only through the subleading interactions in general 7 , while the resonances dominate when γ 1 = γ 2 and γ 1 = 0 (γ 2 = 0), as shown in Fig. 8.

Zero impact parameter
In this section, we discuss the vortex scattering with zero impact parameter. For simplicity, we consider the effective Lagrangian restricted to the subspace of the moduli space given by This subspace is a fixed point set of the following reflection symmetry on the moduli space The leading terms are of order max{e −mea , e −mga }.
where ϕ 0 = 1 2 (ϕ 1 + ϕ 2 ) and ϕ r = ϕ 1 − ϕ 2 . In this case, there exists only one non-vanishing conserved charge which corresponds to the third component ( Q) 3 of the conserved Noether charge. By a Legendre transformation,φ can be eliminated from the effective Lagrangian in favor of Q. As a result, the following potential is induced by the Noether charge This potential is shown in Fig. 9. Then, the equations of motion are given bÿ If the charge Q is sufficiently large compared toφ r ,ṙ and the relative angle ϕ r takes a generic value, two vortices recoil due to the potential V Q , in which case we can continue to use the asymptotic metric. This phenomenon can be seen in the numerical calculation Fig. 4-(a). On the other hand, when Q is small or ϕ r ≈ 0 ( n 1 · n 2 ≈ 1), the vortices can closely approach along the valley of the potential V Q , in which case we can trace the dynamics until the asymptotic metric becomes invalid. Note that the cases with Q ≈ 0 and ϕ r ≈ const. correspond to the resonance at γ 1 = γ 2 .

BPS Dyonic vortices 4.1 Dyonic vortices from a mass deformation
In this section we discuss the dynamics of BPS dyonic vortices in the U(2) gauge theory. In order to obtain the dyonic vortices, we add the adjoint scalar field Σ = Σ 0 t a + Σ a t a and the following kinetic and mass terms [24] to the original Lagrangian Eq. (2.1) where M is a 2-by-2 mass matrix. This is the only consistent mass deformation preserving all supersymmetry, when embedded into the supersymmetric theory with eight supercharges 8 . The mass matrix can be written as a linear combination of the Pauli matrices In this mass deformed model, the BPS bound for the energy density is given by [23,41] The first two terms are the energy density of vortices and the third term is the conserved Noether charge for the unbroken U(1) m symmetry. The forth term is the divergence of the the electric flux in the internal direction specified by the adjoint scalar Σ. We call this quantity simply "electric charge density". The BPS bound is saturated if Eq. (2.9) and the following equations are satisfied Then, Eqs. (4.5), (4.6) become From these equations we find that the BPS dyonic vortex solutions can be obtained by solving Eq. (4.9) with respect to Σ in a static vortex background satisfying Eq. (2.9) and then rotating the orientation as Therefore, the dyonic vortex solutions are stable stationary configurations with rotating orien- tation. An important fact is that the no static force is exerted among the BPS dyonic vortices, that is, the electrostatic interaction Eq. (3.18) is canceled by another interaction induced by the adjoint scalar Σ.
One can find the energy of the BPS saturated configurations by integrating the right-hand side of Eq. (4.4) as where we have used Note that the forth term in Eq. (4.4), as well as the second term, has no contribution to the total energy since the electric field is screened and decays exponentially in the Higgs phase. Note that the dyonic vortices have non-trivial charge distributions (see Fig. 10), although they have no total electric charge. We can also discuss the dyonic configurations in the effective theory of vortices. The deformation term Eq. (4.1) induces the following potential on the moduli space where ξ i is the holomorphic Killing vector for SU(2) defined in Eq.(3.5) and k i is that for U(1) m .
In this deformed effective theory of vortices, the energy is bounded by the Noether charge Q Therefore the solutions of the dyonic vortices, which are trajectories on the moduli space with minimum energy for a given value of m · Q, are determined from the BPS equatioṅ In the rest of this paper, we take the positive sign for BPS dyonic vortices. In the case of m = (0, 0, m), the dyonic vortex solution is given by where z I0 and β I0 are complex constants which are related to the Noether charge by where we have used Eq. (4.15). Again, we find that the orientations of the vortices are rotating with the same period.  The dyonic vortex solution is given by where θ 0 and ϕ 0 are constants. The trajectory corresponding to the dyonic vortex is a small circle at θ 0 , on which the following effective potential is minimized where V Q is the potential induced by the Noether charge Q. Let us consider a small fluctuation δθ from the trajectory (4.20) corresponding to the dyonic vortex: θ = θ 0 + δθ. The effective potential for δθ is given by Therefore, if the dyonic vortex is excited, the angle parameter θ oscillates with frequency ω = 2m cos θ 0 , which is generically of order m = | m|. Thus we find that there exist massive modes around the dyonic vortex configuration.

Dynamics of dyonic vortices
In this section we discuss the dynamics of two dyonic vortices by using the equations of motion Eqs. (3.12) and (3.13) modified by the mass term. Since the action of the Killing vector on the moduli parameters is given by the Killing potential V m can be obtained from the kinetic terms of the effective Lagrangian by droppingż I and replacing˙ n I as˙ The full mass deformed Lagrangian can be found in Appendix A. Then the BPS equations for dyonic vortices can be rewritten as˙ The BPS solution takes the form where v I are vectors such that v I · m = 0.
Let us next see how the equations of motion for the relative position and the orientations are modified. The mass deformation changes α 1 · α 2 in the last term of the equation of motion Eq. (3.12) as where α ′ I is the vectors obtained from α I =˙ n I − i n I ×˙ n I by replacing˙ n I with m × n I α ′ I ≡ m × n I − i n I × ( m × n I ). (4.28) Therefore, if the pair of the vortices are near BPS˙ n I ≈ m × n I , the contributions from the last term of Eq. (3.12) and the mass deformation are small Similarly, we can also show that there exists a similar cancellation in the equation of motion for the orientation Eq. (3.13) In the following, we will see that due to these cancellation, the behavior of the dyonic vortices are quite different from that of vortices before the mass deformation.

Angular momentum, Lorentz Force and a Bound State
Let us first discuss the case in which the relative velocity of the dyonic vortices are small. By using the effective Lagrangian Eq. (A.1), we can write down the conserved angular momentum in terms of the radial coordinates z 12 = re iχ as where we have neglected a small term proportional toχK 0 . The function A χ is given by A χ (r) = − n 1 · Q 2 + n 2 · Q 1 c 2 g K 1 (m g r)m g r. (4.32) The first term in Eq. (4.31) is the ordinary angular momentum of a free particle and the additional term A χ (r) can be interpreted as a "gauge potential" generated by the motion of the orientational moduli.
In the massless case discussed in section 3, the contribution of A χ (r) is small compared with the leading terms. This can be seen by averaging over the rapid motions of the orientations.
Since they rotate around great circles of CP 1 , it follows that The mass deformation drastically changes this situation. Since the orientations rotate around small circles in the case of the dyonic vortices, the average of the gauge potential is non-zero n 1 · Q 2 + n 2 · Q 1 = 2π g 2 m · ( n 1 + n 2 )(1 − n 1 · n 2 ). (4.34) Note that m · n I and n 1 · n 2 are independent of time for BPS dyonic vortices and slowly vary for an interacting pair of vortices. Due to the contribution from the gauge potential A χ , the motions of the dyonic vortices with small relative velocity v = |ż 12 | becomes quite different from that without the mass deformation. For a pair of near BPS dyonic vortices˙ n I ≈ m × n I , the non-trivial gauge potential A χ (r) gives the dominant contribution via the Lorentz force Assuming that the relative velocity v = |ż 12 | and the massive oscillations [δθ in Eq. (4.22)] are small, we can write down the energy of this system as where we have neglected irrelevant terms. For an arbitrary relative distance r = r 0 , we can always adjust L χ so that V L = 0, namely line of the magnetic field similarly to a charged particle in a background magnetic field. The orbit of each dyonic vortex in the z-plane takes the form of an extended coil winding around a contour line (see Fig. 13-(a)). In this sense, a pair of dyonic vortices can form a "bound state" with a large relative distance r 0 ≫ m −1 g . The frequency ω L of the oscillation around the stable point r = r 0 is given by , (4.38) and is generically quite small |ω L | ≪ | m| for well-separated vortices with m g r 0 ≫ 1. The radius of the coil r coil , the velocity of the drift v D and the period of the large circular motion T B are respectively estimated as where we have assumed that the relative velocity v = |ż 12 | is sufficiently small. Note that the dyonic vortices with v larger than a certain critical value run away to infinity as illustrated in  In this analysis, we have neglected subleading terms in the equation of motion. This is valid if the excitation energy is sufficiently small since the subleading terms just change the potential V L (r) slightly for a small relative velocity v. We have also assumed that the motions of the orientations are kept near BPSṅ I ≈ m × n I and fluctuations of the massive modes δθ I are small.
These assumptions are justified if we take the near BPS initial condition with sufficiently small v, since the geodesic force which is independent ofż 12 is almost cancelled by the potential term induced by the mass deformation. Therefore, we expect that the bound state is stable as long as the deviation of the energy of each vortex is sufficiently small.

Scattering of dyonic vortices
In this section, we discuss the scattering of dyonic vortices by using the free motion approximation as in section 3. For simplicity, we restrict the initial conditions for the vortices to the BPS states, that is (vt) 2 + a 2 K 1 m g (vt) 2 + a 2 n 1 · n 2 −2vic 2 g K 0 m g (vt) 2 + a 2 m · ( n 1 + n 2 )(1 − n 1 · n 2 ). (4.41) Here n 1 · n 2 and m · n I are constant since we have assumed that the orientation modes satisfy the BPS equations˙ n I = m × n I . Note that the forces vanish in the limit v → 0 due to the BPS properties of the dyonic vortices. The total change in the relative velocity ∆ż 12 is Similarly, we can calculate the total change in the Q-charge From these results, we find that ∆ż 12 and ∆( m· Q I ) do not vanish even for small relative velocity 9 v ≈ 0. This is because the Lorentz force (the last term in Eq. (4.41)) is proportional to v and its integrated effect is independent of v. Before the mass deformation, the differences ∆ż 12 and ∆ Q I vanish for a pair of slowly moving BPS vortices (v → 0, γ I = 0) as we can see from Eqs. (3.33) and (3.36). Note that in both cases the interaction vanish in the limit v → 0 because of the BPS properties. This is one of the typical differences between BPS vortices before and after the mass deformation.
Since the first two terms in Eq. (4.42) can be obtained by taking the limit γ 1,2 → 0 in Eq. (3.32), they consistently reduce to Eq. (3.32) in the limit | m| → 0. Note that γ I ∝ | Q I | are of order | m| for the BPS dyonic vortices. The effect of the mass deformation lies in the last term, which becomes dominant if m g v ≪ | m|. On the other hand, the free-motion approximation gives quantitatively precise results when the relative velocity v is sufficiently large : m g v ≫ me −mg a .
In such situations 10 , we can show that the impact parameter a should be sufficiently larger than the minimum of the potential r 0 which is given by

Conclusion and Discussion
We have studied the dynamics of non-Abelian vortices by the moduli space approximation using the asymptotic metric of well-separated non-Abelian vortices in the U(N) gauge theory with N Higgs scalar fields in the fundamental representation. Since non-Abelian vortices carry the orientational moduli CP N −1 individually, they can have Noether charges as internal momentum of them, whereas ANO vortices in the Abelian-Higgs model have no such charges.
We have found that the vortices with the same charges repel while those with the opposite charges attract. We have shown that the charges of vortices can change during the scattering process with the total energy conserved; the kinetic energy of orientational moduli can be transfered 10 In the opposite case m g v ≫ | m|, the approximation is always valid, but the effects of the mass deformation become subleading. For instance, there is no minimum in the potential V L (r) in such a case. 11 In the latter case, there would be corrections with resonant behaviors in the scattering angle ∆ż 12 , etc.
to the kinetic energy in real space and vice versa. As results, we have found that in scattering of two non-Abelian vortices, i) the scattering angle depends on the internal orientation, especially parallel orientations give repulsion while anti-parallel orientations give attraction, ii) the energy of real and internal spaces can be transfered, iii) the energy and charge transfer between two vortices occur, and iv) some resonances appears due to synchronization of the orientations.
By introducing the mass deformation into the original theory, the color-flavor symmetry SU(N) is explicitly broken to U(1) N −1 . Noether charges of these U(1) N −1 can be actually regard as U(1) N −1 gauge charges where the Cartan subgroup of U(N) survives in the low energy theory in the large mass limit. We have shown that the dominance of the Lorentz force between vortices gives a bound state with a coiling orbit.
Here we address several discussions.  [45,46] can acquire Q-charges after the mass deformation. In the strong coupling limit e → ∞, they reduce to Q-lumps and their interactions have been studied [19,47].
As future problems, the followings might be interesting.
Our study in this paper has been based on the moduli space metric for well-separated vortices and therefore we cannot discuss vortices with a small separation. One question is what the fate of a pair of vortices with anti-parallel charges is. Will they form a bound state with a very small separation? Or will they reduce to a composite state of coincident vortices (singlet or triplet) [35,38] after radiations of massive particles? The head-on collision of two vortices was studied previously based on the metric on the moduli subspace of two coincident vortices (at zero distance) [39]. However the moduli space metric for two vortices at arbitrary distance is not known, which is needed to answer the above questions. Also, beyond the moduli space approximation, one has to study dynamics by numerically solving the original equations of motion.
We have found that energy transfer and Noether charge transfer between two scattering vortices. This property may support the Boltzman's principle of equality when one considers statistical mechanics of many vortices. Partition function of a gas of non-Abelian vortices was calculated in [48] with assuming the principle of equality so that the calculation is reduced to the integration over the moduli space of vortices. Therefore, one important question is if a large number of scattering of vortices leads to the ergodic theorem so that one can assume the Boltzman's principle of equality.
Non-Abelian vortices are called semi-local if there are more flavors than the number of color.
Non-Abelian semi-local vortices have size moduli and they reduces to local vortices when the size modulus is sent to zero. One of characteristic properties of semi-local vortices is that their profile functions decay polynomially but not exponentially as local vortices. Consequently the size modulus is non-normalizable. Moreover the CP N −1 orientational moduli of a single non-Abelian semi-local vortex are also non-normalizable [49] when the size modulus is non-zero, but they are normalizable only when the size modulus is zero [50]. No metric can be defined for non-normalizable moduli, and dynamics cannot be discussed for those. However we can discuss the dynamics of two semi-local vortices because relative orientations and relative size between them are normalizable. In fact dynamics such as head-on collision was studied for Abelian [46] and non-Abelian [39] semi-local vortices. Since semi-local vortices can be approximated by lump solutions at large distance compared with Compton wave length of massive particles, the dynamics of semi-local vortices at large distance can be approximated by that of lumps [19].
One interesting generalization is changing geometry from a flat space to geometry with nontrivial cycles such as a cylinder [54], a torus [48], and Riemann surfaces with higher genus [55]. For semi-local vortices in compact Riemann surfaces, there is no problem of the non-normalizability and we can discuss their dynamics without introducing any cut-off scale by hand. In fact the moduli space metric was found for well-separated non-Abelian vortices on Riemann surfaces [56]. However dynamics during a long time will be difficult using the asymptotic metric because distances between vortices cannot be kept large for compact spaces.
Another interesting extension is changing gauge groups. Non-Abelian vortices were extended to arbitrary gauge groups G in the form of (U(1) × G)/C(G) with the center C(G) of G [51].
Especially the cases of G = SO(N), USp(2N) have been studied in detail [52,53]. However they are semi-local vortices in general [52] so that dynamics at large separation is the one of lumps [19].
We have studied dynamics of BPS vortices. There exist static force between non-BPS vortices [57]. Superconductors are classified into two types, type I and type II, depending on whether the static force between two vortices is attractive(type I) or repulsive (type II). A classification of non-Abelian superconductors will be more complicated due to the existence of charges of non-BPS non-Abelian vortices [58]. When the couplings are close to the critical coupling (near BPS), dynamics of non-BPS vortices can be studied by the moduli space approximation plus a potential for the static forces, which remains as an important future problem. Inclusion of Chern-Simons terms [59] is also possible extension.
Our studies have been restricted to particle dynamics, namely vortices in 2+1 dimensions. In 3+1 dimensions, vortices are (cosmic) strings which have one spatial dimension in their worldsheet. The moduli space approximation can be applied when the angle between two cosmic strings is small. Collision of two non-Abelian cosmic strings was studied based on the metric on the moduli subspace of two coincident vortices [60,39]. Especially it was found in [39] that orientational moduli of two non-Abelian vortices must be aligned and scatter with 90 degree angle, when they collide in head on, except for a fine-tuned collision. This implies that two non-Abelian cosmic strings reconnect each other. This result was obtained just before and after the collision moment in the linear order in time. On the other hand, Eq. (3.26) shows that orientational moduli of two vortices in head-on collision tend to be aligned. It suggests that two non-Abelian cosmic strings consistently reconnect each other as a long time behavior by feeling attraction between two orientational moduli like a ferromagnet. However we also have found in this paper that two vortices repel when charges induced by the motion of internal orientational moduli of two vortices are opposite. In this case, we expect that two cosmic strings merely reconnect each other.