Local and Semi-local Vortices in Yang-Mills-Chern-Simons model

We study BPS vortex configurations in three dimensional U(N) Yang-Mills theories with Chern-Simons interaction coupled to scalar fields carrying flavor. We consider two kind of configurations: local vortices (when the number of flavors $N_f=N$), and semi-local vortices (when $N_f>N$). In both cases we carefully analyze the electric and magnetic properties and present explicit numerical solutions.


I. INTRODUCTION
Non-Abelian vortices may find their application in a variety of problems ranging from particle physics and cosmology (e.g. confinement, supersymmetric and supergravity models, hot or dense QCD, cosmic strings) to condensed matter physics (e.g. quantum Hall effect). Of particular interest are those vortices solving first order BPS equations, which correspond to the saturation of the Bogomolny bound for the mass and naturally arise in supersymmetric theories (see refs. [1]-( [3] for reviews with complete lists of references).
Non-Abelian BPS equations for vortices have been analyzed both for Yang-Mills-Higgs [4]- [8] and for Chern-Simons-Higgs [9]- [15] models. In the former case the gauge and the (fourth order) Higgs potential coupling constants have to be related in order to pass from the second order equations of motion to a first order BPS system. When the Chern-Simons term dictates the dynamics of the gauge field, one is forced to choose a sixth order Higgs potential in order to find a Bogomolny bound for the vortex mass. Again, coupling constants should be related [16]- [17], [11].
The origin of these requirements can be also understood in the framework of supersymmetry: they are necessary conditions for the existence of an N = 2 supersymmetry extension of the bosonic models. In this context, the firstorder BPS equations arise studying the supersymmetry algebra and looking for supersymmetric states. The resulting selfdual and anti-selfdual solutions break 1/2 of the original supersymmetry [18]- [20].
The mixed case of Yang-Mills-Chern-Simons (YMCSS) vortices was also recently discussed [21]- [22]. As in the abelian case [23], in order to have a Bogomolny bound and first order BPS equations the coexistence of the two terms giving dynamics to the gauge field requires a careful choice of the number and type of scalars. Moreover, the Gauss law, through which the Chern-Simons term enters into the energy, is no more an algebraic equation for A 0 , as in the case when the Yang-Mills term is absent, but a second order differential equation that should be taken into account together with the first order BPS system.
It is the purpose of this work to construct explicit BPS vortex solutions for the the YMCSS model thus completing the analysis presented in [22] where the low-energy vortex dynamics was the main aim of study. We shall take U (N ) as gauge group and include N f scalars in the fundamental representation and one real scalar in the adjoint representation of U (N ). We shall consider both cases: when the number of flavors N f is equal to the number of colors N and also when N f > N , in which case vortices become semi-local [24], [14], [25]- [26].

II. THE MODEL AND THE BPS EQUATIONS
We shall consider the d = 2 + 1 dimensional U(N) Yang-Mills-Chern-Simons model discussed in [22] with dynamics governed by the Lagrangian Here q i are N f scalars with i the flavor index (i = 1, 2, . . . , N f ). Each q i transforms in the fundamental representation of the gauge group U (N ) and φ is a real scalar in the adjoint. We shall first discuss the N f = N case (local vortices) and then extend the analysis to N f > N (semilocal vortices) Whenever it does not lead to confusion summation over flavors is implicit. The gauge field A µ takes values in the Lie algebra of U (N ), The curvature and covariant derivatives are defined as The Chern-Simons coefficient κ must be an integer for N > 1.
The masses m i of the fundamental scalars break the flavor symmetry to U (1) N −1 f . There exists a fully broken Higgs phase in which the scalars take the following expectation values In this vacuum, the U (N ) gauge symmetry and the U (1) N −1 f flavour symmetry are spontaneously broken. There remains only a diagonal symmetry, diag . This corresponds to the combined action of a gauge group element U ab ∈ U (N ) and a flavour transformation where U = V −1 . Let us note that in the e 2 → ∞ limit where the Yang-Mills term and the kinetic energy term for the φ field can be discarded, the adjoint field φ may be eliminated from the Lagrangian, and for m i = 0 one ends up with the sixth order potential which allows one to find BPS equations for the pure CS-Higgs system both in the Abelian [16]- [17] and non-Abelian [11] cases, The energy associated with Lagrangian (1) can be constructed from T 00 , the time-time component of the energy momentum tensor T µν . It takes the form where E α = F 0α , B = F 12 .
Following the Bogomolny procedure of square completion a lower energy bound can be obtained [22] Here n ∈ Z is the topological charge of the configuration, which corresponds to the topological degree associated to the q i component that carries the winding, and Q i are the conserved Noether charges associated with the residual U (1) N −1 flavor symmetry, Using Gauss law one can find the typical Chern-Simons term connection between charge and flux, with Φ the magnetic flux, The bound (7) is saturated whenever the following BPS first-order equations hold It should be signaled that in obtaining the bound, it is necessary to use Gauss' law [22] which should then be considered together with eqs.(12)-(16) when looking for explicit vortex solutions. At the bound, the energy of configurations can be identified with the BPS soliton mass, In what follows we choose the upper sign in eqs.(12)- (16), which corresponds to a non-negative winding. Of course, the opposite choice is equally treatable.

III. THE VORTEX ANSATZ
Starting from the trivial vacuum, a winding can be introduced through a singular gauge transformation generated by We have written the formula above, so as to emphasize that Ω combines an U (1) element with an element of Z N , the center of SU (N ). Then, a configuration of the form with q vac the trivial vacuum (3) will lead to a topologically nontrivial but singular (at the origin) string configuration.
To avoid the singularity the natural ansatz for a regular vortex should be with with η 2 i ≡ v 2 + κm i /2π and q N (ρ) vanishing at ρ = 0. Then, (21) should be supplemented with consistent ansatz for the remaining fields, The complete set of appropriate boundary conditions ensuring finite energy is: Concerning A 0 , equations (14) and (15) require This suggests with C determined by (16): Unless all masses vanish, one cannot set A 0 = −φ. The above equation fixes A 0 in our ansatz:

IV. THE BPS VORTEX SOLUTION
Plugging ansatz (21)-(24) into equations (12), (13) and (17) gives Note that the Gauss Law constraint is at the origin of the second-order derivative in h N in (33). It will be convenient to define where τ = √ βρ. One can use (31) to eliminate ∂ ρ a N in (33) so that equations can be recasted in the form: where α = βγ 2 . The boundary conditions imposed by finite energy read now Concerning the behavior of h at the origin, it should go to a finite constant.
Using (18), the vortex mass for our ansatz takes the form The BPS vortex mass is solely determined by the topological charge, the rôle of η 2 N being just that of a scale. Our result is of course consistent with the re-parametrized form of the energy which after some algebra and integration by parts can be written as follows: The upper sign corresponds to our non-negative winding ansatz. The first three terms in the integral vanish as they are readily identified with the Bogomolny equations, while the last term gives the expected contribution to the vortex mass M = 2πη 2 N n.
Let us note that for m N = 0, eqs.(36)-(39) reduce to the abelian case discussed in [23]. It is indeed typically observed that the non-abelian Z N -vortex equations of a model reduce to the abelian equations when the coupling constants of the U (1)-and the SU (N )-gauge groups are set equal (this choice has been made implicitly here by working with the gauge group U (N )). In the case m N = 0, the only modification with respect to the m N = 0 case arises through the parameter α = e 2 4π κ 2 2πv 2 + κm N −1 . Hence we can obtain the profile functions of any general {m N = 0, κ} vortex from a {m N = 0, κ ′ } solution by setting κ ′ = κ/ 1 + κm N /2πv 2 . However, the behavior of the physical observables depends upon the value of m N . If the Chern-Simons term is absent (κ = 0) the Gauss law is satisfied by h = 0 and our ansatz reduces to the well-honored Abrikosov-Nielsen-Olesen vortex [27]- [28].
To obtain numerical solutions to the BPS equations (36)-(39) we have used a relaxation method, selecting the following four boundary conditions Given our ansatz, the magnetic field B and the electric field E can be defined as They are depicted in Figure 1 where we have set e 2 /v 2 = 1. It is interesting to observe the behaviour of the magnetic field in the case m N = 0. As κ is increased, the magnitude of the magnetic field at the origin initially increases. At large enough κ, however, the B-field starts to decrease at the origin and begins to show a characteristic bump, as encountered in the Abelian case [23]. Any further increase in κ amplifies the size of the bump. The electric field is considerably smaller than the magnetic field for small κ, but it also increases as the CS-coupling becomes important. As in the m N = 0 case, the ratio E max (κ)/B max (κ) increases linearly with κ for small κ, and eventually tends to a constant. We may explain this behavior by observing that we have α(κ) ∝ κ 2 ∀ κ when m N = 0 and α(κ) ∝ κ for κ ≫ 2πv 2 /m N when m N = 0. The limit κ ≫ 2πv 2 /m N is where the effects of non-zero m N become noticeable. The profile functions a, h and q thus depend more sensitively on κ when the mass is zero. Furthermore, when computing the electric and magnetic fields from the functions a and h, their behavior is not explicitly dependent on κ for m N = 0, whereas we have explicit dependence of order O(κ) in B and of order O( √ κ) in E for m N = 0 in the same limit. In the zero-mass case [23], the magnetic field starts to exhibit the typical doughnut-shape as κ is increased, at the same time as its overall strength decreases. The electric field develops the characteristic bump and also becomes weaker. As a consequence, the ratio E max /B max , which vanishes as κ → 0, approaches a finite constant for κ ≫ 2πv 2 /m N . For m N = 0, on the other hand, the explicit κ-dependence of the electric and magnetic fields counter-acts the trend of a and h becoming smaller for larger κ. The maxima of the two fields now increasing for a larger range of κ and do not go to zero as κ → ∞. As a result, the ratio E max (κ)/B max (κ) still tends to a finite constant asymptotically, but it reaches it more slowly as m N becomes larger. It is also worth noting that the radius of the bump in the electric field linearly increases with κ when m N = 0, whereas it approaches a constant when m N = 0 (see Figure 1).
Let us end this section by computing the angular momentum for the vortex solution using the formula (which is actually independent of the ansatz). Given Lagrangian (1), one has Using Gauss' law, which for our ansatz reads we can write or, finally Being κ an integer, also the angular momentum is quantized at the classical level. Note that in view of eq.(10) which in the present case reduces to Q N = κn we see that the angular momentum can be written in terms of the square of the charge (in contrast with the pure Yang-Mills case in which it is proportional to the charge [9]- [10]).

V. SEMI-LOCAL VORTICES
Unlike Abrikosov-Nielsen-Olesen vortices and their non-abelian extensions, the radius of semilocal vortex is not fixed but it becomes a parameter. This kind of vortices emerge when N f > N , that is, when there are N e = N f − N additional fundamental scalars {p e = q N +e }, e = 1, ..., N f − N in comparison with the local vortices arising in the N = N f case.
We then start from Lagrangian (1) now with N f > N and consider minimization of the potential in the general case m i = m j ∀ i = j, with i, j ≤ N . The mass term in the N f > N Lagrangian can only be made to vanish for N non-zero fundamental scalars, since we may only pick N of the diagonal entries in φ to cancel the δ ab m i terms. Then, in order to minimize the potential the remaining q i need to vanish. Without loss of generality, we may choose the {p e = q N +e } to vanish. These fields then lie in the unbroken vacuum. Spontaneous symmetry breaking of U (N ) g × U (1) with unconstrained {α e }. These fields must be topologically trivial. If we adopt the previous ansatz for the original fields, equation (13), which as we explain below is still valid for N f > N requires all components of p a e to vanish identically except for a = N . This suggests the following ansatz for the p e p a e = η N δ aN ξ e (τ ).
Furthermore, it is required by (16) that the masses of any additional (non-trivial) scalar p e be equal to the mass of the field that carries the winding, Equation (6) for the energy in the N f = N case is still valid for N f > N and so is the bound (7) and the BPS equations (12)- (16). Concerning the axially symmetric ansatz, it consist of the one proposed in the local case, eqs.
with |ξ e | 2 = e ξ † e ξ e . Equations (57) and (58) can be used to solve for the profile functions {ξ e } [24]: with χ e ∈ C arbitrary complex constants that parametrize the solutions. Of course, if we set all the χ e parameters to zero the system (56)-(60) coincides with (36)-(39) and the semi-local vortices become ordinary local ones.
The energy of the semi-local vortex is As in the local case, it is straightforward to show that this can be written as: the upper sign corresponding to the non-negative winding vortex. Energy (63) reduces to the lower bound in (7) when the Bogomolny equations are satisfied, In the case of semilocal vortices, one can define a parameter χ, the complexified size of the vortex through the formula We thus see that, as expected, the vortex mass is χ-independent but the behavior of the fields at infinity drastically changes with respect to the local case: the fields have a long range power falloff instead of an exponential one. This can be seen in Figure 2, which shows the numerical solutions to (56)-(60). Indeed, at large distances (ρ ≫ 1/e √ β) and for very large transverse size of the vortex (χ ≫ e √ β) one can see how the asymptotic behavior is no more that of an exponential falloff but a power one. The analytical asymptotic behavior of the fields a N and q N are a n ≈ n |χ| 2 ρ 2|n| , q N ≈ 1 − whereas h N has the same exponential falloff behavior as in the local vortex case: Let us finally note that the presence of the radius χ also reduces the Chern-Simons characteristic bump at the origin.
Concerning the angular momentum for the semi-local vortices, the component T 0ϕ of the relevant energy-momentum tensor component gets just an extra term ∆T 0ϕ with respect to the local case, (D 0 ξ e ) * (D ϕ ξ e ) + h.c. and then Using Gauss' Law we can write where the two first term coincide with the local vortex value for T 0ϕ . Taking into account boundary conditions one now has Thus, unlike the case of local vortex, the angular momentum is not simply quantized in terms of κn 2 but it depends explicitly on the size of the semi-local vortex.

VI. DISCUSSION
A rich spectrum of non-Abelian vortices in theories where the dynamics of the gauge field is governed both by a Yang-Mills and a Chern-Simons action was shown to exist in refs. [21]- [22], where the low-energy vortex dynamics was described in terms of a gauged sigma model on the vortex worldline. Although the BPS equations were obtained, explicit solutions were not presented and this was precisely the main objective of our work. To this end, we proposed an axially symmetric ansatz leading to BPS vortex solutions for a Yang-Mills-Chern-Simons U (N ) gauge theory coupled to scalars when the number of flavors N f ≥ N , analyzing the electric and magnetic properties of the local (N f = N ) and semi-local (N f > N ) vortices.
A first interesting feature of the local vortex solutions concerns the localization of the magnetic and electric fields. As expected, as the Chern-Simons coefficient κ grows, the maximum of magnetic moves away from the vortex center and the electric field, also with an annulus shape, starts to develop. Semi-local vortices exhibit a similar behavior except that B and E have a long range power falloff instead of an exponential one (only the real scalar field keeps its exponential falloff behavior). Another difference between local and semilocal vortices concerns the angular momentum which is a purely topological object in the former case while it depends on the vortex size in the later semilocal case Our investigation started from the Lagrangian proposed in [22] with the scalar potential and constants chosen so as to guarantee the possibility of an N = 2 supersymmetric extension, this ensuring the existence of BPS equations [1]- [3]. Actually, it would be of interest to investigate the properties of the supersymmetric model and to construct the low-energy effective action describing moduli dynamics and analyze its properties both at the classical and quantum level, following the approach presented in ref. [13] for the pure Chern-Simons case. We hope to discuss this issues in a future work.