Aspects of semilocal BPS vortex in systems with Lorentz symmetry breaking

It is shown the existence of a static self-dual semilocal vortex configuration for the Maxwell-Higgs system with a Lorentz-violating CPT-even term. The dependence of the vorticity upper limit on the Lorentz-break term is also investigated.


I. INTRODUCTION
Topological defects arising from the spontaneous symmetry breaking are physical systems of interest in a wide range of theories, from condensed matter to cosmology [1]. These defects may arise from an abelian, as well as non-abelian, symmetry spontaneously broken. The type of the defect depends on the broken symmetry. Among the typical interesting defects are the vortex solutions, whose characteristics were extensively investigated in the literature [2]. On the other hand, extensions of the standard model including Lorentz-violation terms have been greatly studied in recent years [3]. This program includes investigations over all the sectors of the standard model -fermion, gauge, and Higgs sectors (a very incomplete list includes [4]) -as well as gravity extensions [5]. Following this reasoning, the study of topological defects has also entering this framework [6]. Quite recently [7], it was demonstrated that a Maxwell-Higgs systems with a CPTeven Lorentz symmetry violating term yields Bogomol'nyi-Prasad-Sommerfield (BPS) [8] vortex solutions enjoying fractional quantization of the magnetic field.
One of the benchmarks of the vortex theory is the semilocal vortex [9]. Usually, most part of the * Electronic address: ccoronado@feg.unesp.br † Electronic address: hoff@feg.unesp.br ‡ Electronic address: hott@feg.unesp.br § Electronic address: belichjr@gmail.com study of vortex was restricted to the local symmetry. However, the inclusion of a global symmetry, besides the usual local one, may lead to some interesting characteristics in the resulting topological defect as the presence of topological vortex even if the vacuum manifold is simply connected, the presence of infinite defects, and the fact that semilocal strings may end in a cloud of energy. This paper is partially concerned with the demonstration that semilocal vortices may be found in a usual Maxwell-Higgs system plus a CPT-even Lorentz symmetry violating term. In other words, it is possible to combine the generalized vortex solutions found in [7] and the semilocal structure (Section II). As it is well known from the standard properties of the semilocal setup, the minimum of the potential is a three-sphere, which is simply connected. In fact, starting from a SU (2) global ⊗U (1) local symmetry, the symmetry breaks down to U (1) local . Hence, the first homotopy group is trivial, i. e., π 1 (SU (2) global ⊗ U (1) local /U (1) local ) = 1. However, the local symmetry also plays its role. At each point on the three-sphere the local symmetry engenders a circle. In this vein, looking at the local symmetry, one realizes that it is possible to obtain infinitely many vortex solutions, corresponding to the breaking of the local symmetry (π 1 (U (1) local /1) = Z). Since the potential we shall deal with goes as usual, it is possible to say that as in the usual Higgs-Maxwell case [9] , when no Lorentz-violating term is present, the arguments in favor of stable vortices are strong, but not exhaustive. In order to guarantee the existence of semilocal vortices in the Maxwell-Higgs plus Lorentz-violating model, we have to construct the solutions.
It was shown [7] that the presence of the Lorentz symmetry violating term may lead to a peculiar effect in the vortex size. Hence, in view of the aforementioned characteristic of the semilocal vortex, the solution combining both effects may result in a most malleable defect structure, which is shown to be the case. Besides, we show that the Bradlow's limit [10,11] depends on the magnitude of a parameter related to the Lorentz-breaking term, i. e., the vorticity is also affected. In fact, the vorticity increases as the Lorentz violating term becomes more relevant.

II. SEMILOCAL VORTEX WITH A LORENTZ SYMMETRY-BREAKING TERM
We start from the lagrangian density where Φ is given by the SU (2) doublet Φ T = (φ ψ). The covariant derivative is given by D µ = ∂ µ − ieA µ and F µν is the usual electromagnetic field strength, in such a way that the above lagrangian is endowed with the SU (2) global ⊗ U (1) local symmetry. Note that it is similar to the lagrangian investigated in [7], except by the presence of the global symmetry.
The (κ F ) µγαβ term is the CPT-even tensor. It has the same symmetries as the Riemann tensor, plus a constraint coming from double null trace (κ F ) αβ αβ = 0. It may be defined according to from which it is readly verified that As we want to generalize the uncharged vortex solution, it is necessary to set κ 0i = 0, since from the stationary Gauss law obtained from (1) this last condition decouples the electric and magnetic sectors. Hence, considering the temporal gauge A 0 = 0, the energy functional is given by By working with cylindrical coordinates from now on, we implement the standard vortex ansatz The functions g 1 (r), g 2 (r), are regular functions and in the case of a typical vortex solution they have no dynamics as r → ∞. It is quite enough to assure that the coupling of the fields to the gauge field leads to the phases correlation θ 2 = θ + c, being c a constant. Obviously, for a typical vortex solution we shall have the following boundary conditions for a(r) a(r) → n as r → 0 and a(r) → 0 as r → ∞ With the chosen ansatz, the magnetic field is trivially given by Now, taking κ 11 + κ 22 = s it is possible to write By imposing the self-duality condition [7] λ 2 = 2e 2 /(1 − s) -the equivalent to the equality of the scalar and gauge field masses -it is possible to rearrange the terms in (8) after a bit of algebra, such as In the above expression, the linear terms are those which contribute to the minimum energy when the self-dual equations are fulfilled. The first order BPS equations are given by and − 1 er while the energy minimum is given by where in the last equation we used the boundary conditions (6)and the fact that g 1 and g 2 are regular functions. Now we are in position to show that despite the fact that the vacuum manifold is simply connected the field configuration vanishing at the center of the vortex is compatible with the above framework. Introducing g 2 = g 2 1 + g 2 2 , subject to the boundary condition g → 0 as r → 0, one immediately gets where Φ B is the magnetic flux, and The two remaining equations may be bound together as Equations (14) and (15) are identical to the self-dual equations found in [7]. Therefore the same conclusions obtained there are applicable to the present semilocal case. Of particular interest, their numerical results attest the stability of the BPS vortex solutions.
the first order Eqs. (10) and (15), it is easy to see that where i=1,2. Hence the solutions shall obey g ∼ g i and by the constraint g 2 = g 2 1 + g 2 2 we have 1 = f 2 1 + f 2 2 , where f i are numerical factors. Thus, we see that there are plenty configurations satisfying the boundary conditions. Each of this configurations corresponds to a local spontaneous symmetry breaking of the vacuum manifold. In fact, the vacuum manifold associated to the SU (2) global ×U (1) local symmetry may be understood as a three-sphere whose each point (due to the local symmetry) is given by a S 1 circle. It is nothing but the fiber bundle formulation of the vacuum, being the base space that one associated to the SU (2) global (the three-sphere) and the typical fibre performed by the manifold associated to the U (1) local (S 1 circles). The projections are global transformations while the fibre is a gauge transformation. A particular solution of (1 = f 2 1 + f 2 2 ) means a given S 1 → S 1 mapping performed by Φ. The infinitely many possibilities evinced by the equation 1 = f 2 1 + f 2 2 stands for the infinite possibilities of local symmetry breaking, see Figure  1. Finally, the situation from the vacuum manifold is clear: the local symmetry breaking lead to special vortex configurations which can end since the base manifold is simply connected.

III. s PARAMETER AND BRADLOW'S LIMIT
It was demonstrated in [7] that the Lorentz-violating parameter s plays an important role acting as an element able to control both the radial extension and the amplitude of the defect. In summary, the bigger the s parameter the more compact is the vortex in the sense that the scalar field reach its vacuum value (or, equivalently, the gauge field goes to zero) in a reduced radial distance in comparison with the situation when the Lorentz symmetry is preserved, the so-called Abrikosov-Nielsen-Olesen vortices. Thus, if one applies this model to the scenario of type-II superconductors one sees that the Lorentz symmetry-breaking term is responsible to enhance the superconducting phase.
It is insightful to relate this effect with the maximum vorticity which a noninteracting static vortex system may acquire in a given compact base manifold of area A. This upper bound on the vorticity is the so-called Bradlow's limit [10]. Integrating over equation (14), and choosing positive vorticity, it is easy to see that Note that for s = 0 the usual Bradlow's limit is recovered, as expected. As s grows, however, also does the upper limit. In other words it is possible to saturate the manifold with more vorticity.
If we contrast this situation with the information that as s grows the vortex become more compact, we see that these two effects are related: the increasing s the more compact the vortex.
The more compact the vortex, more vortices with vorticity one are allowed within the same base manifold. This may be regarded to the growth in the number of vortices shown in a condensed matter vortex sample under an external (fixed direction) magnetic field, or the reduction of the vortex core size due to an increase in the rotation frequency of an electrically neutral superfluid.
Finally, as s approaches 1 it is possible to see that the Bradlow's limit blows up. Again, it is in consonance with the analysis performed in [7], where this limit means an extremely short-range theory in which the vortex core length goes to zero but the intensity of the magnetic inside the vortex increases. Similarly to what happens in type II-superconductors, a phase transition where the multiplicity of vortices with vorticity one is favored rather than the melting of the condensate might occur. Such behavior of the magnetic field reinforcing the superconducting phase occurs in ferromagnetic materials that have coexisting ferromagnetism with superconductivity [12]. Parenthetically, if one wants to be in touch with quantum field theory bounds, we notice that the bounds s ∈ (−1, √ 2−1) can be obtained by comparison with the results found in the detailed study carried out in [13] for the bounds on the parameter in order to guarantee not only the causality and the unitarity in the dynamical regime, but also the stability of vortexlike configurations (stationary regime) in the Abelian-Higgs model with the CPT-even Lorentz symmetry term in the electromagnetic sector. If we are interested in preserve causality but relax the unitarity of the model, we have to take into account the interval s ∈ ( √ 2 − 1, 1). To verify possible instabilities ( phase transitions ) is interesting study the values of s in this range. As s approaches 1 it is possible to see that the Bradlow's limit blows up. We obtain this domain using the results of ref. [13] and it is adapted to our case.

IV. FINAL REMARKS
Is was shown the existence of semilocal BPS vortices in the Maxwell-Higgs model with a Lorentz symmetry-breaking CPT-even term. The model has, initially, the SU (2) global ⊗ U (1) local and it was demonstrated that the scalar field doublet vanish at the center of the core, even being its vacuum manifold simply connected. As in [9], the vacuum manifold may be understood as a three-sphere pierced by S 1 circles at each point. Hence, there are infinitely many vortices appearing in the local breaking U (1) local → 1. These configurations correspond to the (also infinitely many) possibilities that g may achieve its boundary conditions (remember that g 2 = g 2 1 +g 2 2 ). Going further we studied the effects of the Lorentz symmetry-breaking term on the vorticity, relating it with the analysis performed for the usual vortex solution in this type of system [7].