Aspects of semilocal BPS vortex in systems with Lorentz symmetry breaking

Abstract

The existence is shown 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-symmetry-breaking term is also investigated.

Eur. Phys. J. C (2014) 74:2799DOI 10.1140/epjc/s10052-014-2799-1

Regular Article - Theoretical Physics

Aspects of semilocal BPS vortex in systems with Lorentz symmetry breaking

C. H. Coronado Villalobos1,a, J. M. Hoff da Silva1,b, M. B. Hott1,c, H. Belich2,d

1 Departamento de Fsica e Qumica, UNESP, Univ Estadual Paulista, Av. Dr. Ariberto Pereira da Cunha, 333, Guaratinguet, SP, Brazil

2 Departamento de Fsica e Qumica, Universidade Federal do Esp rito Santo (UFES), Av. Fernando Ferrari, 514, Vitria, ES 29060-900, Brazil Received: 30 November 2013 / Accepted: 25 February 2014 / Published online: 11 March 2014 The Author(s) 2014. This article is published with open access at Springerlink.com

Abstract The existence is shown of a static self-dual semilocal vortex conguration for the MaxwellHiggs system with a Lorentz-violating CPT-even term. The dependence of the vorticity upper limit on the Lorentz-symmetry-breaking term is also investigated.

1 Introduction

The Standard Model (SM) has recently passed its nal test.

The discovery of the Higgs boson has conrmed the last prediction of a model of undisputed success. Despite the tremendous success of this model, it presents a description of massless neutrinos and cannot incorporate gravity as a fundamental interaction.

We expect that new physics may appear if we reach the TeV scale and beyond. But if General Relativity and SM are effective theories, what could be guide concepts to obtain physics beyond SM? The Higgs mechanism is a fundamental ingredient, used in the electroweak unication, to obtain the properties of low-energies physics. The breaking of a symmetry by a scalar eld describing a phase transition is currently used in many branches of sciences. Without going into details, we would say that at the microscopic level an effective eld generated spontaneously can give clues on how to get the fundamental theory. In relativistic systems, the eld that realizes the breaking must be a scalar in order to preserve Lorentz symmetry.

In nonrelativistic quantum systems, phase transitions such as in ferromagnetic systems, the rotation symmetry is broken due to the inuence of a magnetic eld. For relativistic systems, the realization of symmetry breaking can be extended

a e-mail: [email protected]

b e-mail: [email protected]

c e-mail: [email protected]

d e-mail: [email protected]

by considering a background given by a constant 4-vector eld that breaks the symmetry SO(1, 3) and no longer the symmetry SO(3). This new possibility of spontaneous violation was rst suggested by Kostelecky and Samuel [1] in 1989, indicating that, in the string eld theory scenario, the spontaneous violation of symmetry by a scalar eld could be extended to other classes of tensor elds.

This line of research including spontaneous violation of the Lorentz symmetry in the Standard Model is known in the literature as Standard Model Extension (SME) [212], and the breaking is implemented by condensation of tensors of rank >1. This program includes investigations over all the sectors of the standard modelfermion, gauge, and Higgs sectors (a very incomplete list includes [1320])as well as gravity extensions [21]. Following this reasoning, the study of topological defects has also entering this framework [22 26]. Quite recently [27], it was demonstrated that a Maxwell Higgs systems with a CPT-even Lorentz symmetry-violating term yields BogomolnyiPrasadSommereld (BPS) [28, 29] vortex solutions enjoying fractional quantization of the magnetic eld.

Topological defects arising from spontaneous symmetry breaking are physical systems of interest in a wide range of theories, from condensed matter to cosmology [3032]. These defects may arise from an abelian, as well as non-abelian, spontaneously broken symmetry. The type of the defect depends on the broken symmetry. Among the typical interesting defects, vortex solutions are a relevant class and their characteristics have been extensively investigated in the literature [33]. So, an interesting program would be to investigate topological defects in a scenario with the violation of Lorentz symmetry and to identify all those quantities which can be directly affected by this special type of breaking, namely Lorentz-symmetry breaking.

One of the benchmarks of the vortex theory is the semilocal vortex [34]. Usually, most part of the study of vortex was restricted to the local symmetry. However, the inclusion of

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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 innite 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 MaxwellHiggs system plus a CPT-even Lorentz symmetry-violating term. In other words, it is possible to combine the generalized vortex solutions found in [27] and the semilocal structure (Sect. 2). As 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 rst homotopy group is trivial, i.e., 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 innitely many vortex solutions, corresponding to the breaking of the local symmetry (1(U(1)local/1) =

above lagrangian is endowed with the SU(2)global U(1)local

symmetry. Note that it is similar to the lagrangian investigated in [27], except for the presence of the global symmetry.

The (F) term is the CPT-even tensor. It has the same symmetries as the Riemann tensor, plus a constraint coming from double null trace (F) = 0. It may be dened accord

ing to

(F) =

1

2 [parenleftbig] + [parenrightbig] ,(2)

from which it is readily veried that

(F) F F = 2 FF. (3)

As we want to generalize the uncharged vortex solution, it is necessary to set 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 A0 = 0, the energy functional is given by

E =

[integraldisplay] d2x [bracketleftbigg]1

2[(1 tr(i j))ab + ab]Ba Bb + |

D |2

Z). Since the potential we shall deal with goes as usual, it is possible to say that as in the usual HiggsMaxwell case [34] , 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 MaxwellHiggs plus Lorentz-violating model, we have to construct the solutions.

It was shown [27] 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 limit [35,36] 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.

2 Semilocal vortex with a Lorentz symmetry-breaking term

We start from the lagrangian density

L =

14 F F

[bracketrightbigg] . (4)

By working with cylindrical coordinates from now on, we implement the standard vortex ansatz,

= g1(r)ein, = g2(r)ein2,

A =

+

2

4 (2 | |2)2

1er [a(r) n]. (5)

The functions g1(r), g2(r), are regular functions and in the case of a typical vortex solution they have no dynamics as r . It is quite enough to ensure that the coupling of

the elds to the gauge eld leads to the phase correlation 2 = +c, c being 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 . (6)

With the chosen ansatz, the magnetic eld is trivially given by

Bz B =

1 er

14(F) F F + |D |2

2

4 (2 | |2)2, (1)

where is given by the SU(2) doublet T = ( ). The

covariant derivative is given by D = ieA and F is

the usual electromagnetic eld strength, in such a way that the

dadr . (7)

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Now, taking 11 + 22 = s it is possible to write

E =

[integraldisplay] d2r [bracketleftBigg]1

2(1 s)

2

+ 2

1 e2r2

da dr

dg1 dr

2

+2 [parenleftbigg]dg2 dr

+

2 a22

r2 (g21 + g22)

+

42

4 (1 g21 g22)2

[bracketrightBigg] . (8)

By imposing the self-duality condition [27] 2 = 2e2/(1

s)the equivalent to the equality of the scalar and gauge eld massesit is possible to rearrange the terms in (8) after a bit of algebra, such as

E =

[integraldisplay] d2r [bracketleftBigg](1 s)2

[parenleftbigg] 1 er

dadr

e2 (1 s)

(1 g21 g22)

2

+2 [parenleftbigg]

ag1r +

dg1 dr

2

+ 2

[parenleftbigg]

ag2r +

dg2 dr

Fig. 1 The vacuum manifold: the innitely many possibilities of spontaneous local symmetry breaking and the mapping in vortex congurations

The two remaining equations may be bound together as

dgdr =

2

[bracketrightBigg] . (9)

In the above expression, the linear terms are those which contribute to the minimum energy when the self-dual equations are fullled. The rst order BPS equations are given by

dg1dr =

g1a

r ,

2 r

[parenleftBigg]d(ag21) dr

[parenrightBigg]

[parenrightBigg]

2 r

[parenleftBigg]d(ag22) dr

2 r

da dr

dg2dr =

g2a

r (10)

and

gar . (15)

Equations (14) and (15) are identical to the self-dual equations found in [27]. Therefore the same conclusions obtained there are applicable to the present semilocal case. Of particular interest, their numerical results attest to the stability of the BPS vortex solutions.

At this point it would be interesting to say a few words concerning the semilocal solutions. From the rst order equations (10) and (15), it is easy to see that

1 g

dgdr =

1 er

dadr =

e2 (1 s)

(1 g21 g22), (11)

while the energy minimum is given by

Emin = 22n

[parenleftBig]1 g21(0) g22(0)

[parenrightBig] , (12)

where in the last equation we have used the boundary conditions (6) and the fact that g1 and g2 are regular functions.

Now we are in a position to show that despite the fact that the vacuum manifold is simply connected the eld congu-ration vanishing at the center of the vortex is compatible with the above framework. Introducing g2 = g21 + g22, subject to

the boundary condition g 0 as r 0, one immediately

gets

Emin = 22n = 2e| B|, (13)

where B is the magnetic ux, and

B =

1 er

dgidr , (16)

where i = 1, 2. Hence the solutions shall obey g gi and by

the constraint g2 = g21 +g22 we have 1 = f 21 + f 22, where the

fi are numerical factors. Thus, we see that there are plenty congurations satisfying the boundary conditions. Each of this congurations corresponds to 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 of which each point (due to the local symmetry) is given by a S1 circle. It is nothing but the ber bundle formulation of the vacuum, being the base space that one associates to the SU(2)global (the three-sphere)

and the typical ber performed by the manifold associated to the U(1)local (S1 circles). The projections are global transformations while the ber is a gauge transformation. A particular solution of (1 = f 21 + f 22) means a given S1 S1 mapping

performed by . The innitely many possibilities evinced by the equation 1 = f 21 + f 22 stands for the innite possibilities

of local symmetry breaking; see Fig. 1. Finally, the situa-

1 gi

dadr =

e2 (1 s)

(1 g2). (14)

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tion from the vacuum manifold is clear: the local symmetry breaking lead to special vortex congurations which can end since the base manifold is simply connected.

3 s parameter and Bradlow limit

It was demonstrated in [27] 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 larger the s parameter, the more compact is the vortex in the sense that the scalar eld reaches its vacuum value (or, equivalently, the gauge eld goes to zero) in a reduced radial distance in comparison with the situation when the Lorentz symmetry is preserved, the so-called AbrikosovNielsenOlesen 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 enhancing of the superconducting phase.

It is instructive 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 limit [35]. Integrating over Eq. (14) and choosing positive vorticity, it is easy to see that

1 e

2

[integraldisplay]

0

d

r dr 1

r

dadr =

[integraldisplay] d2r e2

1 s

(1 g2), (17)

and then

n

e22

2(1 s)A

. (18)

Note that for s = 0 the usual Bradlow limit is recovered, as

expected. As s grows, however, so 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 vortices become more compact, we see that these two effects are related: the more increasing s, the more compact the vortex. The more compact the vortex, the more vortices with vorticity one are allowed within the same base manifold. This may be regarded as growth in the number of vortices shown in a condensed matter vortex sample under an external (xed direction) magnetic eld, or as reduction of the vortex core size due to an increase in the rotation frequency of an electrically neutral superuid.

Finally, as s approaches 1 it is possible to see that the Bradlow limit blows up. Again, it is in consonance with the analysis performed in [27], where this limit means an extremely short-range theory in which the vortex core length

goes to zero but the intensity of the magnetic eld 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 eld reinforcing the superconducting phase occurs in ferromagnetic materials that have ferromagnetism coexisting with superconductivity [37]. Parenthetically, if one wants to be in touch with quantum eld theory bounds, we notice that the bounds s (1, 2 1) can be obtained by compari

son with the results found in the detailed study carried out in [38] for the bounds on the parameter in order to guarantee not only the causality and the unitarity in the dynamic regime, but also the stability of vortex-like congurations (stationary regime) in the AbelianHiggs model with the CPT-even Lorentz symmetry term in the electromagnetic sector. If we are interested in preserving causality but relaxing the unitarity of the model, we have to take into account the whole interval s (1, 1) (we obtain this domain using the results

of Ref. [38] which was adapted to our case). In verifying possible instabilities, such as those resulting in phase transitions, one has to consider the values of s in this range. As s approaches 1 it is possible to see that the Bradlow limit blows up, signalizing a phase transition.

On the other hand, it is expected that the spontaneous violation of Lorentz symmetry occurs at high-energy (Grand Unication or Planck scale), while at our energy scale it is manifested only very weakly. In [39] the same violating term as used in our article is investigated with 0 < s 1. In

fact, reference [40] presents a table with possible values of Lorentz-symmetry-breaking parameters for a wide class of violating sources in the context of the SME. By considering the results presented in [4042] when the even sector of the SME is taken into account we conclude that |s| < 1014.

Then, for those allowed values of s, we cannot see this transition, once the phase transition might occur for s 1.

4 Final remarks

The existence was shown of semilocal BPS vortices in the MaxwellHiggs model with a Lorentz-symmetry-breaking CPT-even term. The model has, initially, SU(2)global

U(1)local and it was demonstrated that minimum energy congurations are found when the scalar eld doublet vanishes at the center of the core, even its vacuum manifold being simply connected. As in [34], the vacuum manifold may be understood as a three-sphere pierced by S1 circles at each point. Hence, there are innitely many vortices appearing in the local breaking U(1)local 1. These congura

tions correspond to the (also innitely many) possibilities that g may achieve its boundary conditions (remember that g2 = g21 + g22). Going further we studied the effects of the

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Lorentz-symmetry-breaking term on the vorticity, relating it with the analysis performed for the usual vortex solution in this type of system [27].

We would like to remark that, although the gauge structure of the model still retain SUglobal(2) Ulocal(1) invariance, it

is not evident that the modication of the gauge eld kinetic term not necessarily can be always made without spoiling the solution achievement. Particularly, in the case treated here we have resort to the constraint g2 = g21 +g22 and to simple alge

braic procedures to reach Eq. (15) from (10). The point to be stressed is that the gauge eld information, encoded in a(r), must be the same for both parts of the scalar doublet, otherwise the solution cannot be reached. Moreover, among all the possibilities brought about by the Lorentz-symmetry-breaking term, the interesting one, which does not jeopardize the formal construction of the stable BPS solutions, is given when 0i vanishes, leading to the functional form of the energy as in (8). It turns out that, after all, this possibility appears to be appealing, since it possesses quantum eld theory boundaries on its magnitude and, as investigated, leads to an interesting shift in the Bradlow limit, which can be physically interpretable.

It may be instructive to point out a counter example. Suppose a Lagrangian whose Maxwell kinetic term is present, but with another gauge eld ruled by an Abelian ChernSimons term as well (the gauge potential being, then, a simple sum of the Maxwell and ChernSimons standard potentials). The mathematical structure of the action is the same SUglobal(2) Ulocal(1). Therefore, everything would go as

usual. However, if one retains the same scalar eld potential it is not possible to achieve a solution via the binding procedure, and one would not have semilocal vortices.

Usually, the search for stable vortex solutions is restricted to modications of the scalar potential when the gauge eld sector of the model is modied. Within this context, every modication leading to an explicit vortex solution, as well as its semilocal generalization, deserves attention. We believe that the modications (even preserving the mathematical form) on the gauge side of the symmetry are to be treated on the same footing as those in the potential sector, and the solutions must be explicitly constructed and studied.

Acknowledgments The authors thank CAPES and CNPq for the nancial support. JMHS thanks to Aristeu Lima for insightful conversations, and the Niels Bohr Institute where this work was partially done.

Open Access This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.

Funded by SCOAP3 / License Version CC BY 4.0.

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