We consider two self-dual Abelian Higgs systems obtained from Lorentz-breaking symmetry models by dimensional reduction. For the first model, we show that the self-dual equations are identical to those of Nielsen-Olesen vortices. Also, we show that our vortices have electric charge. In the second case we show that self-dual Chern-Simons-Higgs vortices without electric charge are possible.Fil: Sourrouille, Lucas. Universidade Federal do Maranhão; Brasil. Universidade Federal do ABC; Brasil. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Sourrouille, Lucas

English

arXiv

Lucas Sourrouille

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Physical Review D

Note on vortices from Lorentz-violating models

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arXiv:1311.4501v4  [hep-th]  12 Apr 2014A note on vortices from Lorentz-violatingmodelsLucas Sourrouillea,baDepartamento de F́ısica, Universidade Federal do Maranhão,65085-580, São Lúıs, Maranhão, BrazilbCMCC, Universidade Federal do ABC, Santo André, SP, Brazillsourrouille@yahoo.esApril 15, 2014AbstractWe consider two self-dual abelian Higgs systems obtained from Lorentz breakingsymmetry models by dimensional reduction. For the first model, we show that the self-dual equations are identical to those of Nielsen-Olesen vortices. Also, we show that ourvortices have electric charge. In the second case we show that self-dual Chern-Simons-Higgs vortices without electric charge are possible.Keywords:Chern-Simons-like gauge theory, Topological solitons, Lorentz symme-try violation.pacs: 11.10.Kk, 11.10.Lm, 11.27.+d, 12.60.i, 11.30.Cp1 IntroductionThe possible violation of Lorentz invariance has recently received a lot of attention as acandidate for the Planck scale physics [1].The large range of existing phenomenologicaland experimental activities stems from the application of effective field theory[2] andthe construction of the Standard-Model Extension (SME)[3, 4] to catalogue and pre-dict observable effects. The gauge sector of the SME include CPT-even and CPT-oddterms. The CPT-odd, usully called the Carroll-Field-Jackiw term[5], consist on fourparameters, that engenders a parity-odd and birefringent electrodynamics, coupled toChern-Simons dynamics. The CPT-even photon sector is composed by fourth-ranktensor, with Riemann tensor symmetries and a double null trace, which is coupled to agauge term F κλFµν . These models have been studied in different contexts, such as ra-diative corrections [6], nontrivial spacetime topology [7], causality [8], supersymmetry1[9], the cosmic microwave background [10], vacuum Cherenkov radiation[11], generalrelativity [12] and soliton solutions, like monopoles[13] and vortices[14, 15, 16].On the other hand, it is well known that the abelian Higgs model [17] in (2 + 1) di-mensions, support topological vortex solutions of finite energy. With a specific choiceof coupling constants, minimum energy vortex configurations satisfy first order dif-ferential equations [18]. This model admits and N = 2 supersymmetric extensionproviding a connection between BPS equations and supersymmetry [19]. Also the(2 + 1)-dimensional matter field interacting with gauge fields whose dynamics is gov-erned by a ChernSimons term support soliton solutions [20]. When self-interactionsare suitably chosen, vortex configurations satisfy the Bogomolnyi-type equations witha specific sixth order potential [21]. Another important feature of the ChernSimonsgauge field is that inherits its dynamics from the matter fields to which it is coupled,so it may be either relativistic [21] or non-relativistic [22]. In addition the soliton so-lutions are of topological and non-topological nature [23].The purpose of this letter is the study of the self-dual vortices in two Lorentz-violatingmodels. In the first section we will analyze the self-dual equations of a two dimensionalmodel obtained from the CPT -even Lorentz-violating term, coupled to matter field,by dimensional reduction. We will show that this model support self-dual vortex solu-tions with electric charge. In particular, we will show that the self-duality equationsare identical to those of the Nielsen-Olesen vortices. The second section is dedicatedto explore the dimensional reduction of the CPT -odd Lorentz-violating term couple toa Higgs field. We will show, here, that it is possible to obtain exactly the same Bo-gomolnyi equations as the relativistic Chern-Simons-Higgs model. The novelty is thatour vortices have no electric charge unlike the Chern-Simons-Higgs vortex solutions.2 Nielsen-Olesen vortices with electric chargefrom CPT -even Lorentz-violating modelLet us start by considering the (3+1)-dimensional CPT -even Lorentz-violating modelgoverned by the action [3],S =∫d4x(−14kκλµνFκλFµν −AνJν), (1)where the metric tensor is gµν = (1,−1,−1,−1). The coupling kκλµν is real anddimensionless. It has the symmetries of the Riemann tensor and zero double trace, soit contains 19 independent real components. The field strength Fµν is defined byFµν = ∂µAν − ∂νAµ (2)and the term AνJν represents the coupling between the gauge field and an externalcurrent.The equation of motion arising from the Lagrangian (1) iskµαβγ∂αF βγ + Jµ = 0 (3)2In the following we will consider (3+1) dimensional CPT -even Lorentz-violating modelcoupled to Higgs (φ) matterS =∫d4x(−14kκλµνFκλFµν + |Dµφ|2 − V (|φ|))(4)The covariant derivative is defined as Dµ = ∂µ + ieAµ and V (|φ|) is a self-interactingpotential. Since, we will analyze the dimensional reduction of the model (4), it isconvenient to expand the first term of the action,SCPT = −14∫d4xkκλµνFκλFµν= −2∫d4x(k0i0i(F0i)2 + kijij(Fij)2 + k0j0iF0jF 0i +k1µ1νF1µF 1ν + k2µ2νF2µF 2ν + k3µ3νF3µF 3ν + k0ijκF0iF jκ), (5)wherek0i0i(F0i)2 = k0101(F01)2 + k0202(F02)2 + k0303(F03)2kijij(Fij)2 = k1212(F12)2 + k1313(F13)2 + k2323(F23)2k0j0iF0jF 0i = k0102F01F 02 + k0103F01F 03 + k0203F02F 03k1µ1νF1µF 1ν = k1012F10F 12 + k1013F10F 13 + k1213F12F 13k2µ2νF2µF 2ν = k2023F20F 23 + k2021F20F 21 + k2123F21F 23k3µ3νF3µF 3ν = k3031F30F 31 + k3032F30F 32 + k3132F31F 32k0ijκF0iF jκ = 2k0123F01F 23 + 2k0312F03F 12 + 2k0231F02F 31 (6)We will study the soliton solution in a (2 + 1)-dimensional model, obtained from (4)via dimensional reduction. In order to carry out this procedure it is natural to considera dimensional reduction of the action by suppressing dependence on the third spacialcoordinate and renaming A3 as N . Then, the action (4) becomesS(2+1) =∫d3x(SrCPT + |D0φ|2 − |Diφ|2 − e2N2|φ|2 − V (|φ|)), (7)where SrCTP is the reduced CPT -even Lorentz-violating action,SrCPT = −2∫d3x(k0i0i(F0i)2 + kµ3µ3(∂µN)2 + k1212(F12)2 + kµνµ3Fµν∂µN+k0312F12(∂0N) + k0ij3F0i(∂jN) + ki0ijFi0F ij+k0102F01F 02 + k303i(∂0N)(∂iN) + k3132(∂1N)(∂2N)), (8)with (i, j = 1, 2) and (µ, ν = 0, 1, 2).3Here, we are interested in time-independent soliton solutions that ensure the finite-ness of the action (7). These are the stationary points of the energy which for thestatic field configuration readsE(2+1) =∫d2x[2(k0i0i(∂iA0)2 + ki3i3(∂iN)2 + k1212(F12)2 + kiji3Fij∂iN+ki0i3∂iA0∂iN + k0ij3(−∂iA0)(∂jN) + ki0ij(∂iA0)F ij+k0102(∂1A0)(∂2A0) + k3132(∂1N)(∂2N))+ e2|φ|2(N2 −A20)+|(D1 ± iD2)φ|2 ∓ eF12|φ|2 ±12ǫij∂iJj +λ4(|φ|2 − v2)2], (9)where we have used the fundamental identity|Diφ|2 = |(D1 ± iD2)φ|2 ∓ eF12|φ|2 ± ǫij∂iJj (10)The potential term has been chosen so that it coincides with the symmetry breakingpotential of the Higgs modelV (|φ|) =λ4(|φ|2 − v2)2 (11)The term∫d2x12ǫij∂iJj is a surface term and may be dropped with the hypothesis thatφ||x|=∞ = υeiα(θ)Ai||x|=∞ =ie∂i(log φ) (12)This condition also implies that the covariant derivative must vanish asymptotically,which fixes the behavior of the gauge field Ai. Then we haveΦ =∫d2xF12 =∮|x|=∞Aidxi = 2πn (13)where n is a topological invariant which takes only integer values.The formula (9) may be rewritten as a sum of square terms which are boundedbelow by a multiple of the magnitude of the magnetic flux. To achieve this we canregroup the terms of (9) as follows,k0i0i(∂iA0)2 + ki3i3(∂iN)2 + ki0i3∂iA0∂iN = k0i0i(∂iA0 ± ∂iN)2 , (14)where, we have chosen k0i0i = ki3i3 = ±12ki0i3. In addition, if we choose ±ki0ij = kiji3,we havekiji3Fij∂iN + ki0ij∂iA0F ij = ki0ijFij(∂iA0 ± ∂iN) (15)Sincek0ij3(−∂iA0)(∂jN) = −k0123(∂1A0)(∂2N) + k0213(∂2A0)(∂1N) , (16)4we can choose, −k0123 = ±k3132 and k0102 = ±k0213, so thatk3132(∂1N)(∂2N)− k0123(∂1A0)(∂2N) = k0123∂2N(−∂1A0 ∓ ∂1N)k0213(∂2A0)(∂1N) + k0102(∂2A0)(∂1A0) = k0102∂2A0(∂1A0 ± ∂1N) (17)Finally, we can write the following identity:2k1212(F12)2 ∓ eF 12|φ|2 +λ4(|φ|2 − v2)2 =κ2[F 12 ∓eκ(|φ|2 − v2)]2 + (2k1212 −κ2)(F 12)2 +(λ4−e22κ)(|φ|2 − v2)2 ∓ ev2F 12 (18)If the coupling constant λ and κ are chosen such thatλ =e22k1212κ = 4k1212 , (19)the expression (18) takes the simple form2k1212(F12)2 ∓ eF 12|φ|2 +λ4(|φ|2 − v2)2 =2k1212[F12 ∓e4k1212(|φ|2 − v2)]2 ∓ ev2F 12 (20)Using these identities, the energy functional (9) becomesE(2+1) =∫d2x[2(k0i0i(∂iA0 ± ∂iN)2 + ki0ijFij(∂iA0 ± ∂iN)+k0123∂2N(−∂1A0 ∓ ∂1N) + k0102∂2A0(∂1A0 ± ∂1N)+k1212[F12 ∓e4k1212(|φ|2 − v2)]2)+ e2|φ|2(N2 −A20)+|(D1 ± iD2)φ|2 ∓ ev2F 12](21)If we choose A0 = ∓N , then the fields A0 and N are decoupled to the fields Ai and φand we obtain an expression for the energy similar to the well known energy for theMaxwell-Higgs modelE(2+1) =∫d2x(2k1212[F12 ∓e4k1212(|φ|2 − v2)]2 + |(D1 ± iD2)φ|2 ∓ ev2F 12), (22)Thus, the energy (22) is reduced to a sum of square terms which are bounded below bya multiple of the magnitude of the magnetic flux(for positive flux we choose the lowersigns, and for negative flux we choose the upper signs):E(2+1) ≥ ev2|Φ| (23)5The bound is saturated by fields satisfying the first-order Bogomolnyi [18] self-dualityequations:(D1 ± iD2)φ = 0F 12 ∓e4k1212(|φ|2 − v2) = 0 , (24)The equations (24) are identical in form to the Bogomolnyi equations of the abelianMaxwell-Higgs model. The difference lies in the fact that our vortices not only carrymagnetic flux, as in the Higgs model, but also electric charge. This is a consequencethat in our theory the dynamics of gauge field is dictated by a CPT -even Lorentz-violating term instead of a Maxwell term as in Higgs theory.3 Self-Dual Chern-Simons vortices without elec-tric charge from CPT -odd Lorentz-violating modelIn this section, we analyze the CPT -odd Lorentz-violating model [5], coupled to HiggsfieldS =∫d4x[14pαǫαβµνAβFµν + |Dµφ|2 + V (|φ|)], (25)where the coupling coefficient pα is real and has dimensions of mass. The antisymmetrictensor ǫαβµν is the totally antisymmetric tensor such that ǫ0123 = 1.By varying with respect to A3 and A0, we obtain the field equationspα12ǫα3µνFµν + J3 = 0 , (26)pα12ǫα0µνFµν + J0 = 0 , (27)where J0 = −ie[φ∗(D0φ)− φ(D0φ)∗]and J3 = ie[φ∗(D3φ)− φ(D3φ)∗].Here, we are interested in the analysis of the structure of first-order Bogomolnyiself-duality equations for pi = 0 (i = 1, 2, 3) and p0 6= 0. This condition implies areduction of the action (25) and the equations (26), (27)S =∫d4x[14p0ǫ0βµνAβFµν + |Dµφ|2 + V (|φ|)], (28)p0F12 + J3 = 0 , (29)J0 = 0 (30)6We can proceed similarly to previous section and reduce one dimension by assumingthat the fields do not depend on one of the spatial coordinates, say x3, and renamingA3 as N ,Sr =∫d3x(14p0(2A1∂2N − 2A2∂1N + 2NF12)+|D0φ|2 − |Diφ|2 − e2N2|φ|2 + V (|φ|)), (31)Integrating by parts the two first terms of this action, we have,Sr =∫d3x(Np0F12 + |D0φ|2 − |Diφ|2 − e2N2|φ|2 + V (|φ|)), (32)The equation (29) becomesN =p0F122e2|φ|2, (33)Substitution of this equation into the expression of the action (32) yieldsSr =∫d3x( p20F2124e2|φ|2+ |D0φ|2 − |Diφ|2 + V (|φ|)), (34)For the static field configuration the equation (30) reads2e2A0|φ|2 = 0 (35)Thus, the action (34) may be rewritten asSr =∫d2x( p20F2124e2|φ|2− |(D1 ± iD2)φ|2 ± eF12|φ|2 + V (|φ|)), (36)where the identity (10) was used .To uncover the Bogomolnyi-style self-duality, the action is expressed in the formSr =∫d2x( [ p02e|φ|F12 ∓e2p0|φ|(v2 − |φ|2)]2+ |i(D1 ± iD2)φ|2±ev2F12 + V (|φ|) −e4p20|φ|2(v2 − |φ|2)2), (37)Thus, if the potential is chosen to take the self-dual formV (|φ|) =e4p20|φ|2(v2 − |φ|2)2 , (38)the action is bounded below [choosing signs depending on the sign of the flux]Sr ≥ ev2|Φ| (39)7Therefore, when the action reaches its minimum value, the fields will extremise theaction and satisfy the first-order field equations(D1 ± iD2)φ = 0F12 = ±2e3|φ|2p20(v2 − |φ|2) , (40)These equations may be compared with the self-duality equations of the Chern-SimonsHiggs theory. If we choose p0 as the Chern-Simons coupling parameter κ, the equa-tions (40) coincide with the well know Bogomolnyi equations of the relativistic Chern-Simons-Higgs model. Nevertheless, the vortex solutions of the equation (40) are notthe identical to those of the relativistic Chern-Simons-Higgs theory. This is becausethe electric charge density for our solitons is zero, as shown in equation (30).Finally, it is interesting to note, that in our model the Chern-Simons field does not leadto the equation of the type Ei ∝ ǫijJj , which is an equation of the Chern-Simons-Higgstheory. This is due to that we have considered pi = 0.Another difference with the self-dual Chern-Simons-Higgs vortices lies in the fact thatin our model the charge density is not locally proportional to the magnetic field, asshown the equations (29) and (30). Thus, in the case of a static field configuration,we should infer the behavior of the field A0 from the equation (35). The equation (35)shows that A0 may be different from zero if and only if φ(x) = 0. Since, our vortexsolutions are determined by equations (40), we know, from the Chern-Simons-Higgssolutions, that φ(x) has an only one root at x = 0. Therefore, A0 may be differentfrom zero, only, at x = 0. Nevertheless, in order to ensure the regularity of the electricfield at x = 0, we should impose that A0(0) = 0. Thus, our vortex solution has noelectric field.In summary we have discussed the Bogomolnyi framework for two models withLorentz breaking symmetry. In the first case we have analyzed the dimensional reduc-tion of a CPT -even Lorentz-violating term coupled to Higgs field, showing that themodel supports vortex solutions which are identical to those of Maxwell-Higgs model.In the second situation we deal with a reduction of a Carroll-Field-Jackiw term coupledto matter. We have shown, here, that the Bogomolnyi equations of Chern-Simons Higgstheory are obtained as a particular case. It is important to note that in our first model,the vortex solutions has electric charge. This is a difference with the Nielsen-Olesensolutions. Our second model does not have electric charge, being this a difference withrelativistic Chern-Simons-Higgs solitons. Finally, we want to note that vortices withelectric charge in the CPT-even Lorentz-violating model coupled to matter, were foundin Ref.[15]. However the authors use a very strange potential term to generate thesesolutions, and the self-duality equations differ absolutely from our result.Acknowledgements8I would like to thank Rodolfo Casana for useful comments. Also, I would like to thankDmitri Vassilevich for hospitality during the realization of this work. I am grateful tothe referee for his careful reading of the manuscript. This work is supported by CAPES(PNPD/2011).References[1] For review see, V.A. Kostelecky (Ed.), CPT and Lorentz Symmetry III, WorldScientific, Singapore, 2005.V.A. Kostelecky. and S. Samuel, Phys. Rev. D 39, 683 (1989);V.A. Kostelecky and R. Potting, Nucl. Phys. B 359, 545 (1991).[2] V.A. Kostelecki and R. Potting, Phys. Rev. D 51, 3923 (1995).[3] D. Colladay, V.A. Kostelecky, Phys. Rev. D 55, 6760 (1997);D. Colladay, V.A. Kostelecky, Phys. Rev. D 58, 116002 (1998).[4] V.A. Kosteleckf, Phys. Rev. D 69, 105009 (2004).[5] S.M. Carroll, G.B. Field, R. Jackiw, Phys. Rev. D 41 (1990) 1231.[6] R. Jackiw, V.A. Kostelecky, Phys. Rev. Lett. 82, 3572 (1999);M. Perez-Victoria, Phys. Rev. Lett. 83, 2518 (1999);B. Altschul, Phys. Rev. D 70, 101701(R) (2004).[7] F.R. Klinkhamer, Nucl. Phys. B 578, 277 (2000).[8] C. Adam, F.R. Klinkhamer, Nucl. Phys. B 607 (2001).[9] H. Belich, et al., Phys. Rev. D 68, 065030 (2003).[10] B. Feng, et al., Phys. Rev. Lett. 96, 221302 (2006).[11] R. Lehnert, R. Potting, Phys. Rev. Lett. 93 (2004) 110402, hep-ph/ 0406128;R. Lehnert, R. Potting, Phys. Rev. D 70 (2004) 125010, hep-ph/0408285;C. Kaufhold, F.R. Klinkhamer, Nucl. Phys. B 734 (2006) 1, hep-th/ 0508074.[12] R. Jackiw, S.-Y. Pi, Phys. Rev. D 68, 104012 (2003).9[13] M.D. Seifert, Phys. Rev. D 82, 125015 (2010).M.D. Seifert, Phys. Rev. Lett. 105, 201601 (2010)[14] C. Miller, R. Casana, M.M. Ferreira, Jr., E. da Hora Phys.Rev. D 86 065011(2012);[15] C. Casana, M.M. Ferreira, Jr., E. da Hora, C. Miller Phys.Lett. B 718 620-624(2012);[16] R. Casana, L. Sourrouille, Physics Letters B, 726:488492,(2013), [arXiv:hep-th/1307.1734].[17] H.B. Nielsen, P. Olesen, Nucl. Phys. B 61 45 (1973).[18] E. Bogomolyi, Sov. J. Nucl. Phys 24, 449 (1976);H. de Vega and F .A. Schaposnik, Phys. Rev. D 14, 1100 (1976).[19] E. Witten and D. I. Olive, Phys. Lett. B 78 (1978) 97.[20] S. K. Paul and A. Khare, Phys. Lett. B 174, 420 (1986) [Erratum-ibid. 177B,453 (1986)];H. J. de Vega and F. A. Schaposnik, Phys. Rev. D 34, 3206 (1986)[21] R. Jackiw and E. Weinberg, Phys. Rev. Lett. 64 2334 (1990).J. Hong, Y. Kim and P.-Y. Pac, Phys. Rev. Lett. 64 2230 (1990).[22] R. Jackiw and S. Y. Pi, Phys. Rev. Lett. 64, 2969 (1990); R. Jackiw and S. Y. Pi,Phys. Rev. D 42, 3500 (1990). [Erratum-ibid. D 48, 3929 (1993)].[23] R. Jackiw, Ki-Myeong Lee and E.J. Weinberg, Phys.Rev. D 42, 3488 (1990).10

CONICET Digital

arXiv:1311.4501v4  [hep-th]  12 Apr 2014A note on vortices from Lorentz-violatingmodelsLucas Sourrouillea,baDepartamento de F´ısica, Universidade Federal do Maranha˜o,65085-580, Sa˜o Lu´ıs, Maranha˜o, BrazilbCMCC, Universidade Federal do ABC, Santo Andre´, SP, Brazillsourrouille@yahoo.esApril 15, 2014AbstractWe consider two self-dual abelian Higgs systems obtained from Lorentz breakingsymmetry models by dimensional reduction. For the first model, we show that the self-dual equations are identical to those of Nielsen-Olesen vortices. Also, we show that ourvortices have electric charge. In the second case we show that self-dual Chern-Simons-Higgs vortices without electric charge are possible.Keywords:Chern-Simons-like gauge theory, Topological solitons, Lorentz symme-try violation.pacs: 11.10.Kk, 11.10.Lm, 11.27.+d, 12.60.i, 11.30.Cp1 IntroductionThe possible violation of Lorentz invariance has recently received a lot of attention as acandidate for the Planck scale physics [1].The large range of existing phenomenologicaland experimental activities stems from the application of effective field theory[2] andthe construction of the Standard-Model Extension (SME)[3, 4] to catalogue and pre-dict observable effects. The gauge sector of the SME include CPT-even and CPT-oddterms. The CPT-odd, usully called the Carroll-Field-Jackiw term[5], consist on fourparameters, that engenders a parity-odd and birefringent electrodynamics, coupled toChern-Simons dynamics. The CPT-even photon sector is composed by fourth-ranktensor, with Riemann tensor symmetries and a double null trace, which is coupled to agauge term F κλFµν . These models have been studied in different contexts, such as ra-diative corrections [6], nontrivial spacetime topology [7], causality [8], supersymmetry1[9], the cosmic microwave background [10], vacuum Cherenkov radiation[11], generalrelativity [12] and soliton solutions, like monopoles[13] and vortices[14, 15, 16].On the other hand, it is well known that the abelian Higgs model [17] in (2 + 1) di-mensions, support topological vortex solutions of finite energy. With a specific choiceof coupling constants, minimum energy vortex configurations satisfy first order dif-ferential equations [18]. This model admits and N = 2 supersymmetric extensionproviding a connection between BPS equations and supersymmetry [19]. Also the(2 + 1)-dimensional matter field interacting with gauge fields whose dynamics is gov-erned by a ChernSimons term support soliton solutions [20]. When self-interactionsare suitably chosen, vortex configurations satisfy the Bogomolnyi-type equations witha specific sixth order potential [21]. Another important feature of the ChernSimonsgauge field is that inherits its dynamics from the matter fields to which it is coupled,so it may be either relativistic [21] or non-relativistic [22]. In addition the soliton so-lutions are of topological and non-topological nature [23].The purpose of this letter is the study of the self-dual vortices in two Lorentz-violatingmodels. In the first section we will analyze the self-dual equations of a two dimensionalmodel obtained from the CPT -even Lorentz-violating term, coupled to matter field,by dimensional reduction. We will show that this model support self-dual vortex solu-tions with electric charge. In particular, we will show that the self-duality equationsare identical to those of the Nielsen-Olesen vortices. The second section is dedicatedto explore the dimensional reduction of the CPT -odd Lorentz-violating term couple toa Higgs field. We will show, here, that it is possible to obtain exactly the same Bo-gomolnyi equations as the relativistic Chern-Simons-Higgs model. The novelty is thatour vortices have no electric charge unlike the Chern-Simons-Higgs vortex solutions.2 Nielsen-Olesen vortices with electric chargefrom CPT -even Lorentz-violating modelLet us start by considering the (3+1)-dimensional CPT -even Lorentz-violating modelgoverned by the action [3],S =∫d4x(−14kκλµνFκλFµν −AνJν), (1)where the metric tensor is gµν = (1,−1,−1,−1). The coupling kκλµν is real anddimensionless. It has the symmetries of the Riemann tensor and zero double trace, soit contains 19 independent real components. The field strength Fµν is defined byFµν = ∂µAν − ∂νAµ (2)and the term AνJν represents the coupling between the gauge field and an externalcurrent.The equation of motion arising from the Lagrangian (1) iskµαβγ∂αF βγ + Jµ = 0 (3)2In the following we will consider (3+1) dimensional CPT -even Lorentz-violating modelcoupled to Higgs (φ) matterS =∫d4x(−14kκλµνFκλFµν + |Dµφ|2 − V (|φ|))(4)The covariant derivative is defined as Dµ = ∂µ + ieAµ and V (|φ|) is a self-interactingpotential. Since, we will analyze the dimensional reduction of the model (4), it isconvenient to expand the first term of the action,SCPT = −14∫d4xkκλµνFκλFµν= −2∫d4x(k0i0i(F0i)2 + kijij(Fij)2 + k0j0iF0jF 0i +k1µ1νF1µF 1ν + k2µ2νF2µF 2ν + k3µ3νF3µF 3ν + k0ijκF0iF jκ), (5)wherek0i0i(F0i)2 = k0101(F01)2 + k0202(F02)2 + k0303(F03)2kijij(Fij)2 = k1212(F12)2 + k1313(F13)2 + k2323(F23)2k0j0iF0jF 0i = k0102F01F 02 + k0103F01F 03 + k0203F02F 03k1µ1νF1µF 1ν = k1012F10F 12 + k1013F10F 13 + k1213F12F 13k2µ2νF2µF 2ν = k2023F20F 23 + k2021F20F 21 + k2123F21F 23k3µ3νF3µF 3ν = k3031F30F 31 + k3032F30F 32 + k3132F31F 32k0ijκF0iF jκ = 2k0123F01F 23 + 2k0312F03F 12 + 2k0231F02F 31 (6)We will study the soliton solution in a (2 + 1)-dimensional model, obtained from (4)via dimensional reduction. In order to carry out this procedure it is natural to considera dimensional reduction of the action by suppressing dependence on the third spacialcoordinate and renaming A3 as N . Then, the action (4) becomesS(2+1) =∫d3x(SrCPT + |D0φ|2 − |Diφ|2 − e2N2|φ|2 − V (|φ|)), (7)where SrCTP is the reduced CPT -even Lorentz-violating action,SrCPT = −2∫d3x(k0i0i(F0i)2 + kµ3µ3(∂µN)2 + k1212(F12)2 + kµνµ3Fµν∂µN+k0312F12(∂0N) + k0ij3F0i(∂jN) + ki0ijFi0F ij+k0102F01F 02 + k303i(∂0N)(∂iN) + k3132(∂1N)(∂2N)), (8)with (i, j = 1, 2) and (µ, ν = 0, 1, 2).3Here, we are interested in time-independent soliton solutions that ensure the finite-ness of the action (7). These are the stationary points of the energy which for thestatic field configuration readsE(2+1) =∫d2x[2(k0i0i(∂iA0)2 + ki3i3(∂iN)2 + k1212(F12)2 + kiji3Fij∂iN+ki0i3∂iA0∂iN + k0ij3(−∂iA0)(∂jN) + ki0ij(∂iA0)F ij+k0102(∂1A0)(∂2A0) + k3132(∂1N)(∂2N))+ e2|φ|2(N2 −A20)+|(D1 ± iD2)φ|2 ∓ eF12|φ|2 ±12ǫij∂iJj +λ4(|φ|2 − v2)2], (9)where we have used the fundamental identity|Diφ|2 = |(D1 ± iD2)φ|2 ∓ eF12|φ|2 ± ǫij∂iJj (10)The potential term has been chosen so that it coincides with the symmetry breakingpotential of the Higgs modelV (|φ|) =λ4(|φ|2 − v2)2 (11)The term∫d2x12ǫij∂iJj is a surface term and may be dropped with the hypothesis thatφ||x|=∞ = υeiα(θ)Ai||x|=∞ =ie∂i(log φ) (12)This condition also implies that the covariant derivative must vanish asymptotically,which fixes the behavior of the gauge field Ai. Then we haveΦ =∫d2xF12 =∮|x|=∞Aidxi = 2πn (13)where n is a topological invariant which takes only integer values.The formula (9) may be rewritten as a sum of square terms which are boundedbelow by a multiple of the magnitude of the magnetic flux. To achieve this we canregroup the terms of (9) as follows,k0i0i(∂iA0)2 + ki3i3(∂iN)2 + ki0i3∂iA0∂iN = k0i0i(∂iA0 ± ∂iN)2 , (14)where, we have chosen k0i0i = ki3i3 = ±12ki0i3. In addition, if we choose ±ki0ij = kiji3,we havekiji3Fij∂iN + ki0ij∂iA0F ij = ki0ijFij(∂iA0 ± ∂iN) (15)Sincek0ij3(−∂iA0)(∂jN) = −k0123(∂1A0)(∂2N) + k0213(∂2A0)(∂1N) , (16)4we can choose, −k0123 = ±k3132 and k0102 = ±k0213, so thatk3132(∂1N)(∂2N)− k0123(∂1A0)(∂2N) = k0123∂2N(−∂1A0 ∓ ∂1N)k0213(∂2A0)(∂1N) + k0102(∂2A0)(∂1A0) = k0102∂2A0(∂1A0 ± ∂1N) (17)Finally, we can write the following identity:2k1212(F12)2 ∓ eF 12|φ|2 +λ4(|φ|2 − v2)2 =κ2[F 12 ∓eκ(|φ|2 − v2)]2 + (2k1212 −κ2)(F 12)2 +(λ4−e22κ)(|φ|2 − v2)2 ∓ ev2F 12 (18)If the coupling constant λ and κ are chosen such thatλ =e22k1212κ = 4k1212 , (19)the expression (18) takes the simple form2k1212(F12)2 ∓ eF 12|φ|2 +λ4(|φ|2 − v2)2 =2k1212[F12 ∓e4k1212(|φ|2 − v2)]2 ∓ ev2F 12 (20)Using these identities, the energy functional (9) becomesE(2+1) =∫d2x[2(k0i0i(∂iA0 ± ∂iN)2 + ki0ijFij(∂iA0 ± ∂iN)+k0123∂2N(−∂1A0 ∓ ∂1N) + k0102∂2A0(∂1A0 ± ∂1N)+k1212[F12 ∓e4k1212(|φ|2 − v2)]2)+ e2|φ|2(N2 −A20)+|(D1 ± iD2)φ|2 ∓ ev2F 12](21)If we choose A0 = ∓N , then the fields A0 and N are decoupled to the fields Ai and φand we obtain an expression for the energy similar to the well known energy for theMaxwell-Higgs modelE(2+1) =∫d2x(2k1212[F12 ∓e4k1212(|φ|2 − v2)]2 + |(D1 ± iD2)φ|2 ∓ ev2F 12), (22)Thus, the energy (22) is reduced to a sum of square terms which are bounded below bya multiple of the magnitude of the magnetic flux(for positive flux we choose the lowersigns, and for negative flux we choose the upper signs):E(2+1) ≥ ev2|Φ| (23)5The bound is saturated by fields satisfying the first-order Bogomolnyi [18] self-dualityequations:(D1 ± iD2)φ = 0F 12 ∓e4k1212(|φ|2 − v2) = 0 , (24)The equations (24) are identical in form to the Bogomolnyi equations of the abelianMaxwell-Higgs model. The difference lies in the fact that our vortices not only carrymagnetic flux, as in the Higgs model, but also electric charge. This is a consequencethat in our theory the dynamics of gauge field is dictated by a CPT -even Lorentz-violating term instead of a Maxwell term as in Higgs theory.3 Self-Dual Chern-Simons vortices without elec-tric charge from CPT -odd Lorentz-violating modelIn this section, we analyze the CPT -odd Lorentz-violating model [5], coupled to HiggsfieldS =∫d4x[14pαǫαβµνAβFµν + |Dµφ|2 + V (|φ|)], (25)where the coupling coefficient pα is real and has dimensions of mass. The antisymmetrictensor ǫαβµν is the totally antisymmetric tensor such that ǫ0123 = 1.By varying with respect to A3 and A0, we obtain the field equationspα12ǫα3µνFµν + J3 = 0 , (26)pα12ǫα0µνFµν + J0 = 0 , (27)where J0 = −ie[φ∗(D0φ)− φ(D0φ)∗]and J3 = ie[φ∗(D3φ)− φ(D3φ)∗].Here, we are interested in the analysis of the structure of first-order Bogomolnyiself-duality equations for pi = 0 (i = 1, 2, 3) and p0 6= 0. This condition implies areduction of the action (25) and the equations (26), (27)S =∫d4x[14p0ǫ0βµνAβFµν + |Dµφ|2 + V (|φ|)], (28)p0F12 + J3 = 0 , (29)J0 = 0 (30)6We can proceed similarly to previous section and reduce one dimension by assumingthat the fields do not depend on one of the spatial coordinates, say x3, and renamingA3 as N ,Sr =∫d3x(14p0(2A1∂2N − 2A2∂1N + 2NF12)+|D0φ|2 − |Diφ|2 − e2N2|φ|2 + V (|φ|)), (31)Integrating by parts the two first terms of this action, we have,Sr =∫d3x(Np0F12 + |D0φ|2 − |Diφ|2 − e2N2|φ|2 + V (|φ|)), (32)The equation (29) becomesN =p0F122e2|φ|2, (33)Substitution of this equation into the expression of the action (32) yieldsSr =∫d3x( p20F 2124e2|φ|2+ |D0φ|2 − |Diφ|2 + V (|φ|)), (34)For the static field configuration the equation (30) reads2e2A0|φ|2 = 0 (35)Thus, the action (34) may be rewritten asSr =∫d2x( p20F 2124e2|φ|2− |(D1 ± iD2)φ|2 ± eF12|φ|2 + V (|φ|)), (36)where the identity (10) was used .To uncover the Bogomolnyi-style self-duality, the action is expressed in the formSr =∫d2x( [ p02e|φ|F12 ∓e2p0|φ|(v2 − |φ|2)]2+ |i(D1 ± iD2)φ|2±ev2F12 + V (|φ|) −e4p20|φ|2(v2 − |φ|2)2), (37)Thus, if the potential is chosen to take the self-dual formV (|φ|) =e4p20|φ|2(v2 − |φ|2)2 , (38)the action is bounded below [choosing signs depending on the sign of the flux]Sr ≥ ev2|Φ| (39)7Therefore, when the action reaches its minimum value, the fields will extremise theaction and satisfy the first-order field equations(D1 ± iD2)φ = 0F12 = ±2e3|φ|2p20(v2 − |φ|2) , (40)These equations may be compared with the self-duality equations of the Chern-SimonsHiggs theory. If we choose p0 as the Chern-Simons coupling parameter κ, the equa-tions (40) coincide with the well know Bogomolnyi equations of the relativistic Chern-Simons-Higgs model. Nevertheless, the vortex solutions of the equation (40) are notthe identical to those of the relativistic Chern-Simons-Higgs theory. This is becausethe electric charge density for our solitons is zero, as shown in equation (30).Finally, it is interesting to note, that in our model the Chern-Simons field does not leadto the equation of the type Ei ∝ ǫijJj , which is an equation of the Chern-Simons-Higgstheory. This is due to that we have considered pi = 0.Another difference with the self-dual Chern-Simons-Higgs vortices lies in the fact thatin our model the charge density is not locally proportional to the magnetic field, asshown the equations (29) and (30). Thus, in the case of a static field configuration,we should infer the behavior of the field A0 from the equation (35). The equation (35)shows that A0 may be different from zero if and only if φ(x) = 0. Since, our vortexsolutions are determined by equations (40), we know, from the Chern-Simons-Higgssolutions, that φ(x) has an only one root at x = 0. Therefore, A0 may be differentfrom zero, only, at x = 0. Nevertheless, in order to ensure the regularity of the electricfield at x = 0, we should impose that A0(0) = 0. Thus, our vortex solution has noelectric field.In summary we have discussed the Bogomolnyi framework for two models withLorentz breaking symmetry. In the first case we have analyzed the dimensional reduc-tion of a CPT -even Lorentz-violating term coupled to Higgs field, showing that themodel supports vortex solutions which are identical to those of Maxwell-Higgs model.In the second situation we deal with a reduction of a Carroll-Field-Jackiw term coupledto matter. We have shown, here, that the Bogomolnyi equations of Chern-Simons Higgstheory are obtained as a particular case. It is important to note that in our first model,the vortex solutions has electric charge. This is a difference with the Nielsen-Olesensolutions. Our second model does not have electric charge, being this a difference withrelativistic Chern-Simons-Higgs solitons. Finally, we want to note that vortices withelectric charge in the CPT-even Lorentz-violating model coupled to matter, were foundin Ref.[15]. However the authors use a very strange potential term to generate thesesolutions, and the self-duality equations differ absolutely from our result.Acknowledgements8I would like to thank Rodolfo Casana for useful comments. Also, I would like to thankDmitri Vassilevich for hospitality during the realization of this work. I am grateful tothe referee for his careful reading of the manuscript. This work is supported by CAPES(PNPD/2011).References[1] For review see, V.A. Kostelecky (Ed.), CPT and Lorentz Symmetry III, WorldScientific, Singapore, 2005.V.A. Kostelecky. and S. Samuel, Phys. Rev. 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Note on vortices from Lorentz-violating models

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