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Global-String and Vortex Superfluids in a Supersymmetric Scenario C.N. Ferreira 1,∗ J. A. Helayël-Neto 2,† and W.G. Ney 1‡ 1 Nućleo de Estudos em F´ısica, Centro Federal de Educação Tecnológica de Campos Rua Dr. Siqueira, 273, Campos dos Goytacazes, Rio de Janeiro, Brazil, CEP 28030-130 and 2 Centro Brasileiro de Pesquisas F´ısicas, Rua Dr. Xavier Sigaud 150, Urca, Rio de Janeiro, Brazil, CEP 22290-180 (Dated: February 1, 2008) The main goal of thiswork isto investigatethepossibility offindingthesupersymmetricversion 8 of the U(1)-global string model which behaves as a vortex-superfluid. To describe the superfluid 0 phase, we introduce a Lorentz-symmetry breaking background that, in an approach based on su- 0 persymmetry,leads to a discussion on the relation between the violation of Lorentz symmetry and 2 explicit soft supersymmetrybreakings. Wealso studytherelation betweenthestringconfiguration and the vortex-superfluid phase. In the framework we settle down in terms of superspace and su- n perfields,weactually establish adualitybetweenthevortexdegrees offreedom andthecomponent a fieldsof theKalb-Ramondsuperfield. Wemakealso considerations about thefermionic excitations J that may appear in connection with thevortex formation. 4 1 PACSnumbers: 12.60.Jv,11.27.+d ] h t I. INTRODUCTION tron matter. Global strings, which behave as a vortex - p superfluidity states, appear when a discrete symmetry is e Stablevortexstatesmayappearasaninterestingman- broken. Thesestrings,asthelocalstrings[9,10,11],were h most likely produced during phase transitions [12], and ifestation of superfluidity. As one of the motivations, [ appear in some Grand-Unified Gauge Theories. They thesevorticeshavebeenobservedinbosonicorfermionic 3 diluted gases[1, 2]. In the case of the bosonic vortices, carry a large energy density [10]. Both global and lo- v cal strings were mainly studied as a possible mechanism the detection has been confirmed by analyzing the den- 5 for the seed density perturbation which has become a sityvariationsinanexpandingBose-EinsteinCondensate 3 structure of large scale of the Universe we observe today (BEC)[2, 3]. In Fermi systems, we do not expect signif- 9 [13, 14]. 1 icant density variations[4] but, under certain conditions, . the density variations may be induced by the presence Nowadays, the approach considering cosmic string 4 configurations has been revisited in connection with of one or more vortices that can be present in nuclear 0 matter[5]. Theinteriorofaneutronstar,thatistheonly string theory[12, 15, 16] and the Wilkinson Microwave 7 Anisotropy Probe (WMAP)[17]. The importance of this 0 known system close to nuclear and neutron matter, con- : stitutestheappropriatescenarioforthevortexformation context is related to a possible measurement at the level v of string theory, which has supersymmetry (SUSY) as induced by the rotational state of the star. i X Thereareimportantobservationsofastrophysicalrele- one of its main characteristics. SUSY is also related to cosmic strings in other contexts, where one contem- r vancethatmightbeinfluencedbythepresenceofvortices a plates the possibility that the boson-fermion symmetry in the interior of neutron stars; for instance, the pulsar was manifest in the early Universe, but it was broken glitches. The glitchingevents representa directmanifes- approximately at the same time when these topological tation of the presence of superfluid vortices in the inte- defects were formed. rior of the star, the triggering event being an unbalance Many recent works investigate local strings by adopt- between the hydrodynamical forces acting on the vor- ing a supersymmetric framework [18, 19, 20, 21, 22, 23]. tex and the force of interaction of the vortex with nuclei In the context of the star formation [24], SUSY appears presence in the crust, pinning force [6, 7]; but, there are as one of the most interesting mechanisms to describe doubts about the value of the pinning force. cold dark matter[25, 26]. Both, local and global strings, One is related to the value of the energy gap in uni- are also important for their contribution to the gravita- form neutron matter whereas the second problem is due tionalradiationbackground[27]; inthe caseof the global tothe veryoutlinedwayoftreatingvortexstatesinneu- symmetry, instead of radiating gravitationally,the dom- inant radiation mechanism for these strings is the emis- sion of massless Nambu-Goldstone bosons [28]. Global strings which behave as a vortex superfluidity states are ∗Electronicaddress: [email protected] †Electronicaddress: [email protected] connected with a Kalb-Ramond field. In some publica- ‡Electronicaddress: [email protected] tions, it has been shown that, in the low-energy regime, 2 theeffectiveactionthatpresentstheKalb-Ramondfields, wheretheingredientsuperfieldsofthemodelare: achiral that also appear in string theory, provides an accurate scalar supermultiplet, Φ(φ,χ,F), that contains a com- description of the dynamics of global strings[29]. plex scalar field, φ, a spinor, χ , and an auxiliary com- a The Kalb-Ramond field[30, 31] is an antisymmetric plex scalar field, F. The chiral scalar supermultiplet Φ tensor. This tensor, whenever interacting with a mas- can be θ-expanded according to the following expression sive Higgs field, gives us a source. The system may have applicationstosuperfluidheliumandaxioncosmology. A globalvortexbehavesasasuperfluidiftheKalb-Ramond Φ=e−iθσµθ¯∂µ[φ(x)+√2θaχa(x)+θ2F(x)], (2) field breaks Lorentz symmetry in the background. The istheKalb-Ramondfield-strengthsuperfielddefinedin Kalb-Ramondfields in context of the topologicaldefects G terms of the chiral spinor superfield as can be studied in [32], with SUSY framework [20, 23] and associated with Lorentz-symmetry breaking can be studied in [21, 22]. For these implications, in this work, 1 = (DaΣ D¯ Σ¯a˙) (3) we analyze the equivalence of the vortex-superfluids to G 8 a− a˙ globalstringsinasupersymmetriccontext. Thisanalogy where is important to propose alternative models to be considered; the vortex stability and the fermionic and bosonic behaviors of the matter can in the future enlighten us Σ = ψ (x)+θbΩ (x)+θ2 ξ (x)+iσµ ∂ ψ¯a˙(x) howtounderstandthevortexstatesinfermionicmatter. a a ba h a aa˙ µ i The outline of this paper is as follows: in Section 2, iθσµθ¯∂µψ¯a(x) iθσµθ¯θa˙∂µΩa˙a(x) we present some considerations about vortex superfluid −−14θ2θ¯22ψa(x) − models. (4) In Section II, we devote our attention to showing how The chirality condition for this field is D¯a˙Σa =0. the Lorentz-symmetry violation by the Kalb-Ramond The Kalb-Ramond field accommodated in Ωa˙b(x) is background induces explicit SUSY breaking terms. In given by Section III, we focus on the general properties of the supersymmetric model for the vortex and treats some spe- Ω = ǫ ρ(x)+(σµν) (x). (5) cific properties of the supersymmetric superfluid phase ab − ab abBµν in the model we study. In this Section, we also carry with ρ(x) and (x) being complex fields, µν outthesuperfieldidentificationsatzerotemperatureand B start a discussion on the fermionic excitations. Section IV Fermionic excitations are the main issue presented. ρ(x)=P(x)+iM(x), (6) In Section V, we propose a discussion on the non-zero (x)= 1 B iB˜ (x) temperature treatment of the model. Finally, in Section Bµν 4h µν − µν i VI, we draw our General Conclusions. with 1 B˜ (x)= ǫ Bαβ(x) (7) II. THE IMPLICATIONS OF THE µν 2 µναβ LORENTZ-VIOLATING BACKGROUND FOR SOFT SUSY BREAKING ThecomponentsP andψaarecompensatingfieldsand are not present in the θ-expanssion of , as it shall be G explicitly given below. InthisSection,wediscusstheimplicationsofthepres- The superfield (M,ξ,G˜ ), which plays a central role µ enceoftheLorentz-violatingbackgroundforasoftSUSY G in connection with local vortices ([20]), accomodates the breaking. Theideahereistounderstandhowtheexplicit real scalar, M, the fermion ξ and the dual of the Kalb- soft SUSY breaking works to yield mass to some of the RamondfieldstrengthG˜ . Itcanbeθ-expandedaccord- µ field of the model. ing to the following expression: We consider the possibility to get the scalar masses in SUSY theories by working with a Lorentz-symmetry violating background. This framework is important for = 1M + iθaξ + iθ¯a˙ξ¯ + 1θσµ θ¯a˙G˜ a better understanding of the relation between explicit G −2 4 a 4 a˙ 2 aa˙ µ SUSY breaking and Lorentz-symmetry violation. It is 1 1 + θaσµ θ¯2∂ ξ¯a˙ θ2σµ θ¯a˙∂ ξa importanttostressthatthe problemoftheSUSYbreak- 8 aa˙ µ − 8 aa˙ µ ingisaveryimportantmatter,relatedwiththe hiearchy 1 θ2θ¯22M; (8) problem and the mass constraints on the supersymmet- −8 ric particles. In this section, let us start off with the Now, we have all the elements to illustrate how the following supersymmetric Lagrangian: Lorentz-symmetryviolation,signaledby the background ofthe Kalb-Ramondfield, is intimately connectedto the LK =Φ†e4gGΦ|θθθ¯θ¯. (1) appearance of explicit (soft) SUSY breaking terms. 3 We can notice that the superfield carries only some G degreesof freedomof Σ , the fermionic field ψ does not a a appear, ρ appears only through M, and as G˜µ is related LL−SUSY−B =g2ρ|φ|2, (15) to the 2-form, the Kalb-Ramond field, B µν Terms like that may appear as a result of sponta- neousbreakingofSUSY[34]. SoftexplicitSUSYbreaking G˜ = 1ǫ Gναβ. (9) terms are very important in connection with the physics µ µναβ 3! derived from the Minimally Supersymmetric Standard Model (MSSM). In view of that, we try to stress here and ontheconnectionbetweenaLorentz-symmetryviolating backgroundand the appearance of explicit SUSY breaking terms. G =∂ B +∂ B +∂ B (10) µνκ µ νκ ν κµ κ µν III. THE SUPERSYMMETRIC VERSION FOR i A GLOBAL VORTEX AND THE SUPERFLUID L = ∂ φ∗∂µφ+ χ¯γµ∂ χ+g2 φ2G˜ G˜µ BEHAVIOR µ µ µ h 4 | | 1 i i +gG˜µ χ¯γ χ φ¯∂ φ+ φ∂ φ¯ µ µ µ In the present section, we study the supersymmetric (cid:16)4 − 2 2 (cid:17) frameworksettingupthegeneralformalismthatgivesus L (11) Inti thetermstoconstructtheglobalvortexandstudythesu- perfluidbehavior. TheactionstudiedinthepreviousSec- where in this discussion we consider Φ ΦegM. The tion(1)helpsusinunderstandingtheconsequencesofthe → Lagrangian L is the interaction Lagrangian, and its Lorentz-breaking background in connection with SUSY. Int explicit form is not important to show the relation be- Actually, we have to see how the presence of a back- tween the Lorentz and SUSY breakings. ground yielding Lorentz symmetry violation also leads Let us consider the split G as toanexplicitSUSY breaking. Inthe presentSection,let µνλ us adopt the action to study the vortex configuration as in the sequel: Gµνλ =Gµνλ +Gµνλ . (12) (self) (ext) The external Lorentz-symmetry breaking background LK = Φ†e4gGΦ|θ2θ¯2 +S†S|θ2θ¯2 is given by +W|θ2 +W¯|θ¯2. (16) These superfields satisfy a chirality constraint, given Gµνλ =√ρǫ0ijk =√ρǫijk. (13) by the condition D¯ Φ=0 and D¯ =0. The superfield (ext) a˙ a˙ S Φ is defined in (2) and in (8); the superfield S has the G The crucial point here is the justification of why the same properties as Φ and can be θ- expanded according background value of Gµνκ in (13) yields an explicit soft to breakingof SUSY. The whole idea here is that the background for Gµνλ given in (13) lies on a θ-component of G, actually, the θσµθ¯G˜µ in (8), which necessarily signals S =e−iθσµθ¯∂µ[S(x)+√2θaζa(x)+θ2H(x)]. (17) anexplicitSUSYbreaking. Ifthefirstcomponent(theθ -independent one) set up a non-trivial background then In the expression (16), W is the superpotential whose SUSY maynotbackground,then SUSY maynotbe bro- general form is ken; however, whenever the background value sits on a non-trivial θ-component, SUSY is necessarily explicitly W =a Φ +b Φ Φ +c Φ Φ Φ . (18) broken down, and this is the case here. i i ij i j ijk i j k Therelevantbosonicpartofthe(11)importanttoana- The scalar-field potential is given by lyzedthe Lorentz-BreakingrelationwithSUSYbreaking in the backgroundis ∂W V = A¯ A = 2 (19) L=∂ φ∗∂µφ+g2 φ2G˜ G˜µ igG˜µ φ∗∂ φ φ∂ φ∗ Xi i i Xi |∂φi| µ µ µ µ | | − 2 h − i (14) whereA isthe auxiliarycomponentoftheφ -superfield. i i By splitting the Kalb-Ramond fields as (12) and by Let us study the possibility to obtain the supersym- adopting the ansatz of a Lorentz-breaking background, metricversionoftheglobalvortexpotentialaccordingto (13) there emerges a mass term for the bosons, the model discussed in [33]. 4 In the case of a global gauge symmetry, the chiral U(1)symmetry,amasslessGoldstonebosonemergesthat superfield Φ transforms as a phase under the U(1)- yieldsalong-rangeforce. ThebosonicLagrangianresult- i symmetry: ing fromthis supersymmetric model and that is relevant for the superfluid can be written as Φ′ =e−iqiΛΦ (20) i i g2 L =∂ φ†∂µφ+∂ S†∂µS+ φ2G Gµνρ V′+G˜ Jµ. where qi are U(1) global charges and Λ is the rigid U(1) B µ µ 6 | | µνρ − µ rotation angle. The q and Λ are real constants. (26) i Itispossibletobuildupapotentialwithspontaneously forsimplicity,wealsoadopttheredefinitionΦ′ ΦegM. → broken gauge symmetry using three chiral superfields. The current j is given by µ Forcosmicstringswithalocalgaugesymmetry,oneusu- ally needs two charged fields φ , with respective U(1) ± ig charges q±, and a neutral field, Φ0. This mechanism re- Jµ = − φ∗∂µφ φ∂µφ∗ (27) mains the same if we have a global transformation (20); 2 (cid:16) − (cid:17) in this case, the superpotential takes the form: ThebosonicpotentialV′,thatcomesfromtheeq.(22), is given by W(Φ )=µΦ Φ Φ η2 (21) i 0 + − (cid:16) − (cid:17) V′ =g2 φ4+4g2 S 2 φ2. (28) | | | | | | In this approach, the neutral field Φ is important to 0 give us the term responsible for the mass of the scalar We perform the background splitting (12) of (49), so fieldofthetheory,but,inaglobalvortexsuperfluidcon- that the full potential is figuration, we have a Lorentz breaking background, as discussedin[33],andwehaveproven,inthepreviousSec- V =h2 φ4 (g2ρ 4h2 S 2)φ2. (29) tion,thattheLorentzbreakingintroducesmassesforthe | | − − | | | | scalars;then, wecanadopta simpler potential, with one This potential shows us that the U(1)-breaking gives superfield Φ, with charge q , and another superfield, , Φ S massto the moduli field φ while the phase ofthe scalar with charge qS, satisfying the constraint qS = −2qΦ, so field remains massless. In| t|he low-energylimit, the com- that plex scalar field of the Goldstone model can be represented as below: W =h Φ2 (22) S φ=ϕ(r)eiα, (30) Thisformofthesuperpotential,inconnectionwiththe Lorentz-symmetrybreaking(15)ofthe previousSection, where r is the radial coordinate. The boundary condi- leads to the Mexican hat configuration, responsible for tions are given by the global vortex behavior that characterises superfluidity. The SUSY transformations read as below: ϕ(r)=0 to r =δ (31) ϕ(r)=η to r . →∞ i i δM = ǫ¯ ξ¯a˙ ǫaξ , (23) The configuration (30)-(31) is the same as the one a˙ a 2 − 2 of the local vortices, but the long-range interactions of globalstrings,happenduetotheircouplingtoamassless Goldstonefield, causetheir dynamics to be substantially δξ =2σµ ǫ¯a˙ ∂ M iG˜ , (24) a aa˙ (cid:16) µ − µ(cid:17) different from those of the local strings. Spontaneous symmetry breaking requires that ϕ(r) have mass and α be a massless Goldstone boson. This breaking triggered δG˜µ = iǫb(σµν)a∂ ξ + i¯ǫ (σ¯µν)b˙∂ ξ¯a˙ (25) bythesoftSUSYbreakingtermweintroduce,takesplace 2 b ν a 2 b˙ a˙ ν whenever It is important to point out here that the soft supersymmetry breaking terms do not invalidate the super- (g2ρ 4h2 S 2)>0. (32) symmetric transformations; actually, Lorentz symmetry − | | andSUSYarebrokendownbythebackground,butthey Thisrelationshipiscrucialtoensurethestabilityofthe are both symmetries of the action. So, SUSY transfor- potential, for it guarantees the vortex is formed around mations as translations in superspace are not lost. the rightgroundstate. This justifies ourclaim,statedin Global strings appear whenever a U(1) global symme- the previousSection,onthe importanceofthetermthat try is spontaneously broken. After the breaking of the breaks SUSY explicitly (and softly) for the stability of 5 the globalvortex. A solutionto the vortexconfiguration + g2η2√ρεijkG(self)+ 1B Jµν. (36) exists if g2 4h2 S 2. In this configuration, we consider 3 ijk 2 µν ≥ | | the boundary conditions to ϕ, given by (31) and, for the S-field, we consider the ansatz S = s(r)eiΛ. Outside Weusethefactthatarealmasslessscalarfieldinfour- the string, we consider the field S = 0. The global dimensional Minkowski space is equivalent to a rank-2 vortex presents a minimum roll anhdia central maximum anti-symmetric tensor , Bµν[30, 31]; the nature of this characterizes the Mexican hat potential. By analyzing equivalenceinthe caseofthe globalstringscanbe found the potential minimum outside the string, with φ = η in[28]. InSUSY,thisdualitypropertycanbeunderstand and S = 0, we have η = gρ. The ansatz in thehcoire of by superfield identification h i h the string allows us to analyze it in comparisonwith the fermionic Yukawa potential that are the subject of the section IV. Φ†Φ . (37) ∼G The effective Lagrangian Infact,theleftpartthatcontainsthevortexsuperfield givesusthetermϕ2∂ αandtherightsidegivesusaterm µ LB = ∂µϕ∂µϕ+ϕ2∂µα∂µα+∂µs∂µs+s2∂µΛ∂µΛ related with the dual field, G˜ . The identification (37) µ g2 g2 gives us other contributions, related to the scalar field + ϕ2G(self)Gµνρ + √ρεijkϕ2G(self) 6 µνρ (self) 3 ijk M: 1 + B Jµν V. (33) µν 2 − ϕ2 =2ηM (38) Now, let us write, the current J it in terms of the µν Kalb-Ramond field, according to the functional relation The fermionic part is below: 1 √2χ ϕ∗ = iηξ (39) = G˜ Jµd4x= ǫ ∂αBβγJµd4x a − a J Z µ 2Z µαβγ 1 1 = ǫ Bβγ∂αJµd4x= B Jµν (34) 2Z αµβγ 2Z µν √2χ¯a˙ϕ=iηξ¯a˙ (40) where and the vortex identification part ig J = ǫ ∂α φ∗∂βφ φ∂βφ∗ (35) µν µναβ 2 (cid:16) − (cid:17) 1 ϕ2 σµ∂ α+χ¯χ= ηǫ σµ∂νBλρ. (41) µ µνλρ | | 2 The configuration in the core of the string, where the commutator is not zero, [∂ ,∂ ]α=0, in the presence of µ ν 6 We can notice that the fermionic part modifies the avortex. Wecanseethisclearlybyconsideringastraight usual vortex duality relation [33]. If we neglect the vortex along the z- axis, the azimuthal angle, and inte- fermionic contribution, eq.(41) can be written as gratingoveratwo-surfaceorthogonaltothestringyields, [∂ ,∂ ]αdxdy =2π, or [∂ ,∂ ]α= δ(x)δ(y), then, in the x y x y 2π pRresenceofthe vortexthe αisamulti-valuedfunctionof 1 the coordinates and Jµν =0 on the vortex core. ϕ2∂µα= ηǫµνλρ∂νBλρ. (42) 2 6 Outside the string core, φ can be represented as φ ∼ ηexp(iα(x))and S =0;theeffectiveLagrangianforthe Theidentificationofthebosonicpartgivenby(42)has h i Goldstone mode (in the presence of the global strings at the same form as the [33] for the global vortex configu- large distances of the core, which are non-massive exci- ration, but, with the supersymmetric invariance, we can tations) can be written as always have a fermionic part. Theonlyremainingdynamicaldegreeoffreedomisthe scalar (Goldstone boson) field, α. In this approach, we g2η2 L = η2∂ α∂µα+ G(self)Gµνρ have the action for the (global) static string: µ 6 µνρ (self) β 6ρ 1 A= G(Self)Gµνβ +2√ρεijkG(self)+ d4x+ B Jµνd4x (43) Z 6(cid:16) µνβ (Self) ijk β (cid:17) 2Z µν 6 where β =1+g2η2. At this point, it is advisable to remind that an ex- interaction is given as Fi = JjkGjki = √ρǫjkiJjk. The plicit Lorentz-symmetry breaking, as stated above, may bosonic part of the solution has the same form as in the be rephrasedin terms ofa softly explicit SUSY breaking non-supersymmetricmodel,but,inourconstruction,the term as the one we consider here [8]. Now, let us study solution presents the explicit dependence on the param- the solution at long distances compared to the string eters h and g and on the effects of the fermions. In the core, when the interaction of the vortex with the clas- supersymmetric version, the introduction of a Lorentz- sical Goldstone-boson field is described by an effective symmetryviolatingbackgroundgivesusimportantimpli- Lagrangian. The stresstensor inthe backgroundconsid- cationsonthefermionicbackgroundthatweshalldiscuss ers a string at rest point in uˆ direction we have in the next section, when we study the supersymmetric superfluid. T00 =Tii =ρ=p (44) IV. THE ANALYSIS OF THE FERMIONS AND THE SUPERFLUIDITY BEHAVIOR T0i =β√ρG0jkǫi . (45) self jk In a supersymmetric framework, besides the bosonic degrees of freedom, there are fermionic partners in the TheequationofthemotionfortheKalb-Ramondfield theory. In this section, let us analyse the behavior of is the fermions that accompany the bosonic fields. The fermionic action can be written as: 1 ∂ Gµαβ = Jαβ (46) µ β i 1 L = χ¯σµ∂ χ+ B µν +L +L , (49) F µ µν FKR int We obtain the solution 2 2 J where is the fermionic current of the vorticity. The µν J g√ρuîrj uˆjri latter can also be expressed as follows: G0ij = − (47) Self βh r2 1 It yields the stress tensor interaction part given by µν = ǫµναβ∂αχ¯σβχ, (50) J 2 where g (uˆ r)i T0i =2 ρ × (48) h r2 1 = χ¯σµχ. (51) Asinglestraightglobalstringhasalogarithmicallydi- Jµ 2 vergent energy per unit of length. We can think that The Lagrangian L contains the fermionic Kalb- these stringscouldbe ignoredbecausethey appearto be FKR Ramond couplings and reads as below: unphysical. However, following cosmologicalphase transitions, global strings may form loops with finite total energy or open strings with finite energy per horizon. g L = χ¯σµG˜ χ. (52) An interesting application that some authors have been FKR 2 µ envisagingisthepossibilitythatradiativedecayofclosed This Lagrangian (52) amounts to a mass contribution loopsbeconnectedwithdensityfluctuationintheprocess givenbytheLorentz-breakingparameterpresentin(13), of structure formation. This approach, considering the that is, databasis,hasbeenruledoutalone,buttogetherwithin- flationarymodels andconsideringthe noise ofthe exper- imental data, we can still consider them [17]. In the ap- √ρg proachof[33],thatisconsideredhere,thevortexconfigu- LmFKasRs = 2 χ¯χ. (53) rationisstable inthe presenceofthe specialbackground thatbreakstheLorentzinvariance[33]. Thefactthatthe Lint is the interacting Lagrangian, where we include superfluid vortex is immersed in a Lorentz-noninvariant the Yukawa terms which induce masses to the fermions fluidsuggeststhatthecorrectmodelforasuperfluidvor- that couple to the vortex. These Yukawa couplings are texinvolvesthechoiceofaspecialbackground. Therela- collected in: tivistic force law for the responseof a vortexto the local field Gµνρ is analogous to the Lorentz force law in Elec- trodynamics. Theexternalforceduethebackgroundfield Y =g 2φζaχa+2φ∗χ¯aζ¯a+Sχaχa+S∗χ¯aχ¯a . (54) (cid:16) (cid:17) 7 From (54) and (53), there follows an interesting pos- is that the field φ develops an expectation value of mag- sibility. If we choose S = 0 to be zero in the core of nitude (58). The evolution of the phases α of φ and h i the string, the mass (53) does not appear in the core, Λ of S with the temperature is not determined only by the fermionic interaction term vanishes in the core and local physics; their values outside depends on random the fermions and χ and ξ become massless. In this case, fluctuations and α and Λ take different values in differ- where the fermions is not have mass, the fermionic zero- ent regions of space during the evolution. But, since the modespropagatewiththespeedoflightinthez-direction free energyis minimized, these phasesafter the Universe and particles can be ejected from the vortex. Outside expansion can become precedent sections with the tem- the vortex, these particles have masses induced by the perature T = 0. We can define the correlation length, φ-interaction Yukawa term and by (53), as induced by Π(t), to be the length scale above which the values of α the Lorentz-symmetry breaking [35]. and Λ are uncorrelated. The evolution of Π(t) depends on details of the relaxation processes. Indeed, Π(t) has to satisfy the causality bound. The correlation length V. SECOND-ORDER PHASE TRANSITIONS cannot establish scales greater than the causal horizons AND THE RELATION OF THE S-FIELD WITH relatedwiththedistancetravelledbythelightduringthe THE TEMPERATURE life-time of the Universe. For T <T , the scalarfield de- c velopsanexpectationvalue correspondingto some point In this Section, let us study a physical interpretation in the manifold of the minima of the effective poten- M of the field S. Up to now, we know that the field S is tialV. Wecanseein(58)thatthetermthatinourmodel importantforthezero-modes. Now,letusstudyanother is givenbythe softSUSYbreaking,waspresentedinthe interpretation, possibly related with second-order phase high temperature state, but, in the case T Tc, SUSY ≫ transitions. The potential (29) can represent a high- breaking can be neglected, and we can consider the Uni- temperature effective potential[28], that can be written verseasbeinginasupersymmetricphase. Tounderstand as this fact,weneedStringTheoryarguments,notcontem- platedinthiswork. Thisanalysisonlygivesusknowledge aboutthe vortexformation,and it is notable to provide V(φ,T)=m2(T)φ2+h2 φ4 (55) us with information on the Lorentz breaking. But, if we | | | | considerthat,whenthe temperaturebecomeslowvortex where we identify S with the temperature, T. We actu- formation may take place, then Lorentz symmetry may ally consider S 2 =T2, then be violated and there occurs a vortex-superfluid forma- | | tion, as we have analysed throughout this paper. m2(T)=h2(4T2 η2). (56) − VI. GENERAL CONCLUSIONS The term m(T) is the mass for the φ-field, whenever the state is symmetric, φ = 0. This mass vanishes h| |i Inthiswork,wehaveshownthatitispossibletobuild when T =T , and c upastringvortexbymodellingthevortexsuperfluidina supersymmetriccontext. Wehaveanalyzedthepotential η T = . (57) that gives us the correct string vortex configuration, in c 2 zero temperature, it presents a soft SUSY-breaking induced by the hidden sector. We have also analyzed the Another important case occurs for T > T ; the effec- c bosonic aspects of the duality representation of the vor- tive mass m2(T) is positive and the minimum of V is at tex. We also analysed the physical interpretation of the φ = 0. The physical interpretation of this result is that extrafieldS relatedtothepresenceofthezero-modeand theexpectationvalueofφvanishes. Thismeansthatthe we show that we can relate it to the temperature. It is symmetryisrestoredathightemperature. The symmet- advisable to comment here that the violation of Lorentz ric vacuum becomes unstable and φ develops a non-zero symmetry introduced is independent of the soft SUSY expectation value. Minimizing V, as in (55), we obtain, explicit breaking terms. The latter has been considered for T <T , c to be correct taking into account stability aspects of the potential. It would be very interesting to eventually un- 1/2 derstand if the violation of Lorentz symmetry and the |φ|=√2(cid:16)Tc2−T2(cid:17) (58) explicit SUSY breaking could be related to one another. This wouldrender ourproposalmore interesting,inthat Animportantrealizationofthesecondphasetransition wewouldbedealingwithlessarbitraryparameters. Also, is the fact that φ grows continuously from zero, as the it wouldclarify the interplay betweenLorentz-symmetry | | temperature decreasesfromthe criticaltemperature, T . violation (in the sense of particle transformations) and c The cosmological point of view, when the supersym- SUSY explicit breaking that describes mass splittings metric Universe cools through the critical temperature, among bosons and fermions that belong to the same su- 8 permultiplet. This is an issue that could be investigated the parameters of the model. Then, the possibility of better in a future work. The interesting phenomenolog- the Lorentz-symmetry breaking in supersymmetric mat- ical aspect of this discussion would be checking whether terbecomesrelevantforthedarkmatterstabilityaround properties like the masses of the SUSY particles, suchas the stars [25] and particles can then be ejected out of the photino andthe higgsino,wouldnecessarilysignalto these astrophysical structures. In this work, we do not some type of Lorentz-symmetry breaking. In section V, haveanapplicationfor these objects,butwe understand wehaveproposedadiscussioninacosmologicalevolution thatourmodelcanbeanalternativepossibilitytounder- context, but, as already pointed out, these vortices also standsomephenomenainvolvinghighenergies. Thenext appear in star cores. In the case of the neutron stars, as stepisstudyingthefermionicimplicationoftheLorentz- presentedinthe Introduction,the forcethatdictatesthe symmetry breaking Kalb-Ramond background and how vortex stability is induced by the nuclear matter; but, we could find out a mechanismto justify its appearance, we do not eliminate the possibility of the presence of a therelationbetweenLorentzandSUSYbreakingandthe Lorentz-symmetrybreakingtohaveanimportantrolein- origin of the hidden sector represented by soft breaking side the star. Nothing guarantees that the matter inside of global SUSY. the star has a Lorentz-invariant behavior, because the Acknowledgments: highenergyenvolves,inanalogywith highenergyγ-rays fromextragalaticsources[36]. Anotherpointisifwecon- W. Bietenholz is acknowledged for a critical read- siderthat darkmatter ismostly composedby supersym- ing and for many helpful suggestions on an original metric particles, the relation between the Lorentz and manuscript. 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