An Investigation of the K F -type Lorentz-Symmetry Breaking Gauge Models in N=1 -Supersymmetric Scenario
H. Belich, G. S. Dias, J.A. Helayël-Neto, F.J.L. Leal, W. Spalenza
aa r X i v : . [ h e p - t h ] S e p An Investigation of the K F -type Lorentz-Symmetry Breaking Gauge Models in N = 1 -Supersymmetric Scenario. H. Belich a,d,e , G. S. Dias b,f , J.A. Helay¨el-Neto c,d , F.J.L. Leal b,c , W. Spalenza g a Universidade Federal do Esp´ırito Santo (UFES), Departamento de F´ısica e Qu´ımica,Av. Fernando Ferrari S/N, Vit´oria, ES, CEP 29060-900, Brasil, b Instituto Federal do Esp´ırito Santo (IFES), Coordenadoria de F´ısica,Av. Vit´oria 1729, Jucutuquara, Vit´oria - ES, 29040-780, CEP 29040-780, Brasil, c CBPF - Centro Brasileiro de Pesquisas F´ısicas,Rua Xavier Sigaud 150, Rio de Janeiro, RJ, CEP 22290-180, Brasil, d Grupo de F´ısica Te´orica Jos´e Leite Lopes, C.P. 91933, CEP 25685-970, Petr´opolis, RJ, Brasil, e International Institute of Physics, Universidade Federal de Rio Grande do Norte,av. Odilon Gomes de Lima 1722, CEP 59078-400, Natal-RN, Brasil, f Department of Physics, University of Alberta, Edmonton, Alberta, Canada T6G 2J1, and g Instituto Federal do Esp´ırito Santo (Ifes) - Campus Cariacica Rod. Gov. Jos´e Sette s/n ,Cariacica, ES, Brasil, cep 29150-410, Vit´oria, ES, Brasil. ∗ In this work, we present two possible venues to accomodate the K F -type Lorentz-symmetryviolating Electrodynamics in an N = 1-supersymmetric framework. A chiral and a vector superfieldare chosen to describe the background that signals Lorentz-symmetry breaking. In each case, the K µνκλ -tensor is expressed in terms of the components of the background superfield that we chooseto describe the breaking. We also present in detail the actions with all fermionic partners of thebackground that determine K µνκλ . PACS numbers: 11.30.Cp, 12.60.-i.
I. INTRODUCTION
The Standard Model Extension (SME) is the natural framework to investigate properties of Lorentz-violation inphysical systems involving possible extensions of Higgs mechanism. The SME is by-now a very estimulating researcharea and a great deal of results have been attained which include bounds on the photon mass [1], radiative corrections[2], systems with fermions [3], neutrinos [4], topological defects [5], topological phases [6], cosmic rays [7], particledecays [8], and other relevant aspects [9], [10]. The SME has also been used as a set-up to propose Lorentz-symmetryviolation [11] and CPT- probing experiments [12], which have amounted to the imposition of stringent bounds onthe Lorentz-symmetry violating (LV) coefficients.To take into account how this violation is implemented, in the fermion sector of the SME, for example, there aretwo CPT-odd terms, v µ ψγ µ ψ, b µ ψγ γ µ ψ , where v µ , b µ are the LV backgrounds [13], [14]. A similar study has alsobeen developed for the case of a non-minimal coupling with the background, with new outcomes [15]. Atomic andoptical physics are other areas in which Lorentz-symmetry violation has been intensively studied. Indeed, there areseveral works examining Lorentz-violation in electromagnetic cavities and optical systems [16], [17], which contributedto establish upper bounds on the LV coefficients.The appearance of a more complex background, with the existence of an anisotropic vacuum at Planck scale hasdrawn the attention of particle and field theorists in recent years. Taking strings as the fundamental entities at thislevel, the idea that Lorentz-symmetry and CPT invariance might be spontaneously broken in some string theoriesbecame highlly estimulating [18, 21]. Very recently, the surprising result that CPT-violating neutrino mass-squareddifference would be an order of magnitude less than the current uper bound on CPT-violation in the sector of quarksand charged leptons has come to give more support to the proposal of Lorentz-symmetry violation [19], [20]. Actually, ∗ Electronic address: [email protected], [email protected], [email protected], [email protected], [email protected] the idea of a CPT violation in the neutrino sector had been contemplated by Colladay, Kostelecky and Mewes [21].Our main motivation to consider a supersymmetric scenario in connection with the violation of Lorentz-symmetryis based on fact that we adopt the viewpoint, according to [18], that the breaking of Lorentz-symmetry may betriggered by the vacuum condensation of a given tensor field. Nevertheless, at this scale where Lorent-symmetrytakes place, we may still have supersymmetry (SUSY) or, even if SUSY had already been broken down by somemechanism, supersymmetric partners are present and SUSY imprints are not lost. So, by adopting this scenario, weargue that Lorentz-symmetry violation, as originated at a more fundamental level in the string framework, cannotbe dissociated from SUSY. We do not have however elements to decide whether SUSY breaking has occurred beforeLorentz-symmetry violation, though the breaking of the latter signals the violation of the former. However, we wouldlike to point out that, in the work of Ref. [32], Katz and Shadmi, present an interesting discussion in the rˆole ofvacuum expectation values that violate Lorentz symmetry as a possible to realise SUSY breaking by F - and D - termsin the MSSM.To implement effective Lorentz-symmetry-violating actions in supersymmetric models, one may adopt a superspaceformulation with Lorentz and, in some cases, CPT invariances violated by a fixed background in Wess-Zumino-likemodels [22]. In the gauge sector SUSY-preserving theories where discussed in [23]. Also in the gauge sector, theproposed [25] establishes a supersymmetric minimal extension for the Chern-Simons-like term [24] preserving theusual (1 + 3)-dimensional SUSY algebra. The breaking of SUSY will follow the very same route to Lorentz-symmetrybreaking: the statement that v µ is a constant shows thar SUSY is also broken by the fermionic SUSY partner of v µ .Besides the CPT-odd terms, in the gauge sector, we have the CPT-even part, which is represented by a tensor K µναβ that presents the same symmetries of the Riemann tensor, as well as an additional double-traceless condition[26]. In this scenario, we present two possibilities of constructing a supersymmetric version for the K -type models.So, we propose to carry out the supersymmetric extension to the bosonic action below: S = − Z d x K µνκλ F µν F κλ . (1)The “tensor” K µνκλ it’s CPT even, i. e., it does not violate the CPT-symmetry. Despite CPT violation impliesviolation of Lorentz invariance [30], the reverse is not necesarily true. The action above is Lorentz-violanting inthe sense that the “tensor” K µνκλ has a non-zero vacuum expectation value. That “tensor” presents the followingsymmetries: K µνκλ = K [ µν ][ κλ ] , K µνκλ = K κλµν , K µνµν = 0 , (2)as usually appears in the literature, we can reduce the degrees of freedom take into account the ansatz [26]: K µνκλ = 12 ( η µκ ˜ κ νλ − η µλ ˜ κ νκ + η νλ ˜ κ µκ − η νκ ˜ κ µλ ) , (3)˜ κ µν = κ ( ξ µ ξ ν − η µν ξ α ξ α / , (4) κ = 43 ˜ κ µν ξ µ ξ ν , (5)where ˜ κ µν is a traceless “tensor”. Using the ans¨atze (3), (4), in expression (1), we obtain, S = κ Z d x (cid:26) ξ µ ξ ν F µκ F κν + 18 ξ λ ξ λ F µν F µν (cid:27) . (6)The supersymmetrization of the action (6), instead of (1), does not yield higher-spin components in the backgrounddictated by SUSY. To realize the supersymmetrization ,we have two possible routs: to achieve supersymmetrisationof (1) with K µνκλ given by (3), without the particular expression (4) for ˜ κ µν , we get that the background superfieldmust be chiral. On the other hand, if we are to supersymetrise (1) with K µνκλ given by (3) and ˜ κ µν given by (4), weconclude that a vector superfield must be used to accommodate the background vector, ξ λ . The content of partnersis richer in this case than in the case of the chiral background. The first route accommodate the chiral superfield as θσ µ ¯ θ∂ µ S , while the second, is carried out by a vector superfield accommodate this “vector” background as θσ µ ¯ θξ µ .This work is organized as follows: In Section 2, we start with the model of Lorentz breaking proposed as K µνκλ by Kostelek´y. In this Section, we first study the possibility that the background tensor, K µνκλ , be originated from abackground chiral superfield, which imposes a special form - not yet discussed in the literature - for the K − tensor.Then in Section 3, we proceed by investigating the possibility that K µνκλ originates from a (real) vector superfield,which exactly reproduces (1) with the K − tensor defined by means of (3), and (4). Finally, some Concluding Remarksare stated in Section 4. II. FIRST PROPOSAL: LORENTZ-SYMMETRY BREAKING BY A CHIRAL SUPERMULTIPLET.
Based on the work of Ref. [25], we take the idea that the background vector could originate from a chiralsupermultiplet, Ω. As we shall see, this imposes on ξ µ the constraint ξ µ = ∂ µ S , where S is a complex scalar. Indeed,as it will become clear at the end of calculations, this choice of SUSY supermultiplet yields an interesting form for K µνκλ , completely fixed by a complex scalar field. We then show that the superspace action below shall accomplishthe task of yielding the component field extension of the action of eq. (6).Adopting covariant superspace-superfield formulation, we propose the following minimal extension for: S = κ Z d xd θd ¯ θ n ( D α Ω) W α ( D ˙ α Ω) W ˙ α + h.c. o . (7)The superfields W a , V , Ω and the SUSY-covariant derivatives D a , D ˙ a hold the definitions: D a = ∂∂θ a + iσ µa ˙ a ¯ θ ˙ a ∂ µ , (8) D ˙ a = − ∂∂ ¯ θ ˙ a − iθ a σ µa ˙ a ∂ µ ; (9)from D ˙ b W a (cid:0) x, θ, ¯ θ (cid:1) = 0, and D a W a (cid:0) x, θ, ¯ θ (cid:1) = D ˙ a W ˙ a (cid:0) x, θ, ¯ θ (cid:1) , it follows that W a ( x, θ, ¯ θ ) = − D D a V. (10)Its θ -expansion reads as below: W a ( x, θ, ¯ θ ) = λ a ( x ) + iθ b σ µb ˙ a ¯ θ ˙ a ∂ µ λ a ( x ) −
14 ¯ θ θ (cid:3) λ a ( x ) ++ 2 θ a D ( x ) − iθ ¯ θ ˙ a σ µa ˙ a ∂ µ D ( x ) + σ µν ab θ b F µν ( x ) − i σ µν ab σ αb ˙ a θ ¯ θ ˙ a ∂ α F µν ( x ) − iσ µa ˙ a ∂ µ ¯ λ ˙ a ( x ) θ , (11)and V = V † . The Wess-Zumino gauge choice is taken as usually done: V W Z = θσ µ ¯ θA µ ( x ) + θ ¯ θλ ( x ) + ¯ θ θλ ( x ) + θ ¯ θ D, (12)so the action (7) is gauge-invariant. The background superfield is so chosen to be a chiral one. Such a constraintrestricts the maximum spin component of the background to be an s = component-field, showing up as a SUSY-partner for a spinless dimensionless scalar field. Also, one should notice that Ω has dimension of mass to −
1. Thesuperfield expansion for Ω then reads: D ˙ a Ω (cid:0) x, θ, ¯ θ (cid:1) = 0 , consequently Ω (cid:0) x, θ, ¯ θ (cid:1) = S ( x ) + √ θζ ( x ) + iθσ µ ¯ θ∂ µ S ( x ) ++ θ G ( x ) + i √ θ ¯ θ ¯ σ µ ∂ µ ζ ( x ) −
14 ¯ θ θ (cid:3) S ( x ) . (13)With its SUSY transformations given by δS = √ ε α ζ α ,δζ α = √ Gε α + i √ σ µα ˙ α ¯ ε ˙ α ∂ µ S,δG = i √ ε ˙ α ¯ σ µ ˙ αα ∂ µ ζ α . (14)Notice that, if we wish to have ∂ µ S constant (as we shall get in the sequel, a constant ∂ µ S give us a constant K µνκλ ), so that S depends lenearly on x µ , SUSY is also broken by the backgroud, as the expression for δζ shows:if we apply a SUSY transformation on an Ω-superfield for which S = 0, ζ = 0, G = 0, then δζ = 0 whenever ∂ µ S = 0. So, to realise the breaking of Lorentz-symmetry in terms of an Ω with a constant ∂ µ S , then SUSY is notan invariance of the background Ω. However, as it stands, we have an effective model which may descend from amore fundamental theory in which SUSY might be spontaneously broken. At this stage, we have no commitmentwith any specific mechanism for SUSY breaking.The component-wise counterpart for the action (7) is as follows : S = κ Z d x d θ d ¯ θ n ( D α Ω) W α ( D ˙ α Ω) W ˙ α + h.c. o = S boson + S fermion + S coupled , (15) S boson = κ Z d x ( − (cid:20) ∂ λ S∂ µ S ∗ ( F µκ F κλ + F λκ F κµ ) + 18 ∂ λ S∂ λ S ∗ F µν F µν (cid:21) ++ 2 Dε λµτρ ∂ λ S∂ µ S ∗ F τρ − i∂ λ S∂ µ S ∗ ε λτρκ F τρ F µκ + 8 iD∂ λ S∂ µ S ∗ F λµ + − D∂ λ S∂ µ S ∗ ε λµνκ F νκ + 8 D ∂ µ S∂ µ S ∗ + 16 D | G | + h.c. ) , (16) S fermion = κ Z d x ( ∂ λ ζσ µ ∂ µ ¯ ζλσ λ ¯ λ + 12 ∂ λ ζσ µ ¯ λλσ λ ∂ µ ¯ ζ + 2 ∂ µ ζ∂ µ λ ¯ ζ ¯ λ + − ∂ λ ζσ λ ∂ µ ¯ ζλσ µ ¯ λ − ∂ λ ζσ λ ¯ σ µ ∂ µ λ ¯ ζ ¯ λ − λσ λ ¯ σ µ ∂ λ ζ ¯ ζ∂ µ ¯ λ + − ζλ ¯ ζ (cid:3) ¯ λ − ζλ∂ µ ¯ ζ ¯ σ µ σ τ ∂ τ ¯ λ + 12 ζλ∂ µ ¯ λ ¯ σ ν σ µ ∂ ν ¯ ζ ++ 12 ∂ µ ζσ µ ¯ σ ν λ ¯ ζ∂ ν ¯ λ − √ ζ (cid:3) λ ¯ ζ ¯ λ − ζ∂ ν λ∂ µ ¯ λ ¯ σ µ σ ν ¯ λ + − ζ∂ ν λ∂ µ ¯ ζ ¯ σ ν σ µ ¯ λ + ζ∂ µ λ ¯ ζ∂ µ ¯ λ − ζσ µ ∂ µ ¯ λ ¯ ζ ¯ σ ν ∂ ν λ + h.c. ) , (17) the ref. [31] has been used in our calculations S coupled = κ Z d x ( − iDζσ µ ∂ µ ¯ ζ − √ iDG ∗ ζσ µ ∂ µ ¯ λ + 2 √ D∂ ν λσ ν ¯ σ µ ζ∂ µ S ∗ ++ 2 Dζσ ν ∂ µ ¯ ζF ν µ + iDε τρµα ζσ α ∂ µ ¯ ζF τρ + √ G ∗ ζσ µ ∂ ν ¯ λF µν ++ i √ G ∗ ε τρµα ζσ α ∂ µ ¯ λF τρ + √ iζσ τ ¯ σ ν ∂ ν λ∂ µ S ∗ F τ µ + − √ ε τρµα ζσ α ¯ σ ν ∂ µ S ∗ ∂ ν λF τρ − √ iG ∗ Dζσ µ ∂ µ ¯ λ ++ 2 √ Dζ∂ µ λ∂ µ S ∗ − i √ ε µνκτ ζ∂ τ λ∂ µ S ∗ F νκ + 12 √ ε µνκτ ζ∂ τ λ∂ µ S ∗ F νκ + − iD ¯ ζ ¯ σ µ ∂ µ ζ − D ¯ ζ ¯ σ ν ∂ µ ζF ν µ + iDε νκµα ¯ ζ ¯ σ α ∂ µ ζF νκ ++ 2 √ iDG ∗ ∂ µ ζσ µ ¯ λ + 2 D∂ µ ζσ τ ¯ ζF τ µ + iDε τρµα ∂ µ ζσ α ¯ ζF τρ ++ 2 ∂ µ ζσ τ ¯ σ νκ ¯ ζF νκ F τ µ + iε τρµα ∂ µ ζσ α ¯ σ νκ ¯ ζF τρ F νκ + √ ∂ µ ζσ τ ¯ λF τ µ ++ i √ ε τρµα ∂ µ ζσ α ¯ λF τρ − √ iGDλσ µ ∂ µ ¯ ζ + 2 i | G | λσ µ ∂ µ ¯ λ + − i∂ ν λσ ν ¯ σ µ λ∂ µ S ∗ + 4 √ iGD∂ µ λσ µ ¯ ζ − √ iGD ¯ ζ ¯ σ µ ∂ µ λ ¯ ζ + − √ G ¯ ζ ¯ σ µ ∂ τ λF µτ + i √ Gε µντα ¯ ζ ¯ σ α ∂ τ λF µν − i | G | ¯ λ ¯ σ µ ∂ µ λ ++ 2 √ D∂ µ S ¯ ζ∂ µ ¯ λ + √ i∂ µ (cid:0) ¯ λ ¯ ζ (cid:1) ∂ λ SF λµ − √ ε µλτρ ∂ µ (cid:0) ¯ λ ¯ ζ (cid:1) ∂ λ SF τρ + − √ D ¯ λ∂ µ ¯ ζ∂ µ S + 2 √ D ¯ ζ ¯ σ µ σ ν ∂ ν ¯ λ∂ µ S − √ i ¯ ζ ¯ σ ν σ µ ∂ µ ¯ λF νλ ∂ λ S + − √ ε νκλα ¯ ζ ¯ σ α σ µ ∂ µ ¯ λF νκ ∂ λ S + 2 G ∗ ¯ λ ¯ σ µ σ ν ∂ ν ¯ λ∂ µ S − √ ∂ µ S∂ µ D ¯ ζ ¯ λ ++ √ D∂ ν ζσ ν ¯ σ µ λ∂ µ S ∗ − i √ σ λ ∂ λ ζλσ ν ∂ µ S ∗ F ν µ + 12 √ ε µνκα λσ α ¯ σ λ ∂ λ ζ∂ µ S ∗ F νκ + − ∂ λ ζσ λ ¯ λλσ µ ∂ µ ¯ ζ + √ Dλσ λ ¯ σ µ ∂ λ ζ∂ µ S ∗ − i √ ∂ λ ζσ ν ¯ σ λ λ∂ µ S ∗ F ν µ + − √ ε µνκα ∂ λ ζσ α ¯ σ λ λ∂ µ S ∗ F νκ + h.c. ) . (18)We notice that, in trying to supersymmetrise the ξ µ ξ ν F µκ F κν -term, we automatically get the supersymmetrisation ofthe term ξ λ ξ λ F . This is not a simple coincidence, but it can naturally be expected from an analysis of the irreduciblerepresentations of SO (1 , K µνκλ given by [26]: K µνκλ = −
16 ( η kµ ˜ κ νλ − η µλ ˜ κ νκ − η νλ ˜ κ µκ − η kν ˜ κ µλ ) , (19)with, ˜ κ µν = κ (cid:26)(cid:18) ∂ µ S∂ ν S ∗ + ∂ ν S∂ µ S ∗ (cid:19) − ∂ λ S∂ λ S ∗ η µν / (cid:27) , (20)and K µνµν = 0 . (21)Concluding, this special form for ˜ κ is a natural consequence of the assumption that a chiral superfield carries thebackground that breaks Lorentz-symmetry.It is worthy to notice that ˜ κ , and consequently K , depend exclusively on the scalar component S . No ζ -condensateand no G -dependence appear in the expression for the K -tensor. Since the action (15) is quadratic in Ω, one might inprinciple expect that tensor bilinears in ζ could show up as contributions to K µνκλ . However, the K -tensor dependsexclusively on the gradient of S . It becomes clear that a constant ∂ µ S ensures the constancy of K µνκλ , as we hadalready anticipated. In a particular case, where the background fields ∂ µ S = 0, ζ = 0 , and G = 0, we have that theauxiliar field, D , is given by ( we suppose the supersymmetric version of the F µν -term is added up), D = − κi∂ µ S∂ ν S ∗ F µν
16 + 32 κ∂ µ S∂ µ S ∗ . (22)It is interesting to comment that, by virtue of (22), which is valid in the conditions above for the background, thebreaking of Lorentz-symmetry, fixes the auxiliary field to be non-trivial, even, if the gauge potential superfield ( V ) isnot coupled to matter. The backgound, as (22) shows, determines D by the gauge field strenght, F µν . However, ifsupersymmetric matter happens to be minimally coupled to the gauge field, then (22) indicates that charged scalarparticles (selectrons, for example) may acquire a magnetic dipole moment given in terms of the vector ~µ = ~v × ~v ∗ ,where ~v ≡ ~ ∇ S . This investigation is being pursued and we shall soon report on it [29]. III. SECOND PROPOSAL: LORENTZ-SYMMETRY BREAKING FROM A VECTORSUPERMULTIPLET.
Adopting covariant superspace-superfield formulation, we propose the following minimal extension of (6) for: S = κ Z d xd θd ¯ θ n ( D α Ξ) W α ( D ˙ α Ξ) W ˙ α + h.c. o . (23) W α is the superfield strenght of the gauge supermultiplet V W Z , as given in eq. (1). Ξ is the so-called vectorsuperfield, whose θ − expansion is as follows:Ξ (cid:0) θ, ¯ θ (cid:1) = C + θχ + ¯ θ ¯ χ + θ M + ¯ θ M ∗ + θσ µ ¯ θξ µ + θ ¯ θ ¯ ψ + ¯ θ θψ + θ ¯ θ B. (24)In the special case of constant background component fields, the SUSY transformations simplify and acquire theform below: δC = ε α χ α + ¯ ε ˙ α ¯ χ ˙ α ,δχ α = 2 M ε α + σ µα ˙ α ¯ ε ˙ α ξ µ ,δ ¯ χ ˙ α = − ε α σ µα ˙ β ε ˙ β ˙ a ξ µ + 2 M ∗ ¯ ε ˙ α ,δM = ¯ ε ˙ α ¯ ψ ˙ α ,δM ∗ = ε α χ α ,δξ µ = ε α σ µα ˙ α ¯ ψ ˙ α − ¯ ε ˙ α ¯ σ µ ˙ αα ψ α ,δ ¯ ψ ˙ α = 2¯ ε ˙ α B,δψ β = 2 ε β B,δB = 0 . (25)In the general case, B transforms as a total derivative; only for a constant ψ -background, we get δB = 0 . Notice thatwe are taking the full θ -expansion for Ξ . Nothing like a Wess-Zumino gauge can be taken, for Ξ is a fixed backgroundand is not a gauge super-potential.The superaction in component fields is given by, S = κ Z d x d θ d ¯ θ n ( D α Ξ) W α ( D ˙ α Ξ) W ˙ α + h.c. o = S boson + S fermion + S coupled ; (26) S boson = κ Z d x ( (cid:18) ξ λ ξ µ F µκ F κλ + 18 ξ λ ξ λ F µν F µν (cid:19) ++ i ξ λ ξ µ ε λτρκ F τρ F µκ − D ξ λ ξ λ + 8 | M | D + h.c. ) , (27)where we can readily identify the action (1) with K µνκλ given as in (3), and (4). S fermion = κ Z d x ( − iψσ µ ∂ µ ¯ λ ¯ χ ¯ λ + ψλ ¯ ψ ¯ λ + i ∂ µ λσ µ ¯ ψ ¯ χ ¯ λ + ψλ∂ µ λσ µ ¯ χ + − i λσ µ ¯ ψ ¯ χ∂ µ ¯ λ + λ ¯ ψ ¯ λ + iχλ∂ µ λσ µ ψ − iχλψσ µ ∂ µ ¯ λ + h.c. ) , (28) S coupled = κ Z d x ( − iχλξ µ ∂ µ D − χλξ µ ∂ ν F νµ + − i χλε ψνκµ ξ µ ∂ ψ F νκ + 4 χλBD + Dχσ µ ¯ χ∂ ν F νµ ++ i Dε ψνκα χσ α ¯ χ∂ ψ F νκ − iDM ∗ χσ µ ∂ µ ¯ λ + iD∂ ν λσ ν ¯ σ µ χξ µ + − D χψ − i χσ τ ¯ χF τρ ∂ ν F νρ + 14 ε τρκα χσ α ¯ χF τρ ∂ ν F νκ ++ 14 ε λνκρ χσ τ ¯ χF τρ ∂ λ F νκ + i ε ψνκα ε τραβ χσ β ¯ χF τρ ∂ ψ F νκ ++ M ∗ χσ τ ∂ ν ¯ λF τ ν + i M ∗ ε τρµα χσ α ∂ µ ¯ λF τρ + 12 χσ τ ¯ σ ν ∂ ν λξ µ F τ µ ++ i ε τρµα σ α ¯ σ ν ∂ ν λξ µ F τρ − Dχσ τρ λF ρτ + 12 χ∂ µ λ ¯ χ∂ µ ¯ λ ++ iχ∂ µ λξ µ − ¯ χ ¯ σ ν χ∂ µ DF ν µ + i ε νκµα ¯ χ ¯ σ α χ∂ µ DF νκ ++ iM D∂ µ λσ µ ¯ χ + i M λσ κ ¯ χ∂ µ F µκ − M ε µνκα λσ α ¯ χ∂ µ F νκ + − i | M | λσ µ ∂ µ ¯ λ + iM ∂ ν λσ ν ¯ σ µ λξ µ − M Dλ + −
12 ¯ χ ¯ σ ν ∂ µ λF ν µ + i ε νκµα ¯ χ ¯ σ α ∂ µ λF νκ + i M ∗ ∂ µ λσ µ ¯ λ ++ iλσ µ ∂ ν ¯ λξ ν ξ µ − i λσ µ ∂ µ ¯ λξ λ ξ λ − ε λνµα λσ α ∂ ν ¯ λξ λ ξ µ ++ 12 ε νκλα λσ α ¯ ψξ λ F νκ − Bλσ µ ¯ λξ µ + ¯ χ∂ ν ¯ λξ µ F µν ++ i ε τρνλ ¯ χ∂ ν ¯ λξ λ F τρ − i λ ¯ σ ρ λξ λ F λρ − ε λτρα ¯ λ ¯ σ α λξ λ F τρ + − iD∂ µ ¯ λ ¯ σ µ σ λ ¯ χξ λ + 12 ∂ µ ¯ λ ¯ σ µ σ κ ¯ χξ λ F λκ + i ε λνκα ∂ µ ¯ λ ¯ σ µ σ α ¯ χξ λ F νκ + − iM ∗ ∂ µ ¯ λ ¯ σ µ σ λ ¯ λξ λ −
12 ¯ χ ¯ λξ λ ∂ ν F λν − i χ ¯ λε τρνλ ξ λ ∂ ν F τρ + − D ¯ χ ¯ ψ + D ¯ χ ¯ σ νκ ¯ ψF νκ − M ∗ D ¯ ψ ¯ λ − i ψσ τ ¯ λξ µ F τ µ ++ 14 ε τρµα ψσ α ¯ λξ µ F τρ − Bλσ µ ¯ λξ µ + 4 BD ¯ χ ¯ λ + h.c. ) . (29)The terms in the superactions (15) and (26) have the same form with the factor ( D α Ω) W α replaced by ( D α Ξ) W α .Then, we obtain the right combination, i. e., ξ µ ξ ν F µκ F κν + ξ λ ξ λ F µν F µν . To get the term, ξ µ ξ ν F µκ F κν , we couldalso propose the action below: S = κ Z d xd θd ¯ θ (cid:8) ( D α Ξ) ( D α Ξ) W β W β + h.c. (cid:9) . (30)Also in this case, when we supersymmetrise the ξ µ ξ ν F µκ F κν -term, automatically comes out ξ λ ξ λ F . So, we have asecond way to build up the supersymmetric extension of the term (cid:0) ξ µ ξ ν F µκ F κν + ξ λ ξ λ F µν F µν (cid:1) . We just mentionthis possible second way of working out the supersymmetrisation of (6) by a vector superfield background, but we donot exploit here this second possibility. This is why we do not project the superfield action (30) into components.Though we formulate our model in terms of superspace and superfields, the tensor calculus of supersymmetry, wewould like to point out that with a Lorentz-symmetry breaking, in the chiral case given by ∂ µ S = 0, in the vectorcase parametrized by ξ λ = 0, SUSY is readily seen to be broken by the background. But, as it was emphasized, wedo not have elements to decide whether SUSY breaking has occurred before Lorentz-symmetry violation, though thebreaking of the latter signals the violation of the former. IV. CONCLUDING COMMENTS
Our effort in this paper has mainly consisted in finding out possible N = 1 supersymmetric scenarios for the K − tensor- realized Lorentz-symmetry breaking considered in the literature ( see the Refs. quoted throughout the presentwork).We propose two viable descriptions. The first approach is based on a chiral superfield that accommodates thecontent of the background responsible for the breaking of Lorentz-symmetry. In this case, a complex scalar is thesource for K µνκλ . The constancy of the K − tensor is ensured by the linear dependence of S on the x µ − coordinates.An interesting question to be investigated is the analysis of the photon - photino mass splitting in terms of thebackground field components, specially if the fermion component ζ is a non-vanishing constant background. From theexpression for S coupled in eq. (18), there result interesting terms that mix the gauge superfield components ( A µ , λ, and D ) in bilinear terms where the (constant) background fields are also present, as mass parameters. The task of gettingthe < A µ A ν > − < A µ λ > − < λλ > − < λD > − and < DD > − propagators demands special technicalities (Fierzings and spin–projection operators) and, once these tree level 2 − point functions are worked out, we can read offthe propagator poles and discuss in terms of the background components ( S, ζ, and G ). The rˆole of the backgroundfermionic condensates become clear after the propagator poles are identified.This discussion holds through also in the cases of a vector-superfield background that carries ξ µ , from which K µνκλ is expressed. This case exhibits a richer background and fermion condensates that mix χ and Ψ ( see eq. (29)), inaddition to the χ − and Ψ − condensates become important.So, to our mind, in either case, the important question that our study may raise concerns the influence of theLorentz-symmetry violating background on the spectrum and which restrictions it should have so as to avoid theappearance of non-physical excitations, such as tachyons and ghosts.Finally, a non-trivial question that remains to be addressed to is the relation between SUSY and Lorentz-symmetrybreakings. We treat the latter in a supersymmetric formulation and, as we have previously commented, our back-grounds (both Ω and Ξ) are not invariant under SUSY. They simply express the fact that Lorentz-symmetry breakingdoes not support exact SUSY. However, by no means, we are stating that we break Lorentz-symmetry and SUSY atthe same time. SUSY could have been broken before, by some more fundamental mechanism, and Lorentz-symmetrybreaking takes place in an environment which still keeps the inheritance of SUSY through the whole set of supersym-metric partners. This is why approach the deviation from Lorentz-symmetry in connection with SUSY. However, itwould be an interesting task- and we shall soon concentrate on this point - to build up a model such that, when-ever SUSY is spontaneously broken, some tensor field (that belongs to some supermultiplet of the model coupled tothe superfield responsible for the breaking of SUSY) also acquires a non-trivial vacuum expectation value and theLorentz group simultaneously undergoes spontaneous symmetry breakdown. This sets out another possibility, witha contemporary breaking of SUSY and Lorentz-symmetry. We are considering this situation and we shall report onit elsewhere. V. ACKNOWLEDGMENTS
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