Research ArticleA Chargeless Complex Vector Matter Field inSupersymmetric Scenario

L P Colatto1 and A L A Penna23

1CEFETRJ UnED Petropolis 25620-003 Petropolis RJ Brazil2Instituto de Fısica University of Brasılia Brasılia DF Brazil3International Center for Condensed Matter Physics University of Brasılia CP 04513 70919-970 Brasılia DF Brazil

Correspondence should be addressed to L P Colatto lcolattogmailcom

Received 25 April 2015 Revised 25 June 2015 Accepted 26 July 2015

Academic Editor Shaaban Khalil

Copyright copy 2015 L P Colatto and A L A Penna This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited The publication of this article was funded by SCOAP3

We construct and study a formulation of a chargeless complex vector matter field in a supersymmetric framework To this aimwe combine two nochiral scalar superfields in order to take the vector component field to build the chargeless complex vectorsuperpartner where the respective field strength transforms into matter fields by a global 119880(1) gauge symmetry For the aimof dealing with consistent terms without breaking the global 119880(1) symmetry we imposes a choice to the complex combinationrevealing a kind of symmetry between the choices and eliminates the extra degrees of freedom which is consistent with thesupersymmetry As the usual case the mass supersymmetric sector contributes as a complement to dynamics of the model Weobtain the equations of motion of the Procarsquos type field for the chiral spinor fields and for the scalar field on the mass-shell whichshow the same mass as expected This work establishes the first steps to extend the analysis of charged massive vector field in asupersymmetric scenario

1 Introduction

Matter field dynamics was firstly established by Dirac in aconsistent relativistic framework He has studied the freeelectron dynamics where its interaction yields the first stepson QED which was further developed by Feynmann andothers [1 2] These studies were very important for theformulation and the understanding of QFT standard modeland also the strings theory Indeed it has been the basementof all theoretical analysis of any dynamics which is concernedwith integer or half-integer spin particles In fundamentalquantum theory we have classified into two types boson andfermions respectively Fermions usually are the constituentof the matter and bosons are the interaction particle [3] Nev-ertheless if we are treating to the weak force we have to dealwith charged (or not) massive vector (boson) fields which arethe intermediate between the protons and neutrons So wecould interpret as charged vector matter fields

Due to the supersymmetry which plays a fundamentalrole on strings theory fitting together quantum theories ofthe gravitational interactions electroweak and strong forcesthe studies on supersymmetric theories are of great interest tohigh energy physicists such as applications in particle physics[4] and supersymmetry breaking [5] and in the treatment ofclassical supergravity [6]with inclusion of topological Chern-Simons terms [7] Furthermore supersymmetry is requiredto understand the thermodynamics of quantumgravity [8] tobuild new scenarios for the electroweak baryogenesis in highenergies [9 10] as well as establishing superstring theoriescorrectly [11ndash16] Supersymmetry deals with graded Liealgebra in the unique reliable algebra extension which holdsto be consistent with the S-matrix in relativistic quantumfield theory [17ndash20] Recalling that this special symmetrycorrelates fermionic and bosonic fields called superpartnerswhich puts them together in a superfield formulation Itremarks the important role played by the study of matter-like

Hindawi Publishing CorporationAdvances in High Energy PhysicsVolume 2015 Article ID 986570 8 pageshttpdxdoiorg1011552015986570

2 Advances in High Energy Physics

vector fields to construct appropriated supersymmetric mod-els [21 22] Moreover supersymmetric models with chiralsuperfields and global gauge invariance involving matterfields are elegantly constructed [16 20 22] Thus quarksleptons and vector bosons which participate in usual gaugetheories as electroweak theory and chromodynamics in asupersymmetric extension coexist along their superpartnerssquarks sleptons and the fermionic partner of the vectorbosons which are a type of vector matter field For instancethe supersymmetric version of quantum electrodynamicsinvolves a vector supermultiplet whose contents are a mass-less photon and its spin-12 superpartner the photino [1620]

Indeed theoretical formulation of supersymmetric gaugevector field has been largely studied [16 22ndash24] It was shownthat gauge vector field component emerges from nonchiralscalar superfields when one uses some suitable constraint(Wess-Zumino) to remove exceeding nonphysical compo-nents fields [20 22] Nevertheless there is a lack of studiesonmodels that describe supersymmetric vector matter fieldsTherefore one of the aims of this work is the attempt toaddress this lack in order to further study the interactionswhich can be involved To this purpose we construct aformulation in which the vector field 119861120583 is complex andmassive but with no local charge which we simply calledldquochargelessrdquo Indeed as a matter field it transforms by theglobal 119880(1) group [3] that is we would emphasize that 119861120583is not a local gauge field but a free Proca-type one Suchmodels are interesting in order to contribute to the under-standing of the supersymmetric model of electroweak theorythough it contains charged vector particles Furthermore itcan improve our knowledge of the form of nuclear atomicstructure and its interaction at high energy Further takingthis model in a fundamental scenario of the strings theory[25] complex vector fields withmatter symmetry are relevantto the vacuum polarization theory that can be connectedto models which deal with Lorentz symmetry violation inhigh energy physics [26ndash46] Another aim of this work is toobtain the supermultiplet that will accommodate a chargedmatter vector field and its supersymmetric partners and alsoto get the most appropriate supersymmetric action for thisfield which will be the subject of a forthcoming work Tothis purpose we are going to formulate a supersymmetricLagrangian starting from chargeless nonchiral superfieldwhich contains the vector (matter) field The present paperis outlined as follows in Section 2 we present a model thatacommodates two real vector matter fields in Section 3 wecompose the previous model in a complex form and wepresent the Dirac superspinor fieldΨ in Section 4 we presenta general conclusion

2 Two Chargeless Vector MatterSuperfields Model

We are going to present a chargeless (real) formulation forvector matter field To this aim we start from a generalnonchiral scalar superfield which includes in the mattermultiplet a vector field as irreducible representation of

the Lorentz group In order to build more ahead a complexextension we introduce two chargeless real scalar superfieldsdoubling the number of degrees of freedom which arewritten as

Φ(119909120583 120579119886 120579 119886) = 119862 (119909) + 120579

119886120593119886 (119909) + 120579

119886

120593119886(119909)

+ 1205792[119898 (119909) + 119894119899 (119909)]

+ 1205792[119898 (119909) minus 119894119899 (119909)]

+ 1205792120579119886

[120582 119886 (119909) minus119894

2120590120583

119886 119886120597120583120593119886(119909)]

+ 1205792120579119886[120582119886 (119909) minus

119894

2120590120583

119886 119886120597120583120593119886]

+ 120579119886120590120583

119886 119886120579119886

119883120583 (119909)

+ 12057921205792[119863 (119909) minus

14◻119862 (119909)]

Λ (119909120583 120579119886 120579 119886) = 119860 (119909) + 120579

119886120594119886 (119909) + 120579

119886

120594119886(119909)

+ 1205792[120588 (119909) + 119894120591 (119909)]

+ 1205792[120588 (119909) minus 119894120591 (119909)]

+ 1205792120579119886

[120577 119886 (119909) minus119894

2120590120583

119886 119886120597120583120594119886(119909)]

+ 1205792120579119886[120577119886 (119909) minus

119894

2120590120583

119886 119886120597120583120594119886(119909)]

+ 120579119886120590120583

119886 119886120579119886

119884120583 (119909)

+ 12057921205792[119878 (119909) minus

14◻119860 (119909)]

(1)

where the superfieldsΦ andΛ are particular constructions ofmatter vector supermultiplet which include real vector fields119883120583(119909) and 119884120583(119909) with helicity plusmn1 the fields 120593119886(119909) 120582119886(119909)120594119886(119909) and 120577119886(119909) are two-component Weyl fermions withhelicity plusmn12 and the fields 119862(119909) 119863(119909) 119860(119909) 119878(119909) 119898(119909)119899(119909) 120588(119909) and 120591(119909) are real scalar fields with spin-0 It iseasy to verify that to both superfields the number of bosonicand fermionic degrees of freedom is the same We stress thatwe only have applied the reality condition on the superfieldswhich does not spoil the matter structure of these multipletsTherefore the dynamics to chargeless supersymmetric vectorfields can be obtained through suitable field-strengths whichaccommodate the real superfields Φ and Λ

In order to construct the supersymmetric field-strengthsfor the real superfields Φ and Λ which we call charge-less supersymmetric field-strengths we are going to applysupersymmetric covariant derivatives on the above scalarsuperfields which result in chiral superfields in such way that

119882119886 = minus14119863119863119863119886Φ(119909

120583 120579119886 120579 119886)

119882 119886 = minus14119863119863119863 119886Φ(119909

120583 120579119886 120579 119886)

(2)

Advances in High Energy Physics 3

and by similarity for Λ we have that

Ω119886 = minus14119863119863119863119886Λ(119909

120583 120579119886 120579 119886)

Ω 119886 = minus14119863119863119863 119886Λ(119909

120583 120579119886 120579 119886)

(3)

the119882119886 andΩ119886 are chiral spinor superfieldsWe can redefine the superfields in the chiral superspace

coordinates namely Φ(119910120583 120579119886) Φ(119911

120583 120579 119886) Λ(119910

120583 120579119886) and

Λ(119911120583 120579 119886) such that 119910120583 = 119909

120583+ 119894120579120590120583120579 and 119911

120583= 119909120583minus 119894120579120590120583120579

Hence the supersymmetric covariant derivatives are definedas

119863119886 =120597

120597120579119886

+ 2119894120590120583119886 119886120579119886 120597

120597119910120583

119863 119886 = minus120597

120597120579 119886

minus 2119894120579119886120590120583119886 119886120597

120597119911120583

(4)

According to these definitions we can compute the field-strengths119882119886 andΩ119886 and we have

119882119886 (119910 120579) = 120582119886 (119910) + 2120579119886119863(119910) + (120590120583]120579)119886119883120583] (119910)

minus 1198941205792120590120583

119886120597120583120582

(119910)

Ω119886 (119910 120579) = 120577119886 (119910) + 2120579119886119878 (119910) + (120590120583]120579)119886119884120583] (119910)

minus 1198941205792120590120583

119886120597120583120577

(119910)

(5)

and in a similar way we can compute the field-strengths119882 119886 and Ω 119886 So we are in conditions to construct thesupersymmetric model in terms of the superfields 119882119886 andΩ119886 where chargeless vectormatter field is presentThe kineticpart can be written as

119878kin = int1198894119909 119889

2120579 (119882119886119882

119886+Ω119886Ω

119886)

+ 1198892120579 (119882 119886119882

119886

+Ω 119886Ω119886

)

(6)

We have adopted the usual conventions for the spinor alge-bra for the superspace parametrization and the translationinvariance of the integral in the chiral coordinates [1622] Then we obtain that expression (6) has the followingcomponent expansion

119878kin = int1198894119909minus119883120583] (119909)119883

120583](119909) minus119884120583] (119909) 119884

120583](119909)

minus 4119894120582119886 (119909) 120590120583119886120597120583120582

(119909) minus 4119894120577119886 (119909) 120590120583119886120597120583120577(119909)

+ 81198632(119909) + 81198782 (119909)

(7)

Action (7) describes the kinetic part of supersymmetricchargeless vector field However to write the full action

that corresponds to the underlined field theory we can alsoconsider the supersymmetric mass term given by

119878119898 = int1198894119909119889

2120579120572

2[Φ

2+Λ

2] = int119889

4119909120572

2[119862 (119909)119863 (119909)

minus14119862 (119909) ◻119862 (119909) + 120593119886 (119909) 120582

119886(119909) + 120593

119886(119909) 120582

119886

(119909)

minus 119894120593119886120590120583

119886119887120597120583120593119887+119872(119909)

2+119873 (119909)

2+14119883120583119883120583

+14119884120583119884120583 +119860 (119909) 119878 (119909) minus

14119860 (119909) ◻119860 (119909)

+ 120594119886 (119909) 120577119886(119909) + 120594

119886(119909) 120577

119886

(119909) minus 119894120594119886120590120583

119886119887120597120583120594119887

+120588 (119909)2+ 120591 (119909)

2]

(8)

where 1205722 is the mass parameter As usual the ldquomassrdquo part

of the action presents kinetic terms beyond the usual massterms which were eliminated by spinor chirality property ofsuperfields 119882119886 and Ω119886 Furthermore we could infer that themass term in action (8) arises as a dynamical complement tothe supersymmetric vector matter fields Indeed supersym-metric matter-like fields are formulated with chiral spinorsuperfields

3 The Chargeless Complex VectorMatter Superfield Model

We know that the supersymmetric action for two free vectormatter fields might be built through nonchiral scalar super-fields [22] Moreover we can see that the degrees of freedomof this model are compatible with the dynamical free fieldsin complex space C In this section our aim is to derive theappropriated complex superfield to describe supersymmetriccomplex vector fields To this aim we need to strongly definetwo complex nonchiral scalar superfields defined as

Σ (119909120583 120579119886 120579 119886) = 119896 (119909) + 120579119886120585119886 (119909) + 120579

119886

119862 119886 (119909) + 1205792119897 (119909)

+ 1205792119891 (119909)

+ 1205792120579119886

[119866 119886 (119909) minus119894

2120590120583

119886 119886120597120583120585119886(119909)]

+ 1205792120579119886[119877119886 (119909) minus

119894

2120590120583

119886 119886120597120583119862119886

]

+ 120579119886120590120583

119886 119886120579119886

119861120583 (119909)

+ 12057921205792[119889 (119909) minus

14◻119896 (119909)]

(9)

119870(119909120583 120579119886 120579 119886) = 119886 (119909) + 120579119886119879119886 (119909) + 120579

119886

119867 119886 (119909)

+ 1205792119895 (119909) + 120579

2119864 (119909)

+ 1205792120579119886

[119876 119886 (119909) minus119894

2120590120583

119886 119886120597120583119879119886(119909)]

4 Advances in High Energy Physics

+ 1205792120579119886[119880119886 (119909) minus

119894

2120590120583

119886 119886120597120583119867119886

(119909)]

+ 120579119886120590120583

119886 119886120579119886

119885120583 (119909)

+ 12057921205792[V (119909) minus

14◻119886 (119909)]

(10)

and in a similar way we can define the complex conjugatedsuperfields

Superfields (9) and (10) present multiplets with complexvector fields 119861120583(119909) and 119885120583(119909) and spin-1 the 120585119886(119909) 119862 119886(119909)119877119886(119909) 119866 119886(119909) 119879119886(119909) 119867 119886(119909) 119880119886(119909) and 119876 119886(119909) are Weylfermion fields with spin-12 and the 119896(119909) 119889(119909) 119886(119909) andV(119909) are complex scalar fields with spin-0 So the rule thatimplies an invariant mechanism is

119870 = 119894Σdagger (11)

We observe that transformation rule (11) guarantees writinga consistent kinetic term for the complex vector field withoutbreaking the global119880(1) gauge symmetry Another advantagethat came to light is that transformations (11) eliminatethe exceeding fields which does not contribute for thesupersymmetric action which allows bosons and fermionsto have the same physical degrees of freedom Indeed theconstraint relation to the superfields implies the relations ofthe component fields as follows

119886 (119909) = 119894119896lowast(119909)

119879119886 (119909) = 119894119862119886 (119909)

V (119909) = 119894119889lowast(119909)

119867119886 (119909) = 119894120585119886 (119909)

119895 (119909) = 119894119891lowast(119909)

119880119886 (119909) = 119894119866119886 (119909)

119864 (119909) = 119894119897lowast(119909)

119876119886 (119909) = 119894119877119886 (119909)

119885120583 (119909) = 119894119861lowast

120583(119909)

(12)

So we can adjust the complex extension of the chargelesssuperfields Φ and Λ by assuming the equation Σ = Φ + 119894Λwhere we find the following relation of fields

119896 (119909) = 119862 (119909) + 119894119860 (119909)

120585119886 (119909) = 120593119886 (119909) + 119894120594119886 (119909)

119862119886 (119909) = 120593119886(119909) + 119894120594

119886(119909)

119897 (119909) = 119898 (119909) + 119899 (119909) + 119894 (120588 (119909) + 120591 (119909))

119866119886 (119909) = 120582119886 (119909) + 119894120577119886 (119909)

119891 (119909) = 119898 (119909) + 119899 (119909) minus 119894 (120588 (119909) + 120591 (119909))

119877119886 (119909) = 120582119886 (119909) + 119894120577119886 (119909)

119861120583 (119909) = 119883120583 (119909) + 119894119884120583 (119909)

119889 (119909) = 119863 (119909) + 119894119878 (119909)

(13)

To describe the dynamics of the supersymmetric complexvector fields with matter symmetry we need to construct anappropriated complex supersymmetric field-strength modelin order to accommodate superfields Σ and 119870 This can bereached starting from the following definitions

Υ119886 = minus14119863119863119863119886Σ (119909120583 120579119886 120579 119886)

Υ 119886 = minus14119863119863119863 119886Σ

dagger(119909120583 120579119886 120579 119886)

Γ119886 = minus14119863119863119863119886119870(119909120583 120579119886 120579 119886)

Γ 119886 = minus14119863119863119863 119886119870

dagger(119909120583 120579119886 120579 119886)

(14)

where Υ119886 and Γ119886 are charged spinor superfields As a conse-quence of the complex extension procedure we must relatechargeless spinor superfields Ω119886 and 119882119886 with the complexdefinitions (14) which in the simplest way is

Υ119886 = 119882119886 + 119894Ω119886

Υ 119886 = 119882 119886 minus 119894Ω 119886

Γ119886 = Ω119886 + 119894119882119886

Γ 119886 = Ω 119886 minus 119894119882 119886

(15)

and by assuming the spinor identities 119882119886Ω119886= Ω119886119882

119886 and119882 119886Ω

119886

= Ω 119886119882119886 we can find the kinetic supersymmetric

Lagrangian for the complex vector fields

L119896 = 119894 (Υ 119886Γ119886

minusΥ119886Γ119886)

= 119882119886119882119886+119882 119886119882

119886

+Ω119886Ω119886+Ω 119886Ω

119886

(16)

We can observe that the left-hand side of the latterequation is the complex extension of chargeless Lagrangian(7) that was written in terms of charged spinor superfieldsBearing this in mind we can then redefine the kineticLagrangian (16) simply by combining the charged spinorsuperfields Υ119886 and Γ119886 as a ldquoDirac superspinorrdquo Ψ such that

Ψ(119909120583 120579119886 120579 119886) = (

Υ119886

Γ119886) (17)

and also we assume Ψ to be the adjoint Dirac superspinorrepresentation

Advances in High Energy Physics 5

In this case we have that Ψ = Ψdagger1205740= (Γ119886Υ 119886) and so

the supersymmetric action from the kinetic Lagrangian (16)is now given by

119878119896 = int1198894119909119889

2120579119889

2120579 (119894ΨΨ)

= 119894 int 1198894119909119889

2120579119889

2120579 (Υ 119886Γ

119886

minusΥ119886Γ119886)

(18)

We can note that the product of Dirac superspinorsΨΨ obeysmatter symmetry and it presents an interesting analogy tocharged scalar superfield product 119878dagger119878 In this sense we verifythatΨ andΨ represent two chiral supersymmetric extensionsfor the matter vector field which can be transformed under119880(1) global gauge group in the following way

Ψ1015840= 119890minus2119894119902120573

Ψ

Ψ1015840

= Ψ1198902119894119902120573dagger

(19)

where120573 is a global119880(1) gauge parameter and 119902 is the charge ofthe global symmetry We can emphasize that the expressions(19) represent that each of the components of multiplet (17)has the same symmetry So action (18) is then invariantunder transformations (19) In order to obtain the componentLagrangian we can expand the product ΨΨ by consideringthat

Υ119886 (119910 120579) = 119877119886 (119910) + 2120579119886119889 (119910) + (120590120583]120579)119886119865120583] (119910)

minus 1198941205792120590120583

119886120597120583119866

(119910)

Γ119886 (119910 120579) = 119880119886 (119910) + 2120579119886V (119910) + (120590120583]120579)119886119885120583] (119910)

minus 1198941205792120590120583

119886120597120583119876

(119910)

(20)

and similarly for Υ 119886 and Γ 119886We note the presence of the complex matter field-

strengths namely

119865120583] = 120597120583119861] minus 120597]119861120583

119885120583] = 120597120583119885] minus 120597]119885120583

(21)

hence action (18) can be expanded and we obtain

119878119896 = int1198894119909

119894

2119865120583] (119909) 119885

120583](119909) minus

119894

2119865lowast

120583] (119909) 119885lowast120583]

(119909)

minus119877119886(119909) 120590120583

119886120597120583119876

(119909) minus119880119886(119909) 120590120583

119886120597120583119866

(119909)

+119877119886

(119909) 120590120583

119886119887120597120583119876119887(119909) +119880

119886

(119909) 120590120583

119886119887120597120583119866119887(119909)

+ 4119894V (119909) 119889 (119909) minus 4119894Vlowast (119909) 119889lowast (119909)

(22)

In this format we can recognize the dynamical termthat describes the matter vector field as (1198942)119865120583]119885

120583]minus

(1198942)119865lowast120583]119885lowast120583] It involves both 119865120583] and 119885120583] matter tensors

However it does not correspond to the conventional kineticterm for the matter vector field and action (18) shows moredegrees of freedom than necessary In order to get rid of suchfields we must assume the rule of transformation (11) whichis a constraint of half of the degrees and consequently action(18) reaches the correct number of component fields

Applying condition (11) in action (22) we can reach theusual dynamical matter field strength term or

119894

2119865120583]119885120583]minus

119894

2119865lowast

120583]119885lowast120583]

997904rArr minus119865lowast

120583]119865120583] (23)

and so the 119885120583] tensor field is reabsorbed in this action

Likewise and without loss of generality we could have chosenthe inverse relation Σ = 119894119870

dagger what implies reabsorbing the 119865120583]tensor field Then by using the whole relation (12) in action(18) we find that the complex supersymmetric model for thematter vector field can be written as

119878119896 = int1198894119909 minus119865

lowast

120583] (119909) 119865120583](119909)

minus 2119894119877119886 (119909) 120590120583119886120597120583119877

(119909) minus 2119894119866119886 (119909) 120590120583119886120597120583119866

(119909)

+ 8119889lowast (119909) 119889 (119909)

(24)

where expressionminus119865lowast120583](119909)119865120583](119909) represents the usual kinetic

term of the vector matter field while the terms representwith the components 119877

119886 and 119866119886 the fermionic sector and

the last term corresponds to the auxiliary field 119889 term Tocompleteness we are going to introduce the massive actionterm in the model Observing the symmetries of nonchiralfields Σ and 119870 the massive supersymmetric term can besuitable defined as

119878119898 =1205722

2int119889

4119909119889

2120579119889

2120579 [ΣdaggerΣ+119870

dagger119870] (25)

where 1205722 is a mass parameter From nonchiral superfields Σand 119870 we can obtain the massive vector matter field term119861lowast

120583119861120583 as well as their supersymmetric partners In order to

perform it we are going to compute action (25) by employingcondition119870 = 119894Σ

dagger where one has that (12)ΣdaggerΣ+(12)119870dagger119870 =

ΣdaggerΣ and by applying definition (10) the full supersymmetric

matter vector field model can be then obtained from Diracsuperspinor fieldΨ associated to the nonchiral scalar fields Σand119870 in the following form

119878 = 119878119896 + 119878119898 = int1198894119909119889

2120579119889

2120579 (119894ΨΨ+120572

2ΣdaggerΣ) (26)

where the mass part of action can be obtained in componentfields as

119878119898 = 1205722int119889

4119909(119889lowast(119909) 119896 (119909) + 119889 (119909) 119896

lowast(119909)

minus14119896lowast(119909) ◻119896 (119909) minus

14119896 (119909) ◻119896

lowast(119909) + 120585119886 (119909) 119866

119886(119909)

+ 120585 119886 (119909) 119866119886

(119909) +119862119886 (119909) 119877119886(119909) +119862 119886 (119909) 119877

119886

(119909)

6 Advances in High Energy Physics

minus 2119894120585119886

(119909) 120590120583

119886119887120597120583120585119887(119909) minus 2119894119862 119886 (119909) 120590120583 119886119887120597120583119862

119887(119909)

+ 119861lowast

120583119861120583+119891lowast(119909) 119891 (119909) + 119897

lowast(119909) 119897 (119909))

(27)

Here we observe the mass term to 119861120583 119891(119909) and 119897(119909) fields

As in the usual supersymmetric models we note that massaction (27) also contributes to kinetic structure namelywith the terms minus(14)119896lowast(119909)◻119896(119909)minus2119894120585

119886

120590120583

119886119887120597120583120585119887 and

minus2119894119862 119886120590120583 119886119887120597120583119862119887 The action also shows mixing mass scalar

and fermionic terms namely 119889lowast(119909)119896(119909) 119889(119909)119896lowast(119909)

120585119886(119909)119866119886(119909) 120585 119886(119909)119866

119886

(119909)119862119886(119909)119877119886(119909) and 119862 119886(119909)119877

119886

(119909) Byverifying the presence of extra kinetic terms in (27) whatcan suggest that when we particularly treat supersymmetricmatter vector fields the mass action contributes witha ldquodynamic complementrdquo to the kinetic action (24)Furthermore we remark that mass action (27) is importantto match the number of bosonic and fermionic degrees offreedom of the supersymmetric matter vector action (26) forthe consistency of themodelWe can redefine the componentfields absorbing the mass parameter as follows

119896 (119909) 997888rarr1120572119896 (119909)

120585119886 (119909) 997888rarr1120572120585119886 (119909)

119862119886 (119909) 997888rarr1120572119862119886 (119909)

119891 (119909) 997888rarr1120572119891 (119909)

119897 (119909) 997888rarr1120572119897 (119909)

(28)

So the action can be rewritten as

119878ek =12int119889

4119909(minus119865

lowast

120583] (119909) 119865120583](119909)

minus 2119894119877119886 (119909) 120590120583119886120597120583119877

(119909) minus 2119894119866119886 (119909) 120590120583119886120597120583119866

(119909)

minus14119896lowast(119909) ◻119896 (119909) minus

14119896 (119909) ◻119896

lowast(119909)

+ minus2119894120585119886

(119909) 120590120583

119886119887120597120583120585119887(119909) minus 2119894119862 119886 (119909) 120590120583 119886119887120597120583119862

119887(119909)

+ 8119889lowast (119909) 119889 (119909) +119891lowast(119909) 119891 (119909) + 119897

lowast(119909) 119897 (119909))

(29)

analogously the mass action is now given by

119878em = int1198894119909 120572

2119861lowast

120583119861120583+120572119889lowast(119909) 119896 (119909)

+ 120572119889 (119909) 119896lowast(119909) + 120572120585119886 (119909) 119866

119886(119909) + 120572120585 119886 (119909) 119866

119886

(119909)

+ 120572119862119886 (119909) 119877119886(119909) + 120572119862 119886 (119909) 119877

119886

(119909)

(30)

so the complex scalar fields 119889(119909) 119891(119909) and 119897(119909) have nodynamics and arise as auxiliary fields Thus assuming theaction (26) to be the sum of the redefined actions (29) and(30) and rearranging (we adopt the Weyl representation tothe gamma matrices) the (Dirac) spinor fields Θ and Π Thesum of the action (29) and the action (30) results in an off-shell action 119878os written in the following form

119878os = int1198894119909 minus

12119865lowast

120583] (119909) 119865120583](119909) +

14120597120583119896lowast(119909) 120597120583119896 (119909)

minus 119894Θ (119909) 120574120583120597120583Θ (119909) minus 119894Π (119909) 120574

120583120597120583Π (119909)

+ 120572Θ (119909) 1205745Θ (119909) + 120572Π (119909) 120574

5Π (119909) + 4119889lowast (119909) 119889 (119909)

+12119891lowast(119909) 119891 (119909) +

12119897lowast(119909) 119897 (119909) + 120572

2119861lowast

120583119861120583

+120572119889lowast(119909) 119896 (119909) + 120572119889 (119909) 119896

lowast(119909)

(31)

where we denote the Dirac spinors of mass 120572 as

Θ (119909) = (

120585119886

119866119886)

Π (119909) = (

119862119886

119877119886)

(32)

For action (31) we have obtained chiral spinor mass termsgiven by 120572Θ(119909)120574

5Θ(119909) and 120572Π(119909)120574

5Π(119909) From the action

(31) we can obtain the equation of motion for the fields

120597120583119865120583](119909) + 120572

2119861](119909) = 0

120574120583120597120583Θ (119909) minus 120572120574

5Θ (119909) = 0

120574120583120597120583Π (119909) minus 120572120574

5Π (119909) = 0

◻119896 (119909) + 1205722119896 (119909) = 0

(33)

with 119891(119909) = 119897(119909) = 0 Taking off-shell action (31) wenote that it has 16 bosonic degrees of freedom concerningthe matter fields 119861120583(119909) 119896(119909) 119889(119909) 119891(119909) and 119897(119909) and theircomplex conjugated ones as well as 16 fermionic degreesof freedom for the Dirac spinor fields Θ(119909) and Π(119909) andtheir conjugated complex ones which is consistent with thesupersymmetry From equations of motion (33) we note thatthere are three auxiliary complex scalar fields 119889(119909) 119891(119909)and 119897(119909) and massive dynamical complex scalar field 119896(119909)Moreover as expected we have obtained a matter Proca-typeequation for field 119861120583 In this context it is interesting to notethat from the on-shell actionwe can easily extract from action(31) the supersymmetric generalization of matter vector fieldit is only possible if we include two dynamical Dirac chiralspinor fieldsΘ(119909) andΠ(119909) along a massive scalar field 119896(119909)Furthermore a peculiar aspect of the spinor fields Θ(119909) andΠ(119909) in the present case is that their mass terms (33) arisewith chiral structure due to the presence of the matrix 1205745

Advances in High Energy Physics 7

4 Conclusion

In this paper which is part of a program started in a previouswork [32] we propose to formulate and analyze the simpledynamics of chargeless vector matter field in a supersymmet-ric scenarioThere are threemotivations for this proposal thelack of this approach in the literature getting a clue on thepossibility of Lorentz symmetry violation in supersymmetrictheories and the role played by the simplest case of high-spin field in field and string theories To these aims wehave started from real nonchiral scalar superfield in orderto obtain real matter Proca-type field in a supersymmetricLagrangian generalization [32] In a straightforward waythe complex model was obtained extending the real scalarnonchiral superfields to the complex space [47ndash51]

The very interesting point in this enterprise is that inorder to obtain a complex Proca-type term we face anambiguous choice of which is the real or imaginary part of thefield This ambiguity reveals a symmetry which we representas a kind of simple Hodge-duality symmetry 119870 = 119894Σ

dagger Infact despite its apparent simplicity it is essential to match thenumber of degrees of freedom of the fermionic and bosonicsectors and to compose the complex (chargeless) Proca-type dynamical term with global 119880(1) symmetry As thissymmetry appears by construction it represents a differentapproach compared to Ferrara et alrsquos work [24] On theother hand as usual the supersymmetric mass term is alsoimportant to compose the dynamics of the supersymmetricmodel and of its component fields

We can conclude that in addition to the supersymmetricmass term to build supersymmetric matter chargeless vectorfield Lagrangian the Hodge-type duality symmetry is anessential ingredient to be further explored We can alsoobserve that as the usual super-QED model the complexdynamical Proca-type term emerges from a product of Diracsuperspinors ΨΨ We can remark that the Dirac superspinorfieldΨ is also chiral because it is a combination of chiralWeylsuperspinors Γ119886 and Υ

119886 Finally we emphasize that on-shellsupersymmetric action obtained (31) revealing two fermionicDirac fields Θ(119909) and Π(119909) and a massive scalar field 119896(119909)

as supersymmetric partners associated to the complex vectorfield 119861120583(119909) which is the initial step to implement localgauge symmetry charge and interaction we could anticipatethat nonminimal couplings have an essential role in aforthcoming work

Conflict of Interests

The authors declare that they have no conflict of interestsrelated to this paper

References

[1] C Itzykson and J-B Zuber Quantum Field Theory DoverPublications 2005

[2] P RamondFieldTheory AModern PrimerWestviewPress 2ndedition 1990

[3] N N Bogoliubov and D V Shirkov Introduction to the Theoryof Quantized Fields John Wiley amp Sons 3rd edition 1980

[4] P Fayet and S Ferrara ldquoSupersymmetryrdquo Physics Reports vol32 no 5 pp 249ndash334 1977

[5] H PNilles ldquoSupersymmetry supergravity andparticle physicsrdquoPhysics Reports vol 110 no 1-2 pp 1ndash162 1984

[6] P D DrsquoEath ldquoCanonical quantization of supergravityrdquo PhysicalReview D vol 29 no 10 pp 2199ndash2219 1984

[7] J-P Ader F Biet and Y Noirot ldquoSupersymmetric structure ofthe induced W gravitiesrdquo Classical and Quantum Gravity vol16 no 3 pp 1027ndash1037 1999

[8] C Kiefer T Luck and P Moniz ldquoSemiclassical approximationto supersymmetric quantumgravityrdquoPhysical ReviewD vol 72no 4 Article ID 045006 2005

[9] A Riotto ldquoMore relaxed supersymmetric electroweak baryoge-nesisrdquo Physical Review D vol 58 Article ID 095009 1998

[10] A Riotto ldquoSupersymmetric electroweak baryogenesis non-equilibrium field theory and quantum Boltzmann equationsrdquoNuclear Physics B vol 518 no 1-2 pp 339ndash360 1998

[11] I R Klebanov and A A Tseytlin ldquoGravity duals of supersym-metric SU(N)timesSU(N+M) gauge theoriesrdquo Nuclear Physics Bvol 578 no 1-2 pp 123ndash138 2000

[12] P H Chankowski A Falkowski S Pokorski and J WagnerldquoElectroweak symmetry breaking in supersymmetric modelswith heavy scalar superpartnersrdquo Physics Letters Section BNuclear Elementary Particle and High-Energy Physics vol 598no 3-4 pp 252ndash262 2004

[13] J D Edelstein and C Nunez ldquoSupersymmetric electroweakcosmic stringsrdquo Physical Review D vol 55 no 6 pp 3811ndash38191997

[14] L Girardello A Giveon M Porrati and A Zaffaroni ldquoNon-Abelian strong-weak coupling duality in (string-derived) N = 4supersymmetric Yang-Mills theoriesrdquo Physics Letters B vol 334no 3-4 pp 331ndash338 1994

[15] J Wess and J Bagger Supersymmetry and Supergravity Prince-ton University Press Princeton NJ USA 2nd edition 1992

[16] D Bailin and A Love Supersymmetric Gauge Field Theory andString Theory IOP Publishing 1994

[17] S Coleman and J Mandula ldquoAll possible symmetries of the Smatrixrdquo Physical Review vol 159 no 5 pp 1251ndash1256 1967

[18] S J Gates M T Grisaru M Rocek and W Siegel SuperspaceBenjaminCummings Reading 1983

[19] R Haag J T Lopuszanski and M Sohnius ldquoAll possiblegenerators of supersymmetries of the 119878-matrixrdquoNuclear PhysicsB vol 88 pp 257ndash274 1975

[20] S Weinberg The Quantum Theory of Fields vol 3 CambridgeUniversity Press Cambridge UK 2000

[21] J Wess and B Zumino ldquoSupergauge transformations in fourdimensionsrdquo Nuclear Physics B vol 70 no 1 pp 39ndash50 1974

[22] H J W Muller-Kirsten and A Wiedemann SupersymmetryWorld Scientific Singapore 1987

[23] S J Gates Jr ldquoWhy auxiliary fields matter the strange caseof the 4-D 119873 = 1 supersymmetric QCD effective action 2rdquoNuclear Physics B vol 485 no 1-2 pp 145ndash184 1997

[24] S Ferrara L Girardello and F Palumbo ldquoGeneralmass formulain broken supersymmetryrdquo Physical Review D vol 20 no 2 pp403ndash408 1979

[25] V A Kostelecky and S Samuel ldquoSpontaneous breaking ofLorentz symmetry in string theoryrdquo Physical Review D vol 39no 2 pp 683ndash685 1989

[26] V E R Lemes A L M A Nogueira and J A Helayel-NetoldquoSupersymmetric generalization of the tensor matter fieldsrdquo

8 Advances in High Energy Physics

International Journal of Modern Physics A vol 13 no 18 pp3145ndash3156 1998

[27] D Colladay and V A Kostelecky ldquoCPT violation and thestandard modelrdquo Physical Review D vol 55 pp 6760ndash67741997

[28] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standardmodelrdquo Physical ReviewD vol 58 no 11 ArticleID 116002 1998

[29] A A Andrianov and R Soldati ldquoLorentz symmetry breakingin Abelian vector-field models with Wess-Zumino interactionrdquoPhysical Review D vol 51 no 10 pp 5961ndash5964 1995

[30] A A Andrianov and R Soldati ldquoPatterns of Lorentz symmetrybreaking inQEDbyCPT-odd interactionrdquoPhysics Letters B vol435 no 3-4 pp 449ndash452 1998

[31] S M Carrol G B Field and R Jackiw ldquoLimits on a Lorentz-and parity-violating modification of electrodynamicsrdquo PhysicalReview D vol 41 no 4 pp 1231ndash1240 1990

[32] L P Colatto A L A Penna andW C Santos ldquoCharged tensormatter fields and Lorentz symmetry violation via spontaneoussymmetry breakingrdquo European Physical Journal C vol 36 no 1pp 79ndash87 2004

[33] H Belich J L Boldo L P Colatto J A Helayel-Neto and A LMANogueira ldquoSupersymmetric extension of the Lorentz- andCPT-violatingMaxwell-Chern-Simonsmodelrdquo Physical ReviewD vol 68 Article ID 065030 2003

[34] H Belich Jr T Costa-Soares M M Ferreira Jr J A Helayel-Neto and M T D Orlando ldquoLorentz-symmetry violation andelectrically charged vortices in the planar regimerdquo InternationalJournal of Modern Physics A vol 21 no 11 pp 2415ndash2429 2006

[35] M M Ferreira Jr ldquoElectron-electron interaction in a Maxwell-Chern-Simons model with a purely spacelike Lorentz-violatingbackgroundrdquo Physical Review D vol 71 no 4 Article ID045003 9 pages 2005

[36] H Belich T Costa-Soares M M Ferreira Jr and J A Helayel-Neto ldquoNon-minimal coupling to a Lorentz-violating back-ground and topological implicationsrdquo The European PhysicalJournal C vol 41 no 3 pp 421ndash426 2005

[37] M B Cantcheff ldquoLorentz symmetry breaking and planar effectsfrom non-linear electrodynamicsrdquoThe European Physical Jour-nal CmdashParticles and Fields vol 46 no 1 pp 247ndash254 2006

[38] H Belich M A De Andrade and M A Santos ldquoGauge the-ories with Lorentz-symmetry violation by symplectic projectormethodrdquoModern Physics Letters A vol 20 pp 2305ndash2316 2005

[39] T G Rizzo ldquoLorentz violation in extra dimensionsrdquo Journal ofHigh Energy Physics vol 2005 no 9 article 036 2005

[40] M M Ferreira Jr and M S Tavares ldquoNonrelativistic electronndashelectron interaction in a maxwellndashchernndashsimonsndashproca modelendowed with a timelike lorentz-violating backgroundrdquo Inter-national Journal of Modern Physics A vol 22 p 1685 2007

[41] H Belich T Costa-Soares M M Ferreira Jr J A Helayel-Neto and F M O Mouchereck ldquoLorentz-violating correctionson the hydrogen spectrum induced by a nonminimal couplingrdquoPhysical Review D vol 74 no 6 Article ID 065009 6 pages2006

[42] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoOn the consistency of Lorentz invariance violationin QED induced by fermions in constant axial-vector back-groundrdquo Physics Letters B vol 639 no 5 pp 586ndash590 2006

[43] L P Colatto A L A Penna and W C Santos ldquoVector solitonsand spontaneous Lorentz violation mechanismrdquo httparxivorgabshep-th0610096

[44] MM Ferreira Jr andA R Gomes ldquoLorentz violating effects ona quantized two-level systemrdquo httparxivorgabs08034010

[45] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoBare and induced lorentz and cpt invariance violationsin qedrdquo International Journal of Modern Physics A vol 25 no16 pp 3271ndash3306 2010

[46] T G Rizzo ldquoLorentz violation in warped extra dimensionsrdquoJournal of High Energy Physics vol 2010 no 11 article 156 2010

[47] J Helayel-Neto R Jengo F Legovini and S Pugnetti ldquoSup-ersymmetry Complex Structures And Superfieldsrdquo SISSA-3887EP

[48] Y Kobayashi and S Nagamachi ldquoGeneralized complexsuperspacemdashinvolutions of superfieldsrdquo Journal of Mathemati-cal Physics vol 28 no 8 pp 1700ndash1708 1987

[49] P A Grassi G Policastro and M Porrati ldquoNotes on thequantization of the complex linear superfieldrdquo Nuclear PhysicsB vol 597 no 1ndash3 pp 615ndash632 2001

[50] S M Kuzenko and S J Tyler ldquoComplex linear superfield as amodel for Goldstinordquo Journal of High Energy Physics vol 2011no 4 article 57 2011

[51] S J Gates J Hallett T Hubsch and K Stiffler ldquoThe realanatomy of complex linear superfieldsrdquo International Journal ofModern Physics A vol 27 no 24 Article ID 1250143 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

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FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

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GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

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Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

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Soft MatterJournal of

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AerodynamicsJournal of

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PhotonicsJournal of

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Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

2 Advances in High Energy Physics

vector fields to construct appropriated supersymmetric mod-els [21 22] Moreover supersymmetric models with chiralsuperfields and global gauge invariance involving matterfields are elegantly constructed [16 20 22] Thus quarksleptons and vector bosons which participate in usual gaugetheories as electroweak theory and chromodynamics in asupersymmetric extension coexist along their superpartnerssquarks sleptons and the fermionic partner of the vectorbosons which are a type of vector matter field For instancethe supersymmetric version of quantum electrodynamicsinvolves a vector supermultiplet whose contents are a mass-less photon and its spin-12 superpartner the photino [1620]

Indeed theoretical formulation of supersymmetric gaugevector field has been largely studied [16 22ndash24] It was shownthat gauge vector field component emerges from nonchiralscalar superfields when one uses some suitable constraint(Wess-Zumino) to remove exceeding nonphysical compo-nents fields [20 22] Nevertheless there is a lack of studiesonmodels that describe supersymmetric vector matter fieldsTherefore one of the aims of this work is the attempt toaddress this lack in order to further study the interactionswhich can be involved To this purpose we construct aformulation in which the vector field 119861120583 is complex andmassive but with no local charge which we simply calledldquochargelessrdquo Indeed as a matter field it transforms by theglobal 119880(1) group [3] that is we would emphasize that 119861120583is not a local gauge field but a free Proca-type one Suchmodels are interesting in order to contribute to the under-standing of the supersymmetric model of electroweak theorythough it contains charged vector particles Furthermore itcan improve our knowledge of the form of nuclear atomicstructure and its interaction at high energy Further takingthis model in a fundamental scenario of the strings theory[25] complex vector fields withmatter symmetry are relevantto the vacuum polarization theory that can be connectedto models which deal with Lorentz symmetry violation inhigh energy physics [26ndash46] Another aim of this work is toobtain the supermultiplet that will accommodate a chargedmatter vector field and its supersymmetric partners and alsoto get the most appropriate supersymmetric action for thisfield which will be the subject of a forthcoming work Tothis purpose we are going to formulate a supersymmetricLagrangian starting from chargeless nonchiral superfieldwhich contains the vector (matter) field The present paperis outlined as follows in Section 2 we present a model thatacommodates two real vector matter fields in Section 3 wecompose the previous model in a complex form and wepresent the Dirac superspinor fieldΨ in Section 4 we presenta general conclusion

2 Two Chargeless Vector MatterSuperfields Model

We are going to present a chargeless (real) formulation forvector matter field To this aim we start from a generalnonchiral scalar superfield which includes in the mattermultiplet a vector field as irreducible representation of

the Lorentz group In order to build more ahead a complexextension we introduce two chargeless real scalar superfieldsdoubling the number of degrees of freedom which arewritten as

Φ(119909120583 120579119886 120579 119886) = 119862 (119909) + 120579

119886120593119886 (119909) + 120579

119886

120593119886(119909)

+ 1205792[119898 (119909) + 119894119899 (119909)]

+ 1205792[119898 (119909) minus 119894119899 (119909)]

+ 1205792120579119886

[120582 119886 (119909) minus119894

2120590120583

119886 119886120597120583120593119886(119909)]

+ 1205792120579119886[120582119886 (119909) minus

119894

2120590120583

119886 119886120597120583120593119886]

+ 120579119886120590120583

119886 119886120579119886

119883120583 (119909)

+ 12057921205792[119863 (119909) minus

14◻119862 (119909)]

Λ (119909120583 120579119886 120579 119886) = 119860 (119909) + 120579

119886120594119886 (119909) + 120579

119886

120594119886(119909)

+ 1205792[120588 (119909) + 119894120591 (119909)]

+ 1205792[120588 (119909) minus 119894120591 (119909)]

+ 1205792120579119886

[120577 119886 (119909) minus119894

2120590120583

119886 119886120597120583120594119886(119909)]

+ 1205792120579119886[120577119886 (119909) minus

119894

2120590120583

119886 119886120597120583120594119886(119909)]

+ 120579119886120590120583

119886 119886120579119886

119884120583 (119909)

+ 12057921205792[119878 (119909) minus

14◻119860 (119909)]

(1)

where the superfieldsΦ andΛ are particular constructions ofmatter vector supermultiplet which include real vector fields119883120583(119909) and 119884120583(119909) with helicity plusmn1 the fields 120593119886(119909) 120582119886(119909)120594119886(119909) and 120577119886(119909) are two-component Weyl fermions withhelicity plusmn12 and the fields 119862(119909) 119863(119909) 119860(119909) 119878(119909) 119898(119909)119899(119909) 120588(119909) and 120591(119909) are real scalar fields with spin-0 It iseasy to verify that to both superfields the number of bosonicand fermionic degrees of freedom is the same We stress thatwe only have applied the reality condition on the superfieldswhich does not spoil the matter structure of these multipletsTherefore the dynamics to chargeless supersymmetric vectorfields can be obtained through suitable field-strengths whichaccommodate the real superfields Φ and Λ

In order to construct the supersymmetric field-strengthsfor the real superfields Φ and Λ which we call charge-less supersymmetric field-strengths we are going to applysupersymmetric covariant derivatives on the above scalarsuperfields which result in chiral superfields in such way that

119882119886 = minus14119863119863119863119886Φ(119909

120583 120579119886 120579 119886)

119882 119886 = minus14119863119863119863 119886Φ(119909

120583 120579119886 120579 119886)

(2)

Advances in High Energy Physics 3

and by similarity for Λ we have that

Ω119886 = minus14119863119863119863119886Λ(119909

120583 120579119886 120579 119886)

Ω 119886 = minus14119863119863119863 119886Λ(119909

120583 120579119886 120579 119886)

(3)

the119882119886 andΩ119886 are chiral spinor superfieldsWe can redefine the superfields in the chiral superspace

coordinates namely Φ(119910120583 120579119886) Φ(119911

120583 120579 119886) Λ(119910

120583 120579119886) and

Λ(119911120583 120579 119886) such that 119910120583 = 119909

120583+ 119894120579120590120583120579 and 119911

120583= 119909120583minus 119894120579120590120583120579

Hence the supersymmetric covariant derivatives are definedas

119863119886 =120597

120597120579119886

+ 2119894120590120583119886 119886120579119886 120597

120597119910120583

119863 119886 = minus120597

120597120579 119886

minus 2119894120579119886120590120583119886 119886120597

120597119911120583

(4)

According to these definitions we can compute the field-strengths119882119886 andΩ119886 and we have

119882119886 (119910 120579) = 120582119886 (119910) + 2120579119886119863(119910) + (120590120583]120579)119886119883120583] (119910)

minus 1198941205792120590120583

119886120597120583120582

(119910)

Ω119886 (119910 120579) = 120577119886 (119910) + 2120579119886119878 (119910) + (120590120583]120579)119886119884120583] (119910)

minus 1198941205792120590120583

119886120597120583120577

(119910)

(5)

and in a similar way we can compute the field-strengths119882 119886 and Ω 119886 So we are in conditions to construct thesupersymmetric model in terms of the superfields 119882119886 andΩ119886 where chargeless vectormatter field is presentThe kineticpart can be written as

119878kin = int1198894119909 119889

2120579 (119882119886119882

119886+Ω119886Ω

119886)

+ 1198892120579 (119882 119886119882

119886

+Ω 119886Ω119886

)

(6)

We have adopted the usual conventions for the spinor alge-bra for the superspace parametrization and the translationinvariance of the integral in the chiral coordinates [1622] Then we obtain that expression (6) has the followingcomponent expansion

119878kin = int1198894119909minus119883120583] (119909)119883

120583](119909) minus119884120583] (119909) 119884

120583](119909)

minus 4119894120582119886 (119909) 120590120583119886120597120583120582

(119909) minus 4119894120577119886 (119909) 120590120583119886120597120583120577(119909)

+ 81198632(119909) + 81198782 (119909)

(7)

Action (7) describes the kinetic part of supersymmetricchargeless vector field However to write the full action

that corresponds to the underlined field theory we can alsoconsider the supersymmetric mass term given by

119878119898 = int1198894119909119889

2120579120572

2[Φ

2+Λ

2] = int119889

4119909120572

2[119862 (119909)119863 (119909)

minus14119862 (119909) ◻119862 (119909) + 120593119886 (119909) 120582

119886(119909) + 120593

119886(119909) 120582

119886

(119909)

minus 119894120593119886120590120583

119886119887120597120583120593119887+119872(119909)

2+119873 (119909)

2+14119883120583119883120583

+14119884120583119884120583 +119860 (119909) 119878 (119909) minus

14119860 (119909) ◻119860 (119909)

+ 120594119886 (119909) 120577119886(119909) + 120594

119886(119909) 120577

119886

(119909) minus 119894120594119886120590120583

119886119887120597120583120594119887

+120588 (119909)2+ 120591 (119909)

2]

(8)

where 1205722 is the mass parameter As usual the ldquomassrdquo part

of the action presents kinetic terms beyond the usual massterms which were eliminated by spinor chirality property ofsuperfields 119882119886 and Ω119886 Furthermore we could infer that themass term in action (8) arises as a dynamical complement tothe supersymmetric vector matter fields Indeed supersym-metric matter-like fields are formulated with chiral spinorsuperfields

3 The Chargeless Complex VectorMatter Superfield Model

We know that the supersymmetric action for two free vectormatter fields might be built through nonchiral scalar super-fields [22] Moreover we can see that the degrees of freedomof this model are compatible with the dynamical free fieldsin complex space C In this section our aim is to derive theappropriated complex superfield to describe supersymmetriccomplex vector fields To this aim we need to strongly definetwo complex nonchiral scalar superfields defined as

Σ (119909120583 120579119886 120579 119886) = 119896 (119909) + 120579119886120585119886 (119909) + 120579

119886

119862 119886 (119909) + 1205792119897 (119909)

+ 1205792119891 (119909)

+ 1205792120579119886

[119866 119886 (119909) minus119894

2120590120583

119886 119886120597120583120585119886(119909)]

+ 1205792120579119886[119877119886 (119909) minus

119894

2120590120583

119886 119886120597120583119862119886

]

+ 120579119886120590120583

119886 119886120579119886

119861120583 (119909)

+ 12057921205792[119889 (119909) minus

14◻119896 (119909)]

(9)

119870(119909120583 120579119886 120579 119886) = 119886 (119909) + 120579119886119879119886 (119909) + 120579

119886

119867 119886 (119909)

+ 1205792119895 (119909) + 120579

2119864 (119909)

+ 1205792120579119886

[119876 119886 (119909) minus119894

2120590120583

119886 119886120597120583119879119886(119909)]

4 Advances in High Energy Physics

+ 1205792120579119886[119880119886 (119909) minus

119894

2120590120583

119886 119886120597120583119867119886

(119909)]

+ 120579119886120590120583

119886 119886120579119886

119885120583 (119909)

+ 12057921205792[V (119909) minus

14◻119886 (119909)]

(10)

and in a similar way we can define the complex conjugatedsuperfields

Superfields (9) and (10) present multiplets with complexvector fields 119861120583(119909) and 119885120583(119909) and spin-1 the 120585119886(119909) 119862 119886(119909)119877119886(119909) 119866 119886(119909) 119879119886(119909) 119867 119886(119909) 119880119886(119909) and 119876 119886(119909) are Weylfermion fields with spin-12 and the 119896(119909) 119889(119909) 119886(119909) andV(119909) are complex scalar fields with spin-0 So the rule thatimplies an invariant mechanism is

119870 = 119894Σdagger (11)

We observe that transformation rule (11) guarantees writinga consistent kinetic term for the complex vector field withoutbreaking the global119880(1) gauge symmetry Another advantagethat came to light is that transformations (11) eliminatethe exceeding fields which does not contribute for thesupersymmetric action which allows bosons and fermionsto have the same physical degrees of freedom Indeed theconstraint relation to the superfields implies the relations ofthe component fields as follows

119886 (119909) = 119894119896lowast(119909)

119879119886 (119909) = 119894119862119886 (119909)

V (119909) = 119894119889lowast(119909)

119867119886 (119909) = 119894120585119886 (119909)

119895 (119909) = 119894119891lowast(119909)

119880119886 (119909) = 119894119866119886 (119909)

119864 (119909) = 119894119897lowast(119909)

119876119886 (119909) = 119894119877119886 (119909)

119885120583 (119909) = 119894119861lowast

120583(119909)

(12)

So we can adjust the complex extension of the chargelesssuperfields Φ and Λ by assuming the equation Σ = Φ + 119894Λwhere we find the following relation of fields

119896 (119909) = 119862 (119909) + 119894119860 (119909)

120585119886 (119909) = 120593119886 (119909) + 119894120594119886 (119909)

119862119886 (119909) = 120593119886(119909) + 119894120594

119886(119909)

119897 (119909) = 119898 (119909) + 119899 (119909) + 119894 (120588 (119909) + 120591 (119909))

119866119886 (119909) = 120582119886 (119909) + 119894120577119886 (119909)

119891 (119909) = 119898 (119909) + 119899 (119909) minus 119894 (120588 (119909) + 120591 (119909))

119877119886 (119909) = 120582119886 (119909) + 119894120577119886 (119909)

119861120583 (119909) = 119883120583 (119909) + 119894119884120583 (119909)

119889 (119909) = 119863 (119909) + 119894119878 (119909)

(13)

To describe the dynamics of the supersymmetric complexvector fields with matter symmetry we need to construct anappropriated complex supersymmetric field-strength modelin order to accommodate superfields Σ and 119870 This can bereached starting from the following definitions

Υ119886 = minus14119863119863119863119886Σ (119909120583 120579119886 120579 119886)

Υ 119886 = minus14119863119863119863 119886Σ

dagger(119909120583 120579119886 120579 119886)

Γ119886 = minus14119863119863119863119886119870(119909120583 120579119886 120579 119886)

Γ 119886 = minus14119863119863119863 119886119870

dagger(119909120583 120579119886 120579 119886)

(14)

where Υ119886 and Γ119886 are charged spinor superfields As a conse-quence of the complex extension procedure we must relatechargeless spinor superfields Ω119886 and 119882119886 with the complexdefinitions (14) which in the simplest way is

Υ119886 = 119882119886 + 119894Ω119886

Υ 119886 = 119882 119886 minus 119894Ω 119886

Γ119886 = Ω119886 + 119894119882119886

Γ 119886 = Ω 119886 minus 119894119882 119886

(15)

and by assuming the spinor identities 119882119886Ω119886= Ω119886119882

119886 and119882 119886Ω

119886

= Ω 119886119882119886 we can find the kinetic supersymmetric

Lagrangian for the complex vector fields

L119896 = 119894 (Υ 119886Γ119886

minusΥ119886Γ119886)

= 119882119886119882119886+119882 119886119882

119886

+Ω119886Ω119886+Ω 119886Ω

119886

(16)

We can observe that the left-hand side of the latterequation is the complex extension of chargeless Lagrangian(7) that was written in terms of charged spinor superfieldsBearing this in mind we can then redefine the kineticLagrangian (16) simply by combining the charged spinorsuperfields Υ119886 and Γ119886 as a ldquoDirac superspinorrdquo Ψ such that

Ψ(119909120583 120579119886 120579 119886) = (

Υ119886

Γ119886) (17)

and also we assume Ψ to be the adjoint Dirac superspinorrepresentation

Advances in High Energy Physics 5

In this case we have that Ψ = Ψdagger1205740= (Γ119886Υ 119886) and so

the supersymmetric action from the kinetic Lagrangian (16)is now given by

119878119896 = int1198894119909119889

2120579119889

2120579 (119894ΨΨ)

= 119894 int 1198894119909119889

2120579119889

2120579 (Υ 119886Γ

119886

minusΥ119886Γ119886)

(18)

We can note that the product of Dirac superspinorsΨΨ obeysmatter symmetry and it presents an interesting analogy tocharged scalar superfield product 119878dagger119878 In this sense we verifythatΨ andΨ represent two chiral supersymmetric extensionsfor the matter vector field which can be transformed under119880(1) global gauge group in the following way

Ψ1015840= 119890minus2119894119902120573

Ψ

Ψ1015840

= Ψ1198902119894119902120573dagger

(19)

where120573 is a global119880(1) gauge parameter and 119902 is the charge ofthe global symmetry We can emphasize that the expressions(19) represent that each of the components of multiplet (17)has the same symmetry So action (18) is then invariantunder transformations (19) In order to obtain the componentLagrangian we can expand the product ΨΨ by consideringthat

Υ119886 (119910 120579) = 119877119886 (119910) + 2120579119886119889 (119910) + (120590120583]120579)119886119865120583] (119910)

minus 1198941205792120590120583

119886120597120583119866

(119910)

Γ119886 (119910 120579) = 119880119886 (119910) + 2120579119886V (119910) + (120590120583]120579)119886119885120583] (119910)

minus 1198941205792120590120583

119886120597120583119876

(119910)

(20)

and similarly for Υ 119886 and Γ 119886We note the presence of the complex matter field-

strengths namely

119865120583] = 120597120583119861] minus 120597]119861120583

119885120583] = 120597120583119885] minus 120597]119885120583

(21)

hence action (18) can be expanded and we obtain

119878119896 = int1198894119909

119894

2119865120583] (119909) 119885

120583](119909) minus

119894

2119865lowast

120583] (119909) 119885lowast120583]

(119909)

minus119877119886(119909) 120590120583

119886120597120583119876

(119909) minus119880119886(119909) 120590120583

119886120597120583119866

(119909)

+119877119886

(119909) 120590120583

119886119887120597120583119876119887(119909) +119880

119886

(119909) 120590120583

119886119887120597120583119866119887(119909)

+ 4119894V (119909) 119889 (119909) minus 4119894Vlowast (119909) 119889lowast (119909)

(22)

In this format we can recognize the dynamical termthat describes the matter vector field as (1198942)119865120583]119885

120583]minus

(1198942)119865lowast120583]119885lowast120583] It involves both 119865120583] and 119885120583] matter tensors

However it does not correspond to the conventional kineticterm for the matter vector field and action (18) shows moredegrees of freedom than necessary In order to get rid of suchfields we must assume the rule of transformation (11) whichis a constraint of half of the degrees and consequently action(18) reaches the correct number of component fields

Applying condition (11) in action (22) we can reach theusual dynamical matter field strength term or

119894

2119865120583]119885120583]minus

119894

2119865lowast

120583]119885lowast120583]

997904rArr minus119865lowast

120583]119865120583] (23)

and so the 119885120583] tensor field is reabsorbed in this action

Likewise and without loss of generality we could have chosenthe inverse relation Σ = 119894119870

dagger what implies reabsorbing the 119865120583]tensor field Then by using the whole relation (12) in action(18) we find that the complex supersymmetric model for thematter vector field can be written as

119878119896 = int1198894119909 minus119865

lowast

120583] (119909) 119865120583](119909)

minus 2119894119877119886 (119909) 120590120583119886120597120583119877

(119909) minus 2119894119866119886 (119909) 120590120583119886120597120583119866

(119909)

+ 8119889lowast (119909) 119889 (119909)

(24)

where expressionminus119865lowast120583](119909)119865120583](119909) represents the usual kinetic

term of the vector matter field while the terms representwith the components 119877

119886 and 119866119886 the fermionic sector and

the last term corresponds to the auxiliary field 119889 term Tocompleteness we are going to introduce the massive actionterm in the model Observing the symmetries of nonchiralfields Σ and 119870 the massive supersymmetric term can besuitable defined as

119878119898 =1205722

2int119889

4119909119889

2120579119889

2120579 [ΣdaggerΣ+119870

dagger119870] (25)

where 1205722 is a mass parameter From nonchiral superfields Σand 119870 we can obtain the massive vector matter field term119861lowast

120583119861120583 as well as their supersymmetric partners In order to

perform it we are going to compute action (25) by employingcondition119870 = 119894Σ

dagger where one has that (12)ΣdaggerΣ+(12)119870dagger119870 =

ΣdaggerΣ and by applying definition (10) the full supersymmetric

matter vector field model can be then obtained from Diracsuperspinor fieldΨ associated to the nonchiral scalar fields Σand119870 in the following form

119878 = 119878119896 + 119878119898 = int1198894119909119889

2120579119889

2120579 (119894ΨΨ+120572

2ΣdaggerΣ) (26)

where the mass part of action can be obtained in componentfields as

119878119898 = 1205722int119889

4119909(119889lowast(119909) 119896 (119909) + 119889 (119909) 119896

lowast(119909)

minus14119896lowast(119909) ◻119896 (119909) minus

14119896 (119909) ◻119896

lowast(119909) + 120585119886 (119909) 119866

119886(119909)

+ 120585 119886 (119909) 119866119886

(119909) +119862119886 (119909) 119877119886(119909) +119862 119886 (119909) 119877

119886

(119909)

6 Advances in High Energy Physics

minus 2119894120585119886

(119909) 120590120583

119886119887120597120583120585119887(119909) minus 2119894119862 119886 (119909) 120590120583 119886119887120597120583119862

119887(119909)

+ 119861lowast

120583119861120583+119891lowast(119909) 119891 (119909) + 119897

lowast(119909) 119897 (119909))

(27)

Here we observe the mass term to 119861120583 119891(119909) and 119897(119909) fields

As in the usual supersymmetric models we note that massaction (27) also contributes to kinetic structure namelywith the terms minus(14)119896lowast(119909)◻119896(119909)minus2119894120585

119886

120590120583

119886119887120597120583120585119887 and

minus2119894119862 119886120590120583 119886119887120597120583119862119887 The action also shows mixing mass scalar

and fermionic terms namely 119889lowast(119909)119896(119909) 119889(119909)119896lowast(119909)

120585119886(119909)119866119886(119909) 120585 119886(119909)119866

119886

(119909)119862119886(119909)119877119886(119909) and 119862 119886(119909)119877

119886

(119909) Byverifying the presence of extra kinetic terms in (27) whatcan suggest that when we particularly treat supersymmetricmatter vector fields the mass action contributes witha ldquodynamic complementrdquo to the kinetic action (24)Furthermore we remark that mass action (27) is importantto match the number of bosonic and fermionic degrees offreedom of the supersymmetric matter vector action (26) forthe consistency of themodelWe can redefine the componentfields absorbing the mass parameter as follows

119896 (119909) 997888rarr1120572119896 (119909)

120585119886 (119909) 997888rarr1120572120585119886 (119909)

119862119886 (119909) 997888rarr1120572119862119886 (119909)

119891 (119909) 997888rarr1120572119891 (119909)

119897 (119909) 997888rarr1120572119897 (119909)

(28)

So the action can be rewritten as

119878ek =12int119889

4119909(minus119865

lowast

120583] (119909) 119865120583](119909)

minus 2119894119877119886 (119909) 120590120583119886120597120583119877

(119909) minus 2119894119866119886 (119909) 120590120583119886120597120583119866

(119909)

minus14119896lowast(119909) ◻119896 (119909) minus

14119896 (119909) ◻119896

lowast(119909)

+ minus2119894120585119886

(119909) 120590120583

119886119887120597120583120585119887(119909) minus 2119894119862 119886 (119909) 120590120583 119886119887120597120583119862

119887(119909)

+ 8119889lowast (119909) 119889 (119909) +119891lowast(119909) 119891 (119909) + 119897

lowast(119909) 119897 (119909))

(29)

analogously the mass action is now given by

119878em = int1198894119909 120572

2119861lowast

120583119861120583+120572119889lowast(119909) 119896 (119909)

+ 120572119889 (119909) 119896lowast(119909) + 120572120585119886 (119909) 119866

119886(119909) + 120572120585 119886 (119909) 119866

119886

(119909)

+ 120572119862119886 (119909) 119877119886(119909) + 120572119862 119886 (119909) 119877

119886

(119909)

(30)

so the complex scalar fields 119889(119909) 119891(119909) and 119897(119909) have nodynamics and arise as auxiliary fields Thus assuming theaction (26) to be the sum of the redefined actions (29) and(30) and rearranging (we adopt the Weyl representation tothe gamma matrices) the (Dirac) spinor fields Θ and Π Thesum of the action (29) and the action (30) results in an off-shell action 119878os written in the following form

119878os = int1198894119909 minus

12119865lowast

120583] (119909) 119865120583](119909) +

14120597120583119896lowast(119909) 120597120583119896 (119909)

minus 119894Θ (119909) 120574120583120597120583Θ (119909) minus 119894Π (119909) 120574

120583120597120583Π (119909)

+ 120572Θ (119909) 1205745Θ (119909) + 120572Π (119909) 120574

5Π (119909) + 4119889lowast (119909) 119889 (119909)

+12119891lowast(119909) 119891 (119909) +

12119897lowast(119909) 119897 (119909) + 120572

2119861lowast

120583119861120583

+120572119889lowast(119909) 119896 (119909) + 120572119889 (119909) 119896

lowast(119909)

(31)

where we denote the Dirac spinors of mass 120572 as

Θ (119909) = (

120585119886

119866119886)

Π (119909) = (

119862119886

119877119886)

(32)

For action (31) we have obtained chiral spinor mass termsgiven by 120572Θ(119909)120574

5Θ(119909) and 120572Π(119909)120574

5Π(119909) From the action

(31) we can obtain the equation of motion for the fields

120597120583119865120583](119909) + 120572

2119861](119909) = 0

120574120583120597120583Θ (119909) minus 120572120574

5Θ (119909) = 0

120574120583120597120583Π (119909) minus 120572120574

5Π (119909) = 0

◻119896 (119909) + 1205722119896 (119909) = 0

(33)

with 119891(119909) = 119897(119909) = 0 Taking off-shell action (31) wenote that it has 16 bosonic degrees of freedom concerningthe matter fields 119861120583(119909) 119896(119909) 119889(119909) 119891(119909) and 119897(119909) and theircomplex conjugated ones as well as 16 fermionic degreesof freedom for the Dirac spinor fields Θ(119909) and Π(119909) andtheir conjugated complex ones which is consistent with thesupersymmetry From equations of motion (33) we note thatthere are three auxiliary complex scalar fields 119889(119909) 119891(119909)and 119897(119909) and massive dynamical complex scalar field 119896(119909)Moreover as expected we have obtained a matter Proca-typeequation for field 119861120583 In this context it is interesting to notethat from the on-shell actionwe can easily extract from action(31) the supersymmetric generalization of matter vector fieldit is only possible if we include two dynamical Dirac chiralspinor fieldsΘ(119909) andΠ(119909) along a massive scalar field 119896(119909)Furthermore a peculiar aspect of the spinor fields Θ(119909) andΠ(119909) in the present case is that their mass terms (33) arisewith chiral structure due to the presence of the matrix 1205745

Advances in High Energy Physics 7

4 Conclusion

In this paper which is part of a program started in a previouswork [32] we propose to formulate and analyze the simpledynamics of chargeless vector matter field in a supersymmet-ric scenarioThere are threemotivations for this proposal thelack of this approach in the literature getting a clue on thepossibility of Lorentz symmetry violation in supersymmetrictheories and the role played by the simplest case of high-spin field in field and string theories To these aims wehave started from real nonchiral scalar superfield in orderto obtain real matter Proca-type field in a supersymmetricLagrangian generalization [32] In a straightforward waythe complex model was obtained extending the real scalarnonchiral superfields to the complex space [47ndash51]

The very interesting point in this enterprise is that inorder to obtain a complex Proca-type term we face anambiguous choice of which is the real or imaginary part of thefield This ambiguity reveals a symmetry which we representas a kind of simple Hodge-duality symmetry 119870 = 119894Σ

dagger Infact despite its apparent simplicity it is essential to match thenumber of degrees of freedom of the fermionic and bosonicsectors and to compose the complex (chargeless) Proca-type dynamical term with global 119880(1) symmetry As thissymmetry appears by construction it represents a differentapproach compared to Ferrara et alrsquos work [24] On theother hand as usual the supersymmetric mass term is alsoimportant to compose the dynamics of the supersymmetricmodel and of its component fields

We can conclude that in addition to the supersymmetricmass term to build supersymmetric matter chargeless vectorfield Lagrangian the Hodge-type duality symmetry is anessential ingredient to be further explored We can alsoobserve that as the usual super-QED model the complexdynamical Proca-type term emerges from a product of Diracsuperspinors ΨΨ We can remark that the Dirac superspinorfieldΨ is also chiral because it is a combination of chiralWeylsuperspinors Γ119886 and Υ

119886 Finally we emphasize that on-shellsupersymmetric action obtained (31) revealing two fermionicDirac fields Θ(119909) and Π(119909) and a massive scalar field 119896(119909)

as supersymmetric partners associated to the complex vectorfield 119861120583(119909) which is the initial step to implement localgauge symmetry charge and interaction we could anticipatethat nonminimal couplings have an essential role in aforthcoming work

Conflict of Interests

The authors declare that they have no conflict of interestsrelated to this paper

References

[1] C Itzykson and J-B Zuber Quantum Field Theory DoverPublications 2005

[2] P RamondFieldTheory AModern PrimerWestviewPress 2ndedition 1990

[3] N N Bogoliubov and D V Shirkov Introduction to the Theoryof Quantized Fields John Wiley amp Sons 3rd edition 1980

[4] P Fayet and S Ferrara ldquoSupersymmetryrdquo Physics Reports vol32 no 5 pp 249ndash334 1977

[5] H PNilles ldquoSupersymmetry supergravity andparticle physicsrdquoPhysics Reports vol 110 no 1-2 pp 1ndash162 1984

[6] P D DrsquoEath ldquoCanonical quantization of supergravityrdquo PhysicalReview D vol 29 no 10 pp 2199ndash2219 1984

[7] J-P Ader F Biet and Y Noirot ldquoSupersymmetric structure ofthe induced W gravitiesrdquo Classical and Quantum Gravity vol16 no 3 pp 1027ndash1037 1999

[8] C Kiefer T Luck and P Moniz ldquoSemiclassical approximationto supersymmetric quantumgravityrdquoPhysical ReviewD vol 72no 4 Article ID 045006 2005

[9] A Riotto ldquoMore relaxed supersymmetric electroweak baryoge-nesisrdquo Physical Review D vol 58 Article ID 095009 1998

[10] A Riotto ldquoSupersymmetric electroweak baryogenesis non-equilibrium field theory and quantum Boltzmann equationsrdquoNuclear Physics B vol 518 no 1-2 pp 339ndash360 1998

[11] I R Klebanov and A A Tseytlin ldquoGravity duals of supersym-metric SU(N)timesSU(N+M) gauge theoriesrdquo Nuclear Physics Bvol 578 no 1-2 pp 123ndash138 2000

[12] P H Chankowski A Falkowski S Pokorski and J WagnerldquoElectroweak symmetry breaking in supersymmetric modelswith heavy scalar superpartnersrdquo Physics Letters Section BNuclear Elementary Particle and High-Energy Physics vol 598no 3-4 pp 252ndash262 2004

[13] J D Edelstein and C Nunez ldquoSupersymmetric electroweakcosmic stringsrdquo Physical Review D vol 55 no 6 pp 3811ndash38191997

[14] L Girardello A Giveon M Porrati and A Zaffaroni ldquoNon-Abelian strong-weak coupling duality in (string-derived) N = 4supersymmetric Yang-Mills theoriesrdquo Physics Letters B vol 334no 3-4 pp 331ndash338 1994

[15] J Wess and J Bagger Supersymmetry and Supergravity Prince-ton University Press Princeton NJ USA 2nd edition 1992

[16] D Bailin and A Love Supersymmetric Gauge Field Theory andString Theory IOP Publishing 1994

[17] S Coleman and J Mandula ldquoAll possible symmetries of the Smatrixrdquo Physical Review vol 159 no 5 pp 1251ndash1256 1967

[18] S J Gates M T Grisaru M Rocek and W Siegel SuperspaceBenjaminCummings Reading 1983

[19] R Haag J T Lopuszanski and M Sohnius ldquoAll possiblegenerators of supersymmetries of the 119878-matrixrdquoNuclear PhysicsB vol 88 pp 257ndash274 1975

[20] S Weinberg The Quantum Theory of Fields vol 3 CambridgeUniversity Press Cambridge UK 2000

[21] J Wess and B Zumino ldquoSupergauge transformations in fourdimensionsrdquo Nuclear Physics B vol 70 no 1 pp 39ndash50 1974

[22] H J W Muller-Kirsten and A Wiedemann SupersymmetryWorld Scientific Singapore 1987

[23] S J Gates Jr ldquoWhy auxiliary fields matter the strange caseof the 4-D 119873 = 1 supersymmetric QCD effective action 2rdquoNuclear Physics B vol 485 no 1-2 pp 145ndash184 1997

[24] S Ferrara L Girardello and F Palumbo ldquoGeneralmass formulain broken supersymmetryrdquo Physical Review D vol 20 no 2 pp403ndash408 1979

[25] V A Kostelecky and S Samuel ldquoSpontaneous breaking ofLorentz symmetry in string theoryrdquo Physical Review D vol 39no 2 pp 683ndash685 1989

[26] V E R Lemes A L M A Nogueira and J A Helayel-NetoldquoSupersymmetric generalization of the tensor matter fieldsrdquo

8 Advances in High Energy Physics

International Journal of Modern Physics A vol 13 no 18 pp3145ndash3156 1998

[27] D Colladay and V A Kostelecky ldquoCPT violation and thestandard modelrdquo Physical Review D vol 55 pp 6760ndash67741997

[28] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standardmodelrdquo Physical ReviewD vol 58 no 11 ArticleID 116002 1998

[29] A A Andrianov and R Soldati ldquoLorentz symmetry breakingin Abelian vector-field models with Wess-Zumino interactionrdquoPhysical Review D vol 51 no 10 pp 5961ndash5964 1995

[30] A A Andrianov and R Soldati ldquoPatterns of Lorentz symmetrybreaking inQEDbyCPT-odd interactionrdquoPhysics Letters B vol435 no 3-4 pp 449ndash452 1998

[31] S M Carrol G B Field and R Jackiw ldquoLimits on a Lorentz-and parity-violating modification of electrodynamicsrdquo PhysicalReview D vol 41 no 4 pp 1231ndash1240 1990

[32] L P Colatto A L A Penna andW C Santos ldquoCharged tensormatter fields and Lorentz symmetry violation via spontaneoussymmetry breakingrdquo European Physical Journal C vol 36 no 1pp 79ndash87 2004

[33] H Belich J L Boldo L P Colatto J A Helayel-Neto and A LMANogueira ldquoSupersymmetric extension of the Lorentz- andCPT-violatingMaxwell-Chern-Simonsmodelrdquo Physical ReviewD vol 68 Article ID 065030 2003

[34] H Belich Jr T Costa-Soares M M Ferreira Jr J A Helayel-Neto and M T D Orlando ldquoLorentz-symmetry violation andelectrically charged vortices in the planar regimerdquo InternationalJournal of Modern Physics A vol 21 no 11 pp 2415ndash2429 2006

[35] M M Ferreira Jr ldquoElectron-electron interaction in a Maxwell-Chern-Simons model with a purely spacelike Lorentz-violatingbackgroundrdquo Physical Review D vol 71 no 4 Article ID045003 9 pages 2005

[36] H Belich T Costa-Soares M M Ferreira Jr and J A Helayel-Neto ldquoNon-minimal coupling to a Lorentz-violating back-ground and topological implicationsrdquo The European PhysicalJournal C vol 41 no 3 pp 421ndash426 2005

[37] M B Cantcheff ldquoLorentz symmetry breaking and planar effectsfrom non-linear electrodynamicsrdquoThe European Physical Jour-nal CmdashParticles and Fields vol 46 no 1 pp 247ndash254 2006

[38] H Belich M A De Andrade and M A Santos ldquoGauge the-ories with Lorentz-symmetry violation by symplectic projectormethodrdquoModern Physics Letters A vol 20 pp 2305ndash2316 2005

[39] T G Rizzo ldquoLorentz violation in extra dimensionsrdquo Journal ofHigh Energy Physics vol 2005 no 9 article 036 2005

[40] M M Ferreira Jr and M S Tavares ldquoNonrelativistic electronndashelectron interaction in a maxwellndashchernndashsimonsndashproca modelendowed with a timelike lorentz-violating backgroundrdquo Inter-national Journal of Modern Physics A vol 22 p 1685 2007

[41] H Belich T Costa-Soares M M Ferreira Jr J A Helayel-Neto and F M O Mouchereck ldquoLorentz-violating correctionson the hydrogen spectrum induced by a nonminimal couplingrdquoPhysical Review D vol 74 no 6 Article ID 065009 6 pages2006

[42] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoOn the consistency of Lorentz invariance violationin QED induced by fermions in constant axial-vector back-groundrdquo Physics Letters B vol 639 no 5 pp 586ndash590 2006

[43] L P Colatto A L A Penna and W C Santos ldquoVector solitonsand spontaneous Lorentz violation mechanismrdquo httparxivorgabshep-th0610096

[44] MM Ferreira Jr andA R Gomes ldquoLorentz violating effects ona quantized two-level systemrdquo httparxivorgabs08034010

[45] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoBare and induced lorentz and cpt invariance violationsin qedrdquo International Journal of Modern Physics A vol 25 no16 pp 3271ndash3306 2010

[46] T G Rizzo ldquoLorentz violation in warped extra dimensionsrdquoJournal of High Energy Physics vol 2010 no 11 article 156 2010

[47] J Helayel-Neto R Jengo F Legovini and S Pugnetti ldquoSup-ersymmetry Complex Structures And Superfieldsrdquo SISSA-3887EP

[48] Y Kobayashi and S Nagamachi ldquoGeneralized complexsuperspacemdashinvolutions of superfieldsrdquo Journal of Mathemati-cal Physics vol 28 no 8 pp 1700ndash1708 1987

[49] P A Grassi G Policastro and M Porrati ldquoNotes on thequantization of the complex linear superfieldrdquo Nuclear PhysicsB vol 597 no 1ndash3 pp 615ndash632 2001

[50] S M Kuzenko and S J Tyler ldquoComplex linear superfield as amodel for Goldstinordquo Journal of High Energy Physics vol 2011no 4 article 57 2011

[51] S J Gates J Hallett T Hubsch and K Stiffler ldquoThe realanatomy of complex linear superfieldsrdquo International Journal ofModern Physics A vol 27 no 24 Article ID 1250143 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Superconductivity

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ThermodynamicsJournal of

Advances in High Energy Physics 3

and by similarity for Λ we have that

Ω119886 = minus14119863119863119863119886Λ(119909

120583 120579119886 120579 119886)

Ω 119886 = minus14119863119863119863 119886Λ(119909

120583 120579119886 120579 119886)

(3)

the119882119886 andΩ119886 are chiral spinor superfieldsWe can redefine the superfields in the chiral superspace

coordinates namely Φ(119910120583 120579119886) Φ(119911

120583 120579 119886) Λ(119910

120583 120579119886) and

Λ(119911120583 120579 119886) such that 119910120583 = 119909

120583+ 119894120579120590120583120579 and 119911

120583= 119909120583minus 119894120579120590120583120579

Hence the supersymmetric covariant derivatives are definedas

119863119886 =120597

120597120579119886

+ 2119894120590120583119886 119886120579119886 120597

120597119910120583

119863 119886 = minus120597

120597120579 119886

minus 2119894120579119886120590120583119886 119886120597

120597119911120583

(4)

According to these definitions we can compute the field-strengths119882119886 andΩ119886 and we have

119882119886 (119910 120579) = 120582119886 (119910) + 2120579119886119863(119910) + (120590120583]120579)119886119883120583] (119910)

minus 1198941205792120590120583

119886120597120583120582

(119910)

Ω119886 (119910 120579) = 120577119886 (119910) + 2120579119886119878 (119910) + (120590120583]120579)119886119884120583] (119910)

minus 1198941205792120590120583

119886120597120583120577

(119910)

(5)

and in a similar way we can compute the field-strengths119882 119886 and Ω 119886 So we are in conditions to construct thesupersymmetric model in terms of the superfields 119882119886 andΩ119886 where chargeless vectormatter field is presentThe kineticpart can be written as

119878kin = int1198894119909 119889

2120579 (119882119886119882

119886+Ω119886Ω

119886)

+ 1198892120579 (119882 119886119882

119886

+Ω 119886Ω119886

)

(6)

We have adopted the usual conventions for the spinor alge-bra for the superspace parametrization and the translationinvariance of the integral in the chiral coordinates [1622] Then we obtain that expression (6) has the followingcomponent expansion

119878kin = int1198894119909minus119883120583] (119909)119883

120583](119909) minus119884120583] (119909) 119884

120583](119909)

minus 4119894120582119886 (119909) 120590120583119886120597120583120582

(119909) minus 4119894120577119886 (119909) 120590120583119886120597120583120577(119909)

+ 81198632(119909) + 81198782 (119909)

(7)

Action (7) describes the kinetic part of supersymmetricchargeless vector field However to write the full action

that corresponds to the underlined field theory we can alsoconsider the supersymmetric mass term given by

119878119898 = int1198894119909119889

2120579120572

2[Φ

2+Λ

2] = int119889

4119909120572

2[119862 (119909)119863 (119909)

minus14119862 (119909) ◻119862 (119909) + 120593119886 (119909) 120582

119886(119909) + 120593

119886(119909) 120582

119886

(119909)

minus 119894120593119886120590120583

119886119887120597120583120593119887+119872(119909)

2+119873 (119909)

2+14119883120583119883120583

+14119884120583119884120583 +119860 (119909) 119878 (119909) minus

14119860 (119909) ◻119860 (119909)

+ 120594119886 (119909) 120577119886(119909) + 120594

119886(119909) 120577

119886

(119909) minus 119894120594119886120590120583

119886119887120597120583120594119887

+120588 (119909)2+ 120591 (119909)

2]

(8)

where 1205722 is the mass parameter As usual the ldquomassrdquo part

of the action presents kinetic terms beyond the usual massterms which were eliminated by spinor chirality property ofsuperfields 119882119886 and Ω119886 Furthermore we could infer that themass term in action (8) arises as a dynamical complement tothe supersymmetric vector matter fields Indeed supersym-metric matter-like fields are formulated with chiral spinorsuperfields

3 The Chargeless Complex VectorMatter Superfield Model

We know that the supersymmetric action for two free vectormatter fields might be built through nonchiral scalar super-fields [22] Moreover we can see that the degrees of freedomof this model are compatible with the dynamical free fieldsin complex space C In this section our aim is to derive theappropriated complex superfield to describe supersymmetriccomplex vector fields To this aim we need to strongly definetwo complex nonchiral scalar superfields defined as

Σ (119909120583 120579119886 120579 119886) = 119896 (119909) + 120579119886120585119886 (119909) + 120579

119886

119862 119886 (119909) + 1205792119897 (119909)

+ 1205792119891 (119909)

+ 1205792120579119886

[119866 119886 (119909) minus119894

2120590120583

119886 119886120597120583120585119886(119909)]

+ 1205792120579119886[119877119886 (119909) minus

119894

2120590120583

119886 119886120597120583119862119886

]

+ 120579119886120590120583

119886 119886120579119886

119861120583 (119909)

+ 12057921205792[119889 (119909) minus

14◻119896 (119909)]

(9)

119870(119909120583 120579119886 120579 119886) = 119886 (119909) + 120579119886119879119886 (119909) + 120579

119886

119867 119886 (119909)

+ 1205792119895 (119909) + 120579

2119864 (119909)

+ 1205792120579119886

[119876 119886 (119909) minus119894

2120590120583

119886 119886120597120583119879119886(119909)]

4 Advances in High Energy Physics

+ 1205792120579119886[119880119886 (119909) minus

119894

2120590120583

119886 119886120597120583119867119886

(119909)]

+ 120579119886120590120583

119886 119886120579119886

119885120583 (119909)

+ 12057921205792[V (119909) minus

14◻119886 (119909)]

(10)

and in a similar way we can define the complex conjugatedsuperfields

Superfields (9) and (10) present multiplets with complexvector fields 119861120583(119909) and 119885120583(119909) and spin-1 the 120585119886(119909) 119862 119886(119909)119877119886(119909) 119866 119886(119909) 119879119886(119909) 119867 119886(119909) 119880119886(119909) and 119876 119886(119909) are Weylfermion fields with spin-12 and the 119896(119909) 119889(119909) 119886(119909) andV(119909) are complex scalar fields with spin-0 So the rule thatimplies an invariant mechanism is

119870 = 119894Σdagger (11)

We observe that transformation rule (11) guarantees writinga consistent kinetic term for the complex vector field withoutbreaking the global119880(1) gauge symmetry Another advantagethat came to light is that transformations (11) eliminatethe exceeding fields which does not contribute for thesupersymmetric action which allows bosons and fermionsto have the same physical degrees of freedom Indeed theconstraint relation to the superfields implies the relations ofthe component fields as follows

119886 (119909) = 119894119896lowast(119909)

119879119886 (119909) = 119894119862119886 (119909)

V (119909) = 119894119889lowast(119909)

119867119886 (119909) = 119894120585119886 (119909)

119895 (119909) = 119894119891lowast(119909)

119880119886 (119909) = 119894119866119886 (119909)

119864 (119909) = 119894119897lowast(119909)

119876119886 (119909) = 119894119877119886 (119909)

119885120583 (119909) = 119894119861lowast

120583(119909)

(12)

So we can adjust the complex extension of the chargelesssuperfields Φ and Λ by assuming the equation Σ = Φ + 119894Λwhere we find the following relation of fields

119896 (119909) = 119862 (119909) + 119894119860 (119909)

120585119886 (119909) = 120593119886 (119909) + 119894120594119886 (119909)

119862119886 (119909) = 120593119886(119909) + 119894120594

119886(119909)

119897 (119909) = 119898 (119909) + 119899 (119909) + 119894 (120588 (119909) + 120591 (119909))

119866119886 (119909) = 120582119886 (119909) + 119894120577119886 (119909)

119891 (119909) = 119898 (119909) + 119899 (119909) minus 119894 (120588 (119909) + 120591 (119909))

119877119886 (119909) = 120582119886 (119909) + 119894120577119886 (119909)

119861120583 (119909) = 119883120583 (119909) + 119894119884120583 (119909)

119889 (119909) = 119863 (119909) + 119894119878 (119909)

(13)

To describe the dynamics of the supersymmetric complexvector fields with matter symmetry we need to construct anappropriated complex supersymmetric field-strength modelin order to accommodate superfields Σ and 119870 This can bereached starting from the following definitions

Υ119886 = minus14119863119863119863119886Σ (119909120583 120579119886 120579 119886)

Υ 119886 = minus14119863119863119863 119886Σ

dagger(119909120583 120579119886 120579 119886)

Γ119886 = minus14119863119863119863119886119870(119909120583 120579119886 120579 119886)

Γ 119886 = minus14119863119863119863 119886119870

dagger(119909120583 120579119886 120579 119886)

(14)

where Υ119886 and Γ119886 are charged spinor superfields As a conse-quence of the complex extension procedure we must relatechargeless spinor superfields Ω119886 and 119882119886 with the complexdefinitions (14) which in the simplest way is

Υ119886 = 119882119886 + 119894Ω119886

Υ 119886 = 119882 119886 minus 119894Ω 119886

Γ119886 = Ω119886 + 119894119882119886

Γ 119886 = Ω 119886 minus 119894119882 119886

(15)

and by assuming the spinor identities 119882119886Ω119886= Ω119886119882

119886 and119882 119886Ω

119886

= Ω 119886119882119886 we can find the kinetic supersymmetric

Lagrangian for the complex vector fields

L119896 = 119894 (Υ 119886Γ119886

minusΥ119886Γ119886)

= 119882119886119882119886+119882 119886119882

119886

+Ω119886Ω119886+Ω 119886Ω

119886

(16)

We can observe that the left-hand side of the latterequation is the complex extension of chargeless Lagrangian(7) that was written in terms of charged spinor superfieldsBearing this in mind we can then redefine the kineticLagrangian (16) simply by combining the charged spinorsuperfields Υ119886 and Γ119886 as a ldquoDirac superspinorrdquo Ψ such that

Ψ(119909120583 120579119886 120579 119886) = (

Υ119886

Γ119886) (17)

and also we assume Ψ to be the adjoint Dirac superspinorrepresentation

Advances in High Energy Physics 5

In this case we have that Ψ = Ψdagger1205740= (Γ119886Υ 119886) and so

the supersymmetric action from the kinetic Lagrangian (16)is now given by

119878119896 = int1198894119909119889

2120579119889

2120579 (119894ΨΨ)

= 119894 int 1198894119909119889

2120579119889

2120579 (Υ 119886Γ

119886

minusΥ119886Γ119886)

(18)

We can note that the product of Dirac superspinorsΨΨ obeysmatter symmetry and it presents an interesting analogy tocharged scalar superfield product 119878dagger119878 In this sense we verifythatΨ andΨ represent two chiral supersymmetric extensionsfor the matter vector field which can be transformed under119880(1) global gauge group in the following way

Ψ1015840= 119890minus2119894119902120573

Ψ

Ψ1015840

= Ψ1198902119894119902120573dagger

(19)

where120573 is a global119880(1) gauge parameter and 119902 is the charge ofthe global symmetry We can emphasize that the expressions(19) represent that each of the components of multiplet (17)has the same symmetry So action (18) is then invariantunder transformations (19) In order to obtain the componentLagrangian we can expand the product ΨΨ by consideringthat

Υ119886 (119910 120579) = 119877119886 (119910) + 2120579119886119889 (119910) + (120590120583]120579)119886119865120583] (119910)

minus 1198941205792120590120583

119886120597120583119866

(119910)

Γ119886 (119910 120579) = 119880119886 (119910) + 2120579119886V (119910) + (120590120583]120579)119886119885120583] (119910)

minus 1198941205792120590120583

119886120597120583119876

(119910)

(20)

and similarly for Υ 119886 and Γ 119886We note the presence of the complex matter field-

strengths namely

119865120583] = 120597120583119861] minus 120597]119861120583

119885120583] = 120597120583119885] minus 120597]119885120583

(21)

hence action (18) can be expanded and we obtain

119878119896 = int1198894119909

119894

2119865120583] (119909) 119885

120583](119909) minus

119894

2119865lowast

120583] (119909) 119885lowast120583]

(119909)

minus119877119886(119909) 120590120583

119886120597120583119876

(119909) minus119880119886(119909) 120590120583

119886120597120583119866

(119909)

+119877119886

(119909) 120590120583

119886119887120597120583119876119887(119909) +119880

119886

(119909) 120590120583

119886119887120597120583119866119887(119909)

+ 4119894V (119909) 119889 (119909) minus 4119894Vlowast (119909) 119889lowast (119909)

(22)

In this format we can recognize the dynamical termthat describes the matter vector field as (1198942)119865120583]119885

120583]minus

(1198942)119865lowast120583]119885lowast120583] It involves both 119865120583] and 119885120583] matter tensors

However it does not correspond to the conventional kineticterm for the matter vector field and action (18) shows moredegrees of freedom than necessary In order to get rid of suchfields we must assume the rule of transformation (11) whichis a constraint of half of the degrees and consequently action(18) reaches the correct number of component fields

Applying condition (11) in action (22) we can reach theusual dynamical matter field strength term or

119894

2119865120583]119885120583]minus

119894

2119865lowast

120583]119885lowast120583]

997904rArr minus119865lowast

120583]119865120583] (23)

and so the 119885120583] tensor field is reabsorbed in this action

Likewise and without loss of generality we could have chosenthe inverse relation Σ = 119894119870

dagger what implies reabsorbing the 119865120583]tensor field Then by using the whole relation (12) in action(18) we find that the complex supersymmetric model for thematter vector field can be written as

119878119896 = int1198894119909 minus119865

lowast

120583] (119909) 119865120583](119909)

minus 2119894119877119886 (119909) 120590120583119886120597120583119877

(119909) minus 2119894119866119886 (119909) 120590120583119886120597120583119866

(119909)

+ 8119889lowast (119909) 119889 (119909)

(24)

where expressionminus119865lowast120583](119909)119865120583](119909) represents the usual kinetic

term of the vector matter field while the terms representwith the components 119877

119886 and 119866119886 the fermionic sector and

the last term corresponds to the auxiliary field 119889 term Tocompleteness we are going to introduce the massive actionterm in the model Observing the symmetries of nonchiralfields Σ and 119870 the massive supersymmetric term can besuitable defined as

119878119898 =1205722

2int119889

4119909119889

2120579119889

2120579 [ΣdaggerΣ+119870

dagger119870] (25)

where 1205722 is a mass parameter From nonchiral superfields Σand 119870 we can obtain the massive vector matter field term119861lowast

120583119861120583 as well as their supersymmetric partners In order to

perform it we are going to compute action (25) by employingcondition119870 = 119894Σ

dagger where one has that (12)ΣdaggerΣ+(12)119870dagger119870 =

ΣdaggerΣ and by applying definition (10) the full supersymmetric

matter vector field model can be then obtained from Diracsuperspinor fieldΨ associated to the nonchiral scalar fields Σand119870 in the following form

119878 = 119878119896 + 119878119898 = int1198894119909119889

2120579119889

2120579 (119894ΨΨ+120572

2ΣdaggerΣ) (26)

where the mass part of action can be obtained in componentfields as

119878119898 = 1205722int119889

4119909(119889lowast(119909) 119896 (119909) + 119889 (119909) 119896

lowast(119909)

minus14119896lowast(119909) ◻119896 (119909) minus

14119896 (119909) ◻119896

lowast(119909) + 120585119886 (119909) 119866

119886(119909)

+ 120585 119886 (119909) 119866119886

(119909) +119862119886 (119909) 119877119886(119909) +119862 119886 (119909) 119877

119886

(119909)

6 Advances in High Energy Physics

minus 2119894120585119886

(119909) 120590120583

119886119887120597120583120585119887(119909) minus 2119894119862 119886 (119909) 120590120583 119886119887120597120583119862

119887(119909)

+ 119861lowast

120583119861120583+119891lowast(119909) 119891 (119909) + 119897

lowast(119909) 119897 (119909))

(27)

Here we observe the mass term to 119861120583 119891(119909) and 119897(119909) fields

As in the usual supersymmetric models we note that massaction (27) also contributes to kinetic structure namelywith the terms minus(14)119896lowast(119909)◻119896(119909)minus2119894120585

119886

120590120583

119886119887120597120583120585119887 and

minus2119894119862 119886120590120583 119886119887120597120583119862119887 The action also shows mixing mass scalar

and fermionic terms namely 119889lowast(119909)119896(119909) 119889(119909)119896lowast(119909)

120585119886(119909)119866119886(119909) 120585 119886(119909)119866

119886

(119909)119862119886(119909)119877119886(119909) and 119862 119886(119909)119877

119886

(119909) Byverifying the presence of extra kinetic terms in (27) whatcan suggest that when we particularly treat supersymmetricmatter vector fields the mass action contributes witha ldquodynamic complementrdquo to the kinetic action (24)Furthermore we remark that mass action (27) is importantto match the number of bosonic and fermionic degrees offreedom of the supersymmetric matter vector action (26) forthe consistency of themodelWe can redefine the componentfields absorbing the mass parameter as follows

119896 (119909) 997888rarr1120572119896 (119909)

120585119886 (119909) 997888rarr1120572120585119886 (119909)

119862119886 (119909) 997888rarr1120572119862119886 (119909)

119891 (119909) 997888rarr1120572119891 (119909)

119897 (119909) 997888rarr1120572119897 (119909)

(28)

So the action can be rewritten as

119878ek =12int119889

4119909(minus119865

lowast

120583] (119909) 119865120583](119909)

minus 2119894119877119886 (119909) 120590120583119886120597120583119877

(119909) minus 2119894119866119886 (119909) 120590120583119886120597120583119866

(119909)

minus14119896lowast(119909) ◻119896 (119909) minus

14119896 (119909) ◻119896

lowast(119909)

+ minus2119894120585119886

(119909) 120590120583

119886119887120597120583120585119887(119909) minus 2119894119862 119886 (119909) 120590120583 119886119887120597120583119862

119887(119909)

+ 8119889lowast (119909) 119889 (119909) +119891lowast(119909) 119891 (119909) + 119897

lowast(119909) 119897 (119909))

(29)

analogously the mass action is now given by

119878em = int1198894119909 120572

2119861lowast

120583119861120583+120572119889lowast(119909) 119896 (119909)

+ 120572119889 (119909) 119896lowast(119909) + 120572120585119886 (119909) 119866

119886(119909) + 120572120585 119886 (119909) 119866

119886

(119909)

+ 120572119862119886 (119909) 119877119886(119909) + 120572119862 119886 (119909) 119877

119886

(119909)

(30)

so the complex scalar fields 119889(119909) 119891(119909) and 119897(119909) have nodynamics and arise as auxiliary fields Thus assuming theaction (26) to be the sum of the redefined actions (29) and(30) and rearranging (we adopt the Weyl representation tothe gamma matrices) the (Dirac) spinor fields Θ and Π Thesum of the action (29) and the action (30) results in an off-shell action 119878os written in the following form

119878os = int1198894119909 minus

12119865lowast

120583] (119909) 119865120583](119909) +

14120597120583119896lowast(119909) 120597120583119896 (119909)

minus 119894Θ (119909) 120574120583120597120583Θ (119909) minus 119894Π (119909) 120574

120583120597120583Π (119909)

+ 120572Θ (119909) 1205745Θ (119909) + 120572Π (119909) 120574

5Π (119909) + 4119889lowast (119909) 119889 (119909)

+12119891lowast(119909) 119891 (119909) +

12119897lowast(119909) 119897 (119909) + 120572

2119861lowast

120583119861120583

+120572119889lowast(119909) 119896 (119909) + 120572119889 (119909) 119896

lowast(119909)

(31)

where we denote the Dirac spinors of mass 120572 as

Θ (119909) = (

120585119886

119866119886)

Π (119909) = (

119862119886

119877119886)

(32)

For action (31) we have obtained chiral spinor mass termsgiven by 120572Θ(119909)120574

5Θ(119909) and 120572Π(119909)120574

5Π(119909) From the action

(31) we can obtain the equation of motion for the fields

120597120583119865120583](119909) + 120572

2119861](119909) = 0

120574120583120597120583Θ (119909) minus 120572120574

5Θ (119909) = 0

120574120583120597120583Π (119909) minus 120572120574

5Π (119909) = 0

◻119896 (119909) + 1205722119896 (119909) = 0

(33)

with 119891(119909) = 119897(119909) = 0 Taking off-shell action (31) wenote that it has 16 bosonic degrees of freedom concerningthe matter fields 119861120583(119909) 119896(119909) 119889(119909) 119891(119909) and 119897(119909) and theircomplex conjugated ones as well as 16 fermionic degreesof freedom for the Dirac spinor fields Θ(119909) and Π(119909) andtheir conjugated complex ones which is consistent with thesupersymmetry From equations of motion (33) we note thatthere are three auxiliary complex scalar fields 119889(119909) 119891(119909)and 119897(119909) and massive dynamical complex scalar field 119896(119909)Moreover as expected we have obtained a matter Proca-typeequation for field 119861120583 In this context it is interesting to notethat from the on-shell actionwe can easily extract from action(31) the supersymmetric generalization of matter vector fieldit is only possible if we include two dynamical Dirac chiralspinor fieldsΘ(119909) andΠ(119909) along a massive scalar field 119896(119909)Furthermore a peculiar aspect of the spinor fields Θ(119909) andΠ(119909) in the present case is that their mass terms (33) arisewith chiral structure due to the presence of the matrix 1205745

Advances in High Energy Physics 7

4 Conclusion

In this paper which is part of a program started in a previouswork [32] we propose to formulate and analyze the simpledynamics of chargeless vector matter field in a supersymmet-ric scenarioThere are threemotivations for this proposal thelack of this approach in the literature getting a clue on thepossibility of Lorentz symmetry violation in supersymmetrictheories and the role played by the simplest case of high-spin field in field and string theories To these aims wehave started from real nonchiral scalar superfield in orderto obtain real matter Proca-type field in a supersymmetricLagrangian generalization [32] In a straightforward waythe complex model was obtained extending the real scalarnonchiral superfields to the complex space [47ndash51]

The very interesting point in this enterprise is that inorder to obtain a complex Proca-type term we face anambiguous choice of which is the real or imaginary part of thefield This ambiguity reveals a symmetry which we representas a kind of simple Hodge-duality symmetry 119870 = 119894Σ

dagger Infact despite its apparent simplicity it is essential to match thenumber of degrees of freedom of the fermionic and bosonicsectors and to compose the complex (chargeless) Proca-type dynamical term with global 119880(1) symmetry As thissymmetry appears by construction it represents a differentapproach compared to Ferrara et alrsquos work [24] On theother hand as usual the supersymmetric mass term is alsoimportant to compose the dynamics of the supersymmetricmodel and of its component fields

We can conclude that in addition to the supersymmetricmass term to build supersymmetric matter chargeless vectorfield Lagrangian the Hodge-type duality symmetry is anessential ingredient to be further explored We can alsoobserve that as the usual super-QED model the complexdynamical Proca-type term emerges from a product of Diracsuperspinors ΨΨ We can remark that the Dirac superspinorfieldΨ is also chiral because it is a combination of chiralWeylsuperspinors Γ119886 and Υ

119886 Finally we emphasize that on-shellsupersymmetric action obtained (31) revealing two fermionicDirac fields Θ(119909) and Π(119909) and a massive scalar field 119896(119909)

as supersymmetric partners associated to the complex vectorfield 119861120583(119909) which is the initial step to implement localgauge symmetry charge and interaction we could anticipatethat nonminimal couplings have an essential role in aforthcoming work

Conflict of Interests

The authors declare that they have no conflict of interestsrelated to this paper

References

[1] C Itzykson and J-B Zuber Quantum Field Theory DoverPublications 2005

[2] P RamondFieldTheory AModern PrimerWestviewPress 2ndedition 1990

[3] N N Bogoliubov and D V Shirkov Introduction to the Theoryof Quantized Fields John Wiley amp Sons 3rd edition 1980

[4] P Fayet and S Ferrara ldquoSupersymmetryrdquo Physics Reports vol32 no 5 pp 249ndash334 1977

[5] H PNilles ldquoSupersymmetry supergravity andparticle physicsrdquoPhysics Reports vol 110 no 1-2 pp 1ndash162 1984

[6] P D DrsquoEath ldquoCanonical quantization of supergravityrdquo PhysicalReview D vol 29 no 10 pp 2199ndash2219 1984

[7] J-P Ader F Biet and Y Noirot ldquoSupersymmetric structure ofthe induced W gravitiesrdquo Classical and Quantum Gravity vol16 no 3 pp 1027ndash1037 1999

[8] C Kiefer T Luck and P Moniz ldquoSemiclassical approximationto supersymmetric quantumgravityrdquoPhysical ReviewD vol 72no 4 Article ID 045006 2005

[9] A Riotto ldquoMore relaxed supersymmetric electroweak baryoge-nesisrdquo Physical Review D vol 58 Article ID 095009 1998

[10] A Riotto ldquoSupersymmetric electroweak baryogenesis non-equilibrium field theory and quantum Boltzmann equationsrdquoNuclear Physics B vol 518 no 1-2 pp 339ndash360 1998

[11] I R Klebanov and A A Tseytlin ldquoGravity duals of supersym-metric SU(N)timesSU(N+M) gauge theoriesrdquo Nuclear Physics Bvol 578 no 1-2 pp 123ndash138 2000

[12] P H Chankowski A Falkowski S Pokorski and J WagnerldquoElectroweak symmetry breaking in supersymmetric modelswith heavy scalar superpartnersrdquo Physics Letters Section BNuclear Elementary Particle and High-Energy Physics vol 598no 3-4 pp 252ndash262 2004

[13] J D Edelstein and C Nunez ldquoSupersymmetric electroweakcosmic stringsrdquo Physical Review D vol 55 no 6 pp 3811ndash38191997

[14] L Girardello A Giveon M Porrati and A Zaffaroni ldquoNon-Abelian strong-weak coupling duality in (string-derived) N = 4supersymmetric Yang-Mills theoriesrdquo Physics Letters B vol 334no 3-4 pp 331ndash338 1994

[15] J Wess and J Bagger Supersymmetry and Supergravity Prince-ton University Press Princeton NJ USA 2nd edition 1992

[16] D Bailin and A Love Supersymmetric Gauge Field Theory andString Theory IOP Publishing 1994

[17] S Coleman and J Mandula ldquoAll possible symmetries of the Smatrixrdquo Physical Review vol 159 no 5 pp 1251ndash1256 1967

[18] S J Gates M T Grisaru M Rocek and W Siegel SuperspaceBenjaminCummings Reading 1983

[19] R Haag J T Lopuszanski and M Sohnius ldquoAll possiblegenerators of supersymmetries of the 119878-matrixrdquoNuclear PhysicsB vol 88 pp 257ndash274 1975

[20] S Weinberg The Quantum Theory of Fields vol 3 CambridgeUniversity Press Cambridge UK 2000

[21] J Wess and B Zumino ldquoSupergauge transformations in fourdimensionsrdquo Nuclear Physics B vol 70 no 1 pp 39ndash50 1974

[22] H J W Muller-Kirsten and A Wiedemann SupersymmetryWorld Scientific Singapore 1987

[23] S J Gates Jr ldquoWhy auxiliary fields matter the strange caseof the 4-D 119873 = 1 supersymmetric QCD effective action 2rdquoNuclear Physics B vol 485 no 1-2 pp 145ndash184 1997

[24] S Ferrara L Girardello and F Palumbo ldquoGeneralmass formulain broken supersymmetryrdquo Physical Review D vol 20 no 2 pp403ndash408 1979

[25] V A Kostelecky and S Samuel ldquoSpontaneous breaking ofLorentz symmetry in string theoryrdquo Physical Review D vol 39no 2 pp 683ndash685 1989

[26] V E R Lemes A L M A Nogueira and J A Helayel-NetoldquoSupersymmetric generalization of the tensor matter fieldsrdquo

8 Advances in High Energy Physics

International Journal of Modern Physics A vol 13 no 18 pp3145ndash3156 1998

[27] D Colladay and V A Kostelecky ldquoCPT violation and thestandard modelrdquo Physical Review D vol 55 pp 6760ndash67741997

[28] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standardmodelrdquo Physical ReviewD vol 58 no 11 ArticleID 116002 1998

[29] A A Andrianov and R Soldati ldquoLorentz symmetry breakingin Abelian vector-field models with Wess-Zumino interactionrdquoPhysical Review D vol 51 no 10 pp 5961ndash5964 1995

[30] A A Andrianov and R Soldati ldquoPatterns of Lorentz symmetrybreaking inQEDbyCPT-odd interactionrdquoPhysics Letters B vol435 no 3-4 pp 449ndash452 1998

[31] S M Carrol G B Field and R Jackiw ldquoLimits on a Lorentz-and parity-violating modification of electrodynamicsrdquo PhysicalReview D vol 41 no 4 pp 1231ndash1240 1990

[32] L P Colatto A L A Penna andW C Santos ldquoCharged tensormatter fields and Lorentz symmetry violation via spontaneoussymmetry breakingrdquo European Physical Journal C vol 36 no 1pp 79ndash87 2004

[33] H Belich J L Boldo L P Colatto J A Helayel-Neto and A LMANogueira ldquoSupersymmetric extension of the Lorentz- andCPT-violatingMaxwell-Chern-Simonsmodelrdquo Physical ReviewD vol 68 Article ID 065030 2003

[34] H Belich Jr T Costa-Soares M M Ferreira Jr J A Helayel-Neto and M T D Orlando ldquoLorentz-symmetry violation andelectrically charged vortices in the planar regimerdquo InternationalJournal of Modern Physics A vol 21 no 11 pp 2415ndash2429 2006

[35] M M Ferreira Jr ldquoElectron-electron interaction in a Maxwell-Chern-Simons model with a purely spacelike Lorentz-violatingbackgroundrdquo Physical Review D vol 71 no 4 Article ID045003 9 pages 2005

[36] H Belich T Costa-Soares M M Ferreira Jr and J A Helayel-Neto ldquoNon-minimal coupling to a Lorentz-violating back-ground and topological implicationsrdquo The European PhysicalJournal C vol 41 no 3 pp 421ndash426 2005

[37] M B Cantcheff ldquoLorentz symmetry breaking and planar effectsfrom non-linear electrodynamicsrdquoThe European Physical Jour-nal CmdashParticles and Fields vol 46 no 1 pp 247ndash254 2006

[38] H Belich M A De Andrade and M A Santos ldquoGauge the-ories with Lorentz-symmetry violation by symplectic projectormethodrdquoModern Physics Letters A vol 20 pp 2305ndash2316 2005

[39] T G Rizzo ldquoLorentz violation in extra dimensionsrdquo Journal ofHigh Energy Physics vol 2005 no 9 article 036 2005

[40] M M Ferreira Jr and M S Tavares ldquoNonrelativistic electronndashelectron interaction in a maxwellndashchernndashsimonsndashproca modelendowed with a timelike lorentz-violating backgroundrdquo Inter-national Journal of Modern Physics A vol 22 p 1685 2007

[41] H Belich T Costa-Soares M M Ferreira Jr J A Helayel-Neto and F M O Mouchereck ldquoLorentz-violating correctionson the hydrogen spectrum induced by a nonminimal couplingrdquoPhysical Review D vol 74 no 6 Article ID 065009 6 pages2006

[42] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoOn the consistency of Lorentz invariance violationin QED induced by fermions in constant axial-vector back-groundrdquo Physics Letters B vol 639 no 5 pp 586ndash590 2006

[43] L P Colatto A L A Penna and W C Santos ldquoVector solitonsand spontaneous Lorentz violation mechanismrdquo httparxivorgabshep-th0610096

[44] MM Ferreira Jr andA R Gomes ldquoLorentz violating effects ona quantized two-level systemrdquo httparxivorgabs08034010

[45] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoBare and induced lorentz and cpt invariance violationsin qedrdquo International Journal of Modern Physics A vol 25 no16 pp 3271ndash3306 2010

[46] T G Rizzo ldquoLorentz violation in warped extra dimensionsrdquoJournal of High Energy Physics vol 2010 no 11 article 156 2010

[47] J Helayel-Neto R Jengo F Legovini and S Pugnetti ldquoSup-ersymmetry Complex Structures And Superfieldsrdquo SISSA-3887EP

[48] Y Kobayashi and S Nagamachi ldquoGeneralized complexsuperspacemdashinvolutions of superfieldsrdquo Journal of Mathemati-cal Physics vol 28 no 8 pp 1700ndash1708 1987

[49] P A Grassi G Policastro and M Porrati ldquoNotes on thequantization of the complex linear superfieldrdquo Nuclear PhysicsB vol 597 no 1ndash3 pp 615ndash632 2001

[50] S M Kuzenko and S J Tyler ldquoComplex linear superfield as amodel for Goldstinordquo Journal of High Energy Physics vol 2011no 4 article 57 2011

[51] S J Gates J Hallett T Hubsch and K Stiffler ldquoThe realanatomy of complex linear superfieldsrdquo International Journal ofModern Physics A vol 27 no 24 Article ID 1250143 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

4 Advances in High Energy Physics

+ 1205792120579119886[119880119886 (119909) minus

119894

2120590120583

119886 119886120597120583119867119886

(119909)]

+ 120579119886120590120583

119886 119886120579119886

119885120583 (119909)

+ 12057921205792[V (119909) minus

14◻119886 (119909)]

(10)

and in a similar way we can define the complex conjugatedsuperfields

Superfields (9) and (10) present multiplets with complexvector fields 119861120583(119909) and 119885120583(119909) and spin-1 the 120585119886(119909) 119862 119886(119909)119877119886(119909) 119866 119886(119909) 119879119886(119909) 119867 119886(119909) 119880119886(119909) and 119876 119886(119909) are Weylfermion fields with spin-12 and the 119896(119909) 119889(119909) 119886(119909) andV(119909) are complex scalar fields with spin-0 So the rule thatimplies an invariant mechanism is

119870 = 119894Σdagger (11)

We observe that transformation rule (11) guarantees writinga consistent kinetic term for the complex vector field withoutbreaking the global119880(1) gauge symmetry Another advantagethat came to light is that transformations (11) eliminatethe exceeding fields which does not contribute for thesupersymmetric action which allows bosons and fermionsto have the same physical degrees of freedom Indeed theconstraint relation to the superfields implies the relations ofthe component fields as follows

119886 (119909) = 119894119896lowast(119909)

119879119886 (119909) = 119894119862119886 (119909)

V (119909) = 119894119889lowast(119909)

119867119886 (119909) = 119894120585119886 (119909)

119895 (119909) = 119894119891lowast(119909)

119880119886 (119909) = 119894119866119886 (119909)

119864 (119909) = 119894119897lowast(119909)

119876119886 (119909) = 119894119877119886 (119909)

119885120583 (119909) = 119894119861lowast

120583(119909)

(12)

So we can adjust the complex extension of the chargelesssuperfields Φ and Λ by assuming the equation Σ = Φ + 119894Λwhere we find the following relation of fields

119896 (119909) = 119862 (119909) + 119894119860 (119909)

120585119886 (119909) = 120593119886 (119909) + 119894120594119886 (119909)

119862119886 (119909) = 120593119886(119909) + 119894120594

119886(119909)

119897 (119909) = 119898 (119909) + 119899 (119909) + 119894 (120588 (119909) + 120591 (119909))

119866119886 (119909) = 120582119886 (119909) + 119894120577119886 (119909)

119891 (119909) = 119898 (119909) + 119899 (119909) minus 119894 (120588 (119909) + 120591 (119909))

119877119886 (119909) = 120582119886 (119909) + 119894120577119886 (119909)

119861120583 (119909) = 119883120583 (119909) + 119894119884120583 (119909)

119889 (119909) = 119863 (119909) + 119894119878 (119909)

(13)

To describe the dynamics of the supersymmetric complexvector fields with matter symmetry we need to construct anappropriated complex supersymmetric field-strength modelin order to accommodate superfields Σ and 119870 This can bereached starting from the following definitions

Υ119886 = minus14119863119863119863119886Σ (119909120583 120579119886 120579 119886)

Υ 119886 = minus14119863119863119863 119886Σ

dagger(119909120583 120579119886 120579 119886)

Γ119886 = minus14119863119863119863119886119870(119909120583 120579119886 120579 119886)

Γ 119886 = minus14119863119863119863 119886119870

dagger(119909120583 120579119886 120579 119886)

(14)

where Υ119886 and Γ119886 are charged spinor superfields As a conse-quence of the complex extension procedure we must relatechargeless spinor superfields Ω119886 and 119882119886 with the complexdefinitions (14) which in the simplest way is

Υ119886 = 119882119886 + 119894Ω119886

Υ 119886 = 119882 119886 minus 119894Ω 119886

Γ119886 = Ω119886 + 119894119882119886

Γ 119886 = Ω 119886 minus 119894119882 119886

(15)

and by assuming the spinor identities 119882119886Ω119886= Ω119886119882

119886 and119882 119886Ω

119886

= Ω 119886119882119886 we can find the kinetic supersymmetric

Lagrangian for the complex vector fields

L119896 = 119894 (Υ 119886Γ119886

minusΥ119886Γ119886)

= 119882119886119882119886+119882 119886119882

119886

+Ω119886Ω119886+Ω 119886Ω

119886

(16)

We can observe that the left-hand side of the latterequation is the complex extension of chargeless Lagrangian(7) that was written in terms of charged spinor superfieldsBearing this in mind we can then redefine the kineticLagrangian (16) simply by combining the charged spinorsuperfields Υ119886 and Γ119886 as a ldquoDirac superspinorrdquo Ψ such that

Ψ(119909120583 120579119886 120579 119886) = (

Υ119886

Γ119886) (17)

and also we assume Ψ to be the adjoint Dirac superspinorrepresentation

Advances in High Energy Physics 5

In this case we have that Ψ = Ψdagger1205740= (Γ119886Υ 119886) and so

the supersymmetric action from the kinetic Lagrangian (16)is now given by

119878119896 = int1198894119909119889

2120579119889

2120579 (119894ΨΨ)

= 119894 int 1198894119909119889

2120579119889

2120579 (Υ 119886Γ

119886

minusΥ119886Γ119886)

(18)

We can note that the product of Dirac superspinorsΨΨ obeysmatter symmetry and it presents an interesting analogy tocharged scalar superfield product 119878dagger119878 In this sense we verifythatΨ andΨ represent two chiral supersymmetric extensionsfor the matter vector field which can be transformed under119880(1) global gauge group in the following way

Ψ1015840= 119890minus2119894119902120573

Ψ

Ψ1015840

= Ψ1198902119894119902120573dagger

(19)

where120573 is a global119880(1) gauge parameter and 119902 is the charge ofthe global symmetry We can emphasize that the expressions(19) represent that each of the components of multiplet (17)has the same symmetry So action (18) is then invariantunder transformations (19) In order to obtain the componentLagrangian we can expand the product ΨΨ by consideringthat

Υ119886 (119910 120579) = 119877119886 (119910) + 2120579119886119889 (119910) + (120590120583]120579)119886119865120583] (119910)

minus 1198941205792120590120583

119886120597120583119866

(119910)

Γ119886 (119910 120579) = 119880119886 (119910) + 2120579119886V (119910) + (120590120583]120579)119886119885120583] (119910)

minus 1198941205792120590120583

119886120597120583119876

(119910)

(20)

and similarly for Υ 119886 and Γ 119886We note the presence of the complex matter field-

strengths namely

119865120583] = 120597120583119861] minus 120597]119861120583

119885120583] = 120597120583119885] minus 120597]119885120583

(21)

hence action (18) can be expanded and we obtain

119878119896 = int1198894119909

119894

2119865120583] (119909) 119885

120583](119909) minus

119894

2119865lowast

120583] (119909) 119885lowast120583]

(119909)

minus119877119886(119909) 120590120583

119886120597120583119876

(119909) minus119880119886(119909) 120590120583

119886120597120583119866

(119909)

+119877119886

(119909) 120590120583

119886119887120597120583119876119887(119909) +119880

119886

(119909) 120590120583

119886119887120597120583119866119887(119909)

+ 4119894V (119909) 119889 (119909) minus 4119894Vlowast (119909) 119889lowast (119909)

(22)

In this format we can recognize the dynamical termthat describes the matter vector field as (1198942)119865120583]119885

120583]minus

(1198942)119865lowast120583]119885lowast120583] It involves both 119865120583] and 119885120583] matter tensors

However it does not correspond to the conventional kineticterm for the matter vector field and action (18) shows moredegrees of freedom than necessary In order to get rid of suchfields we must assume the rule of transformation (11) whichis a constraint of half of the degrees and consequently action(18) reaches the correct number of component fields

Applying condition (11) in action (22) we can reach theusual dynamical matter field strength term or

119894

2119865120583]119885120583]minus

119894

2119865lowast

120583]119885lowast120583]

997904rArr minus119865lowast

120583]119865120583] (23)

and so the 119885120583] tensor field is reabsorbed in this action

Likewise and without loss of generality we could have chosenthe inverse relation Σ = 119894119870

dagger what implies reabsorbing the 119865120583]tensor field Then by using the whole relation (12) in action(18) we find that the complex supersymmetric model for thematter vector field can be written as

119878119896 = int1198894119909 minus119865

lowast

120583] (119909) 119865120583](119909)

minus 2119894119877119886 (119909) 120590120583119886120597120583119877

(119909) minus 2119894119866119886 (119909) 120590120583119886120597120583119866

(119909)

+ 8119889lowast (119909) 119889 (119909)

(24)

where expressionminus119865lowast120583](119909)119865120583](119909) represents the usual kinetic

term of the vector matter field while the terms representwith the components 119877

119886 and 119866119886 the fermionic sector and

the last term corresponds to the auxiliary field 119889 term Tocompleteness we are going to introduce the massive actionterm in the model Observing the symmetries of nonchiralfields Σ and 119870 the massive supersymmetric term can besuitable defined as

119878119898 =1205722

2int119889

4119909119889

2120579119889

2120579 [ΣdaggerΣ+119870

dagger119870] (25)

where 1205722 is a mass parameter From nonchiral superfields Σand 119870 we can obtain the massive vector matter field term119861lowast

120583119861120583 as well as their supersymmetric partners In order to

perform it we are going to compute action (25) by employingcondition119870 = 119894Σ

dagger where one has that (12)ΣdaggerΣ+(12)119870dagger119870 =

ΣdaggerΣ and by applying definition (10) the full supersymmetric

matter vector field model can be then obtained from Diracsuperspinor fieldΨ associated to the nonchiral scalar fields Σand119870 in the following form

119878 = 119878119896 + 119878119898 = int1198894119909119889

2120579119889

2120579 (119894ΨΨ+120572

2ΣdaggerΣ) (26)

where the mass part of action can be obtained in componentfields as

119878119898 = 1205722int119889

4119909(119889lowast(119909) 119896 (119909) + 119889 (119909) 119896

lowast(119909)

minus14119896lowast(119909) ◻119896 (119909) minus

14119896 (119909) ◻119896

lowast(119909) + 120585119886 (119909) 119866

119886(119909)

+ 120585 119886 (119909) 119866119886

(119909) +119862119886 (119909) 119877119886(119909) +119862 119886 (119909) 119877

119886

(119909)

6 Advances in High Energy Physics

minus 2119894120585119886

(119909) 120590120583

119886119887120597120583120585119887(119909) minus 2119894119862 119886 (119909) 120590120583 119886119887120597120583119862

119887(119909)

+ 119861lowast

120583119861120583+119891lowast(119909) 119891 (119909) + 119897

lowast(119909) 119897 (119909))

(27)

Here we observe the mass term to 119861120583 119891(119909) and 119897(119909) fields

As in the usual supersymmetric models we note that massaction (27) also contributes to kinetic structure namelywith the terms minus(14)119896lowast(119909)◻119896(119909)minus2119894120585

119886

120590120583

119886119887120597120583120585119887 and

minus2119894119862 119886120590120583 119886119887120597120583119862119887 The action also shows mixing mass scalar

and fermionic terms namely 119889lowast(119909)119896(119909) 119889(119909)119896lowast(119909)

120585119886(119909)119866119886(119909) 120585 119886(119909)119866

119886

(119909)119862119886(119909)119877119886(119909) and 119862 119886(119909)119877

119886

(119909) Byverifying the presence of extra kinetic terms in (27) whatcan suggest that when we particularly treat supersymmetricmatter vector fields the mass action contributes witha ldquodynamic complementrdquo to the kinetic action (24)Furthermore we remark that mass action (27) is importantto match the number of bosonic and fermionic degrees offreedom of the supersymmetric matter vector action (26) forthe consistency of themodelWe can redefine the componentfields absorbing the mass parameter as follows

119896 (119909) 997888rarr1120572119896 (119909)

120585119886 (119909) 997888rarr1120572120585119886 (119909)

119862119886 (119909) 997888rarr1120572119862119886 (119909)

119891 (119909) 997888rarr1120572119891 (119909)

119897 (119909) 997888rarr1120572119897 (119909)

(28)

So the action can be rewritten as

119878ek =12int119889

4119909(minus119865

lowast

120583] (119909) 119865120583](119909)

minus 2119894119877119886 (119909) 120590120583119886120597120583119877

(119909) minus 2119894119866119886 (119909) 120590120583119886120597120583119866

(119909)

minus14119896lowast(119909) ◻119896 (119909) minus

14119896 (119909) ◻119896

lowast(119909)

+ minus2119894120585119886

(119909) 120590120583

119886119887120597120583120585119887(119909) minus 2119894119862 119886 (119909) 120590120583 119886119887120597120583119862

119887(119909)

+ 8119889lowast (119909) 119889 (119909) +119891lowast(119909) 119891 (119909) + 119897

lowast(119909) 119897 (119909))

(29)

analogously the mass action is now given by

119878em = int1198894119909 120572

2119861lowast

120583119861120583+120572119889lowast(119909) 119896 (119909)

+ 120572119889 (119909) 119896lowast(119909) + 120572120585119886 (119909) 119866

119886(119909) + 120572120585 119886 (119909) 119866

119886

(119909)

+ 120572119862119886 (119909) 119877119886(119909) + 120572119862 119886 (119909) 119877

119886

(119909)

(30)

so the complex scalar fields 119889(119909) 119891(119909) and 119897(119909) have nodynamics and arise as auxiliary fields Thus assuming theaction (26) to be the sum of the redefined actions (29) and(30) and rearranging (we adopt the Weyl representation tothe gamma matrices) the (Dirac) spinor fields Θ and Π Thesum of the action (29) and the action (30) results in an off-shell action 119878os written in the following form

119878os = int1198894119909 minus

12119865lowast

120583] (119909) 119865120583](119909) +

14120597120583119896lowast(119909) 120597120583119896 (119909)

minus 119894Θ (119909) 120574120583120597120583Θ (119909) minus 119894Π (119909) 120574

120583120597120583Π (119909)

+ 120572Θ (119909) 1205745Θ (119909) + 120572Π (119909) 120574

5Π (119909) + 4119889lowast (119909) 119889 (119909)

+12119891lowast(119909) 119891 (119909) +

12119897lowast(119909) 119897 (119909) + 120572

2119861lowast

120583119861120583

+120572119889lowast(119909) 119896 (119909) + 120572119889 (119909) 119896

lowast(119909)

(31)

where we denote the Dirac spinors of mass 120572 as

Θ (119909) = (

120585119886

119866119886)

Π (119909) = (

119862119886

119877119886)

(32)

For action (31) we have obtained chiral spinor mass termsgiven by 120572Θ(119909)120574

5Θ(119909) and 120572Π(119909)120574

5Π(119909) From the action

(31) we can obtain the equation of motion for the fields

120597120583119865120583](119909) + 120572

2119861](119909) = 0

120574120583120597120583Θ (119909) minus 120572120574

5Θ (119909) = 0

120574120583120597120583Π (119909) minus 120572120574

5Π (119909) = 0

◻119896 (119909) + 1205722119896 (119909) = 0

(33)

with 119891(119909) = 119897(119909) = 0 Taking off-shell action (31) wenote that it has 16 bosonic degrees of freedom concerningthe matter fields 119861120583(119909) 119896(119909) 119889(119909) 119891(119909) and 119897(119909) and theircomplex conjugated ones as well as 16 fermionic degreesof freedom for the Dirac spinor fields Θ(119909) and Π(119909) andtheir conjugated complex ones which is consistent with thesupersymmetry From equations of motion (33) we note thatthere are three auxiliary complex scalar fields 119889(119909) 119891(119909)and 119897(119909) and massive dynamical complex scalar field 119896(119909)Moreover as expected we have obtained a matter Proca-typeequation for field 119861120583 In this context it is interesting to notethat from the on-shell actionwe can easily extract from action(31) the supersymmetric generalization of matter vector fieldit is only possible if we include two dynamical Dirac chiralspinor fieldsΘ(119909) andΠ(119909) along a massive scalar field 119896(119909)Furthermore a peculiar aspect of the spinor fields Θ(119909) andΠ(119909) in the present case is that their mass terms (33) arisewith chiral structure due to the presence of the matrix 1205745

Advances in High Energy Physics 7

4 Conclusion

In this paper which is part of a program started in a previouswork [32] we propose to formulate and analyze the simpledynamics of chargeless vector matter field in a supersymmet-ric scenarioThere are threemotivations for this proposal thelack of this approach in the literature getting a clue on thepossibility of Lorentz symmetry violation in supersymmetrictheories and the role played by the simplest case of high-spin field in field and string theories To these aims wehave started from real nonchiral scalar superfield in orderto obtain real matter Proca-type field in a supersymmetricLagrangian generalization [32] In a straightforward waythe complex model was obtained extending the real scalarnonchiral superfields to the complex space [47ndash51]

The very interesting point in this enterprise is that inorder to obtain a complex Proca-type term we face anambiguous choice of which is the real or imaginary part of thefield This ambiguity reveals a symmetry which we representas a kind of simple Hodge-duality symmetry 119870 = 119894Σ

dagger Infact despite its apparent simplicity it is essential to match thenumber of degrees of freedom of the fermionic and bosonicsectors and to compose the complex (chargeless) Proca-type dynamical term with global 119880(1) symmetry As thissymmetry appears by construction it represents a differentapproach compared to Ferrara et alrsquos work [24] On theother hand as usual the supersymmetric mass term is alsoimportant to compose the dynamics of the supersymmetricmodel and of its component fields

We can conclude that in addition to the supersymmetricmass term to build supersymmetric matter chargeless vectorfield Lagrangian the Hodge-type duality symmetry is anessential ingredient to be further explored We can alsoobserve that as the usual super-QED model the complexdynamical Proca-type term emerges from a product of Diracsuperspinors ΨΨ We can remark that the Dirac superspinorfieldΨ is also chiral because it is a combination of chiralWeylsuperspinors Γ119886 and Υ

119886 Finally we emphasize that on-shellsupersymmetric action obtained (31) revealing two fermionicDirac fields Θ(119909) and Π(119909) and a massive scalar field 119896(119909)

as supersymmetric partners associated to the complex vectorfield 119861120583(119909) which is the initial step to implement localgauge symmetry charge and interaction we could anticipatethat nonminimal couplings have an essential role in aforthcoming work

Conflict of Interests

The authors declare that they have no conflict of interestsrelated to this paper

References

[1] C Itzykson and J-B Zuber Quantum Field Theory DoverPublications 2005

[2] P RamondFieldTheory AModern PrimerWestviewPress 2ndedition 1990

[3] N N Bogoliubov and D V Shirkov Introduction to the Theoryof Quantized Fields John Wiley amp Sons 3rd edition 1980

[4] P Fayet and S Ferrara ldquoSupersymmetryrdquo Physics Reports vol32 no 5 pp 249ndash334 1977

[5] H PNilles ldquoSupersymmetry supergravity andparticle physicsrdquoPhysics Reports vol 110 no 1-2 pp 1ndash162 1984

[6] P D DrsquoEath ldquoCanonical quantization of supergravityrdquo PhysicalReview D vol 29 no 10 pp 2199ndash2219 1984

[7] J-P Ader F Biet and Y Noirot ldquoSupersymmetric structure ofthe induced W gravitiesrdquo Classical and Quantum Gravity vol16 no 3 pp 1027ndash1037 1999

[8] C Kiefer T Luck and P Moniz ldquoSemiclassical approximationto supersymmetric quantumgravityrdquoPhysical ReviewD vol 72no 4 Article ID 045006 2005

[9] A Riotto ldquoMore relaxed supersymmetric electroweak baryoge-nesisrdquo Physical Review D vol 58 Article ID 095009 1998

[10] A Riotto ldquoSupersymmetric electroweak baryogenesis non-equilibrium field theory and quantum Boltzmann equationsrdquoNuclear Physics B vol 518 no 1-2 pp 339ndash360 1998

[11] I R Klebanov and A A Tseytlin ldquoGravity duals of supersym-metric SU(N)timesSU(N+M) gauge theoriesrdquo Nuclear Physics Bvol 578 no 1-2 pp 123ndash138 2000

[12] P H Chankowski A Falkowski S Pokorski and J WagnerldquoElectroweak symmetry breaking in supersymmetric modelswith heavy scalar superpartnersrdquo Physics Letters Section BNuclear Elementary Particle and High-Energy Physics vol 598no 3-4 pp 252ndash262 2004

[13] J D Edelstein and C Nunez ldquoSupersymmetric electroweakcosmic stringsrdquo Physical Review D vol 55 no 6 pp 3811ndash38191997

[14] L Girardello A Giveon M Porrati and A Zaffaroni ldquoNon-Abelian strong-weak coupling duality in (string-derived) N = 4supersymmetric Yang-Mills theoriesrdquo Physics Letters B vol 334no 3-4 pp 331ndash338 1994

[15] J Wess and J Bagger Supersymmetry and Supergravity Prince-ton University Press Princeton NJ USA 2nd edition 1992

[16] D Bailin and A Love Supersymmetric Gauge Field Theory andString Theory IOP Publishing 1994

[17] S Coleman and J Mandula ldquoAll possible symmetries of the Smatrixrdquo Physical Review vol 159 no 5 pp 1251ndash1256 1967

[18] S J Gates M T Grisaru M Rocek and W Siegel SuperspaceBenjaminCummings Reading 1983

[19] R Haag J T Lopuszanski and M Sohnius ldquoAll possiblegenerators of supersymmetries of the 119878-matrixrdquoNuclear PhysicsB vol 88 pp 257ndash274 1975

[20] S Weinberg The Quantum Theory of Fields vol 3 CambridgeUniversity Press Cambridge UK 2000

[21] J Wess and B Zumino ldquoSupergauge transformations in fourdimensionsrdquo Nuclear Physics B vol 70 no 1 pp 39ndash50 1974

[22] H J W Muller-Kirsten and A Wiedemann SupersymmetryWorld Scientific Singapore 1987

[23] S J Gates Jr ldquoWhy auxiliary fields matter the strange caseof the 4-D 119873 = 1 supersymmetric QCD effective action 2rdquoNuclear Physics B vol 485 no 1-2 pp 145ndash184 1997

[24] S Ferrara L Girardello and F Palumbo ldquoGeneralmass formulain broken supersymmetryrdquo Physical Review D vol 20 no 2 pp403ndash408 1979

[25] V A Kostelecky and S Samuel ldquoSpontaneous breaking ofLorentz symmetry in string theoryrdquo Physical Review D vol 39no 2 pp 683ndash685 1989

[26] V E R Lemes A L M A Nogueira and J A Helayel-NetoldquoSupersymmetric generalization of the tensor matter fieldsrdquo

8 Advances in High Energy Physics

International Journal of Modern Physics A vol 13 no 18 pp3145ndash3156 1998

[27] D Colladay and V A Kostelecky ldquoCPT violation and thestandard modelrdquo Physical Review D vol 55 pp 6760ndash67741997

[28] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standardmodelrdquo Physical ReviewD vol 58 no 11 ArticleID 116002 1998

[29] A A Andrianov and R Soldati ldquoLorentz symmetry breakingin Abelian vector-field models with Wess-Zumino interactionrdquoPhysical Review D vol 51 no 10 pp 5961ndash5964 1995

[30] A A Andrianov and R Soldati ldquoPatterns of Lorentz symmetrybreaking inQEDbyCPT-odd interactionrdquoPhysics Letters B vol435 no 3-4 pp 449ndash452 1998

[31] S M Carrol G B Field and R Jackiw ldquoLimits on a Lorentz-and parity-violating modification of electrodynamicsrdquo PhysicalReview D vol 41 no 4 pp 1231ndash1240 1990

[32] L P Colatto A L A Penna andW C Santos ldquoCharged tensormatter fields and Lorentz symmetry violation via spontaneoussymmetry breakingrdquo European Physical Journal C vol 36 no 1pp 79ndash87 2004

[33] H Belich J L Boldo L P Colatto J A Helayel-Neto and A LMANogueira ldquoSupersymmetric extension of the Lorentz- andCPT-violatingMaxwell-Chern-Simonsmodelrdquo Physical ReviewD vol 68 Article ID 065030 2003

[34] H Belich Jr T Costa-Soares M M Ferreira Jr J A Helayel-Neto and M T D Orlando ldquoLorentz-symmetry violation andelectrically charged vortices in the planar regimerdquo InternationalJournal of Modern Physics A vol 21 no 11 pp 2415ndash2429 2006

[35] M M Ferreira Jr ldquoElectron-electron interaction in a Maxwell-Chern-Simons model with a purely spacelike Lorentz-violatingbackgroundrdquo Physical Review D vol 71 no 4 Article ID045003 9 pages 2005

[36] H Belich T Costa-Soares M M Ferreira Jr and J A Helayel-Neto ldquoNon-minimal coupling to a Lorentz-violating back-ground and topological implicationsrdquo The European PhysicalJournal C vol 41 no 3 pp 421ndash426 2005

[37] M B Cantcheff ldquoLorentz symmetry breaking and planar effectsfrom non-linear electrodynamicsrdquoThe European Physical Jour-nal CmdashParticles and Fields vol 46 no 1 pp 247ndash254 2006

[38] H Belich M A De Andrade and M A Santos ldquoGauge the-ories with Lorentz-symmetry violation by symplectic projectormethodrdquoModern Physics Letters A vol 20 pp 2305ndash2316 2005

[39] T G Rizzo ldquoLorentz violation in extra dimensionsrdquo Journal ofHigh Energy Physics vol 2005 no 9 article 036 2005

[40] M M Ferreira Jr and M S Tavares ldquoNonrelativistic electronndashelectron interaction in a maxwellndashchernndashsimonsndashproca modelendowed with a timelike lorentz-violating backgroundrdquo Inter-national Journal of Modern Physics A vol 22 p 1685 2007

[41] H Belich T Costa-Soares M M Ferreira Jr J A Helayel-Neto and F M O Mouchereck ldquoLorentz-violating correctionson the hydrogen spectrum induced by a nonminimal couplingrdquoPhysical Review D vol 74 no 6 Article ID 065009 6 pages2006

[42] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoOn the consistency of Lorentz invariance violationin QED induced by fermions in constant axial-vector back-groundrdquo Physics Letters B vol 639 no 5 pp 586ndash590 2006

[43] L P Colatto A L A Penna and W C Santos ldquoVector solitonsand spontaneous Lorentz violation mechanismrdquo httparxivorgabshep-th0610096

[44] MM Ferreira Jr andA R Gomes ldquoLorentz violating effects ona quantized two-level systemrdquo httparxivorgabs08034010

[45] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoBare and induced lorentz and cpt invariance violationsin qedrdquo International Journal of Modern Physics A vol 25 no16 pp 3271ndash3306 2010

[46] T G Rizzo ldquoLorentz violation in warped extra dimensionsrdquoJournal of High Energy Physics vol 2010 no 11 article 156 2010

[47] J Helayel-Neto R Jengo F Legovini and S Pugnetti ldquoSup-ersymmetry Complex Structures And Superfieldsrdquo SISSA-3887EP

[48] Y Kobayashi and S Nagamachi ldquoGeneralized complexsuperspacemdashinvolutions of superfieldsrdquo Journal of Mathemati-cal Physics vol 28 no 8 pp 1700ndash1708 1987

[49] P A Grassi G Policastro and M Porrati ldquoNotes on thequantization of the complex linear superfieldrdquo Nuclear PhysicsB vol 597 no 1ndash3 pp 615ndash632 2001

[50] S M Kuzenko and S J Tyler ldquoComplex linear superfield as amodel for Goldstinordquo Journal of High Energy Physics vol 2011no 4 article 57 2011

[51] S J Gates J Hallett T Hubsch and K Stiffler ldquoThe realanatomy of complex linear superfieldsrdquo International Journal ofModern Physics A vol 27 no 24 Article ID 1250143 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

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AerodynamicsJournal of

Volume 2014

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PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 5

In this case we have that Ψ = Ψdagger1205740= (Γ119886Υ 119886) and so

the supersymmetric action from the kinetic Lagrangian (16)is now given by

119878119896 = int1198894119909119889

2120579119889

2120579 (119894ΨΨ)

= 119894 int 1198894119909119889

2120579119889

2120579 (Υ 119886Γ

119886

minusΥ119886Γ119886)

(18)

We can note that the product of Dirac superspinorsΨΨ obeysmatter symmetry and it presents an interesting analogy tocharged scalar superfield product 119878dagger119878 In this sense we verifythatΨ andΨ represent two chiral supersymmetric extensionsfor the matter vector field which can be transformed under119880(1) global gauge group in the following way

Ψ1015840= 119890minus2119894119902120573

Ψ

Ψ1015840

= Ψ1198902119894119902120573dagger

(19)

where120573 is a global119880(1) gauge parameter and 119902 is the charge ofthe global symmetry We can emphasize that the expressions(19) represent that each of the components of multiplet (17)has the same symmetry So action (18) is then invariantunder transformations (19) In order to obtain the componentLagrangian we can expand the product ΨΨ by consideringthat

Υ119886 (119910 120579) = 119877119886 (119910) + 2120579119886119889 (119910) + (120590120583]120579)119886119865120583] (119910)

minus 1198941205792120590120583

119886120597120583119866

(119910)

Γ119886 (119910 120579) = 119880119886 (119910) + 2120579119886V (119910) + (120590120583]120579)119886119885120583] (119910)

minus 1198941205792120590120583

119886120597120583119876

(119910)

(20)

and similarly for Υ 119886 and Γ 119886We note the presence of the complex matter field-

strengths namely

119865120583] = 120597120583119861] minus 120597]119861120583

119885120583] = 120597120583119885] minus 120597]119885120583

(21)

hence action (18) can be expanded and we obtain

119878119896 = int1198894119909

119894

2119865120583] (119909) 119885

120583](119909) minus

119894

2119865lowast

120583] (119909) 119885lowast120583]

(119909)

minus119877119886(119909) 120590120583

119886120597120583119876

(119909) minus119880119886(119909) 120590120583

119886120597120583119866

(119909)

+119877119886

(119909) 120590120583

119886119887120597120583119876119887(119909) +119880

119886

(119909) 120590120583

119886119887120597120583119866119887(119909)

+ 4119894V (119909) 119889 (119909) minus 4119894Vlowast (119909) 119889lowast (119909)

(22)

In this format we can recognize the dynamical termthat describes the matter vector field as (1198942)119865120583]119885

120583]minus

(1198942)119865lowast120583]119885lowast120583] It involves both 119865120583] and 119885120583] matter tensors

However it does not correspond to the conventional kineticterm for the matter vector field and action (18) shows moredegrees of freedom than necessary In order to get rid of suchfields we must assume the rule of transformation (11) whichis a constraint of half of the degrees and consequently action(18) reaches the correct number of component fields

Applying condition (11) in action (22) we can reach theusual dynamical matter field strength term or

119894

2119865120583]119885120583]minus

119894

2119865lowast

120583]119885lowast120583]

997904rArr minus119865lowast

120583]119865120583] (23)

and so the 119885120583] tensor field is reabsorbed in this action

Likewise and without loss of generality we could have chosenthe inverse relation Σ = 119894119870

dagger what implies reabsorbing the 119865120583]tensor field Then by using the whole relation (12) in action(18) we find that the complex supersymmetric model for thematter vector field can be written as

119878119896 = int1198894119909 minus119865

lowast

120583] (119909) 119865120583](119909)

minus 2119894119877119886 (119909) 120590120583119886120597120583119877

(119909) minus 2119894119866119886 (119909) 120590120583119886120597120583119866

(119909)

+ 8119889lowast (119909) 119889 (119909)

(24)

where expressionminus119865lowast120583](119909)119865120583](119909) represents the usual kinetic

term of the vector matter field while the terms representwith the components 119877

119886 and 119866119886 the fermionic sector and

the last term corresponds to the auxiliary field 119889 term Tocompleteness we are going to introduce the massive actionterm in the model Observing the symmetries of nonchiralfields Σ and 119870 the massive supersymmetric term can besuitable defined as

119878119898 =1205722

2int119889

4119909119889

2120579119889

2120579 [ΣdaggerΣ+119870

dagger119870] (25)

where 1205722 is a mass parameter From nonchiral superfields Σand 119870 we can obtain the massive vector matter field term119861lowast

120583119861120583 as well as their supersymmetric partners In order to

perform it we are going to compute action (25) by employingcondition119870 = 119894Σ

dagger where one has that (12)ΣdaggerΣ+(12)119870dagger119870 =

ΣdaggerΣ and by applying definition (10) the full supersymmetric

matter vector field model can be then obtained from Diracsuperspinor fieldΨ associated to the nonchiral scalar fields Σand119870 in the following form

119878 = 119878119896 + 119878119898 = int1198894119909119889

2120579119889

2120579 (119894ΨΨ+120572

2ΣdaggerΣ) (26)

where the mass part of action can be obtained in componentfields as

119878119898 = 1205722int119889

4119909(119889lowast(119909) 119896 (119909) + 119889 (119909) 119896

lowast(119909)

minus14119896lowast(119909) ◻119896 (119909) minus

14119896 (119909) ◻119896

lowast(119909) + 120585119886 (119909) 119866

119886(119909)

+ 120585 119886 (119909) 119866119886

(119909) +119862119886 (119909) 119877119886(119909) +119862 119886 (119909) 119877

119886

(119909)

6 Advances in High Energy Physics

minus 2119894120585119886

(119909) 120590120583

119886119887120597120583120585119887(119909) minus 2119894119862 119886 (119909) 120590120583 119886119887120597120583119862

119887(119909)

+ 119861lowast

120583119861120583+119891lowast(119909) 119891 (119909) + 119897

lowast(119909) 119897 (119909))

(27)

Here we observe the mass term to 119861120583 119891(119909) and 119897(119909) fields

As in the usual supersymmetric models we note that massaction (27) also contributes to kinetic structure namelywith the terms minus(14)119896lowast(119909)◻119896(119909)minus2119894120585

119886

120590120583

119886119887120597120583120585119887 and

minus2119894119862 119886120590120583 119886119887120597120583119862119887 The action also shows mixing mass scalar

and fermionic terms namely 119889lowast(119909)119896(119909) 119889(119909)119896lowast(119909)

120585119886(119909)119866119886(119909) 120585 119886(119909)119866

119886

(119909)119862119886(119909)119877119886(119909) and 119862 119886(119909)119877

119886

(119909) Byverifying the presence of extra kinetic terms in (27) whatcan suggest that when we particularly treat supersymmetricmatter vector fields the mass action contributes witha ldquodynamic complementrdquo to the kinetic action (24)Furthermore we remark that mass action (27) is importantto match the number of bosonic and fermionic degrees offreedom of the supersymmetric matter vector action (26) forthe consistency of themodelWe can redefine the componentfields absorbing the mass parameter as follows

119896 (119909) 997888rarr1120572119896 (119909)

120585119886 (119909) 997888rarr1120572120585119886 (119909)

119862119886 (119909) 997888rarr1120572119862119886 (119909)

119891 (119909) 997888rarr1120572119891 (119909)

119897 (119909) 997888rarr1120572119897 (119909)

(28)

So the action can be rewritten as

119878ek =12int119889

4119909(minus119865

lowast

120583] (119909) 119865120583](119909)

minus 2119894119877119886 (119909) 120590120583119886120597120583119877

(119909) minus 2119894119866119886 (119909) 120590120583119886120597120583119866

(119909)

minus14119896lowast(119909) ◻119896 (119909) minus

14119896 (119909) ◻119896

lowast(119909)

+ minus2119894120585119886

(119909) 120590120583

119886119887120597120583120585119887(119909) minus 2119894119862 119886 (119909) 120590120583 119886119887120597120583119862

119887(119909)

+ 8119889lowast (119909) 119889 (119909) +119891lowast(119909) 119891 (119909) + 119897

lowast(119909) 119897 (119909))

(29)

analogously the mass action is now given by

119878em = int1198894119909 120572

2119861lowast

120583119861120583+120572119889lowast(119909) 119896 (119909)

+ 120572119889 (119909) 119896lowast(119909) + 120572120585119886 (119909) 119866

119886(119909) + 120572120585 119886 (119909) 119866

119886

(119909)

+ 120572119862119886 (119909) 119877119886(119909) + 120572119862 119886 (119909) 119877

119886

(119909)

(30)

so the complex scalar fields 119889(119909) 119891(119909) and 119897(119909) have nodynamics and arise as auxiliary fields Thus assuming theaction (26) to be the sum of the redefined actions (29) and(30) and rearranging (we adopt the Weyl representation tothe gamma matrices) the (Dirac) spinor fields Θ and Π Thesum of the action (29) and the action (30) results in an off-shell action 119878os written in the following form

119878os = int1198894119909 minus

12119865lowast

120583] (119909) 119865120583](119909) +

14120597120583119896lowast(119909) 120597120583119896 (119909)

minus 119894Θ (119909) 120574120583120597120583Θ (119909) minus 119894Π (119909) 120574

120583120597120583Π (119909)

+ 120572Θ (119909) 1205745Θ (119909) + 120572Π (119909) 120574

5Π (119909) + 4119889lowast (119909) 119889 (119909)

+12119891lowast(119909) 119891 (119909) +

12119897lowast(119909) 119897 (119909) + 120572

2119861lowast

120583119861120583

+120572119889lowast(119909) 119896 (119909) + 120572119889 (119909) 119896

lowast(119909)

(31)

where we denote the Dirac spinors of mass 120572 as

Θ (119909) = (

120585119886

119866119886)

Π (119909) = (

119862119886

119877119886)

(32)

For action (31) we have obtained chiral spinor mass termsgiven by 120572Θ(119909)120574

5Θ(119909) and 120572Π(119909)120574

5Π(119909) From the action

(31) we can obtain the equation of motion for the fields

120597120583119865120583](119909) + 120572

2119861](119909) = 0

120574120583120597120583Θ (119909) minus 120572120574

5Θ (119909) = 0

120574120583120597120583Π (119909) minus 120572120574

5Π (119909) = 0

◻119896 (119909) + 1205722119896 (119909) = 0

(33)

with 119891(119909) = 119897(119909) = 0 Taking off-shell action (31) wenote that it has 16 bosonic degrees of freedom concerningthe matter fields 119861120583(119909) 119896(119909) 119889(119909) 119891(119909) and 119897(119909) and theircomplex conjugated ones as well as 16 fermionic degreesof freedom for the Dirac spinor fields Θ(119909) and Π(119909) andtheir conjugated complex ones which is consistent with thesupersymmetry From equations of motion (33) we note thatthere are three auxiliary complex scalar fields 119889(119909) 119891(119909)and 119897(119909) and massive dynamical complex scalar field 119896(119909)Moreover as expected we have obtained a matter Proca-typeequation for field 119861120583 In this context it is interesting to notethat from the on-shell actionwe can easily extract from action(31) the supersymmetric generalization of matter vector fieldit is only possible if we include two dynamical Dirac chiralspinor fieldsΘ(119909) andΠ(119909) along a massive scalar field 119896(119909)Furthermore a peculiar aspect of the spinor fields Θ(119909) andΠ(119909) in the present case is that their mass terms (33) arisewith chiral structure due to the presence of the matrix 1205745

Advances in High Energy Physics 7

4 Conclusion

In this paper which is part of a program started in a previouswork [32] we propose to formulate and analyze the simpledynamics of chargeless vector matter field in a supersymmet-ric scenarioThere are threemotivations for this proposal thelack of this approach in the literature getting a clue on thepossibility of Lorentz symmetry violation in supersymmetrictheories and the role played by the simplest case of high-spin field in field and string theories To these aims wehave started from real nonchiral scalar superfield in orderto obtain real matter Proca-type field in a supersymmetricLagrangian generalization [32] In a straightforward waythe complex model was obtained extending the real scalarnonchiral superfields to the complex space [47ndash51]

The very interesting point in this enterprise is that inorder to obtain a complex Proca-type term we face anambiguous choice of which is the real or imaginary part of thefield This ambiguity reveals a symmetry which we representas a kind of simple Hodge-duality symmetry 119870 = 119894Σ

dagger Infact despite its apparent simplicity it is essential to match thenumber of degrees of freedom of the fermionic and bosonicsectors and to compose the complex (chargeless) Proca-type dynamical term with global 119880(1) symmetry As thissymmetry appears by construction it represents a differentapproach compared to Ferrara et alrsquos work [24] On theother hand as usual the supersymmetric mass term is alsoimportant to compose the dynamics of the supersymmetricmodel and of its component fields

We can conclude that in addition to the supersymmetricmass term to build supersymmetric matter chargeless vectorfield Lagrangian the Hodge-type duality symmetry is anessential ingredient to be further explored We can alsoobserve that as the usual super-QED model the complexdynamical Proca-type term emerges from a product of Diracsuperspinors ΨΨ We can remark that the Dirac superspinorfieldΨ is also chiral because it is a combination of chiralWeylsuperspinors Γ119886 and Υ

119886 Finally we emphasize that on-shellsupersymmetric action obtained (31) revealing two fermionicDirac fields Θ(119909) and Π(119909) and a massive scalar field 119896(119909)

as supersymmetric partners associated to the complex vectorfield 119861120583(119909) which is the initial step to implement localgauge symmetry charge and interaction we could anticipatethat nonminimal couplings have an essential role in aforthcoming work

Conflict of Interests

The authors declare that they have no conflict of interestsrelated to this paper

References

[1] C Itzykson and J-B Zuber Quantum Field Theory DoverPublications 2005

[2] P RamondFieldTheory AModern PrimerWestviewPress 2ndedition 1990

[3] N N Bogoliubov and D V Shirkov Introduction to the Theoryof Quantized Fields John Wiley amp Sons 3rd edition 1980

[4] P Fayet and S Ferrara ldquoSupersymmetryrdquo Physics Reports vol32 no 5 pp 249ndash334 1977

[5] H PNilles ldquoSupersymmetry supergravity andparticle physicsrdquoPhysics Reports vol 110 no 1-2 pp 1ndash162 1984

[6] P D DrsquoEath ldquoCanonical quantization of supergravityrdquo PhysicalReview D vol 29 no 10 pp 2199ndash2219 1984

[7] J-P Ader F Biet and Y Noirot ldquoSupersymmetric structure ofthe induced W gravitiesrdquo Classical and Quantum Gravity vol16 no 3 pp 1027ndash1037 1999

[8] C Kiefer T Luck and P Moniz ldquoSemiclassical approximationto supersymmetric quantumgravityrdquoPhysical ReviewD vol 72no 4 Article ID 045006 2005

[9] A Riotto ldquoMore relaxed supersymmetric electroweak baryoge-nesisrdquo Physical Review D vol 58 Article ID 095009 1998

[10] A Riotto ldquoSupersymmetric electroweak baryogenesis non-equilibrium field theory and quantum Boltzmann equationsrdquoNuclear Physics B vol 518 no 1-2 pp 339ndash360 1998

[11] I R Klebanov and A A Tseytlin ldquoGravity duals of supersym-metric SU(N)timesSU(N+M) gauge theoriesrdquo Nuclear Physics Bvol 578 no 1-2 pp 123ndash138 2000

[12] P H Chankowski A Falkowski S Pokorski and J WagnerldquoElectroweak symmetry breaking in supersymmetric modelswith heavy scalar superpartnersrdquo Physics Letters Section BNuclear Elementary Particle and High-Energy Physics vol 598no 3-4 pp 252ndash262 2004

[13] J D Edelstein and C Nunez ldquoSupersymmetric electroweakcosmic stringsrdquo Physical Review D vol 55 no 6 pp 3811ndash38191997

[14] L Girardello A Giveon M Porrati and A Zaffaroni ldquoNon-Abelian strong-weak coupling duality in (string-derived) N = 4supersymmetric Yang-Mills theoriesrdquo Physics Letters B vol 334no 3-4 pp 331ndash338 1994

[15] J Wess and J Bagger Supersymmetry and Supergravity Prince-ton University Press Princeton NJ USA 2nd edition 1992

[16] D Bailin and A Love Supersymmetric Gauge Field Theory andString Theory IOP Publishing 1994

[17] S Coleman and J Mandula ldquoAll possible symmetries of the Smatrixrdquo Physical Review vol 159 no 5 pp 1251ndash1256 1967

[18] S J Gates M T Grisaru M Rocek and W Siegel SuperspaceBenjaminCummings Reading 1983

[19] R Haag J T Lopuszanski and M Sohnius ldquoAll possiblegenerators of supersymmetries of the 119878-matrixrdquoNuclear PhysicsB vol 88 pp 257ndash274 1975

[20] S Weinberg The Quantum Theory of Fields vol 3 CambridgeUniversity Press Cambridge UK 2000

[21] J Wess and B Zumino ldquoSupergauge transformations in fourdimensionsrdquo Nuclear Physics B vol 70 no 1 pp 39ndash50 1974

[22] H J W Muller-Kirsten and A Wiedemann SupersymmetryWorld Scientific Singapore 1987

[23] S J Gates Jr ldquoWhy auxiliary fields matter the strange caseof the 4-D 119873 = 1 supersymmetric QCD effective action 2rdquoNuclear Physics B vol 485 no 1-2 pp 145ndash184 1997

[24] S Ferrara L Girardello and F Palumbo ldquoGeneralmass formulain broken supersymmetryrdquo Physical Review D vol 20 no 2 pp403ndash408 1979

[25] V A Kostelecky and S Samuel ldquoSpontaneous breaking ofLorentz symmetry in string theoryrdquo Physical Review D vol 39no 2 pp 683ndash685 1989

[26] V E R Lemes A L M A Nogueira and J A Helayel-NetoldquoSupersymmetric generalization of the tensor matter fieldsrdquo

8 Advances in High Energy Physics

International Journal of Modern Physics A vol 13 no 18 pp3145ndash3156 1998

[27] D Colladay and V A Kostelecky ldquoCPT violation and thestandard modelrdquo Physical Review D vol 55 pp 6760ndash67741997

[28] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standardmodelrdquo Physical ReviewD vol 58 no 11 ArticleID 116002 1998

[29] A A Andrianov and R Soldati ldquoLorentz symmetry breakingin Abelian vector-field models with Wess-Zumino interactionrdquoPhysical Review D vol 51 no 10 pp 5961ndash5964 1995

[30] A A Andrianov and R Soldati ldquoPatterns of Lorentz symmetrybreaking inQEDbyCPT-odd interactionrdquoPhysics Letters B vol435 no 3-4 pp 449ndash452 1998

[31] S M Carrol G B Field and R Jackiw ldquoLimits on a Lorentz-and parity-violating modification of electrodynamicsrdquo PhysicalReview D vol 41 no 4 pp 1231ndash1240 1990

[32] L P Colatto A L A Penna andW C Santos ldquoCharged tensormatter fields and Lorentz symmetry violation via spontaneoussymmetry breakingrdquo European Physical Journal C vol 36 no 1pp 79ndash87 2004

[33] H Belich J L Boldo L P Colatto J A Helayel-Neto and A LMANogueira ldquoSupersymmetric extension of the Lorentz- andCPT-violatingMaxwell-Chern-Simonsmodelrdquo Physical ReviewD vol 68 Article ID 065030 2003

[34] H Belich Jr T Costa-Soares M M Ferreira Jr J A Helayel-Neto and M T D Orlando ldquoLorentz-symmetry violation andelectrically charged vortices in the planar regimerdquo InternationalJournal of Modern Physics A vol 21 no 11 pp 2415ndash2429 2006

[35] M M Ferreira Jr ldquoElectron-electron interaction in a Maxwell-Chern-Simons model with a purely spacelike Lorentz-violatingbackgroundrdquo Physical Review D vol 71 no 4 Article ID045003 9 pages 2005

[36] H Belich T Costa-Soares M M Ferreira Jr and J A Helayel-Neto ldquoNon-minimal coupling to a Lorentz-violating back-ground and topological implicationsrdquo The European PhysicalJournal C vol 41 no 3 pp 421ndash426 2005

[37] M B Cantcheff ldquoLorentz symmetry breaking and planar effectsfrom non-linear electrodynamicsrdquoThe European Physical Jour-nal CmdashParticles and Fields vol 46 no 1 pp 247ndash254 2006

[38] H Belich M A De Andrade and M A Santos ldquoGauge the-ories with Lorentz-symmetry violation by symplectic projectormethodrdquoModern Physics Letters A vol 20 pp 2305ndash2316 2005

[39] T G Rizzo ldquoLorentz violation in extra dimensionsrdquo Journal ofHigh Energy Physics vol 2005 no 9 article 036 2005

[40] M M Ferreira Jr and M S Tavares ldquoNonrelativistic electronndashelectron interaction in a maxwellndashchernndashsimonsndashproca modelendowed with a timelike lorentz-violating backgroundrdquo Inter-national Journal of Modern Physics A vol 22 p 1685 2007

[41] H Belich T Costa-Soares M M Ferreira Jr J A Helayel-Neto and F M O Mouchereck ldquoLorentz-violating correctionson the hydrogen spectrum induced by a nonminimal couplingrdquoPhysical Review D vol 74 no 6 Article ID 065009 6 pages2006

[42] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoOn the consistency of Lorentz invariance violationin QED induced by fermions in constant axial-vector back-groundrdquo Physics Letters B vol 639 no 5 pp 586ndash590 2006

[43] L P Colatto A L A Penna and W C Santos ldquoVector solitonsand spontaneous Lorentz violation mechanismrdquo httparxivorgabshep-th0610096

[44] MM Ferreira Jr andA R Gomes ldquoLorentz violating effects ona quantized two-level systemrdquo httparxivorgabs08034010

[45] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoBare and induced lorentz and cpt invariance violationsin qedrdquo International Journal of Modern Physics A vol 25 no16 pp 3271ndash3306 2010

[46] T G Rizzo ldquoLorentz violation in warped extra dimensionsrdquoJournal of High Energy Physics vol 2010 no 11 article 156 2010

[47] J Helayel-Neto R Jengo F Legovini and S Pugnetti ldquoSup-ersymmetry Complex Structures And Superfieldsrdquo SISSA-3887EP

[48] Y Kobayashi and S Nagamachi ldquoGeneralized complexsuperspacemdashinvolutions of superfieldsrdquo Journal of Mathemati-cal Physics vol 28 no 8 pp 1700ndash1708 1987

[49] P A Grassi G Policastro and M Porrati ldquoNotes on thequantization of the complex linear superfieldrdquo Nuclear PhysicsB vol 597 no 1ndash3 pp 615ndash632 2001

[50] S M Kuzenko and S J Tyler ldquoComplex linear superfield as amodel for Goldstinordquo Journal of High Energy Physics vol 2011no 4 article 57 2011

[51] S J Gates J Hallett T Hubsch and K Stiffler ldquoThe realanatomy of complex linear superfieldsrdquo International Journal ofModern Physics A vol 27 no 24 Article ID 1250143 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

6 Advances in High Energy Physics

minus 2119894120585119886

(119909) 120590120583

119886119887120597120583120585119887(119909) minus 2119894119862 119886 (119909) 120590120583 119886119887120597120583119862

119887(119909)

+ 119861lowast

120583119861120583+119891lowast(119909) 119891 (119909) + 119897

lowast(119909) 119897 (119909))

(27)

Here we observe the mass term to 119861120583 119891(119909) and 119897(119909) fields

As in the usual supersymmetric models we note that massaction (27) also contributes to kinetic structure namelywith the terms minus(14)119896lowast(119909)◻119896(119909)minus2119894120585

119886

120590120583

119886119887120597120583120585119887 and

minus2119894119862 119886120590120583 119886119887120597120583119862119887 The action also shows mixing mass scalar

and fermionic terms namely 119889lowast(119909)119896(119909) 119889(119909)119896lowast(119909)

120585119886(119909)119866119886(119909) 120585 119886(119909)119866

119886

(119909)119862119886(119909)119877119886(119909) and 119862 119886(119909)119877

119886

(119909) Byverifying the presence of extra kinetic terms in (27) whatcan suggest that when we particularly treat supersymmetricmatter vector fields the mass action contributes witha ldquodynamic complementrdquo to the kinetic action (24)Furthermore we remark that mass action (27) is importantto match the number of bosonic and fermionic degrees offreedom of the supersymmetric matter vector action (26) forthe consistency of themodelWe can redefine the componentfields absorbing the mass parameter as follows

119896 (119909) 997888rarr1120572119896 (119909)

120585119886 (119909) 997888rarr1120572120585119886 (119909)

119862119886 (119909) 997888rarr1120572119862119886 (119909)

119891 (119909) 997888rarr1120572119891 (119909)

119897 (119909) 997888rarr1120572119897 (119909)

(28)

So the action can be rewritten as

119878ek =12int119889

4119909(minus119865

lowast

120583] (119909) 119865120583](119909)

minus 2119894119877119886 (119909) 120590120583119886120597120583119877

(119909) minus 2119894119866119886 (119909) 120590120583119886120597120583119866

(119909)

minus14119896lowast(119909) ◻119896 (119909) minus

14119896 (119909) ◻119896

lowast(119909)

+ minus2119894120585119886

(119909) 120590120583

119886119887120597120583120585119887(119909) minus 2119894119862 119886 (119909) 120590120583 119886119887120597120583119862

119887(119909)

+ 8119889lowast (119909) 119889 (119909) +119891lowast(119909) 119891 (119909) + 119897

lowast(119909) 119897 (119909))

(29)

analogously the mass action is now given by

119878em = int1198894119909 120572

2119861lowast

120583119861120583+120572119889lowast(119909) 119896 (119909)

+ 120572119889 (119909) 119896lowast(119909) + 120572120585119886 (119909) 119866

119886(119909) + 120572120585 119886 (119909) 119866

119886

(119909)

+ 120572119862119886 (119909) 119877119886(119909) + 120572119862 119886 (119909) 119877

119886

(119909)

(30)

so the complex scalar fields 119889(119909) 119891(119909) and 119897(119909) have nodynamics and arise as auxiliary fields Thus assuming theaction (26) to be the sum of the redefined actions (29) and(30) and rearranging (we adopt the Weyl representation tothe gamma matrices) the (Dirac) spinor fields Θ and Π Thesum of the action (29) and the action (30) results in an off-shell action 119878os written in the following form

119878os = int1198894119909 minus

12119865lowast

120583] (119909) 119865120583](119909) +

14120597120583119896lowast(119909) 120597120583119896 (119909)

minus 119894Θ (119909) 120574120583120597120583Θ (119909) minus 119894Π (119909) 120574

120583120597120583Π (119909)

+ 120572Θ (119909) 1205745Θ (119909) + 120572Π (119909) 120574

5Π (119909) + 4119889lowast (119909) 119889 (119909)

+12119891lowast(119909) 119891 (119909) +

12119897lowast(119909) 119897 (119909) + 120572

2119861lowast

120583119861120583

+120572119889lowast(119909) 119896 (119909) + 120572119889 (119909) 119896

lowast(119909)

(31)

where we denote the Dirac spinors of mass 120572 as

Θ (119909) = (

120585119886

119866119886)

Π (119909) = (

119862119886

119877119886)

(32)

For action (31) we have obtained chiral spinor mass termsgiven by 120572Θ(119909)120574

5Θ(119909) and 120572Π(119909)120574

5Π(119909) From the action

(31) we can obtain the equation of motion for the fields

120597120583119865120583](119909) + 120572

2119861](119909) = 0

120574120583120597120583Θ (119909) minus 120572120574

5Θ (119909) = 0

120574120583120597120583Π (119909) minus 120572120574

5Π (119909) = 0

◻119896 (119909) + 1205722119896 (119909) = 0

(33)

with 119891(119909) = 119897(119909) = 0 Taking off-shell action (31) wenote that it has 16 bosonic degrees of freedom concerningthe matter fields 119861120583(119909) 119896(119909) 119889(119909) 119891(119909) and 119897(119909) and theircomplex conjugated ones as well as 16 fermionic degreesof freedom for the Dirac spinor fields Θ(119909) and Π(119909) andtheir conjugated complex ones which is consistent with thesupersymmetry From equations of motion (33) we note thatthere are three auxiliary complex scalar fields 119889(119909) 119891(119909)and 119897(119909) and massive dynamical complex scalar field 119896(119909)Moreover as expected we have obtained a matter Proca-typeequation for field 119861120583 In this context it is interesting to notethat from the on-shell actionwe can easily extract from action(31) the supersymmetric generalization of matter vector fieldit is only possible if we include two dynamical Dirac chiralspinor fieldsΘ(119909) andΠ(119909) along a massive scalar field 119896(119909)Furthermore a peculiar aspect of the spinor fields Θ(119909) andΠ(119909) in the present case is that their mass terms (33) arisewith chiral structure due to the presence of the matrix 1205745

Advances in High Energy Physics 7

4 Conclusion

In this paper which is part of a program started in a previouswork [32] we propose to formulate and analyze the simpledynamics of chargeless vector matter field in a supersymmet-ric scenarioThere are threemotivations for this proposal thelack of this approach in the literature getting a clue on thepossibility of Lorentz symmetry violation in supersymmetrictheories and the role played by the simplest case of high-spin field in field and string theories To these aims wehave started from real nonchiral scalar superfield in orderto obtain real matter Proca-type field in a supersymmetricLagrangian generalization [32] In a straightforward waythe complex model was obtained extending the real scalarnonchiral superfields to the complex space [47ndash51]

The very interesting point in this enterprise is that inorder to obtain a complex Proca-type term we face anambiguous choice of which is the real or imaginary part of thefield This ambiguity reveals a symmetry which we representas a kind of simple Hodge-duality symmetry 119870 = 119894Σ

dagger Infact despite its apparent simplicity it is essential to match thenumber of degrees of freedom of the fermionic and bosonicsectors and to compose the complex (chargeless) Proca-type dynamical term with global 119880(1) symmetry As thissymmetry appears by construction it represents a differentapproach compared to Ferrara et alrsquos work [24] On theother hand as usual the supersymmetric mass term is alsoimportant to compose the dynamics of the supersymmetricmodel and of its component fields

We can conclude that in addition to the supersymmetricmass term to build supersymmetric matter chargeless vectorfield Lagrangian the Hodge-type duality symmetry is anessential ingredient to be further explored We can alsoobserve that as the usual super-QED model the complexdynamical Proca-type term emerges from a product of Diracsuperspinors ΨΨ We can remark that the Dirac superspinorfieldΨ is also chiral because it is a combination of chiralWeylsuperspinors Γ119886 and Υ

119886 Finally we emphasize that on-shellsupersymmetric action obtained (31) revealing two fermionicDirac fields Θ(119909) and Π(119909) and a massive scalar field 119896(119909)

as supersymmetric partners associated to the complex vectorfield 119861120583(119909) which is the initial step to implement localgauge symmetry charge and interaction we could anticipatethat nonminimal couplings have an essential role in aforthcoming work

Conflict of Interests

The authors declare that they have no conflict of interestsrelated to this paper

References

[1] C Itzykson and J-B Zuber Quantum Field Theory DoverPublications 2005

[2] P RamondFieldTheory AModern PrimerWestviewPress 2ndedition 1990

[3] N N Bogoliubov and D V Shirkov Introduction to the Theoryof Quantized Fields John Wiley amp Sons 3rd edition 1980

[4] P Fayet and S Ferrara ldquoSupersymmetryrdquo Physics Reports vol32 no 5 pp 249ndash334 1977

[5] H PNilles ldquoSupersymmetry supergravity andparticle physicsrdquoPhysics Reports vol 110 no 1-2 pp 1ndash162 1984

[6] P D DrsquoEath ldquoCanonical quantization of supergravityrdquo PhysicalReview D vol 29 no 10 pp 2199ndash2219 1984

[7] J-P Ader F Biet and Y Noirot ldquoSupersymmetric structure ofthe induced W gravitiesrdquo Classical and Quantum Gravity vol16 no 3 pp 1027ndash1037 1999

[8] C Kiefer T Luck and P Moniz ldquoSemiclassical approximationto supersymmetric quantumgravityrdquoPhysical ReviewD vol 72no 4 Article ID 045006 2005

[9] A Riotto ldquoMore relaxed supersymmetric electroweak baryoge-nesisrdquo Physical Review D vol 58 Article ID 095009 1998

[10] A Riotto ldquoSupersymmetric electroweak baryogenesis non-equilibrium field theory and quantum Boltzmann equationsrdquoNuclear Physics B vol 518 no 1-2 pp 339ndash360 1998

[11] I R Klebanov and A A Tseytlin ldquoGravity duals of supersym-metric SU(N)timesSU(N+M) gauge theoriesrdquo Nuclear Physics Bvol 578 no 1-2 pp 123ndash138 2000

[12] P H Chankowski A Falkowski S Pokorski and J WagnerldquoElectroweak symmetry breaking in supersymmetric modelswith heavy scalar superpartnersrdquo Physics Letters Section BNuclear Elementary Particle and High-Energy Physics vol 598no 3-4 pp 252ndash262 2004

[13] J D Edelstein and C Nunez ldquoSupersymmetric electroweakcosmic stringsrdquo Physical Review D vol 55 no 6 pp 3811ndash38191997

[14] L Girardello A Giveon M Porrati and A Zaffaroni ldquoNon-Abelian strong-weak coupling duality in (string-derived) N = 4supersymmetric Yang-Mills theoriesrdquo Physics Letters B vol 334no 3-4 pp 331ndash338 1994

[15] J Wess and J Bagger Supersymmetry and Supergravity Prince-ton University Press Princeton NJ USA 2nd edition 1992

[16] D Bailin and A Love Supersymmetric Gauge Field Theory andString Theory IOP Publishing 1994

[17] S Coleman and J Mandula ldquoAll possible symmetries of the Smatrixrdquo Physical Review vol 159 no 5 pp 1251ndash1256 1967

[18] S J Gates M T Grisaru M Rocek and W Siegel SuperspaceBenjaminCummings Reading 1983

[19] R Haag J T Lopuszanski and M Sohnius ldquoAll possiblegenerators of supersymmetries of the 119878-matrixrdquoNuclear PhysicsB vol 88 pp 257ndash274 1975

[20] S Weinberg The Quantum Theory of Fields vol 3 CambridgeUniversity Press Cambridge UK 2000

[21] J Wess and B Zumino ldquoSupergauge transformations in fourdimensionsrdquo Nuclear Physics B vol 70 no 1 pp 39ndash50 1974

[22] H J W Muller-Kirsten and A Wiedemann SupersymmetryWorld Scientific Singapore 1987

[23] S J Gates Jr ldquoWhy auxiliary fields matter the strange caseof the 4-D 119873 = 1 supersymmetric QCD effective action 2rdquoNuclear Physics B vol 485 no 1-2 pp 145ndash184 1997

[24] S Ferrara L Girardello and F Palumbo ldquoGeneralmass formulain broken supersymmetryrdquo Physical Review D vol 20 no 2 pp403ndash408 1979

[25] V A Kostelecky and S Samuel ldquoSpontaneous breaking ofLorentz symmetry in string theoryrdquo Physical Review D vol 39no 2 pp 683ndash685 1989

[26] V E R Lemes A L M A Nogueira and J A Helayel-NetoldquoSupersymmetric generalization of the tensor matter fieldsrdquo

8 Advances in High Energy Physics

International Journal of Modern Physics A vol 13 no 18 pp3145ndash3156 1998

[27] D Colladay and V A Kostelecky ldquoCPT violation and thestandard modelrdquo Physical Review D vol 55 pp 6760ndash67741997

[28] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standardmodelrdquo Physical ReviewD vol 58 no 11 ArticleID 116002 1998

[29] A A Andrianov and R Soldati ldquoLorentz symmetry breakingin Abelian vector-field models with Wess-Zumino interactionrdquoPhysical Review D vol 51 no 10 pp 5961ndash5964 1995

[30] A A Andrianov and R Soldati ldquoPatterns of Lorentz symmetrybreaking inQEDbyCPT-odd interactionrdquoPhysics Letters B vol435 no 3-4 pp 449ndash452 1998

[31] S M Carrol G B Field and R Jackiw ldquoLimits on a Lorentz-and parity-violating modification of electrodynamicsrdquo PhysicalReview D vol 41 no 4 pp 1231ndash1240 1990

[32] L P Colatto A L A Penna andW C Santos ldquoCharged tensormatter fields and Lorentz symmetry violation via spontaneoussymmetry breakingrdquo European Physical Journal C vol 36 no 1pp 79ndash87 2004

[33] H Belich J L Boldo L P Colatto J A Helayel-Neto and A LMANogueira ldquoSupersymmetric extension of the Lorentz- andCPT-violatingMaxwell-Chern-Simonsmodelrdquo Physical ReviewD vol 68 Article ID 065030 2003

[34] H Belich Jr T Costa-Soares M M Ferreira Jr J A Helayel-Neto and M T D Orlando ldquoLorentz-symmetry violation andelectrically charged vortices in the planar regimerdquo InternationalJournal of Modern Physics A vol 21 no 11 pp 2415ndash2429 2006

[35] M M Ferreira Jr ldquoElectron-electron interaction in a Maxwell-Chern-Simons model with a purely spacelike Lorentz-violatingbackgroundrdquo Physical Review D vol 71 no 4 Article ID045003 9 pages 2005

[36] H Belich T Costa-Soares M M Ferreira Jr and J A Helayel-Neto ldquoNon-minimal coupling to a Lorentz-violating back-ground and topological implicationsrdquo The European PhysicalJournal C vol 41 no 3 pp 421ndash426 2005

[37] M B Cantcheff ldquoLorentz symmetry breaking and planar effectsfrom non-linear electrodynamicsrdquoThe European Physical Jour-nal CmdashParticles and Fields vol 46 no 1 pp 247ndash254 2006

[38] H Belich M A De Andrade and M A Santos ldquoGauge the-ories with Lorentz-symmetry violation by symplectic projectormethodrdquoModern Physics Letters A vol 20 pp 2305ndash2316 2005

[39] T G Rizzo ldquoLorentz violation in extra dimensionsrdquo Journal ofHigh Energy Physics vol 2005 no 9 article 036 2005

[40] M M Ferreira Jr and M S Tavares ldquoNonrelativistic electronndashelectron interaction in a maxwellndashchernndashsimonsndashproca modelendowed with a timelike lorentz-violating backgroundrdquo Inter-national Journal of Modern Physics A vol 22 p 1685 2007

[41] H Belich T Costa-Soares M M Ferreira Jr J A Helayel-Neto and F M O Mouchereck ldquoLorentz-violating correctionson the hydrogen spectrum induced by a nonminimal couplingrdquoPhysical Review D vol 74 no 6 Article ID 065009 6 pages2006

[42] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoOn the consistency of Lorentz invariance violationin QED induced by fermions in constant axial-vector back-groundrdquo Physics Letters B vol 639 no 5 pp 586ndash590 2006

[43] L P Colatto A L A Penna and W C Santos ldquoVector solitonsand spontaneous Lorentz violation mechanismrdquo httparxivorgabshep-th0610096

[44] MM Ferreira Jr andA R Gomes ldquoLorentz violating effects ona quantized two-level systemrdquo httparxivorgabs08034010

[45] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoBare and induced lorentz and cpt invariance violationsin qedrdquo International Journal of Modern Physics A vol 25 no16 pp 3271ndash3306 2010

[46] T G Rizzo ldquoLorentz violation in warped extra dimensionsrdquoJournal of High Energy Physics vol 2010 no 11 article 156 2010

[47] J Helayel-Neto R Jengo F Legovini and S Pugnetti ldquoSup-ersymmetry Complex Structures And Superfieldsrdquo SISSA-3887EP

[48] Y Kobayashi and S Nagamachi ldquoGeneralized complexsuperspacemdashinvolutions of superfieldsrdquo Journal of Mathemati-cal Physics vol 28 no 8 pp 1700ndash1708 1987

[49] P A Grassi G Policastro and M Porrati ldquoNotes on thequantization of the complex linear superfieldrdquo Nuclear PhysicsB vol 597 no 1ndash3 pp 615ndash632 2001

[50] S M Kuzenko and S J Tyler ldquoComplex linear superfield as amodel for Goldstinordquo Journal of High Energy Physics vol 2011no 4 article 57 2011

[51] S J Gates J Hallett T Hubsch and K Stiffler ldquoThe realanatomy of complex linear superfieldsrdquo International Journal ofModern Physics A vol 27 no 24 Article ID 1250143 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Advances in High Energy Physics 7

4 Conclusion

In this paper which is part of a program started in a previouswork [32] we propose to formulate and analyze the simpledynamics of chargeless vector matter field in a supersymmet-ric scenarioThere are threemotivations for this proposal thelack of this approach in the literature getting a clue on thepossibility of Lorentz symmetry violation in supersymmetrictheories and the role played by the simplest case of high-spin field in field and string theories To these aims wehave started from real nonchiral scalar superfield in orderto obtain real matter Proca-type field in a supersymmetricLagrangian generalization [32] In a straightforward waythe complex model was obtained extending the real scalarnonchiral superfields to the complex space [47ndash51]

The very interesting point in this enterprise is that inorder to obtain a complex Proca-type term we face anambiguous choice of which is the real or imaginary part of thefield This ambiguity reveals a symmetry which we representas a kind of simple Hodge-duality symmetry 119870 = 119894Σ

dagger Infact despite its apparent simplicity it is essential to match thenumber of degrees of freedom of the fermionic and bosonicsectors and to compose the complex (chargeless) Proca-type dynamical term with global 119880(1) symmetry As thissymmetry appears by construction it represents a differentapproach compared to Ferrara et alrsquos work [24] On theother hand as usual the supersymmetric mass term is alsoimportant to compose the dynamics of the supersymmetricmodel and of its component fields

We can conclude that in addition to the supersymmetricmass term to build supersymmetric matter chargeless vectorfield Lagrangian the Hodge-type duality symmetry is anessential ingredient to be further explored We can alsoobserve that as the usual super-QED model the complexdynamical Proca-type term emerges from a product of Diracsuperspinors ΨΨ We can remark that the Dirac superspinorfieldΨ is also chiral because it is a combination of chiralWeylsuperspinors Γ119886 and Υ

119886 Finally we emphasize that on-shellsupersymmetric action obtained (31) revealing two fermionicDirac fields Θ(119909) and Π(119909) and a massive scalar field 119896(119909)

as supersymmetric partners associated to the complex vectorfield 119861120583(119909) which is the initial step to implement localgauge symmetry charge and interaction we could anticipatethat nonminimal couplings have an essential role in aforthcoming work

Conflict of Interests

The authors declare that they have no conflict of interestsrelated to this paper

References

[1] C Itzykson and J-B Zuber Quantum Field Theory DoverPublications 2005

[2] P RamondFieldTheory AModern PrimerWestviewPress 2ndedition 1990

[3] N N Bogoliubov and D V Shirkov Introduction to the Theoryof Quantized Fields John Wiley amp Sons 3rd edition 1980

[4] P Fayet and S Ferrara ldquoSupersymmetryrdquo Physics Reports vol32 no 5 pp 249ndash334 1977

[5] H PNilles ldquoSupersymmetry supergravity andparticle physicsrdquoPhysics Reports vol 110 no 1-2 pp 1ndash162 1984

[6] P D DrsquoEath ldquoCanonical quantization of supergravityrdquo PhysicalReview D vol 29 no 10 pp 2199ndash2219 1984

[7] J-P Ader F Biet and Y Noirot ldquoSupersymmetric structure ofthe induced W gravitiesrdquo Classical and Quantum Gravity vol16 no 3 pp 1027ndash1037 1999

[8] C Kiefer T Luck and P Moniz ldquoSemiclassical approximationto supersymmetric quantumgravityrdquoPhysical ReviewD vol 72no 4 Article ID 045006 2005

[9] A Riotto ldquoMore relaxed supersymmetric electroweak baryoge-nesisrdquo Physical Review D vol 58 Article ID 095009 1998

[10] A Riotto ldquoSupersymmetric electroweak baryogenesis non-equilibrium field theory and quantum Boltzmann equationsrdquoNuclear Physics B vol 518 no 1-2 pp 339ndash360 1998

[11] I R Klebanov and A A Tseytlin ldquoGravity duals of supersym-metric SU(N)timesSU(N+M) gauge theoriesrdquo Nuclear Physics Bvol 578 no 1-2 pp 123ndash138 2000

[12] P H Chankowski A Falkowski S Pokorski and J WagnerldquoElectroweak symmetry breaking in supersymmetric modelswith heavy scalar superpartnersrdquo Physics Letters Section BNuclear Elementary Particle and High-Energy Physics vol 598no 3-4 pp 252ndash262 2004

[13] J D Edelstein and C Nunez ldquoSupersymmetric electroweakcosmic stringsrdquo Physical Review D vol 55 no 6 pp 3811ndash38191997

[14] L Girardello A Giveon M Porrati and A Zaffaroni ldquoNon-Abelian strong-weak coupling duality in (string-derived) N = 4supersymmetric Yang-Mills theoriesrdquo Physics Letters B vol 334no 3-4 pp 331ndash338 1994

[15] J Wess and J Bagger Supersymmetry and Supergravity Prince-ton University Press Princeton NJ USA 2nd edition 1992

[16] D Bailin and A Love Supersymmetric Gauge Field Theory andString Theory IOP Publishing 1994

[17] S Coleman and J Mandula ldquoAll possible symmetries of the Smatrixrdquo Physical Review vol 159 no 5 pp 1251ndash1256 1967

[18] S J Gates M T Grisaru M Rocek and W Siegel SuperspaceBenjaminCummings Reading 1983

[19] R Haag J T Lopuszanski and M Sohnius ldquoAll possiblegenerators of supersymmetries of the 119878-matrixrdquoNuclear PhysicsB vol 88 pp 257ndash274 1975

[20] S Weinberg The Quantum Theory of Fields vol 3 CambridgeUniversity Press Cambridge UK 2000

[21] J Wess and B Zumino ldquoSupergauge transformations in fourdimensionsrdquo Nuclear Physics B vol 70 no 1 pp 39ndash50 1974

[22] H J W Muller-Kirsten and A Wiedemann SupersymmetryWorld Scientific Singapore 1987

[23] S J Gates Jr ldquoWhy auxiliary fields matter the strange caseof the 4-D 119873 = 1 supersymmetric QCD effective action 2rdquoNuclear Physics B vol 485 no 1-2 pp 145ndash184 1997

[24] S Ferrara L Girardello and F Palumbo ldquoGeneralmass formulain broken supersymmetryrdquo Physical Review D vol 20 no 2 pp403ndash408 1979

[25] V A Kostelecky and S Samuel ldquoSpontaneous breaking ofLorentz symmetry in string theoryrdquo Physical Review D vol 39no 2 pp 683ndash685 1989

[26] V E R Lemes A L M A Nogueira and J A Helayel-NetoldquoSupersymmetric generalization of the tensor matter fieldsrdquo

8 Advances in High Energy Physics

International Journal of Modern Physics A vol 13 no 18 pp3145ndash3156 1998

[27] D Colladay and V A Kostelecky ldquoCPT violation and thestandard modelrdquo Physical Review D vol 55 pp 6760ndash67741997

[28] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standardmodelrdquo Physical ReviewD vol 58 no 11 ArticleID 116002 1998

[29] A A Andrianov and R Soldati ldquoLorentz symmetry breakingin Abelian vector-field models with Wess-Zumino interactionrdquoPhysical Review D vol 51 no 10 pp 5961ndash5964 1995

[30] A A Andrianov and R Soldati ldquoPatterns of Lorentz symmetrybreaking inQEDbyCPT-odd interactionrdquoPhysics Letters B vol435 no 3-4 pp 449ndash452 1998

[31] S M Carrol G B Field and R Jackiw ldquoLimits on a Lorentz-and parity-violating modification of electrodynamicsrdquo PhysicalReview D vol 41 no 4 pp 1231ndash1240 1990

[32] L P Colatto A L A Penna andW C Santos ldquoCharged tensormatter fields and Lorentz symmetry violation via spontaneoussymmetry breakingrdquo European Physical Journal C vol 36 no 1pp 79ndash87 2004

[33] H Belich J L Boldo L P Colatto J A Helayel-Neto and A LMANogueira ldquoSupersymmetric extension of the Lorentz- andCPT-violatingMaxwell-Chern-Simonsmodelrdquo Physical ReviewD vol 68 Article ID 065030 2003

[34] H Belich Jr T Costa-Soares M M Ferreira Jr J A Helayel-Neto and M T D Orlando ldquoLorentz-symmetry violation andelectrically charged vortices in the planar regimerdquo InternationalJournal of Modern Physics A vol 21 no 11 pp 2415ndash2429 2006

[35] M M Ferreira Jr ldquoElectron-electron interaction in a Maxwell-Chern-Simons model with a purely spacelike Lorentz-violatingbackgroundrdquo Physical Review D vol 71 no 4 Article ID045003 9 pages 2005

[36] H Belich T Costa-Soares M M Ferreira Jr and J A Helayel-Neto ldquoNon-minimal coupling to a Lorentz-violating back-ground and topological implicationsrdquo The European PhysicalJournal C vol 41 no 3 pp 421ndash426 2005

[37] M B Cantcheff ldquoLorentz symmetry breaking and planar effectsfrom non-linear electrodynamicsrdquoThe European Physical Jour-nal CmdashParticles and Fields vol 46 no 1 pp 247ndash254 2006

[38] H Belich M A De Andrade and M A Santos ldquoGauge the-ories with Lorentz-symmetry violation by symplectic projectormethodrdquoModern Physics Letters A vol 20 pp 2305ndash2316 2005

[39] T G Rizzo ldquoLorentz violation in extra dimensionsrdquo Journal ofHigh Energy Physics vol 2005 no 9 article 036 2005

[40] M M Ferreira Jr and M S Tavares ldquoNonrelativistic electronndashelectron interaction in a maxwellndashchernndashsimonsndashproca modelendowed with a timelike lorentz-violating backgroundrdquo Inter-national Journal of Modern Physics A vol 22 p 1685 2007

[41] H Belich T Costa-Soares M M Ferreira Jr J A Helayel-Neto and F M O Mouchereck ldquoLorentz-violating correctionson the hydrogen spectrum induced by a nonminimal couplingrdquoPhysical Review D vol 74 no 6 Article ID 065009 6 pages2006

[42] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoOn the consistency of Lorentz invariance violationin QED induced by fermions in constant axial-vector back-groundrdquo Physics Letters B vol 639 no 5 pp 586ndash590 2006

[43] L P Colatto A L A Penna and W C Santos ldquoVector solitonsand spontaneous Lorentz violation mechanismrdquo httparxivorgabshep-th0610096

[44] MM Ferreira Jr andA R Gomes ldquoLorentz violating effects ona quantized two-level systemrdquo httparxivorgabs08034010

[45] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoBare and induced lorentz and cpt invariance violationsin qedrdquo International Journal of Modern Physics A vol 25 no16 pp 3271ndash3306 2010

[46] T G Rizzo ldquoLorentz violation in warped extra dimensionsrdquoJournal of High Energy Physics vol 2010 no 11 article 156 2010

[47] J Helayel-Neto R Jengo F Legovini and S Pugnetti ldquoSup-ersymmetry Complex Structures And Superfieldsrdquo SISSA-3887EP

[48] Y Kobayashi and S Nagamachi ldquoGeneralized complexsuperspacemdashinvolutions of superfieldsrdquo Journal of Mathemati-cal Physics vol 28 no 8 pp 1700ndash1708 1987

[49] P A Grassi G Policastro and M Porrati ldquoNotes on thequantization of the complex linear superfieldrdquo Nuclear PhysicsB vol 597 no 1ndash3 pp 615ndash632 2001

[50] S M Kuzenko and S J Tyler ldquoComplex linear superfield as amodel for Goldstinordquo Journal of High Energy Physics vol 2011no 4 article 57 2011

[51] S J Gates J Hallett T Hubsch and K Stiffler ldquoThe realanatomy of complex linear superfieldsrdquo International Journal ofModern Physics A vol 27 no 24 Article ID 1250143 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

8 Advances in High Energy Physics

International Journal of Modern Physics A vol 13 no 18 pp3145ndash3156 1998

[27] D Colladay and V A Kostelecky ldquoCPT violation and thestandard modelrdquo Physical Review D vol 55 pp 6760ndash67741997

[28] D Colladay and V A Kostelecky ldquoLorentz-violating extensionof the standardmodelrdquo Physical ReviewD vol 58 no 11 ArticleID 116002 1998

[29] A A Andrianov and R Soldati ldquoLorentz symmetry breakingin Abelian vector-field models with Wess-Zumino interactionrdquoPhysical Review D vol 51 no 10 pp 5961ndash5964 1995

[30] A A Andrianov and R Soldati ldquoPatterns of Lorentz symmetrybreaking inQEDbyCPT-odd interactionrdquoPhysics Letters B vol435 no 3-4 pp 449ndash452 1998

[31] S M Carrol G B Field and R Jackiw ldquoLimits on a Lorentz-and parity-violating modification of electrodynamicsrdquo PhysicalReview D vol 41 no 4 pp 1231ndash1240 1990

[32] L P Colatto A L A Penna andW C Santos ldquoCharged tensormatter fields and Lorentz symmetry violation via spontaneoussymmetry breakingrdquo European Physical Journal C vol 36 no 1pp 79ndash87 2004

[33] H Belich J L Boldo L P Colatto J A Helayel-Neto and A LMANogueira ldquoSupersymmetric extension of the Lorentz- andCPT-violatingMaxwell-Chern-Simonsmodelrdquo Physical ReviewD vol 68 Article ID 065030 2003

[34] H Belich Jr T Costa-Soares M M Ferreira Jr J A Helayel-Neto and M T D Orlando ldquoLorentz-symmetry violation andelectrically charged vortices in the planar regimerdquo InternationalJournal of Modern Physics A vol 21 no 11 pp 2415ndash2429 2006

[35] M M Ferreira Jr ldquoElectron-electron interaction in a Maxwell-Chern-Simons model with a purely spacelike Lorentz-violatingbackgroundrdquo Physical Review D vol 71 no 4 Article ID045003 9 pages 2005

[36] H Belich T Costa-Soares M M Ferreira Jr and J A Helayel-Neto ldquoNon-minimal coupling to a Lorentz-violating back-ground and topological implicationsrdquo The European PhysicalJournal C vol 41 no 3 pp 421ndash426 2005

[37] M B Cantcheff ldquoLorentz symmetry breaking and planar effectsfrom non-linear electrodynamicsrdquoThe European Physical Jour-nal CmdashParticles and Fields vol 46 no 1 pp 247ndash254 2006

[38] H Belich M A De Andrade and M A Santos ldquoGauge the-ories with Lorentz-symmetry violation by symplectic projectormethodrdquoModern Physics Letters A vol 20 pp 2305ndash2316 2005

[39] T G Rizzo ldquoLorentz violation in extra dimensionsrdquo Journal ofHigh Energy Physics vol 2005 no 9 article 036 2005

[40] M M Ferreira Jr and M S Tavares ldquoNonrelativistic electronndashelectron interaction in a maxwellndashchernndashsimonsndashproca modelendowed with a timelike lorentz-violating backgroundrdquo Inter-national Journal of Modern Physics A vol 22 p 1685 2007

[41] H Belich T Costa-Soares M M Ferreira Jr J A Helayel-Neto and F M O Mouchereck ldquoLorentz-violating correctionson the hydrogen spectrum induced by a nonminimal couplingrdquoPhysical Review D vol 74 no 6 Article ID 065009 6 pages2006

[42] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoOn the consistency of Lorentz invariance violationin QED induced by fermions in constant axial-vector back-groundrdquo Physics Letters B vol 639 no 5 pp 586ndash590 2006

[43] L P Colatto A L A Penna and W C Santos ldquoVector solitonsand spontaneous Lorentz violation mechanismrdquo httparxivorgabshep-th0610096

[44] MM Ferreira Jr andA R Gomes ldquoLorentz violating effects ona quantized two-level systemrdquo httparxivorgabs08034010

[45] J Alfaro A A Andrianov M Cambiaso P Giacconi and RSoldati ldquoBare and induced lorentz and cpt invariance violationsin qedrdquo International Journal of Modern Physics A vol 25 no16 pp 3271ndash3306 2010

[46] T G Rizzo ldquoLorentz violation in warped extra dimensionsrdquoJournal of High Energy Physics vol 2010 no 11 article 156 2010

[47] J Helayel-Neto R Jengo F Legovini and S Pugnetti ldquoSup-ersymmetry Complex Structures And Superfieldsrdquo SISSA-3887EP

[48] Y Kobayashi and S Nagamachi ldquoGeneralized complexsuperspacemdashinvolutions of superfieldsrdquo Journal of Mathemati-cal Physics vol 28 no 8 pp 1700ndash1708 1987

[49] P A Grassi G Policastro and M Porrati ldquoNotes on thequantization of the complex linear superfieldrdquo Nuclear PhysicsB vol 597 no 1ndash3 pp 615ndash632 2001

[50] S M Kuzenko and S J Tyler ldquoComplex linear superfield as amodel for Goldstinordquo Journal of High Energy Physics vol 2011no 4 article 57 2011

[51] S J Gates J Hallett T Hubsch and K Stiffler ldquoThe realanatomy of complex linear superfieldsrdquo International Journal ofModern Physics A vol 27 no 24 Article ID 1250143 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

High Energy PhysicsAdvances in

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

FluidsJournal of

Atomic and Molecular Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Advances in Condensed Matter Physics

OpticsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstronomyAdvances in

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Superconductivity

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Statistical MechanicsInternational Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

GravityJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

AstrophysicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Physics Research International

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Solid State PhysicsJournal of

 Computational  Methods in Physics

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Soft MatterJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

AerodynamicsJournal of

Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

PhotonicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Biophysics

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

ThermodynamicsJournal of