Supersymmetric Generalization of the Tensor Matter Fields

a r X i v :h e p -t h /9508049v 2 14 A u g 1997

Supersymmetric generalization of the

tensor matter ?elds

V.E.R.Lemes 1,A.L.M.A.Nogueira 2

and

J.A.Helay¨e l-Neto 3C.B.P.F

Centro Brasileiro de Pesquisas F′?sicas,Rua Xavier Sigaud 150,22290-180Urca

Rio de Janeiro,Brazil

CBPF-NF 054/95

Abstract

The supersymmetric generalization of a recently proposed Abelian axial gauge model with antisymmetric tensor matter ?elds is presented.

1Introduction

A few years ago,a line of investigation has been proposed by Avdeev and Chizhov[1] that consists in treating skew-symmetric rank-2tensor?elds as matter rather than gauge degrees of freedom.The model studied in Ref.[1]has been further reassessed from the point of view of renormalization in the framework of BRS quantization[2]. In view of the potential relevance of matter-like tensor?elds for phenomenology[1], it is our purpose in this paper to discuss some facts concerning the formulation of an N=1supersymmetric Abelian gauge model realizing the coupling of gauge?elds to matter tensor?elds and their partners.One intends here to present a superspace formulation of the model and exploit the possible relevance of extra bosonic super-symmetric partners(complex scalars)for the issue of symmetry breaking.We would like to mention that the supersymmetrization of2-forms that appear as gauge?elds is already known in connection with supergravity,and the so-called linear super?elds appear to be the most appropriate multiplets to accomodate the2-form gauge?elds [3].In our case,we aim at the supersymmetrization of matter2-form?elds coupled to Abelian gauge?elds and their supersymmetric partners.

The present work is outlined as follows:in Section2,one searches for the super-multiplet that accomodates the matter tensor?eld and discusses its self-interaction; the coupling to the gauge supermultiplet is pursued in Sections3and4;in Sec-tion5,one couples the well-known O’Raifeartaigh model[5]to the tensor-?eld supermultiplet and discusses some features concerning spontaneous symmetry and supersymmetry breaking.Finally,General Conclusions are drawn in Section6.

2Supersymmetrizing the tensor?eld

Adopting conventions for the spinor algebra and the superspace parametrization of Ref.[6],one?nds that the super?eld accomodating the skew-symmetric rank-2tensor amongst its components is a spinor multiplet subject to the chirality constraint:

Σa=ψa+θbΛba+θ2F a?iθcσμc˙cˉθ˙c?μψa

?iθcσμc˙cˉθ˙c?μθbΛba?1

Σ˙a=F˙a+iθcσμc˙cˉθ˙c?μ

θ2ˉθ2?μ?μ

4

D˙

Σa=D b

b

where ψa and F a are chiral spinors and Λba ,ˉΛ˙b ˙

a are decomposed as:Λba =εba ρ+σμν

ba λμν,ˉΛ˙b ˙a =?ε˙b ˙

a ρ??2εμναβ

T αβ

.Notice also that λμν=iλμνand (λ?μν)=?iλ?μν

,where the twiddle stands for the dual.

The canonical dimensions of the component ?elds read as below:

d (ψ)=d (2

d (ρ)=d (λμν)=1

d (F )=d (

2

.

(2.6)

Based on dimensional arguments,we propose the following superspace action for

the Σa -super?eld:

S =

d 4x d 2θd 2

32

{D a Σa Σ˙

a +q Σa Σa

Σ˙

a }.

(2.7)

To check whether such an action is actually the supersymmetric extension of the

model that treats T μνas a matter ?eld [1],we have now to write down eq.(2.7)in terms of the component ?elds ψ,ρ,λμνand F :

S =

d 4x

+?μρ?μρ??16?μλμν?αλ?αν+i σμ˙aa ?μF a

?i

σμ˙aa ?μ?ν?νψ

a

?q

12

(ρ2

?4λμνλμν)12

(ρ?2

?4λ?αβλ

?αβ

)+4qλμνλμνψ˙

a +4qλ?μνλ?μνF a

ψa

?2q F a ψa ψ˙

a ?qρ

2

ψ˙a ?q (ρ?)2F a ψa +q

ψ˙a ψ˙a ρ?)+4qρψa σμa ˙a ?β(λ?β

μ

ψ˙a )

ψ˙a )

4

(T μν?i T μν

),we arrive at the relation:16?αλαμ?βλ?βμ=2?αT αμ?βT βμ?

1

The action above displays the terms proposed by Avdeev et al.in Ref.[1];besides the anti-symmetric tensor,there appear a complex scalar and a pair of spinors as its supersymmetric partners:ψa,a non-physical fermion,and F a,corresponding to a physical Weyl spinor.The undesirable presence of a spinor with lower canonical

dimension(1

2)generating,as expected,a higher derivative term in

the Lagrangian can be avoided by a reshu?ing of the spinorial degrees of freedom, if one keeps the interactions turned o?.In fact,one can join bothψa and F a in a single fundamental Dirac spinor,Ψ,as follows:

Ψ= F a(x)

Ψγμ?μ(Ψ),provides the kinetical terms forψa and F a,turning the higher derivative term,?iσμ˙aa?μ?2ψa,into a matter of choice for the?eld basis.Nevertheless,this is true only for the free theory.The interaction sector of(2.8)cannot be re-expressed in terms of the Dirac spinorΨ,imposing a dissociation back to Weyl spinorial degrees of freedom.Therefore,it happens that the full theory must carry a higher derivative term,giving birth to a conjecture that this might be the fermionic counterpart of problems concerning the transverse sector of the original-bosonic-model for the tensor matter?eld with interactions, as discussed in[1].

3The gauging of the model

In order to perform the gauging of the model described by eq.(2.7),one proceeds along the usual lines and introduces a chiral scalar super?eld,Λ,to act as the gauge parameter:

Λ=(1?iθaσμa˙a

4

θ2

Λ=(1+iθaσμa˙a

4θ2θ˙

b

θ2

Σread as:

δΣa=ihΛΣa

δΛ

D˙a

D˙aΛ(Σ˙a?ihΛ

in such a way thatΓa transforms like

Γ′a=Γa?D aΛ.(3.6) This yields:

(?aΣa)′=e ihΛ(?aΣa).(3.7) To achieve a U(1)-invariant action,one proposes

S= d4x d2θd2?˙a

Λ?Λ).(3.9) At this point,the gauge sector displays more degrees of freedom than what is actually required to perform the gauging.There are component vector?elds in bothΓa and V.However,we notice that the super?eldΓa is not a true independent gauge potential.Indeed,

Γa=?iD a V(3.10) reproduces correctly the gauge tranformation ofΓa and,at the same time,eliminates the redundant degrees of freedom that would be otherwise present,if we were to keep Γa and V as gauge super?elds.Therefore,the locally U(1)-invariant action takes over the form:

S= d4x d2θ?1D2(e?V D a e V)

θ?1?˙aΣ˙a

?˙a D˙a D˙a V

θ˙aθ˙aσμa˙a Aμ

(3.13)

+θ2λ+λ+θ2γ˙a+θ2?,

where C,λ,

λ=i

b˙a=i

?μ?μ(φ?φ?),δAμ=??μ(φ+φ?),

4

δγa=1w˙a,δ

2

As already known,for the sake of component-?eld calculations,one usually works in the so-called Wess-Zumino gauge,where C,b a andλare gauged away.The expansion for the exponential of the gauge super?eld simpli?es,in this gauge,according to:

e hV=1+hθaσμa˙aθ˙aθ2θaγa+hθ2

4

h2θ2

4

FμνFμν+2?2+iσμ˙aa?μγa+?μρ?μρ??16?μλμν?αλ?αν

+iσμ˙aa?μF a?iσμ˙aa?μ?ν?νψa?i h

2

ρAμ?μρ?

+2hρ?ρ?+h2γ˙a

2

ργaσμa˙a?μ

2

ρ?σμ˙aa?μψa+i hγ˙a?i hψ˙a?hψ˙a +hψ˙aψ˙aψ˙a?hψ˙a

4?νAνψaσμa˙a?μ

2

F aσμa˙a Aμ

8

ψa(σνψ˙a

+hσασν)a˙a Fνα

2γa(σμψ˙a+hσμσα)a˙a

2

?νψa(σνγ˙a

+i h2γ˙a(σβ)˙aa Aμψaλ?αβ?hσβσμ)a˙a?μλαβ

2

γa(σaψ˙a

+hσνσα)a˙a Aν?α

4

?μψa(σνψ˙a

+hσασν)a˙a Aν?αψ˙a+iσμσα)a˙a FμνAα16

ψa(σνψ˙a?i h2σβσμ)a˙aλαβAμ

?i h2σνσα)a˙a AνAα

?μψa(σνψ˙a

8

+i h2σμσα)a˙a?α

ψa AνAμ(σνψ˙a

8

+h3σασμ)a˙a AνAαAμγ˙a

√√F˙a

ψaψa?μ?μ(ψ˙a)

F˙a F˙a

2

?iqρψaσμa˙a?μ(ψ˙a)?4qλμβ?β(ρ?σμ˙aaψa

?16iqλμα?β(λ?βασμ˙aaψa?qh?ψaψaψ˙a?iq hψ˙a

Aμ?μ(ψaψa)ψ˙a?q h2ψ˙aψ˙aγ˙a

2

ψ˙aγ˙aψ˙bψaψa?qhρψaσμa˙a Aμ

σασβ)a˙aλ?αβσασμ)a˙a Aμλαβ

σβσγψ˙a .(4.1)

We should stress here a remarkable di?erence with respect to the case of the chi-ral and anti-chiral scalar

Σto the gauge sector necessarily a?ects theΣ-super?eld self-interaction terms as one reads o?from eq(3.11).The gauging of the U(1)-sym-metry enriches the self-interactions of the tensor?eld not only through its fermionic supersymmetric partners,but also through the introduction of the gauge boson and the gaugino at the level of the matter self-interaction terms(these new couplings are compatible with power-counting renormalizability).This is so because the model presented here is based on a single spinor super?eld.Had we introduced a couple of spinor super?elds,Σa and T a,with opposite U(1)charges:

Σ′a=e ihΛΣa,

(4.2)

T′a=e?ihΛT a,

a self-interacting term of the form(Σa T a T˙a)would automatically be invariant whenever the symmetry is gauged,and there would be no need for introducing the vector super?eld to ensure local invariance.Such a mixed self-interacting term could,in principle,be thought of as a possible source for a mass term for the spinor super?elds,whenever the physical scalar componentρdevelops a non-trivial vacuum expectation value.Nevertheless,by analysing theρ-?eld interactions in the scalar potential,one concludes that there is no room for spontaneous symmetry breaking as induced by such a component?eld(and,similarly,for its counterpart inside T). On the other hand,we could think to introduce a gauge-invariant mass term of the form

S mass= d4x(d2θi mθi mΣ˙a

however,a mixed mass term like the one above introduces two massive excitations of the type k4=m4in the spectrum.So,regardless the sign of m2,a tachyon shall always be present;hence such a mass term is disregarded.

5On Spontaneous Symmetry Breaking

Due to the spinorial character of the super?eldΣa,it cannot be used to accomplish a spontaneous breaking.Indeed,Lorentz invariance is lost wheneverΣa acquires a non-trivial vacuum expectation value.The idea in the present section is to couple, in a gauge-invariant manner,the well-known O’Raifeartaigh model[5]to the spinor super?eldΣa,so as to understand the issue of an eventual mass generation forΣa via spontaneous internal symmetry or supersymmetry breakingdown.For the sake of concreteness,the model we adopt to discuss such a matter is de?ned by the action below:

S= d4x d2θd2φφ+φ?e?hVφ?

+ d4x d2θ 1

θ 1φ2+μφ?+fφφ?+GΣ˙aφ?

+ d4x d2θ?1D2(e?V D a e V)

θ?1?˙aΣ˙a

θ˙a?μ?1θ2?μ?μ)(A+θbξb+θ2b)

(5.2)

θ2

φ+=(1?iθaσμa˙a

4

θ˙a?μ?1θ2?μ?μ)(A?+θbξ?b+θ2b?);

m andμare mass parameters,f has dimension of mass2,whereas g and G are dimensionless coupling constants.Σa andφ?have opposite U(1)-charges.This action,in terms of components,reads:

S= d4x4 4{?μA??μA+?μA???μA?+?μA?+?μA+}+4{b?b+b??b?+b?+b+} +2i{σμ˙aa?μξa+σμ˙aa?μξa?+σμ˙aa?μξa+} +

+ d4x 16hA?+?A++8ihA?+?μAμA++4h2A?+AμAμA+?8hA?+γaξ+a

?8hγ˙a A++4hξ+aσμa˙a Aμ

ξ?˙aξ˙a??16ih?μA??AμA?

+ d4x m(?4bA+ξaξa)+m(?4b?A?+ξ˙a)?4fb?4fb?

+ d4x2 μ(?2b+A??2b?A++ξa+ξa?)+μ(?2b?+A???2b??A?++ξ˙a?) + d4x2 g(?2bA+A??2Ab+A??2AA+b?+Aξa+ξa?+A?ξaξa++A+ξaξa?) +g(?2b?A?+A???2A?b?+A???2A?A?+b??+A?ξ˙a?+A??ξ˙a++A?+ξ˙a?) + d4x4G (?2F aψa+ρ2+4λμνλμν)(A?)2+(?2ψ˙a+(ρ?)2+4λ?μνλ?μν)(A??)2 +2ψbψb(?4b?A?+ξa?ξa?)+2ψ˙b(?4b??A??+ξ˙a?)

ξ˙b?A??)

?2(ρψb?σμνbaλμνψa)(ξb?A?)+2(ρ?σμν˙

b˙a

+ d4x +2hρ?ρ?+2?2+h2

ρandλμνcannot be avoided in a consistent way,just by invoking internal symmetry or supersymmetry breaking as realised by a set of scalar super?elds.

6General Conclusions

The supersymmetrization of the matter tensor?eld?rst investigated in Ref.[1]has been worked out here in terms of a spinor chiral super?eld,Σa,whose kinetic and self-interacting terms have been found in N=1-superspace.The gauging of the model reveals some peculiarities,such as the need of gauge?elds to appear in the matter self-interactions.

Scalar degrees of freedom that accompany the fermionic partners ofλμνcannot be the source for spontaneous symmetry or supersymmetry breaking,as it could in principle be thought.The reason is that Lorentz invariance preventsΣa from developing a non-trivial vacuum expectation value.

A thorough analysis of the coupling betweenΣa and chiral and anti-chiral scalar super?elds indicate that no spontaneous breaking takes place.In components,the scalarρand the tensorλμνdisplay a quartic coupling to the physical scalar com-ponents of the additional matter super?elds,namely,ρ2A2?andλ2μνA2?,and no non-trivial minimum with=0shows up(a non-trivial minimum with non-

zero vacuum expectation value restricted to the neutral scalar A is possible,but has no consequence for mass generation).So,spontaneous breaking does not happen to be a possibility for inducing a mass for the tensorλμν.

As a next step,the analysis of matter tensor?elds in the framework of N=2 extended supersymmetries and the non-Abelian version of the N=1model are to be pursued.

7Conventions

σμ=(1,σ),σμ˙

ba =σμ

a˙b

,(7.1)

whereσ=(σ1,σ2,σ3)are the Pauli matrices

σ1= 0110 ,σ2= 0?i i0 ,σ3= 100?1 .(7.2)

In addition,the matricesσμνand

2

,0)and(0,1

given by

σμa˙a

σμ˙aaσν

a˙b =ημνδ˙a˙

b

?i(

σν˙abσα

b˙b

θθ˙a

θ˙a is de?ned by

?

?

θ˙b=δ˙b˙a.(7.7) Covariant derivatives with respect to the supersymmetry transformations are:

D a=?θ˙a?μ

?

D˙a}=2iσμa˙a?μ;{D a,D b}=0={D˙

b

}.(7.9) The Dirac matrices,γμ,playing a role in the purely physical spinorial kinetic term, i

σμ0 .(7.10) The spinor

Ψ=Ψ?γ0,whereγ0= 0110 .(7.11)

Acknowledgements

We wish to thank Dr.M.A.de Andrade for helpful discussions on an earlier manuscript.The Conselho Nacional de Desenvolvimento Cienti?co e Tecnologico, CNP q-Brasil is gratefully acknowledged for the?nancial support.

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