Lorentz Violation and Spacetime Supersymmetry

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IUHET-454December 2002Lorentz Violation and Spacetime Supersymmetry 1M.S.Berger Physics Department,Indiana University,Bloomington,IN 47405,USA E-mail:berger@https://www.sodocs.net/doc/e98125469.html, Abstract.Supersymmetry and Lorentz invariance are closely related as both are spacetime sym-metries.Terms can be added to Lagrangians that explicitly break either supersymmetry or Lorentz invariance.It is possible to include terms which violate Lorentz invariance but maintain invariance under supersymmetric transformations.I illustrate this with some simple extensions of the original Wess-Zumino model.INTRODUCTION The understanding of spacetime symmetries has grown remarkably in the last century.The mass-momentum behavior is associated with the requirement that theories respect translation invariance,while the intrinsic spin and the existence of antiparticles are consequences of the Lorentz group.The theoretical discovery of supersymmetry is a major achievement of twentieth century physics,and it marked another expansion of our concept of spacetime symmetries.The advent of supersymmetry was followed by its application to gauge theories and the Standard Model and a local version (supergravity)that incorporates general relativity.This is a clear indication that supersymmetry and Lorentz invariance are closely linked.The introduction of fermionic generators that relate fermions and bosons resulted largely from motivations that were theoretical,but

it is clear that supersymmetry has come to dominate particle physics phenomenology.A large part of the history of physics and particle physics in particular involves the introduction of symmetries and the understanding of the spontaneous breaking of these symmetries.The longstanding point of view has been that spacetime symmetry is one of the most fundamental.The familiar symmetries regarding translations,angular momentum,and Lorentz boosts that constitute the Poincarégroup are usually assumed to be unbroken symmetries.While there is great theoretical appeal for these symmetries to be unbroken,this issue is of course a question that can only be decided by experiment.In any case the amount of breaking of the Lorentz symmetry must be very small,if it is indeed nonzero,since it has so far escaped experimental detection.Supersymmetry,

if it exists,comprises another part of the overall spacetime symmetry,and it is clearly broken.In fact,it must be broken badly compared to the scale at which we perform experiments,so much so that we have currently not detected any of the supersymmetric partners to the Standard Model particles.However,from a more theoretical point of view,the breaking of supersymmetry is also very small since the breaking scale is very much suppressed in comparison to the Planck scale(where the true nature of spacetime presumably emerges).

The Poincaréalgebra involves the generator of translations(Pμ)and the generator of rotations and Lorentz boosts Mμνin the following way

Pμ,Pν =0

Pμ,Mρσ =i(ημρPσ?ημσPρ)

Mμν,Mρσ =i(ηνρMμσ?ηνσMμρ?ημρMνσ+ημσMνρ),(1)

Explicit terms can be added to a Lagrangian that violate the Poincaréalgebra.If one wants to preserve energy-momentum conservation,then any Lorentz violation continues to respect the?rst of these equations.The set of terms which can be added to the Standard Model have been categorized[1,2]in an extended Lagrangian.These terms are thought to arise in a more fundamental theory like string theory[3,4]which is nonlocal, but in the context of the Standard Model extension they are viewed as phenomenological parameters.

Supersymmetry involves extending this algebra with a fermionic generator Q2,

Q,Pμ =0

Q,

2The four-component notation for spinors makes the relationship with the Standard Model Lorentz-violating extension[1,2]transparent.

so that one can say the models respect both supersymmetry and translation invariance. Previous discussion of the superPoincaréalgebra has been with regard to its exact realization,or to cases where the subalgebra involving the fermionic generator Q is broken.

This talk is based on work done with V.Alan Kostelecky[5].

LORENTZ-VIOLATING WESS-ZUMINO MODEL

The?rst four-dimensional supersymmetric?eld theory was written down by Wess and Zumino[6].A modest goal is to determine whether it is possible to have an unbroken supersymmetry even in the presence of broken Lorentz symmetry.Can terms that explicitly violate the Lorentz symmetry be added to the Wess-Zumino model that still preserve some version of the supersymmetry?This approach of adding explicit terms is in the same spirit as the Lorentz-violating Standard Model extension in which Lorentz violation is added to the Standard Model3.Consider the following Lagrangian,

L=1

2

?μB?μB+

1

ψ?/ψ+

1

2

G2

+m ?1ψψ+AF+BG

+

g

2 F(A2?B2)+2GAB?

2

ikμν

2

kμνkμρ(?νA?ρA+?νB?ρB),(3)

where one recognizes the?rst three lines as the original Wess-Zumino model.The last two lines involve the Lorentz-violating coef?cients kμνeither linearly or quadratically. These couplings are simply numbers(with Lorentz index labels)that do not change un-der a particle Lorentz transformation,which boosts or rotates local?eld con?gurations within a?xed inertial frame.Since there are an even number of indices these terms do not violate the CPT invariance.Without loss of generality,kμνcan be taken to be a real symmetric,traceless,coef?cient.

Since the supersymmetric transformation relates fermions to bosons,there is a non-trivial relationship between the coupling coef?cients involving the scalars and the fermion.The supersymmetric transformation forces a relationship(namely the common kμν)on the Lorentz-violating terms of Eq.(3)that is similar to the common mass and couplings that are a well-known consequence of supersymmetric theories.

If one modi?es the supersymmetric transformations of the Wess-Zumino model by adding new terms involving the coef?cients kμνcoef?cients,

δA=

εγ5ψ,

δψ=?(i?/+ikμνγμ?ν)(A+iγ5B)ε+(F+iγ5G)ε,

δF=?ˉε(i?/+ikμνγμ?ν)ψ,

δG=ˉε(γ5?/+kμνγ5γμ?ν)ψ,(4) one?nds that the Lagrangian is invariant up to a total derivative.

The commutator of two supersymmetry transformations in Eq.(4)yields

δ1,δ2 =2iε1γμε2?ν,(5)

which involves the generator of translations.A modi?ed supersymmetry algebra there-fore exists,and the lagrangian in Eq.(3)provides an explicit example of an interacting model with both exact supersymmetry and Lorentz violation..

One can also show that a modi?cation of the supersymmetry transformation in Eq.(4) cannot be modi?ed,say by changing the transformation of the scalar?elds A and B,be-cause any such modi?cation would not result in a closure of the supersymmetry algebra. In fact,one can understand the modi?cation of the Lagrangian and the supersymmetric transformations as a global substitution of the form i?μ→i?μ+ikμν?ν.The transla-tion generator Pμcommutes with itself(?rst equation in Eq.(1))and satis?es the?rst equation in Eq.(2),so it then follows that

Q,P2 =0.(6)

Since the superpotential containing the mass and coupling terms is unaffected by the Lorentz violation,analogues should exist for various conventional results on supersym-metry breaking[8,9,10].

One consequence of the supersymmetric Lagrangian is the relationship between the fermionic and scalar propagators.The fermionic propagator is

i

CPT-ODD LORENTZ BREAKING

It is well-known that a local Lorentz-invariant quantum?eld theory preserves the combi-nation CPT where C is charge conjugation,P is parity,and T is time reversal.If Lorentz-violating terms are added to the Lagrangian,however,then this result no longer needs to hold and nonlocality as well as CPT violation is permitted.

A CPT-violating term can be added to the Wess-Zumino model in the following way,

L=1

2

?μB?μB+

1

ψ?/ψ+

1

2

G2 +kμ(A?μB?B?μA)?

1

ψγ5γμψ

+

1

εψ,

δB=i

ψγ5ψ+2AG?2BF ,(13)

cannot be added to the Wess-Zumino theory because it is not invariant under the modi-?ed supersymmetry transformations in Eq.(10).It is not surprising that they cannot be reconciled with supersymmetry since the mass and coupling terms do not respect the ?eld rede?nition.

In addition to the kμνand kμ-dependent terms described above,the simple?eld content of the Wess-Zumino model admits additional renormalizable Lorentz-violating term to its Lagrangian:(A2?B±B2?A),φψσμν?λψ.However,there do not appear to be supersymmetric interpretations for these terms.

SUPERFIELD FORMULATION

The elegant method of superspace[11]allows one to combine all the component?elds of a supersymmetric multiplet into one super?eld.By extending the four dimensions of spacetime to include fermionic dimensions as well,this technique highlights the role of supersymmetry as a spacetime symmetry.The fermionic coordinates,θ,that are added to the usual spacetime coordinates highlight the role of supersymmetry as a spacetime symmetry.The CPT transformation maps the components of a chiral super?eldΦinto themselves,while the parity transformation alone maps left-chiral super?elds into right-chiral super?elds and vice versa.De?ne

φ=1

2(A+iB),F=1

2

(F?iG).(14)

In terms of these complex scalars,the left-chiral super?eld appropriate for the model in

Eq.(3)is

Φ(x,θ)=φ(x)+√θψ

L

(x)+1θ(1?γ5)θF(x)

+1θγ5γμθ(?μ+kμν?ν)φ(x)

?i2θ(?/+kμνγμ?ν)ψL(x)

?1θθ)2(?μ+kμν?ν)2φ(x).(15) Here,the subscript L denotes projection with1

2

mΦ2|F+1

εQΦ(x,θ) where

Q=i?

coupling terms in Eq.(3)in the usual way by projecting out the usual F-term component of the holomorphic functions of the chiral super?eldΦ.The generators Q and Pμ=i?μsatisfy

Q,Pμ =0, Q,

2.Colladay,D.,and Kostelecky,V.A.,Phys.Rev.,D58,116002(1998).

3.Kostelecky,V.A.,and Samuel,S.,Nucl.Phys.,B336,263(1990).

4.Kostelecky,V.A.,and Potting,R.,Phys.Lett.,B381,89–96(1996).

5.Berger,M.S.,and Kostelecky,V.A.,Phys.Rev.,D65,091701(2002).

6.Wess,J.,and Zumino,B.,Nucl.Phys.,B70,39–50(1974).

7.Kostelecky,V.A.,and Lehnert,R.,Phys.Rev.,D63,065008(2001).

8.Wess,J.,and Zumino,B.,Phys.Lett.,B49,52(1974).

9.O’Raifeartaigh,L.,Nucl.Phys.,B96,331(1975).

10.Witten,E.,Nucl.Phys.,B188,513(1981).

11.Salam,A.,and Strathdee,J.,Nucl.Phys.,B76,477–482(1974).