Cosmological Symmetry Breaking, Pseudo-scale invariance, Dark Energy and the Standard Model
aa r X i v : . [ h e p - ph ] J un Cosmological Symmetry Breaking, Pseudo-scaleinvariance, Dark Energy and the Standard Model
Pankaj Jain and Subhadip Mitra
Physics Department, I.I.T. Kanpur, India 208016
Abstract:
The energy density of the universe today may be dominated bythe vacuum energy of a slowly rolling scalar field. Making a quantum expan-sion around such a time dependent solution breaks fundamental symmetriesof quantum field theory. We call this mechanism cosmological symmetrybreaking and argue that it is different from the standard phenomenon ofspontaneous symmetry breaking. We illustrate this with a toy scalar fieldtheory, whose action displays a U(1) symmetry. We identify a symmetry,called pseudo-scale invariance, which sets the cosmological constant exactlyequal to zero, both in classical and quantum theory. This symmetry is alsobroken cosmologically and leads to a nonzero vacuum or dark energy. Theslow roll condition along with the observed value of dark energy leads to avalue of the background scalar field of the order of Planck mass. We alsoconsider a U(1) gauge symmetry model. Cosmological symmetry breaking,in this case, leads to a non zero mass for the vector field. We also show thata cosmologically broken pseudo-scale invariance can generate a wide rangeof masses.
The current cosmological observations [1, 2, 3, 4, 5, 6] suggest that the energydensity of the universe gets a significant contribution from vacuum energy.For a review see [7, 8, 9]. This may be modelled by simply introducing a1osmological constant [7, 8, 10, 11, 12, 13] or dynamically by a scalar fieldslowly rolling towards the true minimum of the potential [14]. If we assumethe existence of such a field then it implies that in the current era its lowestenergy state is not the true vacuum state of the theory. In order to studythe spectrum of this theory one needs to make a quantum expansion arounda time dependent field. This has interesting implications for the physics ofsuch models, not explored so far in the literature. In particular since weare expanding around a time dependent field and not the ground state, theresulting physics need not display the symmetries of the original lagrangian.Hence this may provide us with another method of breaking fundamentalsymmetries. We point out that in general a theory displays the symmetriesof the action provided the ground state of the theory is symmetric underthe corresponding transformations. In the present case, however, the groundstate is irrelevant since the field never reaches this state. The time dependentsolution, around which we are required to make the quantum expansion,need not display the symmetries of the original action. Hence the theorymay display broken symmetries over a large period in the life time of theuniverse.
We consider a simple model of a complex scalar field with a global U(1)symmetry L [Φ( x )] = ∂ µ Φ ∗ ( x ) ∂ µ Φ( x ) − m Φ ∗ ( x )Φ( x ) − λ (Φ ∗ ( x )Φ( x )) . (1)This Lagrangian has global U (1) symmetry, i.e., it is invariant under Φ( x ) → e iθ Φ( x ). So far we have neglected the effect of the background metric whichwill also be included later. We assume that the potential is sufficiently gentle,2ith mass parameter sufficiently small, such that the scalar field is slowlyrolling towards its true minimum. Let η ( t ) be the classical solution to theequations of motion. Here we assume that this solution is independent ofspace and is in general complex.We split Φ( x ) into two parts: Φ( x ) = η ( x ) + φ ( x ). Here η ( x ) is theclassical solution of the equation of motion. Let us assume the classicalsolution depends only on time but is in general complex, i.e, η ( x ) = η ( t ).Hence Φ( x ) = η ( t ) + φ ( x ) and Φ ∗ ( x ) = η ∗ ( t ) + φ ∗ ( x ). L [ η, φ ] = ˙ η ∗ ˙ η + ˙ η ∗ ˙ φ + ˙ η ˙ φ ∗ + ∂ µ φ ∗ ∂ µ φ − m ( φ ∗ φ + ηφ ∗ + η ∗ φ + η ∗ η ) − λ { ( φ ∗ φ ) + ( ηφ ∗ ) + ( φη ∗ ) + ( η ∗ η ) + 2( ηφ ∗ + η ∗ φ ) φ ∗ φ + 4 η ∗ ηφ ∗ φ + 2 η ∗ η ( η ∗ φ + ηφ ∗ ) } . (2)From this we identify the classical Lagrangian. L Classical [ η ( t )] = ˙ η ∗ ˙ η − m η ∗ η − λ ( η ∗ η ) . (3)The classical field η satisfies the equation of motion,¨ η + η ( m + 2 λη ∗ η ) = 0 . (4)If we assume that the quantum fluctuations die sufficiently fast at t = ±∞ ,then we can write Z d x ˙ η ( ˙ φ + ˙ φ ∗ ) = − Z d x ¨ η ( φ + φ ∗ ) = Z d x η ( m + 2 λη )( φ + φ ∗ ) . Hence we get, L = L Classical [ η ( t )] + ∂ µ φ ∗ ∂ µ φ − m φ ∗ φ − λ { ( φ ∗ φ ) + ( ηφ ∗ ) + ( η ∗ φ ) + 2 φ ∗ φ ( ηφ ∗ + η ∗ φ ) + 4 η ∗ ηφ ∗ φ } . (5)3et η = η e iθ and φ ( x ) = ( φ + iφ ) / √
2. This gives L = L Classical [ η ( t )] + 12 ∂ µ φ ∂ µ φ + 12 ∂ µ φ ∂ µ φ − m φ + φ ) − λη φ − φ ) cos 2 θ + 4 φ φ sin 2 θ + 4( φ + φ )] − λ φ + φ ) + 4 η ( φ + φ )( φ cos θ + φ sin θ )] . (6)It is clear that the two modes do not have the same mass. For a complexclassical solution, i.e. with θ = 0, the two modes are coupled. They can bedecoupled by the rotation in internal space φ φ = cos β − sin β sin β cos β φ ′ φ ′ . (7)We assume adiabaticity and hence we ignore the time dependence of rotationmatrix in the kinetic energy term. The rotation angle β is found to be equalto θ and the two mass eigenvalues are found to be m + 6 λη and m + 2 λη .Hence the two modes pick up different masses and the symmetry is broken.We do not find a zero mode, in contrast to the case of spontaneous symmetrybreaking. We call this phenomenon Cosmological Symmetry Breaking. We next consider the complex scalar field model in a background gravita-tional field, which would be considered classically. The action may be writtenas S = Z d x √− g h g αβ ∂ α Φ ∗ ( x ) ∂ β Φ( x ) − m Φ ∗ ( x )Φ( x ) − λ (Φ ∗ ( x )Φ( x )) i . (8)We assume the FRW background metric with the Hubble parameter H ( t )and the expansion factor R ( t ). This model also displays the global U(1)symmetry Φ( x ) → e iθ Φ( x ). We again write Φ( x ) = η ( t ) + φ ( x ), where η ( t ) is4 space independent solution to the classical equations of motion. The realand imaginary parts η and η satisfy d η i dt + 3 H dη i dt + ∂V∂η i = 0 (9)for i = 1 ,
2. Here the potential V ( η ) = m η ∗ η + λ ( η ∗ η ) . The model againdisplays symmetry breaking as long as η ( t ) is different from zero. This isallowed as long as the conditions for slow roll is satisfied. The Hubble pa-rameter is determined by the entire matter content of the universe and herewe shall consider it as an independent function of t . We shall assume it tobe approximately constant as is the case for a vacuum dominated universe.The slow roll condition is satisfied if the mass parameter m << H . Wemay expand the potential around the classical solution and find that themass spectrum is same as found in the earlier section. We, therefore, finda squared mass splitting 4 λη ( t ) , whose scale is determined by the Hubbleparameter. Since the splitting is determined by the value of the Hubble pa-rameter, its value in the current era is quite small. However the splittingneed not be small in the early universe. At that time it may lead to largeobservable consequences.Although the mass splitting is very small, the value of the field η ( t ) canbe large. This depends on our choice of the coupling λ . By choosing λ sufficiently small we can make η ( t ) arbitrarily large and still maintain slowroll conditions. The non-zero value of this field can lead to a wide range ofbreakdown of symmetries, including Lorentz invariance. Lorentz invarianceis broken because the classical field only has time dependence. Furthermoreif we gauge the U(1) invariance then the gauge symmetry will be broken.The mass of the gauge field in this case depends on η ( t ) and can be quitelarge. 5 Dark Energy
So far we have considered a slowly rolling complex scalar field and shownthat it leads to breakdown of symmetries of the original lagrangian. Wenext consider the possibility that the complex scalar field itself leads to darkenergy. We determine the range of allowed values for the background scalarfield in order that it leads to vacuum energy equal to the observed dark energydensity. For this purpose we consider the equation of motion in gravitationalbackground in terms of the two real fields η and η . For orders of magnitudeestimate we may assume η ∼ η . The analysis is easily modified if this is notthe case and does not lead to any essential difference. For slow roll, the secondderivative term is negligible. We may consider two separate cases where eithermass term or the quartic coupling term in the potential dominates. If themass term dominates then the slow roll condition is satisfied if m << H . Ifthe quartic coupling term dominates then we find the condition λ << H /η .We next require that ρ V = V ( η ). In both the cases this leads to the condition, η >> ρ / V /H ≈ s π M PL (10)where M PL is the Planck mass. Here we have used the fact that the vacuumenergy density is almost equal to the critical energy density. Hence we findthat in our model the value of the slow roll scalar field has to be of the orderof the Planck mass or higher.We point out that the value of the coupling constant λ turns out to beextremely small in our model for dark energy. This by itself need not lead toa fine tuning problem since we are free to choose a value for this parameter.Indeed a parameter as small as this is expected due to the widely differentscales of Planck mass and the Hubble constant. The fine tuning problemmay arise if at higher orders we need to adjust this parameter to very high6ccuracy. This may happen if it undergoes large quantum corrections. Thisis an important check of the theory which we shall address in detail in afuture publication. We next gauge the U(1) symmetry considered in the earlier sections. Theresulting Lagrangian, in the presence of background gravity, can be writtenas S = Z d x √− g (cid:20) g αβ ( D α Φ( x )) ∗ D β Φ( x ) − g αβ g κδ F ακ F βδ − m Φ ∗ ( x )Φ( x ) − λ (Φ ∗ ( x )Φ( x )) i (11)where D α = ∂ α − ig A α is the covariant derivative and F αβ the field strengthtensor of the U(1) gauge field A α . This Lagrangian is invariant under Φ( x ) → e iθ ( x ) Φ( x ). We can parameterize Φ( x ) by:Φ( x ) = ( η + ρ/ √
2) exp " i θ + σ √ η ! = η + ρ √ i σ √ . . . ! e iθ . Hence, for small oscillations ρ ( x ) and σ ( x ) are φ ′ ( x ) and φ ′ ( x ) respectively.We define new fields,Φ ′ = exp " − i θ + σ √ η ! Φ = η + ρ/ √ , B µ = A µ − g ∂ µ θ + σ √ η ! . If we neglect the gravitational field, the Lagrangian becomes: L = ( D µ Φ( x )) ∗ ( D µ Φ( x )) − F µν F µν − m ( η + ρ/ √ − λ ( η + ρ/ √ .
7e now follow the same procedure as before. We split the gauge fieldinto two parts - i ) the classical gauge field, β µ ( t ) which depends only on timeand ii ) B µ ( x ), the quantum field i.e., B µ ( x ) = β µ ( t ) + B µ ( x ). So the classicalLagrangian becomes: L Classical = ˙ η + g β µ β µ η − m η − λη − f µν f µν where f µν = ∂ µ β ν − ∂ ν β µ . The equations of motion are: β = 0 , (12)¨ β i = − g β i η , (13)¨ η = − η ( g ~β + m + 2 λη ) . (14)If we assume β µ = 0, then using these equations and dropping the totalderivative terms one can rewrite the Lagrangian as: L = L Classical + 12 ∂ µ ρ∂ µ ρ + g B η + ρ √ ρη ! − ( m + 6 λη ) ρ − λ ρ + 4 √ ρ η ) − F µν F µν (15)where F µν = ∂ µ B ν − ∂ ν B µ . The gauge field has acquired a mass, m B = gη .Hence the gauge invariance is broken in this theory. In the simplest case,discussed in the last section, this mass will be of the order of Planck mass,assuming gauge coupling of order unity and if we require that the scalar fieldvacuum energy gives dominant contribution to dark energy. However if wedo not impose the condition that the vacuum energy associated with the fieldΦ is equal to the observed vacuum energy, then the mass of the gauge fieldis an independent parameter which can be fixed by a suitable choice of theclassical solution. In section 7 below we also provide another generalizationof the lagrangian so as to generate a different mass scale.8 Pseudo-scale Invariance
We have so far introduced a new method of breaking symmetries of a fieldtheory. The procedure is found to naturally lead to dark energy in the formof the vacuum energy of a slowly rolling scalar field. However so far wenot addressed the question of why the cosmological constant is so small.The problem is ofcourse well known. Quantum field theory in general pro-duces cosmological constant many orders of magnitude larger than what isobserved. In the absence of any symmetry, which may demand absence ofcosmological term in the action, this is a serious problem in fundamentalphysics. In this section we identify a symmetry which eliminates cosmologi-cal constant both at classical and quantum level.We first consider actions which are scale invariant. It is clear that at theclassical level scale invariance eliminates all dimensionful parameters fromthe action, including a cosmological constant term. However in the quantumtheory, it is well known that scale invariance is anomalous and hence maygenerate a cosmological constant. In very interesting papers Cheng [15] andCheng and Kao [16, 17] have argued that scale transformations can be brokeninto a general coordinate transformation and what is refered to as the pseudo-scale transformations. Under the pseudo-scale transformations x → x Φ → Φ / Λ g µν → g µν / Λ A µ → A µ . (16)The matter part of the action is invariant under this transformation. Thegravitational action is not invariant but, as explained in Ref. [15, 16], it can9e easily generalized so that it is invariant. One simply replaces [16],14 πG R → β Φ ∗ Φ R (17)where G is the gravitational constant, R the Ricci scalar and β a dimen-sionless constant. Next assuming a slow role scalar field discussed in earliersections, with its value of order the Planck mass, we find that the resultingaction will have predictions identical to Einstein’s gravity, at leading order.The cosmological constant term is not invariant under pseudo-scale invari-ance and hence eliminated in the classical action. The theory now has exactlyzero cosmological constant as long as the symmetries of the action are notbroken.The pseudo-scale invariance is, however, broken through our cosmologicalsymmetry breaking mechanism, discussed earlier. Hence this mechanism willgenerate a nonzero cosmological constant, or dark energy. We can directlyborrow the results obtained in section 4 with the mass parameter m set tozero. The theory discussed in section 4 now displays pseudo-scale invariance,besides invariance under the U(1) transformations. Both of these symmetriesare broken cosmologically and the slow roll condition, along with the valueof the observed dark energy density sets the scale of the classical scalar fieldof the order of Planck mass.We next consider a regulated action, assuming dimensional regularization.Here we focus only on scalar fields. In this case we consider the scalar fieldaction in n dimensions, S = Z d n x √− g h g αβ ∂ α Φ ∗ ( x ) ∂ β Φ( x ) − λ ( √− g ) ( n − /n (Φ ∗ ( x )Φ( x )) i . (18)The action is invariant under the transformation x → x → Φ / Λ a ( n ) g µν → g µν / Λ b ( n ) (19)where b ( n ) = 4 a ( n ) / ( n −
2) and we may choose a ( n ) to be any functionof n . This generalizes the pseudo-scale transformations to n dimensions.The action displays exact symmetry under pseudo-scale transformations in n dimensions. However the action has general coordinate invariance only in4 dimensions. In dimensions other than 4 the potential term violates generalcoordinate invariance. Here we take the point of view that the fundamentalquantum theory may obey a more general transformation law rather thangeneral coordinate invariance. We are guided primarily by data and theabsence of general coordinate invariance in dimensions other than four willgive modified predictions only at very high energy scale of the order of Planckmass. At this scale the theory is so far untested and we cannot rule out ouraction.To summarize, we find that we can impose pseudo-scale invariance as anexact symmetry. The symmetry is not anomalous. This symmetry prohibitsus to introduce a cosmological constant, both at the classical and quantumlevel. Hence it may provide an explanation for why the cosmological con-stant is so small. Alternative approaches to solve the cosmological constantproblem are described in Ref. [10, 18, 19, 20, 21, 22, 23, 24, 25]. The basic problem of generating realistic masses of the observed particles,however, still remains in our theory. Pseudo-scale invariance prohibits anymass terms in the action. Hence the standard Higgs mechanism is not appli-cable and all the Standard Model fields, for example, will remain massless.11he pseudo-scale invariance is ofcourse broken cosmologically and hence onemay expect that we may be able to break the standard model gauge sym-metry also by a slowly rolling scalar field. However we have to do this suchthat the mass of the scalar field is sufficiently large and not ruled out exper-imentally. In the construction so far, the mass of the scalar field has beenfound to be very small. One possibility is that this scalar boson is eliminatedfrom the spectrum by gauging the pseudoscale invariance [15, 16]. In thiscase the Higgs boson will be eliminated from the spectrum. An alternateconstruction, which does not involve gauging the pseudo-scale invariance, isdescribed below.We next construct a toy model such that, besides generating dark energy,it also breaks another U (1) symmetry with a sufficiently large mass of thescalar field. This is a toy model which can be generalized to construct anacceptable Standard Model of particle physics. We consider a model withtwo complex scalar fields Φ and Ψ. Here Φ will be considered as a slowlyrolling field which gives rise to dark energy. We construct an action suchthat it is invariant under the transformation Φ → e iθ Φ and Ψ → e iξ Ψ aswell as the pseudo-scale transformations. Here we restrict ourselves to fourdimensions. S = Z d x √− g " g αβ ∂ α Φ ∗ ( x ) ∂ β Φ( x ) + g αβ ∂ α Ψ ∗ ( x ) ∂ β Ψ( x ) − λ (Φ ∗ ( x )Φ( x )) − λ (Ψ ∗ Ψ − λ Φ ∗ Φ) . (20)Here λ is taken to be of order unity. Its precise value will be fixed by the massof the Ψ particle. The coupling λ << → e iξ Ψ.We again expand these two fields as Φ = η ( t ) + φ and Ψ = ζ ( t ) + ψ ,12here η ( t ) and ζ ( t ) are the time dependent classical fields and φ and ψ arethe quantum fluctuations. The classical fields satisfy, d η i dt + 3 H dη i dt + λ ( η + η ) η i + λ λ [ ζ + ζ − λ ( η + η )] η i = 0 , (21) d ζ i dt + 3 H dζ i dt + λ [ ζ + ζ − λ ( η + η )] ζ i = 0 . (22)We consider a slow roll solution such that all the second derivative termsare negligible. We set ζ i = (1 + δ ) λ η i , where δ <<
1. We can determine δ perturbatively by solving the differential equations. As we have seen earlier,slow roll condition requires that λ << H /η . Here we have an additionalterm proportional to λ in the equation of motion for η i , eq. 21. Substituting ζ i = (1 + δ ) λ η i in eq. 22, we get an estimate of dη i /dt . Substituting this ineq. 21 we find δ ≈ λ λ λ . (23)We want to choose λ such that ζ i is of order of the Weinberg Salam symmetrybreaking scale. It is clear that in this case δ <<
1, which is required for theself consistency of the perturbative solution.We can now determine the contribution of the term proportional to λ tothe equation for η . We find that it gives a contribution of order λ λ ( η + η ) η i , which is much smaller compared to leading order term λ ( η + η ) η i .Hence the term proportional to λ can be treated perturbatively. We alsofind the λ term gives a correction of order ( λ/λ ) λφ to the vacuum energy.Since λ/λ <<
1, this correction is negligible. We, therefore, find that thenew term in the lagrangian, proportional to λ , can be ignored at the leadingorder in the equation of motion for η and also gives negligible correction tothe vacuum energy density. Furthermore the complete solution in its presencecan be determined by treating this term perturbatively.13inally we estimate the mass of the Ψ particle. For this we expand thepotential in terms of the fields φ and ψ and collect terms which are secondorder in these fields. We find V = λ (Φ ∗ Φ) + λ (Ψ ∗ Ψ − λ Φ ∗ Φ) = λ (cid:20) η φ + η φ + 2 η η φ φ + 12 ( η + η )( φ + φ ) (cid:21) + λ λ ( η φ + η φ ) − δλ λ ( η + η )( φ + φ )+ λ [( ζ ψ + ζ ψ ) + δλ ( η + η )( ψ + ψ ) − λ ( ζ ψ + ζ ψ )( η φ + η φ )] + . . . (24)where we have only displayed the quadratic terms in the fluctuations. Wenow need to diagonalize the mass matrix. The form of these terms suggeststhat we define the fields φ + = φ cos θ + φ sin θ φ − = − φ sin θ + φ cos θ ψ + = ψ cos θ + ψ sin θ ψ − = − ψ sin θ + ψ cos θ (25)where cos θ = η / q η + η and sin θ = η / q η + η . In terms of the rotatedfields the potential can be written as V = ( (cid:20) λ + (1 − δ ) λ λ (cid:21) φ + λ − δλ λ ! φ − + λ (1 + 3 δ ) λ ψ + δλ λ ψ − − λ λ (1 + δ ) φ + ψ + ) ( η + η ) . (26)We find that the states φ + and ψ + mix with one another. The mass matrixcan be diagonalized and we find the four states ψ ′ + = ψ + cos θ − φ + sin θ φ ′ + = ψ + sin θ + φ + cos θ (27)14 − and φ − with mass squared eigenvalues, 2 λ λ ( η + η ) , λ ( η + η ) , λ ( η + η ) and λ ( η + η ) respectively. The mixing angle θ << λ . We find one particle with relatively large mass of order2 λ λ ( η + η ). By adjusting λ we can choose this to be of order of 100GeV and hence can model the Higgs particle. The remaining three particleshave very small masses. If we gauge one of the U(1) symmetry then one ofthese particles will be eliminated from the spectrum and will instead give riseto a massive gauge boson. The theory then predicts two very light weaklycoupled particles.In this section we have considered a toy model which illustrates that wecan generate any mass scale by cosmological symmetry breaking in a theorywith pseudoscale invariance. The precise gauge group used U (1) × U (1) isnot essential for this purpose and the construction can be easily generalizedto the standard model. It is ofcourse important to check that quantumcorrections do not lead to acute fine tuning problems. We postpone thisnecessary check to future research. We point out that an alternative to theconstruction in this section is to simply gauge the pseudo-scale invariance[15, 16, 26, 27, 28, 29, 30, 31, 32, 33]. This eliminates the Higgs boson fromthe particle spectrum and instead predicts a new vector boson with mass ofthe order of Planck mass. We have shown that a slowly rolling solution to scalar field theories, leadsto breakdown of symmetries of the action. We call this phenomenon cos-mological symmetry breaking and show that it is intrinsically different fromspontaneous symmetry breaking. We argue that if we impose pseudo-scale15nvariance on the action then it sets the cosmological constant to zero bothin the classical and the quantum theory. The pseudo-scale invariance is alsobroken cosmologically, leading to a slowly varying cosmological constant. Wefurther show that cosmologically broken pseudo-scale invariance can lead toa wide range of particle masses and it appears possible to impose this sym-metry on the full action of fundamental particle physics.
Acknowledgements:
We thank S. D. Joglekar for useful discussions.
References [1] A. G. Riess et al., Astron. J. , 1009 (1998).[2] P. M. Garnavich et. al., ApJ , 74 (1998).[3] S. Perlmutter et al., ApJ , 565 (1999).[4] J. L. Tonry et al., ApJ , 1 (2003).[5] B. J. Barris et al., ApJ , 571 (2004).[6] A. G. Riess et al, ApJ , 665 (2004).[7] P. J. E. Peebles and B. Ratra, Rev. Mod. Phys. , 559 (2003).[8] T. Padmanabhan, Phys. Rep. , 235 (2003).[9] E. J. Copeland, M. Sami and S. Tsujikawa, Int. J. Mod. Phys. D 15 ,1753 (2006).[10] S. Weinberg, Rev. Mod. Phys. , 1 (1989).[11] S. M. Caroll, W. H. Press and E. L. Turner, Ann. Rev. Astron. As-trophys. , 499 (1992). 1612] V. Sahni and A. A. Starobinsky, Int. J. Mod. Phys. D 9 , 373 2000.[13] J. R. Ellis, Phil. Trans. Roy. Soc. Lond.
A 361 , 2607 (2003).[14] R. R. Caldwell, R. Dave and P. J. Steinhardt, Phys. Rev. Lett. ,1582 (1998).[15] H. Cheng, Phys. Rev. Lett. , 2182 (1988).[16] H. Cheng and W. F. Kao, MIT preprint Print-88-0907 (1988).[17] W. F. Kao, Phys. Lett. A 154 , 1 (1991).[18] A. Aurilia, H. Nicolai and P.K. Townsend, Nucl. Phys.
B 176 , 509(1980).[19] J. J. Van Der Bij, H. Van Dam and Y. J. Ng, Physica
A 116 , 307(1982).[20] M. Henneaux and C. Teitelboim, Phys. Lett.
B 143 , 415 (1984).[21] J. D. Brown and C. Teitelboim, Nucl. Phys.
B 297 , 787 (1988).[22] W. Buchmuller and N. Dragon, Phys. Lett.
B 223 , 313 (1989).[23] M. Henneaux and C. Teitelboim, Phys. Lett.
B 222 , 195 (1989).[24] A. Daughton, J. Louko and R. D. Sorkin, Talk given at 5th Cana-dian Conference on General Relativity and Relativistic Astrophysics(5CCGRRA), Waterloo, Canada, 13-15 May 1993, published inCanadian Gen. Rel. 0181, (1993).[25] D. E. Kaplan and R. Sundrum, JHEP , 042 (2006).[26] D. Hochberg and G. Plunien, Phys. Rev.
D 43 , 3358 (1991).1727] W.R. Wood and G. Papini, Phys. Rev.