Nonlinear Realizations of Supersymmetry and Other

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arXiv:hep-th/0510187v1 21 Oct 2005 charges nldsteuboe pc-iegenerators space-time unbroken the includes olna elztoso uesmer n te Symmetries Other and Supersymmetry of Realizations Nonlinear r radw struhteueo ffcieLgagasbase Lagrangians realizations[1]-[2]. effective of use the through is breakdown try spon the by mandated constraints dynamical indepen the model sapsulating useful, extremely An Goldstino. N the a fermion, includes necessarily dynamics mu its breaking Consequently, consist the to neous. supergravity, able a be producing to thus order symmetry in the Moreover, symmetry. broken a as 1 -alades [email protected] address: e-mail h uesmer gae)algebra (graded) supersymmetry The fsprymty(UY st eraie nntr,te tm it then nature, in realized be to is (SUSY) supersymmetry If er ncnucinwt te pnaeu n/repiil brok explicitly and/or spontaneous including other symmetries with conjunction in super metry broken spontaneously of realizations nonlinear Simultaneous n h uecnomlsmere sreviewed. is symmetries superconformal the and Q α , Q ¯ [ [ { α ˙ ,Q R, P Q n the and µ α Q , , α Q ¯ α = ] etLfyte N47907-2306 IN Lafayette, West α ˙ [ = ] } 2 = Q R R eateto Physics of Department α P charge. ymty lblcia ymty dilatations symmetry, chiral global symmetry, σ µ udeUniversity Purdue [ ; α , µ Q α ˙ ¯ P ..Love S.T. α R, ˙ Abstract µ [ = ] Q ¯ ; 1 α ˙ P = ] { µ Q R , α 1 − Q , 0 = ] Q ¯ P α β ˙ µ } , ln ihbt h SUSY the both with along = { Q ¯ α ˙ , Q ¯ β ˙ ambu-Goldstone } aeu symme- taneous et a fen- of way dent, nnonlinear on d 0 = tb sponta- be st nl gauge ently sym- en s be ust (1) Consider some unspecified underlying theory in which both the supersymmetry (SUSY) and the R symmetry are spontaneously broken. This al- lows for the possibility that the dynamics producing the SUSY breaking has a different origin than that producing the R symmetry breaking and moreover that the scales at which the symmetry breakings occur could be independent. The low energy degrees of freedom comprising the relevant effective Lagrangian include the Goldstino, the Nambu-Goldstone fermion of spontaneously broken supersymmetry[3]-[4] described by the two-component Weyl α spinor fields λ , λ¯α˙ and the R-axion, a, the (pseudo-) Nambu-Goldstone boson of spontaneously and soft explicitly broken R symmetry. If the spontaneously broken supersymmetry is gauged to form a supergravity, the erst- while Goldstino degrees of freedom are absorbed to become the longitudinal (spin 1/2) modes of the spin 3/2 gravitino via the super-Higgs mechanism. As such the dynamics of those modes are given by that of the Goldstino. In fact, it is through the Goldstino degrees of freedom that SUSY might be most readily revealed experimentally[5]. This brief review focuses on the nonlinear realization of spontaneously broken N = 1 SUSY[6] in four flat space-time dimensions[7] in the presence of other spontaneously and/or explicitly broken symmetries. Included in this class are R symmetry, superconformal symmetry, global chiral symmetries and dilatations. One method[2] for constructing nonlinear realizations of spontaneously broken internal symmetries employs the construction of the coset group element parametrized by the coset space coordinates which are the associated Nambu-Goldstone bosons and then extracting the changes in these coordinates under group multiplication. Here this very general procedure is ap- plied to construct nonlinear realizations of SUSY and R symmetry[3],[8],[9]. Since these are spontaneously broken spacetime symmetries, the motion in the coset space, whose coordinates are the R axion and the Goldstinos, is

2 accompanied by motion in spacetime. Thus it is useful to include the product of the unbroken translation group element along with the coset group elements and deﬁne

i α ¯ ¯α˙ −ix P µ 2 (λ (x)Qα+λα˙ (x)Q ) i a(x)R Ω(x, λ, λ,¯ a)= e µ e fs e fa . (2) where fs and fa are the SUSY and R-symmetry breaking scales respectively. To extract the motion in the coset space, consider the product gΩ where α ¯ ¯α˙ g = ei(ξ Qα+ξα˙ Q )eiρR is a group element parametrized by the space-time α independent 2-component Weyl spinors ξ , ξ¯α˙ for the SUSY transformations and ρ for the R-transformation. This product of group elements again takes the form of a product of a translation and a coset group element but with translated spacetime points and coset coordinates so that

′ ′ ′ ′ ˙ ′ i α ¯ ¯α i ′ ′ −ix P µ 2 (λ (x )Qα+λα˙ (x )Q ) a (x )R ′ ′ ′ ′ gΩ(x, λ, λ,¯ a)= e µ e fs e fa = Ω(x , λ , λ¯ , a ). (3) The total (∆) variation of a generic ﬁeld, φi(x), is deﬁned as ∆φi(x)= φi′(x′) − φi(x). Under nonlinear SUSY, one extracts

¯ α 2 α ¯ ¯ 2 ¯ ¯ ∆Q(ξ, ξ)λ (x)= fs ξ ; ∆Q(ξ, ξ)λα˙ (x)= fs ξα˙ ; ∆Q(ξ, ξ)a(x) = 0 , (4) while under R transformations,

α α ∆R(ρ)λ (x)= iρλ (x); ∆R(ρ)λ¯α˙ (x)= −iρλ¯α˙ (x); ∆R(ρ)a(x)= faρ . (5)

The inhomogeneous terms in the various transformation are characteristic of the Nambu-Goldstone realization of the associated symmetries. The ac- companying movement in spacetime, ∆xµ = x′µ − xµ, is given by

¯ µ i µ ¯ µ ¯ µ ¯ µ ∆Q(ξ, ξ)x = 2 [λ(x)σ ξ − ξσ λ(x)] ≡−Λ (ξ, ξ); ∆R(ρ)x = 0 . (6) fs It also proves useful to deﬁne the intrinsic (δ) variations of the ﬁelds as

i i′ i i µ i δφ (x)= φ (x) − φ (x)=∆φ (x) − ∆x ∂µφ (x) . (7)

3 Any ﬁeld, φi, whose intrinsic variation under nonlinear SUSY is given by i µ i δQ(ξ, ξ¯)φ (x) = Λ (ξ, ξ¯)∂µφ is said to carry the standard realization. From Eq. (3), it follows that the algebra-valued Maurer-Cartan 1-form, Ω−1dΩ, is a total (∆) invariant under translations, SUSY and R transformations: (Ω−1dΩ)′(x′) = (Ω−1dΩ)(x). Expanding in terms of the translation, SUSY and R generators as

1 1 1 −1d i ωµ x P ωα x Q ω x Q¯α˙ ω x R , Ω Ω= [− P ( ) µ + 2 Q( ) α + 2 ¯Q¯ α˙ ( ) + R( ) ] (8) fs fs fa and employing the Baker-Campbell-Hausdorﬀ formula, the coeﬃcient one- forms are extracted as

↔ ν µ ν i ν ¯ µ ν ωP (x) = dx [ηµ + 4 λ ∂ µ σ λ] ≡ dx Aµ (x) fs − i a(x) µ − i a(x) µ α µ fa α fa α α ωQ(x) = dx e ∂µλ (x)= ωP (x)e Dµλ (x)= ωP (x)∇µλ (x) i a(x) µ i a(x) µ fa ¯ fa ¯ ¯ ω¯Q¯ α˙ (x) = e ∂µλα˙ (x)= ωP (x)e Dµλα˙ (x)= ωP (x)∇µλα˙ (x) µ µ ωR(x) = dx ∂µa(x)= ωP (x)Dµa(x) . (9)

µ As a consequence of the ∆ invariance of the Maurer-Cartan form, ωP and the covariant derivatives ∇µλ, ∇µλ,¯ Dµa are all invariant under the total ∆- variations, while the covariant derivatives transform as the standard realization under the intrinsic nonlinear SUSY variations while being R invariant. ′ µ ′ µ ∂x µ ν ν µ Using the ∆ invariance of ωP along with dx = ∂xν dx = dx (ην − µ ν µ ′ ν ′ −1 ρ ν ∂ν Λ ) ≡ dx Gν , it follows that Aµ (x ) = Gµ (x)Aρ (x) and thus the product d4x det A is ∆ invariant: d4x′ det A′(x′) = d4x det A(x). Con- ↔ ν ν i ν sequently, Aµ = η + 4 λ ∂ µ σ λ¯ can be viewed as a “vierbein” and an µ fs action invariant under both nonlinear SUSY and R-transformations can be constructed as

Γ= d4xL = d4x(det A) O(Da, ∇λ, ∇λ¯) (10) Z Z

4 with O is any Lorentz singlet function. The leading terms in a derivative expansion of the Lagrangian are

f 4 1 L = − s det A − det A D aDµa . (11) 2 2 µ

In the absence of the R axion, Eq. (11) reduces to the Akulov-Volkov action[3]. A nonlinearly SUSY invariant but soft R-symmetry breaking mass term can also be included producing the Lagrangian

f 4 1 1 L = − s det A − det A D aDµa − m2 det A a2 . (12) 2 2 µ 2 a

For any phenomenological application, it is necessary to determine the coupling the Nambu-Goldstone modes to generic matter fields (ψi)[10] and a to gauge fields Bµ[11]-[12]. The matter fields transform as a standard realization under nonlinear SUSY and carry R weight nψ. The nonlinear i SUSY covariant derivative for the matter fields is then defined as (∇µψ) = i i − f nψa i − f nψa −1 ν i e a Dµψ = e a Aµ ∂νψ , so that it also transforms as the standard realization under nonlinear SUSY while being R invariant. A combined nonlinear SUSY and gauge covariant derivative can then be introduced as a i −inψ f −1 ν i a a j i (Dµψ) = e a (A )µ [∂µψ + TijBµψ ]. So doing, (Dµψ) also transforms as the nonlinear SUSY standard realization provided the vector poten- ¯ a ρ ¯ a ρ ¯ a tial has the SUSY transformation δQ(ξ, ξ)Bµ = Λ (ξ, ξ)∂ρBµ +∂µΛ (ξ, ξ)Bρ . a −1 ν a Alternatively, one can introduce the redefined gauge field Vµ = (A )µ Bν which itself transforms as the standard realization and in terms of which the SUSY and gauge covariant derivative takes the form

i i − f nψa i a a j (Dµψ) = e a [(Dµψ) + TijVµ ψ ] . (13)

In a similar fashion, the gauge covariant ﬁeld strength can also be brought a −1 α −1 β a into the standard realization by deﬁning Fµν = (A )µ (A )ν Fαβ, where

5 a a a b c Fαβ = ∂αBβ − ∂βBα + ifabcBαBβ is the usual field strength having the non- ¯ a ρ ¯ a ρ ¯ a linear SUSY transformation δQ(ξ, ξ)Fµν = Λ (ξ, ξ)∂ρFµν + ∂µΛ (ξ, ξ)Fρν + ρ ¯ a ∂ν Λ (ξ, ξ)Fµρ . Using as basic building blocks the gauge singlet Goldstino and R-axion nonlinear SUSY covariant derivatives, the matter fields and their SUSY- gauge covariant derivatives and the standard realization field strength tensor, a nonlinear SUSY, R and gauge invariant action can be constructed as

Γ = d4x (detA) O(∇ λ, ∇ λ,¯ D a, ψi, D ψi, F ) (14) Z µ µ ν µ µν where O is any Lorentz and gauge invariant function. Included in O is the Standard Model Lagrangian or its supersymmetric extension modified only by the replacement of all derivatives by nonlinear SUSY covariant derivatives and field strengths by standard realization field strengths. These invariant action terms dictate the couplings of the Goldstino and R-axion which carry the residual consequences of the spontaneously broken supersymmetry and R-symmetry. For linearly realized SUSY, a considerable amount of the underlying theoretical structure can be gleaned from the symmetry currents. A general Noether current (composed of generic fields φi) has the form

∂L J µ = − ∆φi − ∆xνT µ . (15) ∂∂ φi ν Xi µ where ∂L T µ = − ∂ φi + ηµL (16) ν ν ∂∂ φi ν Xi µ is the canonical energy-momentum tensor. Here ηµν is the Minkowski metric tensor with signature (−, +, +, +). Application of Noether’s theorem then

6 gives the current divergence

δΓ ∂ J µ = −∆L− (∂ ∆xµ)L + δφi , (17) µ µ δφi Xi where the last term on the RHS is simply the Euler-Lagrange derivative δΓ ∂L ∂L i = i − ∂ i which vanishes on shell. δφ ∂φ µ ∂∂µφ For the model described by the Lagrangian of Eq. (12), the conserved canonical energy-momentum tensor[8],[13] is simply

µ −1 µ ρ −1 µ T ν = A ν L + (det A) D aAρ Dν a (18)

Note that the energy-momentum tensor starts as a positive vacuum energy, 4 00 fs < 0|T |0 >= 2 , associated with the spontaneous SUSY breaking. Using the Noether construction, it is straightforward to obtain the form of the supersymmetry and R currents along with their (non-) conservation laws. The conserved supersymmetry currents are

µ ¯ α µ ¯µ ¯α˙ µ ν ¯ Q (ξ, ξ)= ξ Qα + Qα˙ ξ = 2T νΛ (ξ, ξ) , (19)

µ 2i µ ν µ µ ν with Q = 2 T ν (σ λ¯)α ; Q¯ = −2iT ν (λσ )α˙ , while the R current α fs α˙

µ −1 µ ρ 2 µ ν ¯ R = fa(det A) Aρ D a − 4 T ν(λσ λ) (20) fs has (on-shell) divergence

µ 2 ∂µR = mafa(det A) a (21)

µ µ displaying the soft R symmetry breaking. Note that R = fa∂ a + ... and µ 2 µ ¯ ¯µ 2 µ µ Qα = ifs (σ λ)α + ... ; Qα˙ = −ifs (λσ )α˙ ) + ... so that R interpolates for the R-axion ﬁeld a, while the supersymmetry currents interpolate for the

Goldstino ﬁelds λα, λ¯α˙ .

7 For linearly realized supersymmery, the various currents are components of a supercurrent[14] and are related via SUSY transformations. Under nonlinear supersymmetry, the currents transform[8],[13] as

¯ µ α µ ¯µ ¯α˙ ρ ¯ µ µ ¯ ρ µ ¯ ρ δQ(ξ, ξ)R = i(ξ Qα − Qα˙ ξ )+ ∂ρ[Λ (ξ, ξ)R − Λ (ξ, ξ)R ] + Λ (ξ, ξ)∂ρR ¯ µ ν ¯ µ ν ¯ µ µ ¯ ν µ ¯ ν δQ(ξ, ξ)Qα = 2i(σ ξ)αT ν + ∂ν [Λ (ξ, ξ)Qα − Λ (ξ, ξ)Qα] + Λ (ξ, ξ)∂νQα ¯ ¯µ ν µ ν ¯ ¯µ µ ¯ ¯ν µ ¯ ¯ν δQ(ξ, ξ)Qα˙ = −2i(ξσ )α˙ T ν + ∂ν[Λ (ξ, ξ)Qα˙ − Λ (ξ, ξ)Qα˙ ] + Λ (ξ, ξ)∂ν Qα˙ µ ρ µ µ ρ µ ρ δQ(ξ, ξ¯)T ν = ∂ρ[Λ (ξ, ξ¯)T ν − Λ (ξ, ξ¯)T ν] + Λ (ξ, ξ¯)∂ρT ν . (22)

Being algebraically divergenceless, the total derivative terms on the RHS are Belinfante improvements which can absorbed into deﬁning improved µ µ ¯µ currents. Thus, under the nonlinear SUSY, R transforms into Qα, Qα˙ and µ ¯µ µ ¯ Qα, Qα˙ transforms into T ν. This holds even though a and λ, λ need not be SUSY partners in the underlying theory, the dynamics responsible for SUSY breaking and R breaking may have diﬀerent origins and fs need not equal fa. Since they are related by SUSY transformations, there will be relations among R and SUSY current correlators. Moreover, as these currents interpolate for the R-axion and Goldstinos, the current correlator relations will translate into relations among Green functions containing R- axions and Goldstinos. For linear realizations of supersymmetry, not only are the R current, supersymmetry currents and the energy momentum tensor related by supersymmetry transformations, but the explicit breakings of the R and dilatation symmetries are also related via SUSY transformations to the breaking of the SUSY conformal symmetry. To examine the analog of this connection for nonlinear SUSY, we need the variations under the full superconformal algebra[16],[17],[8] which, in addition to the SUSY algebra generators, contains the dilatation generator, D, the special conformal generators, Kµ the

SUSY conformal generators, Sα, S¯α˙ as well as generators of angular momen-

8 tum, M µν. The total variations of the Goldstino and R-axion ﬁelds under these transformations are secured[8],[15] as

1 1 ∆ λα = − λα ; δ λ¯α˙ = − λ¯α˙ ; δ a = 0 D 2 D 2 D 4i ˙ ˙ 2i ˙ 3f λβ λαλβ λ¯β f 2xνσβα λαλ¯β a a λ ∆Sα = 2 ; ∆Sα = − s ¯ν − 2 ; ∆Sα = 2 α fs fs fs ¯ β 2 ν αβ˙ 2i ¯α˙ β ¯ ¯β˙ 4i ¯α˙ ¯β˙ ¯ 3fa ¯α˙ ∆S¯α˙ λ = fs x σ¯ν + 2 λ λ ; ∆S¯α˙ λ = − 2 λ λ ; ∆S¯α˙ a = 2 λ fs fs fs α 2i ¯ α ν α 3fa ¯ ∆Kµλ = (xµ − 4 λσµλ)λ − ix (σµν λ) ; ∆Kµa = − 4 λσµλ fs fs ¯α˙ 2i ¯ ¯α˙ ν ¯ α˙ ∆Kµλ = (xµ − 4 λσµλ)λ − ix (¯σµν λ) , (23) fs while the space-time point moves as

µ µ µ ν 2 ν ν 1 ¯¯ ∆Dx = −x ; ∆Kν x = −[ηµx − 2x xµ + ηµ 8 (λλ)(λλ)] fs µ i ν µ 2 µ ¯ ∆Sαx = 2 (σ σ¯ λ)αxν − 6 λα(λσ λ) fs fs ¯ µ i ¯ µ ν 2 ¯ µ ¯ ∆S¯α˙ x = − 2 (λσ¯ σ )α˙ xν − 6 λα˙ (λσ λ) . (24) fs fs The intrinsic variations are then readily obtained using Eq. (7) and the above total variations. Since the dilatation variations form a linear represen- tation, the scale symmetry is not spontaneously broken. The scaling weights 1 are extracted as dλ = dλ¯ = − 2 ; da = 0. In fact, the Nambu-Goldstone par- ticle associated with any spontaneously broken symmetry which commutes with the dilatation charge is constrained to have scaling weight zero[20] as is the case here for the R axion. With the variations in hand, the dilatation current is constructed as µ ν µ D = x T ν (25) while its (on-shell) divergence

µ ∂ ∂ ∂ ∂µD = [fs + fa + ma ]L (26) ∂fs ∂fa ∂ma

9 exhibits the explicit scale symmetry breaking. Note that there are independent breakings arising from the spontaneous SUSY breaking scale, fs, the spontaneous R symmetry breaking scale, fa, and the soft R symmetry breaking mass term, ma. The special conformal transformations, although nonlinear, have no constant terms. Thus the special conformal tranforma- tions are also not spontaneously broken in this realization. Since the S¯(S) variations of λ(λ¯) start as constants, it follows that the SUSY conformal symmetries are also spontaneously broken. However, the α ¯α˙ associated Nambu-Goldstone fermions, λS , λS are not independent dynami- α 1 µ ¯ α ¯α˙ cal degrees of freedom. Rather, one finds that λS = − 4 (σ ∂µλ) +... ; λS = ˙ ˙ 1 ∂ λσµ α˙ ... λβ f 2δβ ... ¯ λ¯β f 2δβ ... − 4 ( µ ) + so that ∆Sα S = s α + ; ∆Sα˙ S¯ = − s α˙ + . The fact that there can be spontaneously broken spacetime symmetries without independent Nambu-Goldstone fields[18] is not in conflict with Goldstone’s theorem[19] which guarantees an independent Nambu-Goldstone field for every spontaneously broken global symmetry. Applying Noether’s theorem, the superconformal currents are then computed as

µ ν α ¯ ν α˙ ¯ 3 µ 2 µ ν ¯ S (η, η¯) = [(¯ησ¯ ) Qµα + Qµα˙ (¯σ η) ]xν + (ηλ +η ¯λ)[ 4 R + 6 T ν(λσ λ)] . fs fs Under nonlinear SUSY, the divergence of these currents transform as

µ µ µ δQ(ξ, ξ¯)∂µS (η, η¯) = 3(ηξ +η ¯ξ¯)∂µR + 2i(ηξ − η¯ξ¯)∂µD νρ νρ µ µ ν − (ξσ η +η ¯σ¯ ξ¯)∂µM νρ + ∂µ[Λ (ξ, ξ¯)∂ν S (η, η¯)] (27) where

µ µ µ 1 λ ¯ λ¯ µ M νρ = T νxρ − T ρxν − 4 [λσ σ¯νρλ + λσνρσ λ]T λ (28) 2fs is the conserved angular momentum tensor. Thus the divergence of the R-symmetry current and the divergence of the dilatation current is related to the divergence of the superconformal current through a nonlinear SUSY

10 transformation. Since the divergences of these currents give the explicit breakings of these symmetries in the action, these breakings are also related via the nonlinear SUSY transformation. Now suppose the dynamics also generates a spontaneously broken internal global chiral symmetry with associated Nambu-Goldstone boson ﬁelds πi which transform under nonlinear SUSY as the standard realization. A nonlinear SUSY invariant action which also contains a soft explicit chiral symmetry breaking mass term is readily constructed as f 4 f 2 1 Γ= d4x{− s (det A) − a (det A) D aDµa − m2f 2(det A) a2 Z 2 2 µ 2 a a 2 ¯ Fπ † µ < ψψ > † − (det A) Tr DµU D U + (det A) Tr mU + Um } 4 h i 2 h i i i where U(π)= e2iT π /fπ with T i the fundamental representations of the broken generators of the internal symmetry group, fπ is the Nambu-Goldstone boson decay constant, m is the mass matrix characterizing the soft explicit chiral symmetry breaking and ψ a chiral fermion of the underlying theory whose condensation is responsible for the spontaneous chiral symmetry breaking. Finally, consider the case where in addition to the spontaneously broken SUSY and spontaneous and softly broken R symmetry and chiral symmetries, there is a spontaneously broken dilatation symmetry. This results in appearance of the dilaton, ϕ, the Nambu-Goldstone boson of spontaneously broken scale symmetry which transforms as a standard realization under the nonlinear SUSY and as a singlet under both R and the chiral symmetry. Un- ϕ f der dilatations, the combination, Σ(ϕ) = e D transforms as ∆D(ǫ)Σ = ǫΣ which leads to an inhomogeneous scale transformation of the dilaton given by ∆D(ǫ)ϕ = ǫfD. The prescription for nonlinearly realizing the scale symmetry is then to introduce a factor of Σ(ϕ) for every dimensionful parameter appearing in the action. This procedure makes the scaling weight of each

11 term in the action equal to four. Note that the Akulov-Volkov vierbein has scaling weight zero. So doing, the leading terms in an eﬀective Lagrangian which nonlinearly realizes SUSY, R-symmetry, chiral symmetry and dilatations takes the form[21]

f 4 f 2 L = − s (det A)Σ4 − a (det A)Σ2D aDµa 2 2 µ 1 2 2 µ † 1 2 − fπ (det A)Σ T r DµU(π)D U (π) − fD (det A) DµΣDµΣ 4 h i 2 ¯ 2 < ψψ > 3−γ † fa 2 2 2 + (det A)Σ Tr mU + Um − ma (det A)Σ a . 2 h i 2

Here the last line includes the soft explicit R and chiral breakings while γ is the anamolous dimension of the underlying fermion field. Note, however, that since det A = 1+..., this Lagrangian contains an Σ4 potential term for the Nambu-Goldstone dilaton which in turn produces a 4 (3−γ) ¯ destabilizing linear in ϕ term unless fs = 2 < ψψ > T r[m] in which case it is cancelled against the linear in ϕ term arising from the soft expicit chiral symmetry breaking. But then the Goldstino decay constant, fs, vanishes in chiral limit. Since the Goldstino decay constant is the matrix element of the supersymmetry current between the vacuum and the Goldstino state, its vanishing signals the nonviability of the Goldstone realization. Conse- quently, scale symmetry and SUSY symmetries cannot be simultaneously nonlinearly realized and the spectrum cannot contain both a dilaton and a Goldstino as Nambu-Goldstone particles. Note that there is no difficulty in constructing an effective Lagrangian invariant under both nonlinear SUSY and chiral symmetry transformations nor under nonlinearly realized scale symmetry and chiral symmetry[22]. It is only when both SUSY and scale symmetry are to be realized nonlinearly that one encounters the inconsis- tency.

12 This work was supported in part by the U.S. Department of Energy under grant DE-FG02-91ER40681A29 (Task B). I thank T. Clark for many enjoyable discussions.

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