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Once more on the θ-vacua in 2 + 1

arXiv:hep-ph/9207250v1 21 Jul 1992IASSNS 91/76November 1991Once more on the θ-vacua in 2 + 1dimensionalQED and 3 + 1dimensional gluodynamics.A.R.Zhitnitsky 1Institute for Advanced Study, Princeton, N.J. 08540,USA.andInstitute of Nuclear Physics, Academy of Sciences of the USSR, 630090 Novosibirsk, USSR.AbstractTwo different but tightly connected problems, U(1) and strong CP violation problems, arediscussed in two different models which exhibit both asymptotic freedom and confinement. One ofthem is the 3d Polyakov’s model of compact QED and the other is 4d gluodynamics.

It is shownthat although both these models possess the long range interactions of the topological charges, onlyin the former case physics does not depend on θ; while the latter exhibits an explicit θ- dependence.The crucial difference is due to the observation, that the pseudoparticles of 4d gluodynamicspossess an aditional quantum number, apart of the topological charge Q .1e-mail address is zhitnita@vxcern.cern.ch1

1.Introduction.In the present letter I investigate some problems related to the θ vacua in the SU(2) gauge theories.As is known one can add to the standard Lagrangian the so called θ term:∆L = θQ(1)where Q is the topological charge which can be written for the 4d gluodynamics and for 2+1 QEDcorrespondingly as follows:Q =132π2Zd4xGaµν ˜Gaµν =132π2Zd4x∂µKµ,a = 1, 2, 3. µ, ν = 1, 2, 3, 4. (2)Q = 18πZd3xǫijk∂i(Gajkφa) = 14πZBkdSk,i, j, k = 1, 2, 3.

(3)Here Gaµν is the field strength tensor and φa is the scalar field which is present in the definition of theinstanton solution in the 2+1 theory (a monopole, in the terminology of the 3+1 theories ) [1],[2]. Inthe unitary gauge φa becomes constant φa →(0, 0, 1) at large distances and the combination ǫijkGajkφareduces to Bi.

This is just the reason why this theory is called the 2+1 QED [3].As is known , the θ term preserves renormalizability of the theory but is P and T odd. As it can beseen from (2,3) the θ term is a full divergence, therefore it is equivalent to certain boundary conditions .The standard line of reasoning in this case is as follows.

Because the boundary conditions can influenceto the physics far from the boundary in the ordered phase only, and because confinement occurs in adisordered phase , one can think that the all θ vacua are equivalent. If so, the nonvanishing surfaceintegral (2,3) due to a separate topological solution could be an artifact of the dilute gas approximationwhich ignores completely the most profound features of the considering models - confinement.

Indeedthe surface integral does not vanish since the topological field falls offtoo slowly at large distancesdue to the presence of massless particles in the original Lagrangian , which is not present, however, inthe physical spectrum. Therefore one might hope that the confinement effects would make the surfaceintegral vanishes.

Such a viewpoint based on the analysis 2+1 QED [4](see also the recent paper [5]onthis topic), was strongly advocated by Polyakov.2

Indeed , Vergeles has shown [4] that as soon as quasiparticles interaction becomes strong enough,the θ dependence disappear from the effective long distance Lagrangian and physics becomes θ inde-pendent. We will shortly reproduce this result in the next section ,and emphasize generality of thisresult and its independence on the space dimensions.At the same time , as is known [6],[7] ,[8] the θ- dependence of physics is linked to the U(1)problem.

If we believe that the resolution of the U(1) problem appears within the framework of thepapers [6],[7] ,we must assume that the correlatorK = iZd4x < 0|T132π2 Gaµν ˜Gaµν(x),132π2 Gaµν ˜Gaµν(0)|0 >(4)is nonzero. It means that the vacuum expectation value (vev) of the topological density< |132π2 Gaµν ˜Gaµν| >= 12Kθ,θ ≪1.

(5)is nonzero too. But as was shown in ref.

[8] the nonzero vev < G ˜G ≯= 0 does imply the CP -violation inphysical transition and leads , for example ,to the mixing of the heavy quarkonium levels with JP = 0+and JP = 0−in terms of < G ˜G >. Only dispersion relations are used to translate < G ˜G ≯= 0 into aproof of CP- violation in physical effects.So, if we believe that U(1) problem is solved in the framework of Witten-Veneziano approach [6],[7]we automatically get K ̸= 0 (4) and therefore, the nontrivial θ dependence.With introduction of light quarks (u,d,s) the value of correlator K(4) changes 2,but θ is stillexperimentally observable quantity.

Moreover, a few strict results ( computation of η →ππ decay [8]and electrical dipole moment [9]) were obtained using the soft meson technique.I will not consider in this papers the theory of light quarks and confine myself by considerationof pure gluodynamics.I will discuss the resolution of the aforementioned apparent paradox (thecoexistence of the strong quasiparticle interaction with the nontrivial θ dependence) within frameworkof the dynamical toron approach which was discussed early in context of different field theories, see2It is obviously that K ∼m1q irrespectively to the number of light flavours [7].3

ref. [10] and references therein.

I would like to recall that in all known cases, the toron calculationsgive, at least, selfconsistent results. Thus, one may expect that apparent puzzle should be solved inan automatic way.We discuss the statistical ensemble of quasiparticles which, presumably [11], describes the grandpartition function of the 4d YM theory and which possesses long distances strong interaction.

It willbe shown that this ensemble describes the system with nontrivial θ- dependence irrespectively to thestrength of quasiparticle interaction . Crucial point, in compare with analogous calculation of ref.

[4]in Polyakov’s model, is a very nontrivial algebraic structure of the quasiparticle interaction.Let us now recall some basic facts about the toron’s role in two and four dimensional physics.Besides that we give some arguments why quasiparticles with fractional topological charge should beconsidered in those cases and what physical effects arise due to fluctuations with fractional Q.The self dual ”toron” solution with fractional topological charge was first considered by ’t Hooft[12](but in quite different context). It is defined in a box of size Lµ, smeared over this box and exists whenthe sizes Lµ satisfy certain relations.

The calculation of gluino condensate < λ2 > in the supersym-metric YM theory (SYM) based on the ’t Hooft solution was carried out in ref.[13]. However by manyreasons (in particular, difficulties with introduction of fields in the fundamental representation andconsideration of the ensemble interacting quasiparticles) the ’t Hooft solution [12] and correspondingcalculation [13],[14] can be considered as only illustrative example with fractional topological charge.Nevertheless,I believe that solutions with a fractional Q may play an important role in theory, butthese solutions should be formulated in another way (see[10] and references therein).The correspondingpseudoparticle can be understood as a point defect when the regularization parameter( which is presentin the definition of the solution) goes to zero.

We keep the term ’toron’ introduced in ref.[12]. Bythis we emphasize the fact that the new solution also minimizes the action and carries the topologicalcharge Q = 1/2,so it possesses all the characterestics ascribed to the standard toron[12].All calculations, based on the solution [10] demonstrate its very nontrivial role for different field4

theories. Most glaringly these effects appear in supersymmetric variants of a theory.

In particular,in the supersymmetric CP N−1-theories, the torons (point defects) can ensure a nonvanishing valuefor the < ¯ψψ >∼exp(2iπk/N + iθ/N) with right θ-dependence.Such behavior is in agreementwith the value of the Witten index which equals N[15] and in agreement with the large N-expansion[16]. In analogous way, the chiral condensates can be obtained for 4d theories: supersymmetric YM(SYM), [13], supersymmetric QCD (SQCD).

In these cases a lot of various results are known fromindependent consideration (such as the dependence of condensates on parameters m, g; the Konishianomaly equation and so on...). [18].

Toron approach is in agreement with these general results. Thesame approach can be used for physically interesting theory of QCD with Nf = Nc.

In this casean analogous calculation of < ¯ψψ > does possible because of cancellation of nonzero modes, like insupersymmetric theories. For this theory the contribution of the toron configurations to the chiralcondensate has been calculated and is equal to: < ¯ψψ >= −π2 exp(5/12)24Λ3 [10] (see also E. Cohen[17]).

As is well known in any consistent mechanism for chiral breaking a lot of problems, such as:the U(1)-problem, the number of discrete vacuum states, the θ-puzzle, low energy theorems and soon, must be solved in an automatic way. We have checked that all these properties [10] are consistentwith the toron calculation.To give some insight about the algebraic structure of the quasiparticle interaction, I would liketo cite some results for 2dCP N−1 model which is known [16] possesses nontrivial θ-dependence.

Itturns out that the toron gas contribution to grand partition function in 2dCP N−1 model reduces tothe classical Coulomb system(CCS) [11]:Z =∞Xk=0λk1+k2(k1)!(k2)!Xµα,qαk1+k2Yi=1d2xiexp(−ǫint.),(6)ǫint. = −4Xi>jqi ⃗µiln(xi −xj)2qj ⃗µj + 2 ln L2(Xiqi ⃗µi)2,λ = cM0f(M0)exp(−πf(M0)).where N different kinds of torons classified by the weight ⃗µα of fundamental representation of theSU(N) group and qi is the sign of the topological charge.

Besides that, in formula (6) the valuef(M0) is the bare coupling constant and M0 is ultraviolet regularization, so that in eq. (6) there5

appears the renormalization invariant combination λ. As can be seen from (6) the configurations onlysatisfying the neutrality requirementXiqi ⃗µi = 0(7)are essential in thermodynamic limit L →∞.

However we will consider the system in the box withsize L and so we keep this term for the future analysis.Using the correspondence between CCS and the Toda field theoryZθ =ZD⃗φexp(−Zd2xLeff. ),(8)Leff = 1/2(∂µ⃗φ)2 −X⃗µαλ2 exp(i4√π ⃗µα⃗φ + iθ/N) −X⃗µαλ2 exp(−i4√π ⃗µα⃗φ −iθ/N).the expectations of different values( the vacuum energy, the topological density, the Wilson line ...)was calculated.

All results ( confinement, right dependence on θ/N and so on ) are precisely what oneobtains from the large N expansion. In this effective field theory ,⃗φ represents the N −1 componentscalar potential , the sum over ⃗µα in (8) is over the N weights of the fundamental representation ofSU(N) group.

Note, that the first interaction term is related to torons and the second one to anti-torons. Besides that, since we wish to discuss the θ dependence , we also include a term proportionalto the topological charge densityθ4πǫµνFµν to the starting lagrangian (formula, analogous to eqs.

(1,2,3)), and corresponding track from this to the effective lagrangian(8).The most important result from ref. [11] is the nontrivial dependence on θ of the topological densityand susceptibility, the values which are relevant for the solution of the U(1) problem :< ǫµνFµν4π>θ∼sin( θN ).

(9)Zd2x < ǫµνFµν4π(x), ǫµνFµν4π(0) >∼1N cos( θN ). (10)Here Fµνǫµν4πis the topological density in 2dCP N−1 model.

The vacuum characteristics listed aboveare analogous to eqs. (4,5) in 4d YM theory.The reason to have the nontrivial θ- dependence (9,10) as well as the strong quasiparticle interaction∼ln(xi−xj)2 (6) in this 2dCP N−1 model is the presence of the nontrivial algebraic structure ∼qi ⃗µiqj ⃗µj6

in the expression for ǫint (6). Just this fact was crucial in the analysis of the 2dCP N−1 model.

As weobserved early, [11] the same structure for quasipartcle interaction take place in the 4d YM theory inthe striking contrast with 2+1 Polyakov’s model [3], where the interaction energy proportional to thetopological (magnetic) charges (3) of quasiparticles only:ǫint ∼qiqj|xi −xj|. (11)Just this difference leads, as will be shown bellow, to the existence of the nontrivial θ dependence in4d YM theory in spite of the fact of the strong quasiparticle interaction ∼ln(xi −xj)2.2.Comparative analysis of the 2+1d QED and 3+1d gluodynamics.We start from the short review of the results of ref.

[4] to demonstrate the independence of θ thepartition function in Polyakov’s model [3] . To do this let us consider the contribution to the partitionfunction of the configurations with fixed numbers of n1 monopoles and n2 antimonopoles.

As has beenshown by Polyakov [3] the corresponding contribution reduces to the following expression :ZNQθ≡exp(iQθ) exp(−FNQ) = exp(iQθ)1n1!n2!NYi=1d3xiλN exp(−NXi̸=jqiqj|xi −xj|)(12)Q = n1 −n2,N = n1 + n2,qi = ±1,i = 1, NHere , λ is fugacity of the obtained Coulomb gas and it is determined by the ultraviolet behaviour ofthe theory.If the Coulomb interaction could be ignored, then (12) yields for free energy Fθ the θ- dependentexpression.This is a well known result obtaining in the framework of dilute noninteracting gasapproximation.In our case the partition function with a fixed topological chargeZQθ ≡exp(iQθ) exp(−FQ) =XNZNQθ→ZdNZNQθ(13)can be obtained by the steepest descent method with respect N. With L ,the linear size of the system,going to infinity, we get the following expression for the noninteracting case:exp(−FQ) =ZdN exp[N ln(L3λ) −N + Q2(lnN + Q2−1) −N −Q2(lnN −Q2−1)] ∼(14)7

exp(N0) exp(−Q2N0),where we have used formula ln(k!) ∼k ln k −k and N0 is to be determined from the equation :ln(L3λ) = 12ln(N 20 −Q24),QN ≪1,N0 ≃2L3λ.

(15)Now the partition function for the dilute noninteracting gas is evaluated in the following way:Zθ ≡exp(−Fθ) ∼exp(2L3λ)ZdQ exp(iθQ) exp(−Q24λL3 )∼exp[2λL3(1 −θ22 )](16)Fθ ≃−2λL3(1 −θ22 ),θ ≪1As it can be seen from(16) the essential terms are those which have the charges|Q| ∼λL3θ,|Q|N ∼θ(17)and our approximation Q/N →0 can be justified in the limit θ →0. So, as expected , the free energyFθ does depend on θ explicitly.Now we return to the Polyakov’s model.

It is known that the Coulomb interaction in plasmacannot be ignored. Hence, formula (16) is not true.

To give some estimation in this case let us placeour system to the box with size L. In this case, the excessive charge deposits on the wall of the boxand the free energy is the same as for neutral Debye plasma plus the Coulomb energy [4]:exp(−FQ) ∼ZdN exp[N ln(L3λ) −N + Q2(lnN + Q2−1) −N −Q2(lnN −Q2−1) −Q2L ](18)Now the partition function with a fixed charge can be obtained by the same way as before (14). Theestimates of Fθ is now less trivial:Zθ ≡exp(−Fθ) ∼exp(2 L3λ)ZdQ exp(iθQ) exp(−Q24λL3 ) exp(−Q2L )(19)Here the new factor exp(−Q2L ) is due to the Coulomb interaction,and now the essential configurationshave charges:|Q| ∼Lθ,QN ∼θλL2 →0(20)8

and therefore :Fθ ∼−2λL3 + Lθ2 = −2λL3(1 −θ22λL2 ). (21)Thus, at L →∞the free energy Fθ in the Polyakov’s model does not depend on θ and coincideswith the corresponding expression at θ = 0.

Let me emphasize that this result is due to the stronginteraction of quasiparticles.At first sight this derivation looks rather general. Moreover, one can suspect that as soon as wehave a strong enough interaction and corresponding trace of it in the formula analogous to (19),we willobtain the expression for the free energy, like (21),which does not exhibit any θ dependence at L →∞.This is indeed the case for the simplest algebraic structure ∼qiqj for the quasiparticle interaction.Moreover, the arguments, given above are in perfect agreement with intuitive picture ( discussed inIntroduction) connecting θ independence of the effective lagrangian and confinement.Let us now proceed to the analysis of 4d gluodynamics.

In this case, like in 2dCP 1 model, a toronclassified by two numbers : the sign of the topological charge qi and the isotopic projection Ii:|qi, Ii >(22)The details are in the original paper [11] , and I would like only mention here the definition of thetoron isotopic projection.Let us introduce along with papers [19] an additional object into the theory, so called measuringoperator C such thatDµC(x) = 0(23)and consider the integralI =Ztr(CFµν)dσµν(24)where the toron lies in a plane σµν. In this case the different choice of C obeing eq.

(23) yelds differentisospin directions. It is useful to keep in mind some anology with monopole’s classification.

In thiscase the role of C plays a Higgs field [20], which far from the core satisfies the same equation (23). Let9

us note, that an each quasiparticle is classified by I3 proection and so has a nontrivial transformationproperties under the SU(2) gauge group. However due to the neutrality condition in the infinite volumelimit (7), the vacuum is the singlet state under the gauge transformation.

Moreover,the answer doesnot depend on the choice of the z axis (the axis of quantization)when the sum over all possible isospinsfor all quasiparticles will be done.With this in mind let us consider in more details the grand partition function for the 4d YM theory[11] which looks like (6) with some trivial changes :λCP N−1 →λY M ∼( M11/30g4(M0) exp(−4π2g2(M0)))311(25)d2x →d4x,−4 ln(xi −xj)2 →−23 ln(xi −xj)2,−4 ln L →−23 ln L.The last replacements in (25) is the direct consequence of renorminvariance of the theory [11]. Nowlet us examine the contribution to the partition function of the terms with fixed numbers of n1(m1)torons (antitorons) with isospin up and n2(m2) torons (antitorons) with isospin down.

In this casethe integration over N can be done as before by the steepest descent method with respect N ( let usnote that the term related with interaction does not depend on N) and so the formula analogous toeq. (19) for the free energy Fθ in 4d gluodynamics looks as follows:Zθ ≡exp(−Fθ) ∼exp(2L4λ)ZdQ exp(iθ2Q) exp(−Q24λL4 )Zd(Q1 −Q2) exp(−23(Q1 −Q2)2 ln L) (26)Q ≡(Q1 + Q2),n1 + n2 = 1/2(N + Q1 + Q2),m1 + m2 = 1/2(N −Q1 −Q2)Here the 12Q1 = 12(n1 −m1) is the topological charge carried by the quasiparticles with isospins up and12Q2 = 12(n2 −m2) is the same for quasiparticles with isospins down.

While obtaining (26) we tookinto account the form of the last term in eq. (6) (with corresponding replacement (25) for transitionfrom CP N−1 model to 4d gluodynamics):(Xiµiqi)2 ln L →(n1 −m1 −n2 + m2)2 ln L = (Q1 −Q2)2 ln L.(27)The trace of this interaction is the appearance of the last term in eq.

(26) analogous to the correspondingcontribution ∼exp(−Q2L ) in eq. (19) for 2+1 QED.

Besides that, we have the factor (iθ2) in eq. (26)10

instead of(iθ) in eqs. (16,19): it is the direct consequence of the fractional value for the toron topologicalcharge equals one half.Now we see the crucial difference between the expressions for free energy in the 4d gluodynamics(26) and 2+1d QED (19).

If the topological quasiparticles were classified by topological charge only(it would corresponds Q2 = 0 in the formula (26)) we would obtain the θ independent expression forfree energy, just as it has happened in 2+1d QED , see eq. (21).But fortunately , we have less trivial expression for 4d gluodynamics:The last term does not depend on combination Q1 + Q2 and is factorized out .

The nontrivialon θ integral over d(Q1 + Q2) is reduced to the noninteraction gas case (16) and does depend on θexplicitly at L →∞:Fθ ≃−2λL4(1 −12(θ2)2),θ ≪1(28)Let me emphasize that this result has been obtained due to the relevant new quantum number clas-sifying the quasiparticle. Besides that I interpret this result as the expansion ofFθ ∼−| cos(θ2)|(29)obtained in [11] by quite different method.

As discussed in that paper the reason for such θ dependenceis existence of the two vacuum solutions (for SU(2) group) minimizing vacuum energy. Althogh eachof the separate solutions has a θ period of 4π, the overall minimum has a θ period of 2π because thesolutions jump from one value to another at θ = π.

Very important is that two solutions have beenprepaired for such jumps from the very beginning. Indeed, as soon as we allowed one half topologicalcharge,the number of the classical vacuum states is increased by the same factor two in compare witha standard classification, counting only integer winding numbers |n >.Of course,vacuum transitions eliminate this degeneracy.

However the trace of enlargement numberof the classical vacuum states does not dissapear. Vacuum states now classified by two numbers :0 ≤θ < 2π and k = 0, 1.

Let me repeat that origin for this is our main assumption that fractional11

charge is admitted and therefore the number of classical vacuum states is multiplied by a factor two.I have to note that the same situation takes place in the supersymmetric YM theory, but in thiscase the vacuum states are still degenerate after vacuum transitions. The number k in this case justnumerates different vacua at the same θ.

However, the gauge classification for the winding vacuabefore perturbation for both models (supersymmetric and nonsupersymmetric one) is the same.We close this discussion by a remark that the analogous θ/N dependence was discovered in glu-odynamics at large N [6],[7], [21]. In these papers was argued that the vacuum energy at large Nappears in the form E ∼E(θ/N).

Such a function can be periodic in θ with period 2π only if thereare many vacuum states for given values of θ. Indeed, gluodynamics can be understood as a QCD withvery large quark’s mass.

In this limit effective lagrangian can be founded [21] and it turns out thatthe number of vacua is of order N at N →∞. This fact actually is coded in the effective lagrangiancontaining the multi-branched logarithm log det(U).

In the Veneziano approach [7] the same fact canbe seen from the formula for multiple derivation of the topological density Q with respect to θ atθ = 0.∂2n−1∂θ2n−1 < Q(x) >∼( 1N )2n−1 , n = 1, 2...(30)So, eq. (29) definitely does not contradict to results for large N expansion.It is instructive to understand this result in terms of the effective field description .

Just like in2dCP N−1 model, the grand partition function for toron gas in 4d gluodynamics can be re-expressed interms of effective field theory very similar to eq. (6).

The difference with 2dCP N−1 model only in thekinetic term, which has the standard form ∼(∂µφ)2 in 2dCP N−1 model and looks more complicatedin 4d gluodynamics ∼(✷φ)2 [11]. This difference, however, does not influence on the arguments givenbellow.

Only the interaction part of the effective lagrangian , depending on θ, is relevant for ouranalysis.For Polyakov’s model the effective lagrangian has the same Sine-Gordon form (8), but with acrucial difference. Namely , the grand partition function (12) for 2+1d QED can be re-expressed in12

terms of a field theory with lagrangianLeff = 12(∂µφ)2 −λ cos(φ + θ)(31)but without any additional summing overµα 3. After shift φ →φ −θ, the θ dependence in the Leffdisappear in according with our formula (21),[4].In contrast with it the situation in the 2dCP N−1 model and 4d gluodynamics is quite differentbecause in these cases the θ parameter can not be removed from effective lagrangian (8) in agreementwith formula (28).3.Final remarks.The main point of this Letter is that the dynamical resolution of the U(1) problem in gluodynamicslooks very naturally in terms of ensemble of the quasiparticles with the fractional topological charges(torons),which have a nontrivial long range interactions .

This approach is in a perfect agreement withWitten-Veneziano solution of the U(1) problem in framework of large N-expansion [6],[7]. Besides thatthis approach demonstrates a self-consistency of the different calculation in a various field theories[10],where the results are known beforehand.We have discovered that the algebraic structure of the quasiparticle interaction is just what isneeded for the solution of U(1) problem irrespectively to the strength of interaction.This resultin gluodynamics is in a striking contrast with that for the 2+1 QED, where strong quasiparticleinteraction eliminates the θ dependence in the theory.The other point which I would like to mention here is as follows.

Although we have discussed thedynamical solution of the U(1) problem within the toron framework , the obtaining results have a moregeneral origin. Indeed , we have shown that the nontrivial θ dependence does appear in the effectivelagrangian 4 in spite of the fact of strong quasiparticle interaction, provided the quasiparticles are3Let us note, that in 4d gluodynamics this summing reproduces in the effective Lagrangian the well known ZN Weylsymmetry of the original YM theory.4This is an essential requirement for the solution of the U(1) problem13

classified by the new quantum number ,the weight of representation of the group.AcknowledgmentsI am very grateful to Frank Wilczek for stimulating discussions and for hospitality at IAS, Prince-ton, where this manuscript was finished. I thank I.B.Khriplovich for several instructive discussionsduring the course of this work and S.Samuel for sending me his recent preprint[5] .References[1] A.M.Polyakov, JETP.Lett.20,194 (1988).

[2] G’t Hooft, Nucl.Phys. B79 ,276 (1974).

[3] A.Polyakov, Phys.Lett. B59,82 (1975); Nucl.Phys.B120,428 (1977).

[4] S.Vergeles, Nucl.Phys. B152,330(1979).

[5] S.Samuel, Preprint CCNY-HEP-91/5,May 1991. [6] E.Witten, Nucl.Phys.B156,269 (1979).

[7] G.Veneziano, Nucl.Phys.B159,213 (1979). [8] M.Shifman , A.Vainshtein and V.Zakharov , Nucl Phys.B166 ,493(1980).

[9] R.Crewther et al. Phys.Lett.

B88,123 (1979). [10] A.R.Zhitnitsky, Nucl.Phys.B340,56 (1990).

[11] A.R.Zhitnitsky, Nucl.Phys.B374,183(1992). [12] G’t Hooft, Commun.Math.Phys.81(1981)267.

[13] E. Cohen and C. Gomez, Phys.Rev.Lett.52(1984)23714

[14] C. Gomez, Phys.Lett.B 141 (1984),366. [15] E.Witten, Nucl.Phys.B202,253 (1982).

[16] A.D’Adda,P.Di Vecchia and M.Luscher, Nucl.Phys.B14,63 (1978).E.Cremer and J.Scherk, Phys.Lett.B74,341 (1978).H.Eichenherr, Nucl.Phys.B146,215 (1978).E.Witten, Nucl.Phys.B149,285 (1979). [17] , E.Cohen, 1984, communication from referee.

[18] D.Amati et al., Phys.Rep.162,169(1988). [19] H.G.Loos,Ann.

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[21] E.Witten,Ann.Phys.(N.Y. )128,(1980),363.15

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