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AS A HIGH TEMPERATURE LATTICE GAUGE THEORY

arXiv:hep-th/9212074v1 11 Dec 1992DFTT 71/92December 1992THE KAZAKOV-MIGDAL MODELAS A HIGH TEMPERATURE LATTICE GAUGE THEORYM. Caselle, A. D’Adda and S. PanzeriIstituto Nazionale di Fisica Nucleare,Sezione di TorinoDipartimento di Fisica Teorica dell’Universit`a di Torinovia P.Giuria 1, I-10125 Turin,ItalyAbstractWe show that the Kazakov-Migdal (K-M) induced gauge model in d di-mensions describes the high temperature limit of ordinary lattice gauge theo-ries in d + 1 dimensions.

The matter fields are related to the Polyakov loops,while the spatial gauge variables become the gauge fields of the K-M model.This interpretation of the K-M model is in agreement with some recent resultsin high temperature lattice QCD.⋄email address: Decnet=(31890::CASELLE,DADDA,PANZERI)internet=CASELLE(DADDA)(PANZERI)@TORINO.INFN.IT1

1. IntroductionRecently V.Kazakov and A.Migdal proposed a new lattice gauge model, in whichthe gauge self-interaction is induced by scalar fields in the adjoint representation [1].The action they propose is defined on a generic d-dimensional lattice and has thefollowing form:S =XxNTr[m2φ2(x) −Xµφ(x)U(x, x + µ)φ(x + µ)U†(x, x + µ)](1)where φ(x) is an Hermitian N × N matrix defined on the sites x of the lattice andU(x, x + µ) is a unitary N × N matrix, defined on the links (x, x + µ), and playsthe role, as in the usual lattice discretization of Yang-Mills theories, of the gaugefield.

Integrating over the scalar field φ one can induce an effective action for thegauge field,ZDUDΦexp(−S) ∼ZDUexp(−Sind[U])(2)with:Sind[U] = −12XΓ|TrU[Γ]|2l[Γ](2m2)l[Γ],(3)where l[Γ] is the length of the loop Γ, U[Γ] is the ordered product of link matricesalong Γ and the summation is over all closed loops.The model turns out to be solvable in the large N limit, for any space-timedimension d [1, 2, 3] and exactly solvable, for any value of N in d=1 [4]. A furtherinteresting feature of the model is its deep connection with the theory of non-critical strings, in particular it can be shown that in the d = 1 case the K-Mmodel is equivalent to the vortex free sector of the d=1 string [5].

Despite all thesenice features the model seems to miss the original goal of Kazakov and Migdal,since the induced gauge theory (3), due to the fact that the matter fields are inthe adjoint representation, has a super-confining behaviour [6] and seems to havethe wrong perturbative vacuum [7, 8]. Several improvement of the original K-Maction have been proposed to avoid these problems [8, 9, 10, 11], but even if manyinteresting results have been obtained the problem of the identification with a purelattice gauge theory of the ordinary type (namely with the ordinary confinementbehaviour) while keeping exact solvability is still open.In this letter we suggest a new point of view on the problem and show thatthe K-M model can be identified before integrating over the matter fields with anordinary SU(N) lattice gauge theory in the high temperature limit, the matterfields being related to the Polyakov loops, while the spatial gauge variables becomethe gauge fields of the K-M model.This letter is organized as follows: after a short introduction (just to set nota-tions) on finite temperature lattice gauge theory (sect.2), we describe and discussthe equivalence with the K-M action in sect.3 and 4.Sect.5 is devoted to thecontinuum limit and sect.6 to some concluding remarks.2

2. Finite temperature lattice gauge theoriesLet us consider a pure gauge theory with gauge group SU(N) defined on a d+1dimensional cubic lattice.

In order to describe a finite temperature LGT we musttake periodic boundary conditions in one direction (which we shall call from nowone “time-like” direction), while the boundary conditions in the other d direction(which we shall call “space-like”) can be chosen freely. Let us take a lattice of Nt(Ns) spacings in the time (space) direction.

The theory will contain only gaugefields described by the link variables Un;i ∈SU(N) where n ≡(⃗x, t) denotes thespace-time position of the link and i its direction. It is useful to choose differentcouplings in the time and space directions.

Let us call them βt and βs respectively.Let us take the simplest choice for the lattice gauge action, namely the Wilsonaction:SW =XnN ReβtXitr(Un;0i) + βsXi

We can solve the above equationsin terms of ǫ as follows:βt = 2g2ǫad−3(7)βs = 2ǫg2ad−3 . (8)In a finite temperature discretization it is possible to define gauge invariant vari-ables which are topologically non-trivial loops, closed due to the periodic boundaryconditions in the time directions.

The simplest choice is the Polyakov loop definedas follows:P(⃗x) = trNtYt=1(U⃗x,t;0)(9)1 Notice that the coupling constant g is rescaled here by a factor N with respect for instanceto the notations of ref. [12]3

where x labels the space coordinates of the lattice sites. Moreover, an importantfeature of the finite temperature theory with respect to the zero temperature caseis that it has a new global symmetry (independent from the gauge symmetry) withsymmetry group the center C of the gauge group (in our case ZN).

The Polyakovloop turns out to be a natural order parameter for this symmetry.It is possible to obtain, just from the definition itself of the model, some generalproperties in the high temperature regime (see for instance [12]). In this region thesymmetry with respect of the center of the group is broken, the theory is deconfined,the Polyakov loop has a non-zero expectation value and, what is more important,it is an element of the center of the gauge group.

Physically this means that thelinks in the time direction fluctuate around one of the Nth roots of unity of theZN group. We can lift this degeneracy by adding to the Wilson action a “magneticterm”Sm = N hX⃗xRe trNtYt=1(U⃗x,t;0)(10)and eventually send h →0.

This procedure selects the vacuum in which< P(⃗x) >= 1.The spatial links are static up to gauge transformations and this fact tells usthat the spatial degrees of freedom of the model behave as if they belonged to a zerotemperature model in one dimension less and with coupling constant g2T. If onecould integrate out these spatial degrees of freedom the resulting effective action forthe Polyakov loops would be that of a d- dimensional spin model with short rangespin-spin interaction [12].3.

The K-M model as an high temperature LGT.Let us take the d + 1 dimensional LGT described above (with gauge groupSU(N)) and consider first a very special case, namely Nt = 1. We will be interestedin the T →∞(hence ǫ →0) limit.Let us call St (Ss) the contribution of the temporal (spatial) plaquettes to theaction, with a “magnetic term” of the type (10) included in St. Due to the boundaryconditions and to the Nt = 1 position we can rewrite St as:St = N βtX⃗x,iRe trnU⃗x;iU⃗x+ˆi;0U†⃗x;iU†⃗x;0o+ N hX⃗xRe (U⃗x;0)(11)This model is interesting in itself and represents the generalization of the K-Mmodel to “matter” fields described by unitary matrices.

Notice also that since thespace contribution Ss to the action is depressed as βs ∼1/T, in the T →∞limitthis model should capture most of the features of high temperature LGT.4

Following Svetitsky and Yaffe [12] one can argue at this point that in the hightemperature limit each temporal plaquette fluctuates around the identity . It iseasy to see that the fluctuations are of order 1/√βt .

In fact if we put Upl = eiφ√βtwith φ a traceless hermitian matrix we have:ZdUpleNβttr Re Upl = eN2βt√βtZdφe−N trφ2[1 + O(1/βt)](12)Correspondingly the link variables in the time direction fluctuate around one ofthe Nth roots of unity of the ZN group with fluctuations still of order 1/√βt. Whenthe magnetic term is switched on the vacuum is forced to be U⃗x;0 = 1 and we canexpand U⃗x;0 in the following way:U⃗x;0 ≡ei φ(⃗x)√βt = 1 + iφ(⃗x)√βt−φ2(⃗x)2βt+ · · ·(13)By inserting (13) in (11) we find:St = N βt tr(d Ωx1 + 1βtX⃗x −m2φ(⃗x)2 +dXi=1U⃗x;iφ(⃗x)U†⃗x;iφ(⃗x +ˆi)!

)(14)where m2 = d+ h2βt and Ωx is the volume of the d dimensional space. In ref.

[12] onlythe constant leading contribution of order βt was considered; here we recognize thatthe contributions of the fluctuations, of order 1, coincide, apart from an irrelevantconstant, with the K-M action with arbitrary m2 (eq. (1)) and in the limit of zeromagnetic field with the K-M action at the critical point m2 = d. The expansion(13) and in particular the choice of the expansion parameter1√βt is also justifieda posteriori by the analysis of the distribution of the eigenvalues of φ(⃗x) resultingfrom the action (14).

It was shown in [3] that at least in the large N limit and form2 > d the eigenvalue distribution of φ(⃗x) is semicircular with a finite , non-zeroradius. This implies that the eigenvalues of the original unitary matrix U⃗x;0 arerestricted to a region of order1√βt around 1 on the unit circle, thus justifying theexpansion (13).

For d > 1 such distribution of the eigenvalues survives also in thelimit h →0 (with h > 0) namely m2 →d, whereas for d = 1 the radius of theeigenvalue distribution goes to infinity as h →0 (m2 →1) . This shows that in theone dimensional case the critical point can only be described in the unitary matrixformulation given by (11).Let us consider now the case of an arbitrary Nt.

The action (without the con-tribution of spatial plaquettes) is:St = NβtX⃗x,iNtXt=1Re tr{U⃗x,t;iU⃗x+ˆi,t;0U†⃗x,t+1;iU†⃗x,t;0} + N hX⃗xReP(⃗x)(15)5

where P(⃗x) is the Polyakov loop given in (9). It is possible to choose the gaugein such a way that U⃗x,t;0 = 1 for t = 1, 2, .., Nt −1 so that the only non triviallinks in the time direction are U⃗x,Nt;0 , related to the Polyakov loop by the relationP(⃗x) = trU⃗x,Nt;0.

The plaquettes at t < Nt then reduce to U⃗x,t;iU†⃗x,t+1;i . It followsthat in the large βt limit the spatial link variables at different values of t coincidesup to fluctuations of order 1/√βt.

So we can putU⃗x,t;i = U⃗x;iexp{iψi(⃗x, t)√βt}(16)where we are free to choose ψi(⃗x, 1) = 0. In complete analogy with the case Nt = 1we also have:U⃗x,Nt;0 = exp{iφ(⃗x)sNtβt}(17)where the factor √Nt in the exponent at the r.h.s.

of (17) is needed , as we shallsee, for a smooth Nt →∞limit. By inserting equations (16) and (17) into theaction (15) and by expanding in powers of 1/βt up to terms of order 1 , one obtainsan action quadratic in the ψi(⃗x, y)’s:St=N trX⃗x;i{Ntβt +Nt−1Xt=2[ψi(⃗x, t)ψi(⃗x, t + 1) −ψ2i (⃗x, t)] −ψ2i (⃗x, Nt) −−qNtψi(⃗x, Nt)[φ(⃗x +ˆi) −U†⃗x;iφ(⃗x)U⃗x;i] −Nt2 [φ(⃗x +ˆi) −−U†⃗x;iφ(⃗x)U⃗x;i]2 −NhNtβtφ2(⃗x)}(18)The fluctuations ψi(⃗x, t) can be eliminated by performing the corresponding gaus-sian integrals and the final result coincides with the K-M model as given by eq.

(14)2.The same result can be obtained also by noticing that, as the self interaction termis neglected, each spatial link variable U⃗x,t;i belongs to just two temporal plaquettesand that the corresponding integral can be done by using character expansion inanalogy to lattice QCD in two dimensions. This allows to define a “renormalized”coupling βeff through the relation:If(βeff) ="If(β)I0(β)#Nt(19)where If and I0 are the coefficients of the fundamental and identity characters in thecharacter expansion of the action.

Then it can be shown that the large β behaviourof βeff(β) is as expected βeff ∼βNt (see for instance ref. [13]).The final result eq.

(14) is then independent from the number Nt of lattice spacingin the time direction provided the field φ(⃗x) is normalized as in (17). Notice that2Notice that Ωx in (14) is now replaced by the volume of the whole space-time lattice6

the r.h.s. of ( 17) is also independent of Nt according to the definition of βt givenin (7).

So the continuum limit in the time dimension Nt →∞is well defined andleads to the K-M model.Hence we can conclude thatfor d > 1 the d-dimensional K-M model is a good description of the smallfluctuations of the Polyakov loops around their minimum (frozen) position in thehigh temperature limit of a pure lattice gauge theory in d + 1 dimensions with theWilson action.4. Comments and remarksThe integration over the φ fields in (14) leads to the induced gauge action eq.

(3)which can now be interpreted as describing the spatial degrees of freedom of a pureLGT at T = ∞. Since, as mentioned above, the space degrees of freedom in thehigh temperature limit behave as a zero temperature d-dimensional gauge theorywith coupling constant g2eff = g2T, the K-M model corresponds to the geff = ∞point of the theory thus leading to the superconfinement behaviour.At T < ∞one has to add a small (order 1/T) gauge self-interaction term.

Thisterm obviously destroys the exact solvability of the model, but it can be treated asa small perturbation around the K-M solution. The main outcome of our analysisis probably in the fact that we now understand that such a perturbative analysiswould be an high-temperature expansion for the LGT.Let us consider for instance the expectation value of a non backtracking Wilsonloop which is zero at T = ∞due to the superconfinement.

Its first non trivialperturbative contribution in the 1/T expansion is obtained by filling the loop byelementary plaquettes 3. The resulting “filled Wilson loops” is invariant under thelocal ZN symmetry and has to be evaluated using the measure of the pure M-Kinduced action.

This was done in ref. [7] where it was shown that the “filled Wilsonloop” has an area law behaviour.

It is also remarkable and not quite understoodyet that in this context intriguing connections with 2d spin models emerge. Ourinterpretation is that the “filled Wilson loop ” is the first non vanishing term ina large temperature expansion of the vacuum expectation value of the ordinaryWilson loop.The K-M model is only apt to describe fluctuations around one given vacuum ofthe high temperature LGT; in other words it can only describe finite temperatureLGT in the broken ZN phase.

However, even in the framework of the K-M model3In more than two spatial dimension one has to sum over the contributions of all surfaces madeof elementary plaquettes and having the original loop as a boundary. These contributions are oforder T −A where A is the area of the surface, so that the surface of minimal area dominates athigh temperature.7

one has at least a qualitative understanding of the restoring of the ZN symmetrybelow the critical temperature. In fact the expectation value of the Polyakov loop< P(⃗x) >=< Treiφ(⃗x)qNtβt >(20)can be evaluated in the large N limit by replacing φ(⃗x) with its classical value givenin ref.

[3]. At the critical point m2 = d this is given by a semicircular distributionof eigenvalues of radius r =q 2d−1d(d−1).At high temperature, more precisely forrqNtβt ≪π, the eigenvalues are peaked around one vacuum and the ZN symmetryis broken.

However for values of βt such that rqNtβt ≈π the eigenvalues ofqNtβt φ(⃗x)are distributed over several vacua on most of the unit circle and so the symmetryis eventually restored.Without a more detailed understanding of the symmetry restoration nothingnew can be said in this context on the Svetitsky-Yaffe conjecture that the criticalbehavior of a (d+1)-dimensional finite temperature gauge theory is the same as thatof a d-dimensional spin model with the same ZN symmetry. At this stage at leastthe emergence of spin models [7] in the expectation value of the Wilson loop doesnot appear to be correlated with the Svetitski-Yaffe conjecture.

It should be noticedhowever that if the contribution of the spatial plaquettes is neglected, an effectivetheory for the Polyakov loop valid also in the region of large φ(⃗x) could in principlebe obtained directly from (15) by integrating over the spatial links variables. Thiswould require the use of a generalization of the Itzykson-Zuber formula describedin [7] which is exact for N = 2 and valid as an asymptotic formula for N > 2.The instability which occurs in the m2 < d region, corresponding to negativevalues of h in (15), can be understood from the high temperature LGT point ofview as a consequence of the fact that with our “magnetic field” in this regime weare actually pushing the system out of the chosen vacuum which becomes unstable.Similarly the instability due to the presence of a linear term [14] which occurs ifone looks at U(N) instead of SU(N) K-M models, can be understood as a signatureof the Goldstone modes of the broken U(1) symmetry which one has in this case.Let us finally consider the case d = 1.

Spatial plaquettes are absent in thiscase and eq. (15) gives the complete action irrespective of the value of βt.

This isjust two dimensional QCD on a cylinder or , in case the spatial dimension is alsocompactified, on a torus. We know from ref.

[3] that in the d = 1 case at the criticalpoint m2 = 1 the distribution of the eigenvalues of U⃗n,Nt;0 is not confined to a smallregion of the unit circle. As a consequence we are always in the unbroken phaseand the description of the system in terms of the K-M model is not valid.

On theother hand by expanding both spatial and temporal links in a fashion similar toeq. (16) one can reduce the action of eq.

(15) with periodic boundary conditions inboth space and time to an action containing only one space and one time-like link,in agreement with general result on lattice QCD in two dimensions [11].8

5. Taking the continuum limitIn the previous sections we have derived the K-M model as a high temperaturelimit of lattice QCD.

However while in ordinary lattice QCD one has a well definedcontinuum limit, this is not the situation in the K-M model with a quadratic poten-tial at the critical point m2 = d. It was pointed out in [3] that if one approachesm2 = d from below there exists a solution of the master field equation correspond-ing to a semicircular distribution of eigenvalues whose radius r goes to infinity asm2 →d, a signal of the existence of a critical point. Such configuration however isa local maximum of the free energy; moreover the free energy itself is unboundedfrom below for m2 < d. This can be cured by adding a higher order term to theaction, for instance a quartic termλ4Trφ4.From the point of view of the hightemperature expansion this is not such an ad hoc adjustment as it might appear atfirst, in fact the continuum limit corresponds to the limit where the radius of theeigenvalue distribution goes to infinity and in that limit higher order terms in φ(⃗x)are expected to be relevant.

Therefore in the continuum limit the high temperatureQCD is most likely described by a K-M model with a higher order potential inagreement, as discussed below, with some recent results on lattice QCD [16, 17].It has indeed been shown in [15] that, with the addition of a quartic term λ4Trφ4to the action, a second order critical point can be reached, at least in the case ofSU(2) in d=4. The analysis of the phase diagram can be done rather easily withina mean field approximation: in the λ, m2 plane one finds a line of first order tran-sitions starting from the K-M critical point and ending with a second order criticalpoint located at λ = 2.57 and m2 = 2.264.

Moreover direct Montecarlo simulationsshow that these mean field results are quite accurate and even the critical pointlocation is essentially confirmed by the computer simulation. We have made a sim-ilar mean field analysis in the d = 3 case, in which we are interested, showing asimilar scenario: a line of first order phase transitions starting from m2 = 3 andending with a second order critical point located at λ ∼2.2, m2 ∼1.56.

In [15]it was stressed that the corresponding continuum theory had nothing to do withordinary SU(2) gauge theory. Our suggestion is that it should instead describe thehigh temperature, deconfined phase of SU(2) (with one more space-time dimen-sion).

This conjecture is in remarkable agreement with some recent results on thehigh temperature behaviour of lattice gauge theories, both within a perturbativeapproach [16], and with montecarlo simulations [17]. The main idea behind [16, 17]is that (3+1) dimensional gauge theories at high temperature undergo a peculiarform of dimensional reduction.

The original picture [18] of a complete dimensionalreduction (namely a complete decoupling of the degrees of freedom in the compacti-fied time direction, which would lead to an effective three-dimensional theory of theordinary type) does not occur in general and one finds instead a three dimensionalgauge theory coupled with a scalar field φ in the adjoint representation with a non4Notice that there is a factor of two between our normalization of the mass parameter and thatof [15]9

trivial potential V (φ). This potential has a quadratic term with the wrong sign tobe identified as a mass term and a quartic contribution which stabilizes the action,exactly as in our mean field solution.

Moreover the action used in [17] (eq. (20)of [17]) for the montecarlo simulation is exactly that of a K-M model in the instable(m2 < 3) region, with a quartic term (like the above described mean field solution)plus the gauge self-interaction term.

Taking into account the obvious uncertaintiesof our mean field estimate the numerical agreement on the coefficient of the quarticcontribution between our result and that of [17] is impressive.6. ConclusionsIn this letter we have shown that, besides the usual identification with a LGTcoupled with adjoint matter in the strong coupling regime, the Kazakov Migdalmodel can also be related to the behaviour of a pure lattice gauge theory (withWilson action) in the limit of very high temperature.

In particular we have seenthat the φ fields describe the small fluctuations around the frozen position of thePolyakov loops.We think that this new point of view is important if we want to understandsome of the peculiar features of the K-M model (superconfinement, lack of ordinaryperturbative vacuum, presence of an unstable phase for m2 < d and of a finite gap inthe free energy), but besides this we think that there are two other reasons of inter-est. First the master field solution of the K-M model could give us a powerful toolto do both analytical and numerical calculations in pure LGT at high temperature.Second, more ambitious point: the K-M model is related to non critical strings,although this relation is still not quite clear for d > 1 (see however [19] and [20]for some recent progress on this subject).

Our interpretation might lead (at leastin the high temperature, large N limit) to a first description in terms of strings ofLGT, a hope which justifies by itself further efforts toward a better understandingof the K-M model.Note addedWhile completing this letter we received a new interesting paper by Dobroliubov,Kogan, Semenoffand Weiss [21], in which the interpretation of the “filled Wilsonloop” as the result of the perturbative expansion of gauge self-interaction term isdiscussed. The two regimes discussed in [21] may be identified within our approachas the high temperature deconfined regime and the low temperature confined phase10

respectively, and the coupling λ of [21] with our ratio βsβt5.AcknowledgmentsOne of us (A.D.) thanks J. Ambjørn and D. Boulatov for useful discussion andin particular J. Ambjørn for pointing out to us the relevance of ref. [16] and [17].References[1] V.A.

Kazakov and A.A. Migdal, “Induced QCD at Large N”, preprint PUPT- 1322, LPTENS -92/15, May, 1992. [2] A.A. Migdal, “Exact Solution of Induced Lattice Gauge Theory at large N”,preprint PUPT - 1323,Revised, June, 1992;[3] D.Gross,Phys.

Lett.B293 (1992) 181[4] M. Caselle, A. D’Adda and S.Panzeri, Phys. Lett.

B293 (1992) 161[5] D.Gross and I. Klebanov,Nucl. Phys.B344 (1990)4 75 and B354 (1991) 459.

[6] I.I. Kogan, G.W.

Semenoffand N.Weiss, “Induced QCD and Hidden Local ZNSymmetry”, preprint UBCTP 92-22, June, 1992. [7] I.I.Kogan et al.

“Area Law and Continuum Limit in “Induced QCD”, preprintUBCTP 92-26, ITEP M6/92, July 1992[8] I.I.Kogan, A.Morozov, G.W.Semenoffand N.Weiss, Continuum limits of ‘in-duced QCD’: lessons of the gaussian model at D = 1 and beyond, UBC preprint92-27 (August, 1992)[9] A.A.Migdal, Mixed model of induced QCD, Paris preprint LPTENS-92/23 (Au-gust, 1992)[10] S.Khokhlachev and Yu.Makeenko, “Adjoint fermions induce QCD”, preprintITEP-YM-7-92, August, 1992. [11] B.Ye.

Rusakov,Mod. Phys.

Lett.A5 (1990)693. [12] B. Svetitsky and L.Yaffe,Nucl.

Phys.B210 (1982) 423.5Notice, in order to avoid confusion that our high temperature phase is what is called in [21]the low temperature one and is characterized by a K-M behaviour and by λ ≡βsβt →011

[13] See for instance: J.M.Drouffe and J.B.Zuber, Phys. Rep. 102 (1983) 1[14] M.Caselle, A. D’Adda and S.Panzeri, “Kazakov–Migdal induced gauge theoryand the coupling of 2d quantum gravity to d=1 matter”, preprint DFTT 65/92,To be published on the Proceedings of the Lat92 conference.

[15] A.Gocksch and Yu.Shen, “The phase diagram of the N=2 Kazakov-Migdalmodel” preprint BNL47829, August, 1992.S.Aoki, A.Gocksch and Yu.Shen, “A study of the N = 2 Kazakov–MigdalModel”, preprint UTHEP-242, August, 1992. [16] See for instance: S.Nadkarni, Phys.

Rev. D38 (1988) 3287, and referencestherein[17] P.Lacock, D.E.Miller and T.Reisz, Nucl.

Phys. B369 (1992) 501, and referencestherein[18] See for instance: T.Appelquist and R.D.Pisarski Phys.

Rev. D23 (1981) 2305[19] D.V.

Boulatov, Infinite-tension strings at d > 1, preprint NBI-HE-92-78(November 1992). [20] S. Dalley and I. Klebanov, String spectrum of 1+1- dimensional large N QCDwith adjoint matter, preprint PUPT-1342 (S ept.

1992). [21] M.I.Dobroliubov, I.I.Kogan, G.W.Semenoffand N.Weiss, Induced QCD withoutlocal confinement, UBCTP preprint 92-32 (December, 1992)12

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