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The Spectrum of the Kazakov-Migdal Model

arXiv:hep-lat/9212027v1 20 Dec 1992December, 1992UTHEP-250The Spectrum of the Kazakov-Migdal ModelS. AokiInstitute of Physics, University of TsukubaTsukuba, Ibaraki 305, JapanandAndreas GockschPhysics Department, Brookhaven National LaboratoryUpton, NY 11973, USAAbstractGross has found an exact expression for the density of eigenvalues in the simplestversion of the Kazakov-Migdal model of induced QCD.

In this paper we computethe spectrum of small fluctuations around Gross’s semi-circular solution. By solvingMigdal’s wave equation we find a string-like spectrum which, in four dimensions, cor-responds to the infinite tower of mesons in strong coupling lattice QCD with adjointmatter.

In one dimension our formula reproduces correctly the well known spectrumof the hermitean matrix model with a harmonic oscillator potential. We comment onthe relevance of our results to the possibility of the model describing extended objectsin more than one dimension.

IntroductionThe original Kazakov-Migdal (KM) model of induced QCD [1] consists of a scalar in theadjoint representation of SU(N), covariantly coupled to gauge fields on a hypercubic latticeof spacing a. There is no kinetic term for the gauge fields and the action of the model isS = NXx"trU(Φ(x)) −trXµΦ(x)Uµ(x)Φ(x + µa)U†µ(x)#.

(1)The hope was that this model could perhaps induce QCD in the sense that a distancesmuch larger than a a kinetic term for the gauge fields would be generated through scalarinteractions at distances on the order of the cutoffa. By carefully tuning the parameters ofthe potential it might then be possible to reach an ‘asymptotic freedom domain’ in whichthe continuum limit could be taken.

The mass of the scalar, which is kept heavy in thislimit, would act as an effective cutofffor the resulting continuum (QCD?) theory.The beauty of this idea of course lies in the fact that the model given in Eq.

(1) isanalytically tractable in the limit of an infinite number of colors N [2].The model isespecially simple in the case of a purely quadratic potential and initially there was clearlyno a priori reason for why the objective of inducing QCD, if possible at all, could not beachieved by just tuning the mass of the scalar. This simplest of all possibilities is by nowruled out.

The reason is as follows: D. Gross [3] has found an exact solution to the saddlepoint equations describing the translationally invariant eigenvalue distribution of the Φ-fieldin the N = ∞limit. This solution has no continuum limit in four dimensions.

It ratherdescribes the large N, infinite coupling (remember, there is no plaquette term for the gaugefields in (1)) limit of lattice QCD with adjoint matter. This in by itself is not sufficient toprove ‘non-induction’ for there could be other saddle points.

However computer simulationsof the SU(2) theory [4] also show that there is no critical point in the case of a purely gaussianpotential. Assuming that the large N limit is smooth the theorem is proved.

Incidentally,the simulations also showed that by adding a quartic term a critical point could be reached.The nature of the resulting continuum theory is not clear at the moment. Migdal [5] hasmade considerable progress in the investigation of the theory with a general potential butlately he has concentrated his efforts on the study of an extended version of Eq.

(1), the socalled ‘mixed model’ of induced QCD [6].In the mixed model [6] nf ≪N heavy fermions in the fundamental representation areadded to the action in order to break the additional local symmetries [3] of the model, inparticular the local ZN symmetry Uµ(x) →Zµ(x)Uµ(x) first discussed by Kogan et al. [7].As pointed out by these authors, due to this symmetry the Wilson loop cannot acquire anexpectation value.

The adjoint loop of course does, but it is always screened and cannotserve as an order parameter for confinement. It is Migdal’s hope that there is a phase of themixed model in which the center symmetry is broken and Wilson loops show an area law.In the present paper we will have nothing more to say about the question of whetherand how QCD can be induced.

Rather we will extend the very pretty work of Ref. [3] onthe original Gaussian KM model by calculating the spectrum of small fluctuations around1

the Gross saddle point. We believe that there are many reasons for why this is interesting.In four dimensions our result gives the masses of the mesons of the infinite coupling latticegauge theory to leading order in 1/N.

The large N, strong coupling spectrum of the theorywith fundamental fermionic matter has been known for over a decade now [8] and the adjointscalar spectrum nicely complements this old work. Furthermore, the model in Eq.

(1) canbe viewed as a gauged matrix model. Hermitean matrix models in D ≤1 dimensions areintimately related to discretized versions of the Polyakov string [1] and as such describe 2Dgravity coupled to matter fields.

In particular, a case of great current interest is the onedimensional case, describing c = 1 matter coupled to 2D gravity. In more than one dimensionordinary matrix models become very complicated [9] and one might hope that the gaugedversion in Eq.

(1) might be useful as theory of extended objects in more than one dimension.It has also recently been suggested [10] that the KM model can be viewed in D dimensionsas the high temperature limit of the (D + 1) dimensional Wilson action. If this interstingconjecture is indeed correct then our result in three dimensions should describe the spectrumof fluctuations of the 4D-Polyakov line around one of it’s ZN minima at infinite temperature.Finally, our calculation can serve as a testing ground for the methods developed by Migdal.In some sense the present problem represents the ‘hydrogen atom’ of induced QCD; we canactually work out the eigenvalues and the eigenfunctions of Migdal’s wave equation in closedform.

In the process we also found a term in the wave equation which had been omitted inRef. [11].The Gross SolutionIn this section we will quickly review Gross’s solution of induced QCD with a quadraticpotential.

In the process we will also establish notation. We will follow Migdal’s approach[11] to the problem and express the action in Eq.

(1) in terms of the density of eigenvaluesρ(λ) = 1NXiδ(λ −λi). (2)This is easily done by first integrating out the gauge fields using the Itzykson-Zuber integral[12], which producesS = NXx,iU(λi(x)) −Xx,i̸=jln |λi(x) −λj(x)| −Xx,µln[I(λ(x), λ(x + µa)](3)where the logarithm comes from two factors of the Vandermond determinant and I denotesthe Itzykson-Zuber determinant.

Now using Eq. (2) the action can be written in terms of ax-dependent density asS[ρ] = N2 XxZdλρx(λ)U(λ) −N2 XxZdλρx(λ)Zdλ′ρx(λ′) ln |λ −λ′| −Xx,µln[I(ρx, ρx+µa)].

(4)Note that the effective action for ρ will also receive contributions from a Jacobian due tothe change of variables from the eigenvalues to the density [11]. Since everything we haveto say is independent of this term however we will just drop it here.2

An equation for the eigenvalue distribution ρx can be found by looking for stationarypoints of the effective action in Eq. (4).

In performing infinitesimal variations of ρx onemust be careful though not to change the normalization of ρx in Eq.(2). This can beachieved by taking variations of the formδρx(λ) = −1Ndψ(λ)dλ(5)or equivalentlyδδψ(λ) = 1Nddλδδρ(λ)(6)where ψ(λ) vanishes at the end point of the support of ρ(λ).

The saddle point equationfor (a translationally invariant) ρ follows by setting the first variation of the effective actionδSδψ(λ) equal to zero. One obtains, denoting the logarithmic derivative of the Itzykson-Zuberdeterminant by F(λ),F(λ) = −2ReV ′(λ) + U′(λ)2D(7)whereV ′(λ) =Zdλ′ ρ(λ′)λ −λ′.

(8)This equation by itself does not determine ρ since F is unknown. The function F(λ) howeversatisfies a set of Schwinger-Dyson equations which were derived by Migdal in Ref.

[2]. Usingthese equations one finally ends up with an integral equation for ReV ′(λ) whose imaginarypart determines ρ:ReV ′(λ) = PZ dλ′π arctanπρ(λ′)λ −R(λ′)(9)whereR(λ) = D −1DReV ′(λ) + U′(λ)2D .

(10)In one dimension the gauge field can be gauged away and using U(Φ) = 12m2Φ2 the actionin Eq. (1) is just that of a free scalar field.

In this case it is well known that ρ is semicircularin shape, i.e.ρ(λ) = ρ0(λ) = 1πsµ −µ2λ24(11)and µ =√m4 −4. Gross [3] realized that the semicircular form (11) for ρ actually solvesEqs.

(9,10) in any number of dimensions. He showed that in D dimensionsµ± =m2(D −1) ± Dqm4 −4(2D −1)2D −1(12)3

and also computed the free energy1:F = 12m22µ + 12 ln µ −D2 [s1 + 4µ2 −1 −ln(12 + 12s1 + 4µ2)](13)The following things are important to note: µ+ is a minimum of F and vanishes at m2 = 2Din D ≤1 implying that in this case the continuum limit can be taken. In D > 1 on theother hand, µ+ never vanishes and instead µ−→0 as m2 →2D−.

However the saddle pointat µ = µ−is a local maximum which, as we shall see, has important consequences for thespectrum of the theory.To summarize this section: For a quadratic potential the distribution of eigenvalues issemicircular in shape in any number of dimensions. In one or less dimensions a continuumlimit can be constructed and we expect a physical spectrum since one is perturbing around alocal minimum of the free energy.

In more than one dimensions on the other hand one expectstachyons in the continuum limit. The spectrum of fluctuations around µ+ will describe thespectrum of mesons made out of adjoint matter in the strong coupling lattice theory.

Wewill now go on to show all this explicitly.Migdal’s Wave EquationIn order to get at the spectrum of the theory in leading order in 1N one must work out theeffective action describing the fluctuations around an extremum of the action (4). This hasbeen done by Migdal [11].

Writingδρx(λ) = ρx(λ) + δρx(λ)(14)with δρ as in Eq. (5) and ρx(λ) a solution of (7), one obtainsS2[ρ] = −XxZdλZdλ′"12η(λ, λ′)ψx(λ)ψx+aµ(λ′) + [1(λ −λ′)2 + Dσ(λ, λ′)]ψx(λ)ψx(λ′)#.

(15)In the case at hand here we have ρx(λ) = ρ0(λ). Now using plane waves, ψx(λ) = ψ(λ)eiP x,one immediately obtains Migdal’s wave equation for the particle spectrum:Ω2Zdλ′η(λ, λ′)ψ(λ′) = −Zdλ′[1(λ −λ′)2 + Dσ(λ, λ′)]ψ(λ′).

(16)In Eqs. (15,16) we have introduced Migdal’s notation for the derivatives of F(λ), i.e.σ(λ, λ′) = ddλ′δF(λ)δρx(λ′)(17)1It is interesting to note that the the free energy obtained by Kawamoto and Smit [8] can be broughtinto the same form if one makes the curious substitution λ = −4µ2 and M = im28 , where λ is the square ofthe meson field and M is the bare quark mass.4

andη(λ, λ′) = ddλ′δF(λ)δρx+aµ(λ′). (18)Also, in Eq.

(16) we have defined Ω2 = Pµ cos(Pµ).Migdal [11] has derived Integralequations for the functions η and σ. We have checked his derivation of these equations andagree up to an additional term in the equation for σ.

The origin of this term is explained inAppendix 1. Using the correct equations for η and σ from the Appendix we obtain, denotingddλ′ψ(λ′) ≡H(λ′) and integrating Eq.

(16) over λ, the following equation:Ω2Zdλ′ H(λ′)λ′ −λ0=ZdλZdλ′H(λ′)K(λ0, λ)"(1 −D)λ −λ′ + DG(λ0, λ′)G(λ0, λ)1λ −λ′#−DZdλ′H(λ′)G(λ0, λ′). (19)Several comments are in order.

First, all the integrals in Eq. (19) are defined in the principalvalue sense.

Second, Migdal’s functions K(λ0, λ) and G(λ0, λ) are given byK(λ0, λ) =ρ(λ)π2ρ2(λ) + (λ0 −R(λ))2(20)andG(λ0, λ) =1λ0 −R(λ)Re exp"+Zdλ′π(λ′ −λ) arctanπρ(λ′)λ0 −R(λ′)#. (21)Third, the range of the integrals is over the support of ρ(λ) which in our case is the interval[−2√µ, + 2√µ].

Fourth, the last term in Eq. (19) is the one absent in Migdal’s paper [11].In principal the calculation of the spectrum of fluctuations around the Gross saddle pointis now straight forward: Compute G(λ0, λ) using it’s definition Eq.

(21), do the λ-integralin (19) and finally solve the resulting (singular) integral equation. However there are severalsubtleties in the calculation which we thought make it worthwhile to present the calculationof G(λ0, λ) in some detail.

This is done in Appendix 2. The interested reader will be ableto check all the other integrals relevant here by using the method used in the Appendix.Suffice it to say here, that boundary conditions are important.

For example, the branch ofthe ‘arctan’ in Eqs. (19,21) is chosen by a boundary condition at infinity [2].

Hence, in thecalculation λ0 must be taken outside the support of ρ0.We have obtained the following expression for G(λ0, λ):G(λ0, λ) =1(R −µ2) · (Rλ0 −λ) −qµ2λ204−µ(λ0 −Rλ)2 + µ −µ2λ24(22)where R(λ) = Rλ =12D(m2 + µ(D −1))λ is linear in λ for the semicircular solution. Toderive our final result for the wave equation below, we have repeatedly used the importantidentityR2 −µ24 = 1(23)5

satisfied by µ±. Finally, for completeness we also give the results of the λ-integrals in Eq.(19).

We obtainedZdλK(λ0, λ)λ −λ′=µ32(R −xy)4√x2 −1[(x −Ry)2 + µ24 (1 −y2)](24)andZdλK(λ0, λ)G(λ0, λ′)G(λ0, λ)1λ −λ′ = µ2 (1 −q(Rx −µ2√x2 −1)2 −1(Rx −y) −µ2√x2 −1 )(25)where we have introduced the new variables x =√µ2 λ0 and y =√µ2 λ′.Calculation of SpectrumUsing the results of the previous section we obtained the following integral equation deter-mining the spectrum of the theory to leading order in 1/N:Ω2Z +1−1 dy h(y)y −x =Z +1−1 dyh(y)A(x) −B(x)y(x −Ry)2 + µ24 (1 −y2)(26)whereA(x) = (1 −D)µR2√x2 −1 −Dx(27)andB(x) = (1 −D)µx2√x2 −1 −DR. (28)To bring the wave equation into this form we have assumed that H(λ′) depends on λ′ onlythrough y, i.e.

H(λ′) = h(y).To determine the spectrum of the theory means to determine Ω2 in Eq. (26).

Due to thecompactness of the integration interval we expect a discrete spectrum. To solve the equationwe follow the time honored method of expansion in a complete, orthonormal set of functions.In particular, due to the range of the integration interval and the form of the integrand onthe left hand side of the equation, Chebyshev polynomials are a natural set of functions touse here2.

Hence we make the ansatzh(y) =XncnTn(y)√1 −y2. (29)Using this ansatz the integral on the left hand side of Eq.

(26) can be done immediately(remembering though that |x| > 1 in the calculation) and one obtainsΩ2Z +1−1 dy h(y)y −x = −πΩ2 Xncn(x −√x2 −1)n√x2 −1. (30)2We use the conventions of Ref.

[13] for the definition of the Chebyshev polynomials.6

On the right hand side we first writeA(x) −B(x)y(x −Ry)2 + µ24 (1 −y2)=f(x)y −y++g(x)y −y−(31)wherey± = Rx ± µ2√x2 −1. (32)In the last equation the identity Eq.

(23) has been used. Now the integral on the right handside of the wave equation (26) can also be done and one obtains after some algebra,Z +1−1 dyh(y)A(x) −B(x)y(x −Ry)2 + µ24 (1 −y2)=−π2Xncn{[(D −1) + D](R + µ2 )−n+[D −(D −1)](R + µ2)n}(x −√x2 −1)n√x−1.

(33)Now comparing Eqs. (30) and (33) we finally obtain for the eigenfunctions and eigenvalueshn(y) =Tn(y)√1 −y2(34)andΩn2 = (D −1)2[(R + µ2 )−n −(R + µ2 )n] + D2 [(R + µ2 )n + (R + µ2)−n].

(35)The last two equations constitute the main result of this paper. We will discuss it’s sig-nificance in the next and final section.

It is noted that the fluctuations themselves (at agiven momentum Pµ) are of course given by δρ(y) = −1NTn(y)√1 −y2 which properly satisfiesZ 1−1 dyδρ(y) = 0 for n = 1, 2, · · ·, but ̸= 0 for n = 0. Therefore h0(y) should be excludedfrom the eigenfunctions.Discussion and ConclusionAs expected we have found a discrete spectrum with an infinite number of states.Thequantum number n labels states according to their parity under y →−y.

The eigenfunctionshn(y) are even (odd) under this operation for n even (odd). Let us begin our considerationsof the spectrum now by looking at the one dimensional case in some detail.

To this end itis useful to defineM = ln(R + µ2 )(36)and write Eq. (35) asΩn2 = D cosh(nM) −(D −1) sinh(nM)(37)7

In one dimension we obtaincos(Pn) = cosh(n · arccosh(m22 )) = Tn(m22 )(38)The interpretation of this result is straightforward. As we mentioned before, in one dimensionthe gauge fields play no role and the theory is free with propagatorGij,kl(P) = 1N1m2 −2 cos(P)δilδjk.

(39)Clearly, the ‘pole’ of the propagator is at cos(P) = m22 which agrees with (38) in the casen = 1. In position space this corresponds to an exponential falloffof the propagator withmass M. The states with n > 1 correspond to ‘mesons’ made out of n Φ’s.

The x-spacepropagator for these objects decays exponentially with mass n·M which in momentum spacegives Eq. (38).In one dimension µ =√m4 −4 vanishes as m2 →2+ and we can take the continuumlimit.

Writing m2 = 2 + m20a2 and P = P0a one obtains−P 20 ≡E2n = n2 · m20. (40)Hence in the continuum limit we get a linearly rising spectrum of states of mass nm0.

Note,that for m0 = 0 we get a massless boson in one dimension as opposed to the massless bosonin 1 + 1 (one spacelike, one timelike) obtained in the double scaling limit of the continuummodel [14]. In less than one dimension µ still vanishes at m2 = 2D and one obtains expandingEq.

(37) at m2 = 2D + m20a2E2n = n · m20(41)in the continuum limit.Contrary to the case in one dimension, the particle masses areproportional to √n.Let us now go on and discuss the case D > 1. As we pointed out before the stable solutionis µ = µ+ in Eq.

(12) which is always greater than zero. In this case the interpretation ofthe spectrum in Eq.

(37) in four dimensions is that of the mesons in strong coupling latticeQCD with adjoint matter. Note that asymptotically in n, Ω2n = (R + µ/2)n. The sameformula holds in the limit D →∞for all n. It is interesting to see what happens if insteadwe use µ = µ−.

Setting m2 = 2D −m20a2 in order to approach the critical point from below,one gets−P 20 ≡E2n = −n · m20(42)in the continuum limit. Although infinitely many states appear in the continuum limit, theyare all tachyonic as expected.

The spectrum in D > 1 could be improved if an upside-downquadratic potential[15] is used for U(Φ) in Eq. (1).

The continuum limit in this case isequivalent to setting m2 = −2D −m20a2 in Eq. (37), which givesE2na2 = 2D ((−1)n −1) + (−1)n · n · m20a2.

(43)For n even, we obtain En = n · m20 > 0 though En = −∞for n odd.8

What do the above results tell us? In one dimension we got the expected result.

In morethan one dimension we find an infinite tower of states which certainly suggests a ‘stringy’interpretation of the spectrum. Makeenko [16] has written down Schwinger-Dyson equationsfor the functionsG(Cxy) =< 1N tr(Φ(x)U(Cxy)Φ(y)U(Cyx)) >(44)where U(Cxy) denotes a string of links along the path Cxy.

For the quadratic potentialMakeenko has obtained the solution for G(Cxy) and it would be interesting to see whetherby summing his result over all paths our spectrum can be reproduced [6]3.If it is aninterpretation of our result in terms of the excitations of a string of flux seems natural.AcknowledgementsWe thank V. Kazakov and A. Migdal for valuable discussions. This work was supported inpart by a DOE grant at Brookhaven National Laboratory (DE-AC02-76CH00016).Appendix 1.

Derivation of the extra term in eq. (19)In this Appendix we show how the last term in eq.

(19) is derived. In the Appendix of Ref.

[11] Migdal has derived the following equation (78)Zdµρ(µ)gλ(φ′, µ)z −µ= Tλ(z)Zdφ ρ(φ)z −φ(σ(φ′, φ) +1Gλ(φ)ddφ′Gλ(φ′)−Gλ(φ)(φ′−φ))(λ −R(φ))2 + π2ρ2(φ)(45)wheregλ(φ′, φ) = ddφ′δGλ(φ)δρx(φ′) . (46)Setting z →∞in the above formula, we findZdµρ(µ)gλ(φ′, µ) =Zdφρ(φ)(σ(φ′, φ) +1Gλ(φ)ddφ′Gλ(φ′)−Gλ(φ)(φ′−φ))(λ −R(φ))2 + π2ρ2(φ).

(47)From eq. (59) in Ref.

[11]Zdµρx(µ)Gλ(µ) =Zdν ρx+aµ(ν)λ −ν,(48)we obtainZdµρ(µ)gλ(φ′, µ) = −ddφ′Gλ(φ′) ,(49)3In one dimension one just obtains the square of the x-space propagator for a free particle by thisprocedure, i.e one only obtains the lightest ‘meson’.9

which differs from eq. (77) in Ref.

[11]. Inserting (49) into (47), we finally find equation forσ(φ′, φ)0 = ddφ′Gλ(φ′) +Zdφρ(φ)(σ(φ′, φ) +1Gλ(φ)ddφ′Gλ(φ′)−Gλ(φ)(φ′−φ))(λ −R(φ))2 + π2ρ2(φ).

(50)Appendix 2. Calculation of G(λ0, λ)The definition of G(λ0, λ) isG(λ0, λ) =1λ0 −R(λ)Re"exp[Zdλ′π(λ′ −λ) arctanπρ(λ′)λ0 −R(λ′)]#(51)where R(λ) = Rλ for the semicircular solution πρ(λ) =qµ −µ2λ2/4.

We first calculate thefollowing integralJ ≡Z+dλ′π(λ′ −λ) arctanqµ −µ2λ2/4λ0 −Rλ′. (52)Integration by parts together with the obvious rescaling of variables givesJ = µxπZ 1−1dy√1 −y2 ln(y −z) [f+y −y++f−y −y−](53)where x =√µ2 λ0, z =√µ2 λ, y =√µ2 λ′, f± = ±y± −R/xy+ −y−, and y± are given in Eq.

(32) inthe text. Using the following properties for Chebyshev polynomials Tn(y)ln(y −z) = −T0(y) ln(2tz) −2∞Xn=1Tn(y) (tz)nn(54)1y −y±= −2t±1 −t2±( T0(y) + 2∞Xn=1Tn (t±)n )(55)Z 1−1dy√1 −y2 Tn(y) · Tm(y) = δnm ×(π/2n ̸= 0πn = 0(56)where tz = z −i√1 −z2 and t± = (R ∓µ2)(x −√x2 −1), we perform the integral in Eq.

(53) and obtainJ = −ln [ 1 −tzt+1 −tzt−] . (57)From the above result G becomesG(λ0, λ) =1λ0 −RλRe [ 1 −tzt+1 −tzt−] ,(58)which finally gives eq.

(22) in the text.10

References[1] V.A. Kazakov and A.A. Migdal, Induced QCD at large N, Princeton preprint PUPT-1322 (1992).

[2] A.A. Migdal, Exact solution of induced Lattice Gauge Theory at large N, Princetonpreprint PUPT-1323 (1992). [3] D. Gross, Phys.

Lett. 293B, 181 (1992).

[4] A. Gocksch and Y. Shen, Phys. Rev.

Lett. 69, 2747 (1992); S. Aoki, A. Gocksch and Y.Shen, A study of the N=2 Kazakov-Migdal model, preprint UTHEP-242 (1992).

[5] A.A. Migdal, Phase transitions in induced QCD, Ecole Normale preprint LPTENS-92/22 (1992). [6] A.A. Migdal, Mixed model of induced QCD, Ecole Normale preprint LPTENS-92/23(1992).

[7] I.I. Kogan, V.W.

Semenoffand N. Weiss, UBC preprint UBCTP 92-022 (1992). [8] N. Kawamoto and J. Smit, Nucl.

Phys. B192, 100 (1981).

[9] A useful review on the subject is: V. Kazakov, Bosonic strings and string field theoryin one dimensional target space, preprint LPTENS 90/30 (1990). [10] M. Caselle, A. D’Adda and S. Panzeri, The Kazakov-Migdal model as a high tempera-ture limit of lattice gauge theory, Torino preprint DFTT 71/92.

[11] A.A. Migdal, 1N Expansion and particle spectrum in induced QCD, Princeton preprintPUPT-1332 (1992). [12] C. Itzykson and J. Zuber, J.

Math. Phys.

21, 411 (1980). [13] I.S.

Gradshteyn and I.M. Ryzhik, Table of integrals, series and products, AcademicPress, New York (1980).

[14] S. R. Das and A. Jevicki, Mod. Phys.

Lett. A5, 1639 (1990).

[15] V. A. Kazakov, D-dimensional induced gauge theory as a solvable matrix model, CERNpreprint, CERN-TH-6754/92. [16] Yu.

Makeenko, Large N reduction, master field and loop equations in the Kazakov-Migdal model, preprint ITEP-YM-6-92 (1992).11

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