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SYMMETRY AND OBSERVABLES IN INDUCED QCD

arXiv:hep-th/9303056v1 9 Mar 1993SYMMETRY AND OBSERVABLES IN INDUCED QCDGordon W. Semenoffand Nathan WeissDepartment of Physics, University of British Columbia,Vancouver, British Columba, Canada V6T 1Z1Proceedings ofMathematical Physics, String Theory and Quantum GravityRhakov, Ukraine, October, 1992AbstractWe review some of the basic features of the Kazakov-Migdal model of inducedQCD. We emphasize the role of ZN symmetry in determining the observable prop-erties of the model and also argue that it can be broken explicitly without ruiningthe solvability of induced QCD in the infinite N limit.

We outline the sort of criticalbehavior which the master field must have in order that the model is still solvable.We also review some aspects of the D = 1 version of the model where the partitionfunction can be obtained analytically.I. INDUCING QCDQuantum chromodynamics is presently accepted as the only viable theory of the stronginteractions.

It describes many of the quantitative features of the interactions of hadronsat high energies. However, at energies lower than the hadronic scale it gives only qualita-tive information.

Part of the reason is that perturbative QCD has only one dimensionlessconstant, the gauge coupling which is changed by renormalization from a dimenisonless con-stant into a mass scale. The perturbative regime is where all external momenta of Feynmandiagrams are greater than this mass scale.

It is thus impossible to address the infrared struc-ture of QCD, such as details of its spectrum and low energy interactions using conventionalperturbation theory.1

Another parameter of QCD which could be varied is the number of colors of quarks.QCD is known to simplify somewhat in the limit where the number of colors, N, is large[2]. In this limit, only planar graphs contribute to scattering amplitudes which consequentlyexhibit some of the qualitative features of the strong interactions.

However, so far no explicitsolution of QCD in dimensions greater than 2 is available in the large N limit and it hasthus led to very few quantitative results.There are some features of the large N limit which are particularly appealing. First, thereis every indication that the large N limit is quite smooth and that it exhibits confinementand dynamical chiral symmetry breaking which is found in the actual QCD for N=3.

Also,1/N2 = 1/9 is an expansion parameter which is less than one, so some large N results shouldhave reasonable accuracy. Furthermore, the parameter 1/N is not dependent on the massscale, so once a solution is found it is good at both high and low energies.

This might alsoshed some light on the relationship between lattice and continuum QCD.Recently, Kazakov and Migdal [3] have proposed a novel approach to the large N limitwhich has the hope of providing an exact solution of infinite N QCD. They consider inducedQCD which is obtained by integrating over the scalar fields in the lattice gauge theory withthe partition functionZKM =Zdφ[dU] exp−NXxTrV [φ(x)] + NXTrφ(x)U(xy)φ(y)U†(xy)(1)where φ(x) are N×N Hermitean matrices which reside on lattice sites x, U(xy) are unitaryN×N matrices which reside on links < xy > between neighboring sites x and y, dφ is theEuclidean integration measure for Hermitean matrices, [dU] is the invariant Haar measurefor integration over the unitary group U(N) and V [φ] is a potential for the scalars.

Thismodel is invariant under the gauge transformationsφ(x) →ω(x)φ(x)ω†(x)(2)U(xy) →ω(x)U(xy)ω†(y)(3)where ω(x) is an element of U(N). By restricting the trace of φ to zero and the determinantof U to one in (1) we could also consider a model with SU(N) gauge symmetry.The partition function of the Kazakov-Migdal model (1) can be regarded as the 1/g2 →∞limit of lattice scalar QCD with actionZ =Zdφ[dU] exp−NXxTrV [φ(x)] + NXTrφ(x)U(xy)φ(y)U†(xy)++Ng2X✷(TrU(✷) + TrU†(✷))!

(4)This action differs from that of the Kazakov-Migdal model by the addition of the Wilsonterm,X✷TrU(✷) + TrU†(✷)(5)2

The Wilson term is the trace of a product of link operators around an elementary plaquetteof the lattice.This term is the naive latticization of the continuum Yang-Mills action,TrFµνFµν. In (4) ✷denotes plaquettes of the lattice and U(✷) a product of the link operatorson the links on the boundary of ✷.Asymptotic freedom implies that the continuum limit of the lattice theory (4) is otainedby taking the bare coupling constant to zero,1g2 →∞(6)In fact, if instead of a lattice cutoffwe had a large momentum cutoffΛ the bare couplingwhich would be necessary to insure one loop renormalizability of QCD is1g2 =1148π2 ln(Λ2/µ2)(7)The hypothesis of the Kazakov-Migdal model is that the scalar QCD might still find a wayto arrange things so that a continuum limit exists in the opposite limit, where the barecoupling constant is infinite1g2 →0(8)in [3] they present a naive argument to show how this might be possible.

They begin withQCD coupled to scalars and without a kinetic term for the gluon field. The Yang-Millsaction is induced by the vacuum polarization of the scalar fields in the cutofftheory.

Theone-loop result is1g2ind=196π2 ln(Λ2/m2)(9)where m is the scalar mass. This can produce the g2 in (9) necessary to obtain a continuumlimit for the gauge field sector of the theory with the ultraviolet cutoffreplaced by the scalarmass if we take the mass of the scalar to bem2 = µ2 Λ2µ2!1/21(10)In this way, by giving up on finiteness of the scalar mass we can, at least at one-loop order,induce a renormalizable action for QCD.

Of course this is only a rough argument. Higherorder corrections from hard gluons will change this result significantly.

They can only becompensated by some strong self-interactions of the scalar field. The resulting picture isone of a complicated, strongly interacting theory.

It also requires that we have the abilityto arrange that the scalar mass goes to infinity with a slower exponent than the cutoffinthe continuum limit. This is possible if the scalar field theory has a socond order phasetransition and the accompanying critical behavior.

The appealing feature of this model isthat one may be able to solve it in the large N limit.Recently we have proposed a slight modification of this idea [1]. The solvability of thelarge N limit comes about through the absence of a kinetic term for the gauge fields in the3

bare Lagrangian. To leading order in N this property is also there if the kinetic term is notzero but is sub-leading in large N, or1g2 = λN(11)where λ ∼1 and 1/g2 ∼1/N as N →∞In [1] we argued that, by tuning λ appropriatelywe could still produce QCD with a string tension which is finite in the continuum limit.The key to the solvability of the Kazakov-Migdal model is the fact that the single–linkItzykson–Zuber integral can be done analytically [25,26]IIZ =Z[dU]eN Pφiχj|Uij|2 = det(ij) eφiχj∆[φ]∆[χ](12)where∆[φ] = detij (φi)j−1 =Yidetij eNφi(x)φj(y)∆[φ(x)]∆[φ(y)](14)Here, the integral over U matrices can be obtained explicitly only because the Wilsonkinetic term is absent from the action.

If the action has a Wilson term with coefficient λwhich is of order one in the infinite N limit, the Wilson term is of order N whereas all otherterms in the action are of order N2. Then, the Wilson term can be ignored and the effectiveaction for the scalar field, to leading order in N is still given by (14).The eigenvalues φi behave like a master field since the large N limit in (14) is theclassical limit and the integral can be performed by saddle point approximation.

Migdal [4]has derived integral equations which are obeyed by the eigenvalue density and has given anexpression for the asymptotics of the solution. Corrections to the classical behavior and thespectrum of elementary excitations can also be computed [5].

(This is of course neglectingthe corrections which would arise from the presence of the Wilson term, which should beginto contribute at this order.) This model has been considered further in [6] - [22].If the model (1) has a second order phase transition and if the fluctuations in the vicinityof the critical point are non-Gaussian, one might expect that the critical behavior shouldbe represented by QCD, the only known nontrivial four dimensional field theory with non-Abelian gauge symmetry.Since the rough argument leading to (10) indicates that thescalar mass should scale to infinity slower than the lattice scale in the continuum limit, itis necessary that the scalar field exhibits critical behavior, i.e.

that the effective scalar fieldtheory in (14) has a second order phase transition itself.In order to familiarize the reader with the Kazakov-Migdal model, we shall begin byreviewing how it can be solved in the simple case of a lattice with a single site and subse-quently the case where the lattice is one-dimensional and periodic. These simple models areinteresting in that one can obtain critical behavior for the scalar fields when the scalar fieldaction is quadratic.4

II. KAZAKOV-MIGDAL MODEL ON A SINGLE SITEIt is instructive to consider the Kazakov–Migdal model on a single site.

To this endconsider the following integralZ =ZDφDU e−m2Tr (φ2)+Tr (φUφU−1)(15)The evaluation of this integral follows very closely the method of D’Adda et. al.

[10]. Theintegral over U can be done using the Itzykson–Zuber formula [26], [25] and the resultdepends only on the eigenvalues of φ.

We thus have thatZ ∝Z Yidφi ∆2(φ) e−m2 Pi φ2i deteφiφj∆2(φ)(16)The determinant can be written explicitly as a sum over permutationsZ ∝Xσ∈SNǫ(σ)Z Yidφi e−m2 Pi φ2i ePi φiφσ(i)(17)where ǫ(σ) is the sign of the permutation σ. This integral is a Gaussian integral which canbe done explicitly.

The integrand can be simplified by introducing two real quantities a andb such thata2 + b2 = m2andab = 1(18)so thatZ ∝Xσ∈SNǫ(σ)Z Yidφi e−12(aφi+bφσ(i))2(19)anda, b = m2 ±√m4 −1(20)Note that Eq. (19) is symmetric under the interchange of a and b.The calculation proceeds by a change of variables from the eigenvalues φi toξi = aφi + bφσ(i)(21)The transformation is linear and the Jacobian in independent of φ.

To evaluate the Jacobianfirst note that any permutation σ can be written as a product of r-cycles of the formc1, c2...cr →c2, c3...cr, c1(22)Consider a fixed permutation σ and write σ as a product of n1 cycles of size 1, n2 cycles ofsize 2, ... nk cycles of size k, ... . ClearlyNXj=1jnj = N(23)5

It is also straightforward to prove by recursion that the Jacobian for an r-cycles isdeta−b0...0a−b0......0a−b−b0...0a= ar −br(24)where the above is an r × r matrix.Using the change of variables (21) in Eq. (19) we find thatZ ∝Xσ∈SNǫ(σ) Zσ(25)withZσ =Z Yidφi e−ξ2i /2 =ZYcyclesYi∈cyclesdξi1ar −br e−ξ2i /2(26)Performing the Gaussian integral we see thatZσ = πN/2NYi=11ai −bini∝∞Yi=11ai −bini(27)since nk=0 for k>N.The next step is to do the sum over permutations (25).

To this end note that there areN!NYj=11nj!1jnj(28)distinct permutations which can be described the a given set (n1, ...nN, ...) (with nk=0 fork>N). ThusZ ∝∞Xn1=0· · ·∞Xnk=0· · ·δ(N −∞Xk=1knk)∞Yj=11aj −bjnj(−1)(j+1)nj1knk1nk!

(29)where we have used the fact thatǫ(σ) =∞Yk=1(−1)(k+1)nk(30)Using the integral representation of the δ function δ(x) =R 2π0dθ exp(−ixθ) we find thatZ ∝Zdθ e−iNθ∞Yk=1∞Xnk=0eiθknk (−1)(k+1)nk[k(ak −bk)]nk nk!(31)The sum in Eq. (31) can be summed to an exponential.The remaining product thenbecomes a sum over the exponents so that6

Z ∝Zdθ e−iNθ exp" ∞Xk=1(−aeiθ)kk11 −a2k#(32)where we use have used the fact that b=1/a. Without loss of generality we assume that a<1(i.e.

a=m2 −√m4 −1. The term 1/(1 −a2k) can now be expanded in a geometric seriesand the resulting sum over k sums to a logarithm.

ThusZ ∝Zdθ e−iNθ∞Yn=0exphlog1 + eiθ a2n+1i=Zdθ e−iNθ∞Yn=01 + eiθ a2n+1(33)The final step is to perform the integration over θ. To do this note that only terms inthe product which are proportional to exp(iNθ) will give a nonzero integral.

ThusZ ∝aN∞Xn1=0∞Xn2=n1+1· · ·∞XnN=nN+1a2n1+...nN(34)The sums can be done one at a time. They are all geometric series.

The result isZ ∝1aNNYk=1(a2)k1 −(a2)k =a2N2/2NYk=111 −(a2)k(35)In fact if we define q=a2= (m2 −√m4 −1)2 thenZ ∝qN2/2 NYk=111 −qk! (36)Notice that apart from the factor qN2/2 this is just the expression for a q-factorial.

It isinteresting to note that a similar expression is obtained for the partition function of a systembosons on a circle at nonzero temperature. In fact the partition function for a bosonic stringon a circle (suitably restricted to the singlet sector) is precisely given by the above formula.III.

KAZAKOV-MIGDAL MODEL ON A CIRCLEIt is straightforward to generalize the above calculation to the evaluation of the partitionfunction of the Kazakov–Migdal Model on a circle. This calculation is discussed in Ref.

[10]and [7]. We consider the partition functionZ =ZLYx=1DφxLYx=1DUx,x+1 e−m2 Pi Tr (φ2x)+Px Tr (φxUx,x+1φx+1U−1x,x+1)(37)where the fields φx live on the sites x of a circle with periodic boundary conditions φL+1 = φ1and the Ux,x+1 live on the links of the circle.

There are two distinct ways of calculating thispartition function.The first method is to eliminate almost all the Ux,x+1’s by a gaugetransformation leaving only one U which cannot be eliminated.This can be chosen tobe the U on the first link i.e. U=U1,2.

The result is a partition function which involvesan integral over only a single link variable, and a Gaussian integral over all N Hermetian7

matrices. The second method is to integrate over all the link variables Ux,x+1 explicitlyusing the same formula as for the single link case above.

This second method generalizesthe method presented in the previous section for the single link integral. The details of thiscalculation are discussed by D’Adda and Panzeri [10].

The basic idea is to perform all theU integrations in Eq. (37).

The result isZ ∝ZLYx=1 NYi=1dφix!∆2(φx)LYx=1det eφixφjx+1∆(φx)∆(φx+1)(38)where φix is the i’th eigenvalue of the matrix φx. Note that the Vandermonde determinants∆(φ) precisely cancel leaving us withZ ∝ZLYx=1det eφixφjx+1 e−m2 PNi=1(φix)2(39)This is now a Gaussian integral of precisely the same form as the integral for the K-M modelon a point.

It is evaluated in Ref. [10].

The result isZ ∝qLN2/2 NYk=111 −qLk! (40)withq = m2 −√m4 −1(41)Note that this is obtained from the single link integral by a simple replacement of q by qL.IV.

SYMMETRY AND OBSERVABLESIt was pointed out in [6] that, like all adjoint lattice models, the Kazakov-Migdal model(1) has an extra gauge symmetry which is not a symmetry of continuum QCD. The actionin (1) is invariant under redefining any of the gauge matrices by an element of the center ofthe gauge group,U(xy) →z(xy)U(xy)(42)φ(x) →φ(x)(43)where z(xy) ∈U(1) if the gauge group is U(N) and z(xy) ∈ZN if the gauge group is SU(N).

(We shall call the symmetry a ZN gauge symmetry in either case.) It was subsequentlypointed out by Gross [11] and by Boulatov [19] that there is a larger symmetry of this kind:one could redefine U and U† by any element which commutes with the matrix φ.

The ZNsymmetry in (43) is the maximal subgroup of the transformations discussed by Gross andBoulatov which can be implemented with field independent elements z(xy). Because of thissymmetry the conventional Wilson loop observables of lattice gauge theory have vanishingaverage unless they have either equal numbers of U and U† operators on each link or else,in the case of SU(N), unless they have an integer multiple of N U’s or N U†’s.8

In conventional QCD, the expectation value of the Wilson loop operator gives the freeenergy for a process which creates a heavy quark-antiquark pair, separates them for sometime and lets them annihilate.From the asymptotics for large loops, one extracts theinteraction potential for the quarks. If the expectation value of the Wilson loop behavesasymptotically like e−αA where A is the area of a minimal surface whose boundary is the loop,the quark-antiquark potential grows linearly with separation at large distances and quarksare confined.

The parameter α is the string tension. On the other hand if the expectationvalue of the Wilson loop goes like the exponential its perimeter then the potential is notconfining.In the Kazakov-Migdal model (1), due to the ZN symmetry, the expectation value of theWilson Loop is identically zero for all loops with non-zero area.

We can interpret this asgiving an area law with infinite string tension, α = ∞, and no propagation of colored objectsis allowed at all. (An exception is the baryon (UN) loops in the case of SU(N) where thecorrect statement is that N-ality cannot propagate.) It is for this reason that the originalKazakov-Migdal model has difficulty describing pure gluo-dynamics.The ZN symmetry of the pure Kazakov-Migdal model is broken explicitly by the intro-duction of a Wilson term in (4).

However, if the Wilson term has vanishingly small coefficientand is negligible in the large N limit, one might expect the problem of ZN symmetry toremain - the string tension would still be infinite. It has been argued in [1] that this neednot be so.

It was shown that, if the scalar fields exhibit a particular kind of critical behavior,it is possible that the presence of an infinitesimally small Wilson term is sufficient to give afinite string tension.There are currently several other points of view on how to avoid the constraints ofZN symmetry. In [6] it was suggested that if there is a phase transition so that the ZNsymmetry is represented in a Higgs phase, the resulting large distance theory would resembleconventional QCD.

This approach has been pursued in [9], [12], [13]. An alternative, whichwas advocated in [6,7], is to use unconventional observables such as filled Wilson loops whichreduce to the usual Wilson loop in the naive continuum limit but which are invariant underZN.

The third possibility is to break the ZN symmetry explicitly. This was suggested byMigdal [15] in his mixed model in which he breaks the ZN symmetry by introducing intothe model heavy quarks in the fundamental representation of the gauge group.In [1] we considered explicit ZN symmetry breaking using a Wilson term.

We showedthat it is equivalent to using the filled Wilson loop observables. We used that fact that thefilled Wilson loops arise naturally from ordinary Wilson loops in a modified version of theKazakov-Migdal model which has additional explicit symmetry breaking terms.

We arguedthat one version of this modified model should be solvable in the large N limit.V. FILLED WILSON LOOPSWe begin with a brief review of the properties of the filled Wilson loop operators whichwere introduced in [6] and discussed in detail in [7].

These are a special class of correlationfunctions which survive the ZN symmetry of the original Kazakov-Migdal model (1). Theyare defined by considering an oriented closed curve Γ made of links of the lattice.

Theordinary Wilson loop operator on Γ is given by9

W[Γ] = TrY∈ΓU(xy). (44)For any surface S which is made of plaquettes such that the boundary of S is the curve Γwe defineWF[Γ, S] = W[Γ]Y✷∈SW †[✷](45)where ✷denotes an elementary plaquette in the surface S. The filled Wilson loop for Γ isnow defined asWF[Γ] =XSµ(S)WF[Γ, S](46)where the sum is over all surfaces S whose boundary is the loop Γ with some (yet to bespecified) weight function µ(S).

Notice that for each plaquette ✷∈S we have inserted thenegatively oriented Wilson loop W †[✷]. Thus for arbitrary weight functional µ(S) the filledWilson loop operator is invariant under the local ZN gauge symmetry since it has equalnumbers of U and U† operators on each link.

Although we have assumed that the loop isfilled with elementary plaquettes this can be easily generalized to other fillings (the otherextreme case being the adjoint loop W[Γ]W †[Γ]). We can also define the ‘filled correlator”of more than one loop by summing over all surfaces whose boundary is given by those loops.In ref.

[7] it was shown that computing the expectation value of WF[Γ] is equivalent tocomputing the partition function of a certain statistical model on a random two–dimensionallattice. When computing ZN gauge invariant correlation functions of U–matrices in themaster field approximation the φ–integral is evaluated by substituting the master field ¯φ =diag(¯φ1, .

. .

, ¯φN) for the eigenvalues of φ.< Ui1j1 . .

. U†k1l1 .

. .

>=R dφ[dU]e−Tr(PV [φ]−PφUφU†)Ui1j1 . .

. U†k1l1 .

. .R dφ[dU]e−Tr(PV [φ]−PφUφU†)(47)≈R d¯φ[dU]eTr(P ¯φU ¯φU†)Ui1j1 .

. .

U†k1l1 . .

.R d¯φ[dU]eTr(P ¯φU ¯φU†)In this integral, the scalar field is written at φ = V φDV † with φD a diagonal matrix. Theeigenvalues of φ are fixed at the value of the master field ¯φ and, in order to obtain gaugeinvariance of the correlator, the angular matrices V are still integrated,d¯φ ≡∆(¯φ)DV(48)In any gauge invariant correlator, the matrices V can be absorbed by redefining U.If we consider for the moment surfaces which are not self-intersecting so that the filledWilson loop correlator has at most one UU† pair on any link we need to consider only thetwo field correlator < UijU†kl >.

Gauge invariance implies that [7]< UijU†kl >= CijδilδjkwithCij =R d¯φ[dU]eN PTr(¯φU ¯φU†)|Uij|2R d¯φ[dU]eN PTr(¯φU ¯φU†)(49)10

Thus, in the master field approximation, the expectation value of the filled Wilson loop isgiven by< WF[Γ] >=XSµ(S)Ysitesx∈SNXi(x)=1Ylinks∈SCi(x) j(x)(50)This is a generalized Potts model on a random surface in which N–component spins resideat each site and the Boltzmann weights Cij for the bonds are correctly normalized to beconditional probabilities; Pi Cij=1, Pj Cij=1.VI. COMPUTING CORRELATION FUNCTIONSTechniques for evaluating Cij for general N and for arbitrary ¯φ are presented in [17] and[18].

An explicit formula for SU(2) is given in [7]. Although the general formula for Cij inSU(N) is quite difficult to deal with, it is still possible to estimate the surface dependenceof the statistical model partition function in (50) when the master field ¯φ is homogeneousby considering two different limits.A.

Low TemperatureFirst, consider the case where ¯φi are large. We also assume that the eigenvalues ¯φi arenot too close to each other in the sense thatXi̸=j1¯φi −¯φj2 << N(51)(Note that this is a single sum over j for fixed i.).

The integral in (12) is known to beexact in the semi-classical approximation (see [7] for a discussion). The classical equationof motion is[U ¯φU†, ¯φ] = 0(52)which, since ¯φ is diagonal, is solved by any U of the form U0 = DP where D is a diagonalunitary matrix and P is a matrix which permutes the eigenvalues,(P ¯φP)ij = δij ¯φP (i)(53)Also, when N is large and ¯φ is not too small the identity permutation gives the smallestcontribution to the action in (12) and therefore is the dominant classical solution.

In thiscase we use this minimum to evaluate the correlators,I−1IZZ[dU]eN P ¯φi ¯φj|Uij|2Ui1j1 . .

. UinjnU†k1l1 .

. .

U†knln = δi1j1 . .

. δknlnSi1···ink1...kn(54)where we have written the normalized integral over diagonal matrices11

Si1···ink1...kn =R QℓdθℓQp

. in is a permutation of k1 .

. .

kn0otherwise(55)We have decomposed the integration over unitary matrices into an integration over thediagonals and an integration over the unitary group modulo diagonals [27]. The diagonals arethe ‘zero modes’ for the semiclassical integral and must be integrated exactly.

The unitarymodulo diagonal integral is damped by the integrand and is performed by substituting theclassical configuration. Of course, to get the next to leading order the latter integrationmust be done in a Gaussian approximation.

It can be done for the first few correlators, Theresult isCii = 1 −1NXk̸=i1¯φi −¯φk2when i ̸= jCij = 1N1¯φi −¯φj2We remark that similar calculations can be easily done for correlators of more than twoU’sCij,kl = I−1IZZ[dU]eN P ¯φi ¯φj|Uij|2|Uij|2|Ukℓ|2(56)i = j ,k = ℓCii,kk = 1 −1NXn̸=i"1(¯φi −¯φn)2 +1(¯φk −¯φn)2#i ̸= j ,k = ℓCij,kk = 1N1(¯φi −¯φj)2(57)i ̸= j ,k ̸= ℓCij,kl = 1N21(¯φi −¯φj)21(¯φk −¯φℓ)2 (1 + δikδjℓ) ,where the next corrections will be of the form1N21¯φi −¯φj2Xm̸=k1¯φk −¯φm2 . (58)We call this limit of large ¯φ the “Low Temperature” limit since in this limitCLTij = δij + .

. .

(59)and the value of the spin at each site is equal. In this case the Cij represent the Bolzmanweight for a perfectly ordered system.

These two cases lead to profoundly different behaviorfor the filled Wilson loop. We shall assume that, by choosing the potential for the scalarfield in (14) appropriately, either of these limits could be obtained (the eigenvalue repulsiondue to the Vandermonde determinants in (14) and the possibility of adding repulsive centralpotentials makes the low temperature limit more natural).12

B. High TemperatureThe other limit is where ¯φ is small.

There, we can obtain the correlators by Taylorexpansion,Cij =Z[dU]1 + NTr¯φU ¯φU† + . .

.|Uij|2 = 1N +¯φi ¯φjN+ . .

. (60)We call this the “High Temperature” limit since in this limit is independent of i and j.It thus represents the Bolzman weights for a highly disordered system.C.

Renormalization of the String TensionWe begin by estimating the value of the filled Wilson loop for a fixed surface S. In the“high temperature” case the statistical model is disordered. The sums over configurationsat the various sites are independent and they contribute an overall factor NV (where V isthe number of vertices on the surface) to the expectation value of the filled Wilson loop.Furthermore each link contributes a factor Cij=1/N so that the links contribute a totalfactor of N−L where L is the total number of links.

It follows that the expectation value ofthe filled Wilson loop goes like< WF[Γ, S] >HT∼NV −L = N2−2g(S)N−A(S)(61)where A(S) is the area and g(S) is the genus of the surface S (i.e. the number of plaquettescomprising S) and we have used Euler’s theorem, χ ≡2 −2g = V −L + A.

We thus get therenormalization of the string tension δαHT = log N. Notice also that higher genus surfacesare suppressed and that the loop (genus) expansion parameter is 1/N2. This is preciselywhat is obtained in the conventional strong coupling expansion of Wilson’s lattice gaugetheory which is known to describe a string theory with extra degrees of freedom associatedwith self-intersections of the string [23].In the “low temperature” case, the statistical system is ordered.

The spins on all thesites are frozen at a uniform value. In this case the partition function is proportional to thedegeneracy of the ground state,< WF[Γ, S] >LT= N(62)Note that in this case the statistical model gives no contribution to the string tension(δαLT ≈0) and there is no suppression of higher genus surfaces.In order to proceed to the evaluation of the filled Wilson loop we need to choose a weightfunction µ(S) in order to perform the sum over surfaces.

The most reasonable criterion forchoosing such a weight function is our desire to get a finite physical string tension in thecontinuum limit. In order to accomplish this goal we must choose a weight function µ(S)which depend on the area of the surface differently in the low and in the high temperaturecases.

It is known that the number of closed surfaces with a given area grows exponentiallyasn(A) ∼Aκ(g)eµ0A(63)13

where κ(g) is a universal constant which depends only on the genus of the surface and µ0 is anon-universal, regulator dependent constant [24] which will lead to a renormalization of thestring tension. In our case, although the surfaces are open, the above formula should still bevalid for surfaces whose area is much larger than the area of the minimal surface boundedby Γ.

If the continuum limit of our theory is realized in the “high temperature” phase weshould use the weight function µHT(S) ∼NA(S)e−µ0A(S). This leads to a vanishing stringtension in the lattice theory which is a necessary condition for having a finite string tensionin the continuum limit.

To accomplish the same goal in the “low temperature” phase weshould use µLT(S) ∼e−µ0A(S). Although these choices of µ(S) give the desired result, it israther unnatural to have to choose µ(S) in such an ad hoc fashion.D.

Filled Wilson Loops from the Wilson ActionFortunately there is a very natural way to obtain the sum over surfaces in (50). Considerthe following expectation value< WF[Γ] >= < W[Γ]eλP✷(W [✷]+W †[✷]) >< eλ P✷(W (✷)+W †[✷]) >(64)where W[Γ] is the conventional Wilson loop.

Remember that the average is weighted by theKazakov–Migdal action as in (47): In the master field limit it is computed by integrating onlyover U–matrices with φ = ¯φ and with the Kazakov–Migdal action. Note that the exponentin (64) is simply the conventional Wilson kinetic term for the gauge fields in lattice gaugetheory.

If we expand the right hand side of (64) in λ the non–vanishing terms are all of thosesurfaces which fill the Wilson loop. The result is thus a filled Wilson loop with a surfaceweight µ(S) = λA(S).It is clear that we would obtain exactly the same expression (in the master field ap-proximation) by evaluating the expectation value of the ordinary Wilson loop operator inthe modified version of the Kazakov–Migdal model (4) in which a conventional Wilson term(λ P✷W(✷) + W †[✷]) is added to the action.

This term breaks the ZN gauge symmetryexplicitly and allows Wilson loop operators with non–zero area to have non–zero expecta-tion values. We would expect that it is necessary to keep λ small if one is to maintain thesuccesses of the Kazakov–Migdal model.

We shall now argue that in the “low temperature”limit this picture is self–consistent in the sense that the physical string tension is finite whenλ is small and consequently the saddle point solution of the original model is unchanged.We shall also see that this is not the case in the “high temperature” phase.Let us begin by determining how λ should behave in the continuum limit if we are tohave a finite physical string tension. As discussed above a necessary condition for havinga finite physical string tension is that the string tension in lattice units should vanish.

Itis thus necessary for the bare string tension −ln λ to be chosen so as to precisely cancelthe renormalization of the string tension due to both the statistical model to the sum oversurfaces. It is straightforward to check that in the “high temperature” phase we must chooseλHT = Ne−µ0, whereas in the “low temperature” phase we must chooseλLT = e−µ0.

Noticethat this λ is proportional to N in the “high temperature” phase and thus cannot be assumedsmall in large N.14

In the large N limit of conventional lattice gauge theory the coefficient of the Wilsonterm must be proportional to N if one is to obtain a consistent large N expansion. In ourcase we see that this is true for the “high temperature” phase in which case the Wilson termis of the same order as the Kazakov–Migdal term and it thus plays an important role in theinfinite N limit.

One can say that in this phase we have ordinary QCD. Unfortunately it isimpossible to preserve the master field solution of the Kazakov-Migdal model in this limitsince the Wilson term, being of order N, would modify the large N solution, ruining theself-consistency of the mean-field approximation as described here.The situation is much more appealing in the “low temperature” phase.

In this case therequired coefficient of the Wilson term is of order one. It is subdominant and thereforenegligible in the large N limit.

Thus, Migdal’s solution [4] of the Kazakov-Migdal modelin the large N limit should still apply to our proposed modification of the action. In factthe only reason that the Wilson term is important at all in the large N limit of the “lowtemperature” phase is related to the collective phenomenon which orders the statisticalsystem on the surfaces.

It effectively makes the statistical model’s contribution to the stringtension much smaller than would be expected from naive counting of powers of 1/N and atruly infinitesimal breaking of the ZN gauge symmetry (λLT/N →0 as N →∞) is sufficientto make the averages of Wilson loop operators non-vanishing. The self-consistency of thispicture can also be demonstrated by computing the contribution of the Wilson term to thefree energy.

This can be computed in a small λ expansion. For a cubic lattice the result is:Z =< eλLT(P✷Tr(W [✷]+W †[✷])) >= ZKM exp NV D(D −1)2λ2LT + 2λ6LT + .

. .

! (65)is of order N (where V is the volume, D is the dimension).

This should be compared withthe free energy in the pure Kazakov-Migdal model which is proportional to N2. Here, thefirst term in the free energy is the contribution of the doubled elementary plaquette andthe second term is due to the two orientations of the elementary cube.

It is interestingthat, to order 6, there is no energy of interaction of doubled elementary plaquettes witheach other. We conjecture that the interaction energy of surfaces is absent to all orders andthe free energy obtains contributions from all possible topologically distinct surfaces whichcan be built from elementary plaquettes.

This suggests a free string picture of the “lowtemperature” limit of the Kazakov-Migdal model at lattice scales.E. Self-Intersecting SurfacesWe have thus far neglected the self-intersecting surfaces in the sum (50) which are gen-erated by the expansion of (64) in λ.

In order to evaluate the contribution of these surfaceswe need to compute the correlator of n UU† pairs on the same link. The computation ofthese correlators in full generality is quite complicated.

In the Appendix we compute themin the “low temperature” (ordered) phase. We find that< Ui1j1 .

. .

UinjnU†k1l1 . .

. U†knln >= δi1j1 .

. .

δknlnSi1...ink1...kn(66)where Si1...ink1...kn is the tensor which is one if i1 . .

. in is a permutation of k1 .

. .

kn and is zerootherwise. It is now evident that in this limit the U–matrices are replaced by unit matrices15

which freeze together the spin degrees of freedom on the various intersecting surfaces. As aspecial case we can consider a single, connected, self–intersecting surface.

In this case all thespin indices on the surface are equal and since Si1...k1... = 1 when all arguments are equal thepartition function of the statistical model corresponding to that surface is simply N just asit was for a non-intersecting surface. Thus just as the statistical model does not contributeto the string tension it also does not contribute to the interaction energy of self–intersectingsurfaces.

This implies that in the “low temperature” limit, the sum over connected surfaceswhich have a common boundary behaves like a Nambu-Goto string theory with no internaldegrees of freedom.VII. DISCUSSIONIn summary, the self-consistency of the “low temperature” limit leads us to a new largeN limit of the conventional lattice gauge theory coupled to scalars:Z =Zdφ[dU] exp−NXxTrV [φ(x)] + NXTrφ(x)U(xy)φ(y)U†(xy)++λX✷(W(✷) + W †(✷))!

(67)The conventional large N limit occurs when λ is of order N and describes scalar QCD.The other limit occurs when N →∞with λ of order one. This model is soluble using theKazakov–Migdal approach.It is the latter case in which λ remains constant that is of special interest to us.

In thiscase we saw that the large N expansion corresponds to a string theory with some unusualfeatures. The partition function and the Wilson loop expectation value can be described asa sum over surfaces.

What is unusual is that the genus of the surfaces is not suppressed inthe large N limit, as it is in continuum QCD. (We do of course expect the higher correctionsin 1/N to suppress higher genus terms.) For a continuum string theory this sum over thegenus is badly divergent.

This, together with the presence of tachyons, suggests that the trueground state of the string theory is some sort of condensate. This could pose a complicationfor the present version of the Kazakov-Migdal model in the continuum limit and deservesfurther attention.

It is still a mystery to us how the sum over all surfaces at the lattice scaleshould turn into the sum over planar diagrams in the continuum theory of QCD.An alternative to the model presented here is the mixed model which was inventedby Migdal [15] to solve the problem of ZN symmetry.It contains heavy quarks in thefundamental representation of the gauge group. Despite the obvious differences between ourmodel and Migdal’s mixed model they have many features in common.

As in all cases whenthere are fields in the fundamental representation, the asymptotics of the Wilson loops inthe mixed model exhibit a perimeter law. In conventional QCD one would expect that ifthe quarks are heavy enough, there is an area law for small enough loops, i.e.

there wouldexist a size scale which is far enough into the infrared region that the quark potential islinear but the interaction energy is not yet large enough that it is screened by producingquark-antiquark pairs. Thus, in QCD we expect that adding heavy quarks would not ruinthe area law for Wilson loops smaller then some scale.16

The mixed model has just the opposite scenario, it is possible to get an area law only whenthe heavy quarks are light enough. This is a result of the fact that, in the Kazakov-Migdalmodel, no Wilson loops are allowed at all unless the ZN symmetry is explicitly broken.

Inthe mixed model, the ZN charge of links in a Wilson loop must be screened by the heavyquarks. This can happen in two ways.

First, the Wilson loop can just bind a heavy quarkto form an adjoint loop - giving a perimeter law for the free energy of the loop. This isthe leading behavior if the fermion mass, M, is large.

The free energy would go like 1/MPwhere P is the perimeter. The only way an area law might arise is when the fermions arelight enough that their propagators could from a filled Wilson loop with free energy 1/M2Lwhere L ≈2A is the number of links.

Then, since the entropy for filled loops is much largerthan that for adjoint loops, these configurations would be important if M4 < eµ0. Then, theasymptotics behavior of the Wilson loop would still have a perimeter law but there wouldbe loops with 4A −P < µ0/ ln M where there would be an approximate area law.Acknowledgement This work is supported in part by the Natural Sciences and Engi-neering Research Council of Canada and by the National Science Foundation grant # NSFPHY90-21984.17

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[27] Despite the simple form of S its tensor structure is quite complicated. For exampleSi1i2k1k2 = δi1k1δi2k2 + δi1k2δi2k1 −δi1i2δk1k219

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