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SOME NEW/OLD APPROACHES TO QCD

arXiv:hep-th/9212148v1 24 Dec 1992LBL 33232PUPT 1355November, 1992SOME NEW/OLD APPROACHES TO QCDDAVID J. GROSS†LBLABSTRACTI discuss some recent attempts to revive two old ideas regarding ananalytic approach to QCD–the development of a string representationof the theory and the large N limit of QCD.⋆Talk delivered at the Meeting on Integrable Quantum Field Theories, VillaOlmo and at STRINGS 1992, Rome, September 1992. This work was supportedin part by the Director, Office of Energy Research, Office of High Energyand Nuclear Physics, Division of High Energy Physics of the U.S. Departmentof Energy under Contract DE-AC03-76SF00098 and in part by the NationalScience Foundation under grant PHY90-21984.†On leave from Princeton University, Princeton, New Jersey.

DisclaimerThis document was prepared as an account of work sponsored bythe United States Government. Neither the United States Governmentnor any agency thereof, nor The Regents of the University of California,nor any of their employees, makes any warranty, express or implied, orassumes any legal liability or responsibility for the accuracy, complete-ness, or usefulness of any information, apparatus, product, or processdisclosed, or represents that its use would not infringe privately ownedrights.

Reference herein to any specific commercial products process, orservice by its trade name, trademark, manufacturer, or otherwise, doesnot necessarily constitute or imply its endorsement, recommendation,or favoring by the United States Government or any agency thereof, orThe Regents of the University of California. The views and opinions ofauthors expressed herein do not necessarily state or reflect those of theUnited States Government or any agency thereof of The Regents of theUniversity of California and shall not be used for advertising or productendorsement purposes.Lawrence Berkeley Laboratory is an equal opportunity employer.2

1. IntroductionIn this lecture I shall discuss some recent attempts to revive someold ideas to address the problem of solving QCD.

I believe that it istimely to return to this problem which has been woefully neglected forthe last decade. QCD is a permanent part of the theoretical landscapeand eventually we will have to develop analytic tools for dealing withthe theory in the infra-red.

Lattice techniques are useful but they havenot yet lived up to their promise. Even if one manages to derive thehadronic spectrum numerically, to an accuracy of 10% or even 1%, wewill not be truly satisfied unless we have some analytic understandingof the results.

Also, lattice Monte-Carlo methods can only be used toanswer a small set of questions. Many issues of great conceptual andpractical interest–in particular the calculation of scattering amplitudes,are thus far beyond lattice control.

Any progress in controlling QCD inan explicit analytic fashion would be of great conceptual value. It wouldalso be of great practical aid to experimentalists, who must use ratherad-hoc and primitive models of QCD scattering amplitudes to estimatethe backgrounds to interesting new physics.I will discuss an attempt to derive a string representation of QCDand a revival of the large N approach to QCD.

Both of these ideas havea long history, many theorist-years have been devoted to their pursuit–so far with little success. I believe that it is time to try again.

In partthis is because of the progress in the last few years in string theory.Our increased understanding of string theory should make the attemptto discover a stringy representation of QCD easier, and the methodsexplored in matrix models might be employed to study the large N limitof QCD. For both political and intellectual reasons I fervently urge stringtheorists to try their hand at these tasks.3

2. QCD as a String TheoryIt is an old idea that QCD might be represented as a string the-ory.

This notion dates back even before the development of QCD. In-deed, string theory itself was stumbled on in an attempt to guess sim-ple mathematical representations of strong interaction scattering ampli-tudes which embodied some of the features gleamed from the experi-ments of the 1960’s.

Many of the properties of hadrons are understand-able if we picture the hadrons as string-like flux tubes. This pictureis consistent with linear confinement, with the remarkably linear Reggetrajectories and with the approximate duality of hadronic scatteringamplitudes.Within QCD itself there is internal, theoretical support for this idea.First, the1N expansion of weak coupling perturbation theory can beinterpreted as corresponding to an expansion of an equivalent stringtheory in which the string coupling is given by 1N .

This is the famousresult of ’t-Hooft’s analysis of the1N expansion of perturbative QCD[1]. The same is true for any matrix model–i.e.

a model invariant underSU(N) or U(N), in which the basic dynamical variable is a matrixin the adjoint representation of the group.The Feynman graphs insuch a theory can be represented as triangulations of a two dimensionalsurface. This is achieved by writing the gluon propagator as a doubleindex line and tiling the graph with plaquettes that cover the closedindex loops.

’t-Hooft’s principal result was that one can use1N to pickout the topology, i.e. the genus=number of handles, of the surface, sincea diagram which corresponds to a genus G Riemann surface is weightedby ( 1N )2G−2.

The leading order in the expansion of the free energy inpowers of 1N is proportional to N2 (reasonable since there are N2 gluons,and is given by the planar graphs of the theory.Another bit of evidence comes from the strong coupling lattice for-mulation of the theory. The strong coupling expansion of the free energycan indeed be represented as a sum over surfaces [5].

Again there is anatural large N expansion which picks out definite topologies for thesesurfaces. This result is an existence proof for a string formulation ofQCD.

However, the weights of the surfaces are extremely complicated4

and it is not at all clear how to take the continuum limit.⋆From quite general considerations we expect that the large N limitof QCD is quite smooth, and should exhibit almost all of the qualitativefeatures of theory. Thus an expansion in powers of 13 or (13)2 mightbe quite good.

The longstanding hope has been to find an equivalent(dual) description of QCD as some kind of string theory, which wouldbe useful in to calculate properties of the theory in the infrared.The problems with this approach are many. First, if QCD is de-scribable as a string theory it is not as simple a theory as that employedfor critical strings.

It appears to be easier to guess the string theoryof everything than to guess the string theory of QCD. Most likely theweights of the surfaces that one would have to sum over will dependon the extrinsic geometry of the surface and not only its intrinsic ge-ometry.

We know very little about such string theories. Also there arereasons to believe that a string formulation would require many (per-haps an infinite) new degrees of freedom in addition to the coordinatesof the string.

Finally, there is the important conceptual problem–howdo strings manage to look like particles at short distances. The onething we know for sure about QCD is that at large momentum transferhadronic scattering amplitudes have canonical powerlike behavior in themomenta, up to calculable logarithmic corrections.

String scattering, onthe other hand, is remarkably soft. Critical string scattering amplitudeshave, for large momentum transfer, Gaussian fall-off[3].

How do QCDstrings avoid this?†⋆There is also the problem that for large N there is typically a phase transitionbetween the strong and weak coupling regimes [2].† Recently there have been some interesting speculations regarding this problem[4].5

3. Two Dimensional QCDTwo dimensional QCD (QCD2) is the perfect testing ground for theidea that gauge theories can be equivalent to string theory.

First, manyfeatures of the theory are stringier in two than in four dimensions. Forexample, linear confinement is a perturbative feature which is exact atall distances.Most important is that the theory is exactly solvable.This is essentially because in two dimensions gluons have no physical,propagating degrees of freedom, there being no transverse dimensions.In fact QCD2 is the next best thing to a topological field theory.

Thecorrelation functions in this theory will depend, as we shall see, only onthe topology of the manifold on which formulate the theory and on itsarea. For this reason we will be able to solve the theory very easily andexplicitly.Consider for example the expectation value of the Wilson loop forpure QCD2 , TrR PeHC Aµdxµ, for any contour, C, which does not inter-sect itself.

Choose an axial gauge, say A1 = 0, then the Lagrangian isquadratic, given by 12 Tr E2, where E = ∂1A0 is the electric field. TheWilson loop describes a pair of charged particles propagating in time.This source produces, in two dimensions, a constant electric field.

TheWilson loop is then given by the exponential of the constant energy ofthe pair integrated over space and time. This yields,TrR PeHC Aµdxµ = e−g2C2(R)A ,(3.1)where g is the gauge coupling, C2(R) the quadratic Casimir operator forrepresentation R and A the area enclosed by the loop.

The expectationvalue of more complicated Wilson loops that do self intersect can also becalculated. Kazakov and Kostov worked out a set of rules for such loopsin the large N limit [6].

They are quite complicated. QCD2 with quarksis also soluble, at least in the large N limit.

The meson spectrum wassolved for N →∞by ’t Hooft. It consists of an infinite set of confinedmesons with masses mn that increase as m2n ∼n.

This provides onewith a quite realistic and very instructive model of quark confinement[1], [7].Is QCD2 describable as a string theory? The answer is not known,although there is much evidence that the answer is yes.

I shall describebelow a study that I have carried out to investigate this issue [8].6

To simplify matters I shall discard the quarks and consider the puregauge theory. This would correspond to a theory of closed strings only,quarks are attached to the ends of open strings.

We shall consider thepartition function for a U(N) or SU(N) gauge theory, on an arbitraryEuclidean manifold M,ZM =Z[DAµ]e−14g2RM d2x√g Tr F µνFµν . (3.2)One might think that in the absence of quarks the theory is totallytrivial, since in two dimensions there are no physical gluon degrees offreedom.

This is almost true, however the free energy of the gluons willdepend non-trivially on the manifold on which they live. In fact, onecannot simply gauge the gluons away.

If, for example, M contains anon-contractible loop C, then if Tr PeHC Aµdxµ ̸= 1, one can not gaugeAµ to zero along C. Thus, the partition function will be sensitive to thetopology of M.Although non-trivial the theory is extremely simple, almost as sim-ple as a topological theory. It is easy to see that the partition functionwill only depend on the topology and on the area of the manifold M.This is because the theory is invariant under all area preserving diffeo-morphisms.

To demonstrate this note that the two-dimensional fieldstrength can be written as Fµν = ǫµνf, where ǫµν is the anti-symmetrictensor and f a scalar field. Thus the action is S =RTr f2dµ, wheredµ = √gd2x is the volume form on the manifold.

This action is inde-pendent of the metric, except insofar as it appears in the volume form.Therefore the theory is invariant under area preserving diffeomorphisms(W∞). The partition function can thus only depend on the topologyand on the area of the manifold M,ZM = Z[G, g2, A, N] = Z[G, g2A, N] ,(3.3)where G is the genus of M.Now we can state the conjecture that the logarithm of this partitionfunction, the free energy, is identical to the partition function on somestring theory, with target space M, where we would identify the stringcoupling with 1N and the string tension with g2N,7

lnZ[G, g2A, N]= ZStringTargetSpace M[gst = 1N , α′ = g2N] . (3.4)As a candidate for the type of string theory I am thinking of considerthe Nambu action, whereinZStringM=Xh=genus(gst)2h−2ZDxµ(ξ)eRd2ξ√g ,(3.5)where g is the determinant of the induced metricg = det[gαβ] = det[∂xµ∂ξα∂xν∂ξβGµν(x)] ,(3.6)and Gµν(x) is the metric on the manifold M. This string theory, whenthe target space is two-dimensional, is indeed invariant under area pre-serving diffeomorphisms of the target space.To see this note that√g = |∂xµ∂ξα |√G, which is obviously unchanged by a map xµ →x′µas long as | ∂xµ∂x′ν | = 1.⋆Unfortunately the only way we know to quan-tize this theory is to transform it into the Polyakov action, which uponquantization yields the standard non- critical string [10].

This is notwhat we want to do here, since the resulting theory is not even Lorentzinvariant. Is there another quantization of the Nambu string that differsfrom the Polyakov quantization in two- dimensions?

The answer is notknown.⋆Actually the Polyakov action with a two-dimensional target space also has aW∞symmetry, although is is realized in a very nonlinear fashion. One mightspeculate that this is related to the well known W∞symmetry of the c = 1string theory [9].8

3.1. Evaluation of the Partition FunctionThe partition function for QCD2 can easily be evaluated by meansof the following idea, originally due to Migdal [11].The trick is touse a particular lattice regularization of the theory which is both exactand additive.

For the lattice we take an arbitrary triangulation of themanifold and define the partition function asZM =Z YLdULYplaq.ZP [UP] ,(3.7)where UP = QL∈plaq. UL, and ZP[UP] is some appropriate lattice ac-tion.

Any action will do as long as it reduces in the continuum limitto the usual continuum action. Instead of the Wilson action, ZP (U) =e−1g2 Tr(U+U †), we shall choose the heat kernel action,ZP =XRdR χR(UP)e−g2C2(R)AP,(3.8)where the sum runs over representations R of SU(N) (or U(N)), dR isthe dimension of R, χR(UP) the character of UP in this representation,C2(R) the quadratic Casimir operator of R and AP the area of theplaquette.It is easy to see, using the completeness of the characters to ex-pand about ZPUL→1+iAµdxµ→PR dRχR(UP) = δ(UP −1) + .

. ., that inthe continuum limit of this theory reduces to ordinary Yang-Mills the-ory.

What is special about the heat kernel action is that it is additive.Namely, we can integrate over each link on the triangulation, say U1,which appears in precisely two triangles, using the orthogonality of thecharacters,RdV χa(XV )χb(V †Y ) = δabda χa(XY ), and obtain,ZdU1ZP1(U2U3U1)ZP2(U†1U4U5) = ZP1+P2(U2U3U4U5) . (3.9)This formula expresses the unitarity of the action, since in fact ZP (U) =⟨U|e−g2A∆|1⟩, where ∆is the Laplacian on the group.9

We can use this remarkable property of the heat kernel action toargue that the lattice representation is exact and independent of thetriangulation. This is because we can use (3.9)in reverse to add asmany triangles as desired, thus going to the continuum limit.

On theother hand we can use (3.9) to reduce the number of triangles to the bareminimum necessary to capture the topology of MG. A two- dimensionalmanifold of genus G can be described by a 4G-gon with identified sides:a1b1a−11 b−11. .

. aGbGa−1G b−1G .

The partition function can be written usingthis triangulation as,ZMG =XRdRe−g2C2(R)AZ YDUiDViχR[U1V1U†1V †1 . .

. UGVGU†GV †G] .

(3.10)We can now evaluate the partition function using the orthogonality ofthe characters and the relation,RDUχa[AUBU†] =1daχa[A]χa[B], toobtain [12], [13],ZMG =XRd2−2GRe−λAN C2(R) ,(3.11)where λ ≡g2N is kept fixed. Thus we have an explicit expression forthe partition function.

It depends, as expected, only on the genus andthe area of the manifold.3.2.The Large N expansionThe formula (3.11) for the partition function is quite complicated,being written as a sum over all representations of SU(N). The repre-sentations of SU(N) or U(N) are labeled by the Young diagrams, withm boxes of length n1 ≥n2 ≥n3 ≥.

. .

nm ≥0. Such a representationhas,10

C2(R) = NmXi=1ni +mXi=1(ni + 1 −2i);dR = ∆(h)∆(h0),hi = N + ni −1,h0i = N −i∆(h) =Y1≤i

Consider theexpansion in powers of 1N of the free energy,ln[ZMG] =∞Xg=01N2g−2fGg (λA) . (3.13)If this were given by a sum over maps of a two-dimensional surface ofgenus g onto a two-dimensional surface of dimension G we would expectthat fGg (λA) ∼( 1N )2g−2e−λAn, where n is the winding number of themap, i.e.

the topological index that tells us how many times the mapx(ξ) covers M. This is the integral of the Jacobian of the map ξ →x,Rd2ξ det[∂xµ∂ξi ], which differs from the Nambu area,Rd2ξ| det[∂xµ∂ξi ]|, sincethe surface can fold over itself.Now there is a minimum value that G can take, given the genus Gof the target space and the winding number n. Thus for example thereare no smooth maps of a sphere onto a torus or a torus onto a genus twosurface. Similarly there are no smooth maps of a genus g surface ontoa genus g surface that wind around it more than once.

To get an ideaof the bound consider holomorphic maps, in which case the Riemann-Hurwitz theorem state states that 2(g −1) = 2n(G −1) + B, where Bis the total branching number. In the case of smooth maps there seemsto be the following bound [14],2(g −1) ≥2n(G −1) .

(3.14)Thus if QCD2 is described by a string theory we would expect that11

fGg (λA) =Xn 0if (g −1) < n(G −1)e−nλAωng (A)otherwise. (3.15)We can use these inequalities as tests of whether our conjecture is cor-rect.

To do this we need to expand (3.11) in powers of 1N .The hardest case is that of the sphere (G = 0), since the sum overrepresentations blow up rapidly and it is not even evident that there ex-ists a tamed large N expansion. We can break up the sum in (3.11) intoa sum over representations with n boxes in the Young tableaux since,for large N, C(Rn) N→∞→N Pi ni = Nn.

Thus,ZG=0 =XRd2Re−λAN C2(R) N→∞→XnXRnd2Rne−nλA(1 + . .

.) .

(3.16)To evaluate this we need to evaluate the following sum, PRn d2Rn. Thiscan be done using a method of discrete orthogonal polynomials [8], yield-ing,XRnd2Rn =N2 + n −1n.

(3.17)Then it follows that,ZG=0 →exp[−N2ln(1 −e−λA) + 2λAe−2λA(1 −e−λA)2 + . .

.+O(N0) + . .

.] .

(3.18)Here there are no constraints implied by the inequality (3.15) , but thestructure of the expansion is very interesting.The case of the torus, (G = 1), is some what simpler. One can easilyderive that (for SU(N)) [8],12

ZG=1 =XRe−λAN C2(R) →exp[−N0 ln η(−e−λA)+λAN2∞Xn=1e−nλA[Xab=na2b +Xab+cd=nac] + . .

.] ,(3.19)where η(x) = Q∞n=1(1−xn)−1.

This is totally consistent with the boundg ≥1.Most interesting is the case of G > 1, where the inequalities arequite stringent. In this case one can easily derive [8],ZG →Xn( 1N )2n(G−1)e−nλAXr=rep of Snn!dr2(G−1) ,(3.20)where the sum is over representations of the symmetric group Sn and dris the dimension of the rth representation of Sn.

Not only is this in totalaccord with our expectations, but one can also show that ωng (A) =, forg = 1 + n(G −1), is precisely the number of topologically inequivalentmaps on the genus g manifold onto the genus G manifold with windingnumber n[15].So the large N expansion of QCD2 looks precisely like what we wouldexpect from string considerations. What remains to be understood arethe all the rational numbers that appear as coefficients of the powersof e−λA and of1N in terms of the counting of maps of Mg onto MG.Some of these are understood, but not all.

Then it remains to constructa string action that reproduces these counting rules.13

4. Induced QCD4.1.

The Large N Limit of QCDQCD is hard to solve since it is a theory with no free, adjustableor small parameters. In pure QCD (no quarks) the only parameter wecan adjust is the number of colors, N. Luckily, in the large N limitQCD simplifies enormously, and this limit remains the best hope toyield an exact or controllable treatment of the theory.

We know thatas N = ∞only planar graphs survive. More generally we know that interms of the appropriate variables the large N limit of gauge invariantobservables is given, for N = ∞by the master field, namely a solutionof an appropriate classical equation of motion [16].

The large N limit isin the nature of a semi-classical expansion, with N playing the role ofPlanck’s constant. Unlike the running coupling N does not vary withmomentum and we expect the large N limit to be qualitatively correctfor all momenta, to correctly capture the small distance asymptoticfreedom of the theory as well as exhibit confinement at large distances.In the N = ∞we should have an infinite spectrum of stable mesonsand glueballs.

Even baryons, bound states of N quarks, are describable,in this limit, as solitons of the effective Lagrangian for the master field[17]. Thus the hope has survived that we could find an exact solution ofQCD for N = ∞, which would yield the hadronic spectrum, and wouldbe the starting point for a systematic large N expansion which couldallow us to calculate scattering amplitudes.The standard method of solving a theory in the large N limit isto find an appropriate saddlepoint for the partition function.

In thecase of QCD this is difficult. Consider the standard (Wilson) latticeformulation of the theory,ZQCD =Z YLDULe−Pplaq.Ng2(a) TrQL UL+h.c..(4.1)The integrand behaves as the exponential of an action that is of orderN2, thus one might hope to evaluate it by saddlepoint techniques.

How-ever, the measure is also of order cN 2 and therefore one must somehowget rid of N2 degrees of integration before this can be done. The reasonQCD is not yet solved in the large N limit is that no one knows how toreduce the theory to N variables per site.14

Another theory which is also insoluble in the large N limit is thenon-critical string with c > 1. Following the recent success of the matrixmodel solutions of string theory [18], we can construct such strings ifwe could deal with the large N limit of a scalar matrix model in Ddimensions, sayZstringD=Z YiDφie−N Pi Tr U(φi)+N Pi,µ=1...D Tr(φiφi+µ) ,(4.2)which describes a scalar field on a D- dimensional lattice.

The connec-tion with string theory is made in the usual way, the Feynman diagramsof the perturbative expansion of (4.2) correspond, in an expansion inpowers of 1N , to triangulations of two-dimensional surfaces. The scalarfields correspond to matter on this surface and thus, (4.2) , could yield,at the appropriate critical point where the mean number of trianglesdiverges, a c = D string theory.

The standard approach to the largeN-limit of such a theory is to diagonalize the matrices φ, i.e. to pass toradial coordinates, φi = ΩiλiΩ†i, where λi is diagonal.

In terms of thesevariables,ZstringD=Z YiDλiDΩi∆2(λi)e−N Pi Tr U(λi)+N P TrλiVijλjV †ij,(4.3)where Vij = ΩiΩ†j and ∆(φ) = Qi DVijQplaq. δ(1 −QL Vij).

Theconstraints arise since the Vij’s are pure gauge fields.If not for theconstraints we could perform the integral over the Vij’s and reduce theintegral to one over N variables per site that could be evaluated bysaddlepoint techniques. It is these constraints that have prevented theconstruction of strings with c > 1.Now let us combine these two models to consider QCD with adjointscalar matter,15

ZadjQCD =Z YLDULYiDφie−N Pi Tr U(φi)+N Pi,µ=1...D Tr(φiUµφi+µU †µ)e−Ng2(a)Pplaq.Tr(QL UL+h.c.). (4.4)This theory is invariant under standard gauge transformations, φi →ViφiV †i ;Uµ →ViUµV †i+µ,which allow us to diagonalize the φ’s.

Howeverthe presence of the Wilson action prevents us from handling this theoryfor large N. If set the gauge coupling to zero, we recover the previousmodel, since in this limit we can drop the Wilson action term, as long aswe enforce the constraints, tr[UP] = 1. However if we take the oppositelimit, i.e.

set g = ∞, then we can simply drop the Wilson action andthe model will be soluble in the large N limit. This is induced QCD[19].Induced QCD has the one great advantage of being soluble, or atleast reducible to a well defined master field equation.

This is becausethe integral over the link matrices can now be performed. This is thefamous Itzykson- Zuber integral [20],I(φ, χ) ≡ZDUeN TrφUχU †= deteNφiχj∆(φ)∆(χ).

(4.5)This formula is very profound, underlies all the analysis of the c = 1,matrix model, and can be derived in many ways. One is the demonstra-tion that the integral is given exactly by the WKB approximation, andthe answer is simply the sum over the N!

saddlepoints, for which arethe U are permutation matrices.Although soluble this model appears to be very far from QCD, sinceasymptotic freedom instructs us to set the lattice coupling to zero, notinfinity, in the continuum limit. However, Kazakov and Migdal arguedthat even though there is no kinetic term for the gauge field, it couldbe induced at large distances [19].

They argued that if one integratesout the scalar mesons (even in the case of noninteracting scalars withU(Φ) =12m2Φ2), then at distances large compared to a, one wouldinduce in four dimensions an effective gauge interaction,16

Seff(U) ∼N96π2 ln(1m2a2) Tr F 2µν + finite as a →0 . (4.6)This is simply the one loop vacuum graph for the scalars in a backgroundgauge field, which is logarithmically divergent in four dimension.

Nowthis looks very much like the ordinary Yang-Mills action,1g2(a) tr FµνF µν,if we recall that asymptotic freedom tell us that1g2(a) = 11N48π2 ln(1M 2g a2),where Mg is a mass scale for QCD, say the glueball mass. We can there-fore identify these two (the fact that there are N2 scalars is crucial, asis the sign of the effective action which is due to the non-asymptoticfreedom of the scalars.) If we do so then we find that, M2g = m2311a122.Thus in the continuum limit the adjoint scalars become infinitely mas-sive and decouple, but not before they have drive1g2 up, from zero atdistance a to the large QCD value atdistance 1m, where1Mg >> 1m >> a.The basic idea is that the infrared slavery of the scalars, at the size ofthe lattice spacing, produces an effective gauge theory at a larger scale(much larger than the inverse scalar mass), which then produces theusual asymptotically free fixed point theory.There are many problems with this idea.

For one the hard gluonsare not absent and their contribution will overwhelm that of the scalarsat short distances. Their asymptotic freedom is more powerful thanthe infrared slavery of scalars.

Another issue is that the above theorypossesses a much larger symmetry than the SU(N) gauge symmetry ofthe usual lattice action. It is not difficult to see that, in D dimensions,it is invariant under (D −1) × (N −1) extra local U(1)-gauge symme-tries.

This is because the transformation Uµ(x)→V †µ(x)Uµ(x)Vµ(x+µa),leaves the action invariant as long as Vµ(x) is a unitary matrix that com-mutes with Φ(x). If Vµ(x) were independent of µ then this would be theordinary gauge invariance.

Thus we have D −1 new gauge symmetries,which are of course isomorphic to the special unitary transformationsthat commute with Φ [21]. Thus Vµ(x) = Dµ(x)Ω(x), where Ω(x) is theunitary matrix that diagonalizes Φ and Dµ(x) is diagonal.A subset of this symmetry is the, field independent, local ZN sym-metry, Uµ(x)→ZµUµ(x)Z†µ, where Zµ is an element of the center of thegroup.

This symmetry alone prevents the Wilson loop from acquiringan expectation value. A Wilson loop contains different links, and thus17

W(C) = ⟨QL∈C UL⟩→QL ZµW(C) ⇒W(C) = 0. This symme-try must be broken if we are to recover the QCD fixed point from thisformulation [22].Finally, as we shall see, the simple Gaussian model is soluble andthe answer is very simple and not equivalent to QCD [21].

However,there are interesting attempts to save the model and furthermore evenif it does not yield a solution of QCD it might provide some interestingsoluble matrix models which could yield new solutions of new stringtheories. Induced QCD is a matrix model and thus it corresponds tosome kind of sum over surfaces.

If we look at the Itzykson-Zuber integralwe note that it could be expressed as,I(φ, χ) = exp12 Tr φ2 Tr χ2 +a Tr φ4 Tr χ4 + bN2(Tr φ2)2(Tr χ2)2 +. .

..(4.7)These terms will affect the structure of the large N expansion of theFeynman diagrams, and can be interpreted as yielding extra weightswhen the two-dimensional surfaces intersect [23]. Thus this model corre-sponds, perhaps, to some kind of string theory with weights that dependon the extrinsic geometry.4.2.

Solution of the Gaussian modelTo try to solve the model of induced QCD we first integrate out heUL’s, then look for extrema of the effective action,S[φi] = N2 1N TrXiU(φi)+ 1N2Xi,µln I(φi, φi+µ)+ 1N2Xiln ∆2(φi). (4.8)In the large N limit the integral will be dominated by a translation-ally invariant saddlepoint for the density of eigenvalues of the matricesΦi, ρ(x) ≡1NPNa=1 δ(x −φa).Migdal has derived the master fieldequation for the saddlepoint, using the Schwinger-Dyson equations thatare satisfied by I(φ, χ) [24].

These are consequences of the fact thatI satisfies tr[( 1N∂∂φ)k]I = tr(χ)kI. The net result is that one derivesan equation for the function F(z) ≡Rdz ρ(ν)z−ν , whose imaginary part isImF(ν) = −πρ(ν),18

ReF(λ) = PZdν2πi ln[λ −12DU′(ν) −D−1D ReF(ν) + iπρ(ν)λ −12DU′(ν) −D−1D ReF(ν) −iπρ(ν)]. (4.9)This equation is much more complicated than the usual Riemann-Hilbertproblem that one obtains for simple matrix models.

It is sufficientlynon-linear and complex that one might imagine that it describes QCD.The master field equation simplifies dramatically for D = 1. This isbecause in one dimension the gauge field can be gauged away completely,thus the model is equivalent to a scalar field on a one-dimensional lattice.The large N limit of this model describes the c = 1 string on a discretetarget space, a model which has been solved in the double- scaling limitfor small lattice spacing [25].

It undergoes a phase transition at a finitelattice spacing and it might be very instructive to use (4.9) to explorethis phenomenon.In particular for the quadratic potential the path integral is Gaus-sian,Z =Z YnDΦne−N Pn Tr{ m22 Φ2n−ΦnΦn+1} . (4.10)Thus the eigenvalues of Φ will be given by the semi-circular distribution,namely πρ(ν) =qµ −µ2ν24 , where µ is determined by the mean of thesquares of the eigenvalues, ⟨1N Tr(Φ2)⟩= 1µ.

It is therefore sufficient tocalculate the expectation value of1N Tr(Φ2), which is given by the oneloop integral,1N Tr(Φ2) =πZ−πdp2π1m2 + 2cosp =1√m4 −4≡1µ . (4.11)It is easy to verify that this solves (4.9), using the fact thatF(z) = µz2 −rµ2z24−µ; ReF(ν) = 12µν,(4.12)However, if we return to (3.8), we see that the integral involved is of thesame form for any D, as long as ℜV ′(ν) is linear in ν.

This suggests19

that we can find a solution of (4.9) with a semi-circular distribution ofeigenvalues for a quadratic potential in any dimension [21].Indeed, one can see that a semi-circular distribution of eigenvaluessatisfies (4.9) for any D as long as,µ±(D) = m2(D −1) ± Dpm4 −4(2D −1)2D −1. (4.13)This solution is much too trivial to describe QCD.

In particular, forD > 1 there is no sensible continuum limit of the model.Is the solution unique? To see that it is note that in the masterfield equation the dimension of space-time enters only via the number ofnearest neighbors of a given site, the coordination number of the lattice.⋆The translationally invariant background scalar field is the same for anylattice with the same coordination number.

The observables, say thescalar propagator, will of course depend on the full structure of thelattice, but not he background field. Therefore we can choose anothersimpler lattice with the same coordination number, say a Bethe lattice,which contains no closed loops.

For such a lattice, as in the case of theD = 1 model, the gauge field can be eliminated completely, and themodel is equivalent to,ZBetheLattice =ZDφie−N Pi Tr m22 φ2i +N P Tr[φiφj] . (4.14)This model is easily soluble.

We define Z(φ) to be the partition functionof a branch of the Bethe lattice with coordination number 2D, so thatZ =RDφZ(φ)2De−m22 N Tr φ2. Z(φ) satisfies the equation,Z(φ) =ZDφ′Z(φ′)2D−1e−m22 N Tr φ′2+N Tr[φφ′] .

(4.15)These equations are easily soluble. Take Z(φ) to have the form Z(φ) =ce−N α2 Tr φ2, then (4.15) determines α to equal α = −m2±√m4−4(2D−1)2(2D−1).Then (4.14) can be used to determine 1N Tr φ2 =1m2+2αD, which agreesprecisely with 1µ as given by (4.13).⋆I thank C. Bachas for emphasizing this point to me.20

4.3. ProspectsThe simplest Gaussian model fails, but all hope is not lost.

It iscertainly possible to induce QCD if one introduces enough flavors ofmatter. The problem is that one then loses solubility.

It might be thatthe self interactions of the scalars could be adjusted to drive the theorytowards the asymptotically free fixed point. This hope has been pursuedwith great vigor by Migdal, who has also considered adding fermions,not too many so that the model remains soluble, so as to break theZN symmetry [26].

Time will tell whether this will succeed. Even ifit does not these model might yield a new class of interesting solublematrix models which could teach us something about new classes ofstrings, perhaps strings that depend on extrinsic geometry.

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