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AZPH-TH/93-01, COLO-HEP/303

arXiv:hep-th/9212158v3 15 Jan 1993AZPH-TH/93-01, COLO-HEP/303hep-th/9212158January 1993Difficulties in Inducing a Gauge Theory at Large NB. S. Balakrishna1Department of PhysicsUniversity of Colorado, Boulder, CO 80309andDepartment of PhysicsUniversity of Arizona, Tucson, AZ 85721ABSTRACTIt is argued that the recently proposed Kazakov-Migdal model of inducedgauge theory, at large N, involves only the zero area Wilson loops that areeffectively trees in the gauge action induced by the scalars.

This retains only aconstant part of the gauge action excluding plaquettes or anything like them andthe gauge variables drop out.1Address for correspondence. Email: bala@haggis.colorado.edu

Recently, Kazakov and Migdal made an attempt to induce large N QCD from a latticemodel[1]. This model is far simpler than the usual lattice QCD as it has no kinetic termfor the gauge variables.

The hope was that the scalars present in the model may somehowgenerate this kinetic term and lead to a nontrivial gauge theory. The simplicity of the modellet the authors of [1] to solve the model exactly at large N. This exact solution was expectedto correspond to large N QCD near some critical point.

Initial attempts involving a quadraticpotential for the scalar variables did not succeed[2], but the hope remained that a nontrivialpotential may perhaps induce large N QCD. Numerical[3] and analytical[4] means suggestedthat there do exist critical points in the model exhibiting nontrivial scaling behavior.A significant amount of work[5] has already been done in this regard, but it is still notclear what the large N result really corresponds to.

Is it a theory of scalars interactingwith the gauge fields? If so, can the scalars be made heavy enough to decouple from thegauge variables to induce QCD?

The effective action obtained by integrating away the scalarsshould hold a clue to these questions. It is an action for the gauge variables involving theirgauge invariant combinations, the Wilson loops, of all sorts.

But, there is a danger that thenontrivial loops of interest might get suppressed for large N, leaving only those bounding zeroarea. Because this excludes plaquettes or anything like them, the gauge variables decouple.It would still be a highly nontrivial matter to solve the model at large N. Here, we presentarguments that support this unfortunate scenario.The Kazakov-Migdal model is defined by the following action defined on a D dimensionalhypercubic lattice:S = N2 XxtrV (Φ(x)) −N2 X⟨xy⟩trhΦ(x)U(xy)Φ(y)U†(xy)i.

(1)The lattice sites are labeled by x and the links by ⟨xy⟩. Gauge variables are representedby N × N unitary matrices, U(xy)’s, associated with the links.

For large N, this could beregarded as a model based on U(N) rather than SU(N). Φ is an N × N hermitian matrixrepresenting a scalar in the adjoint representation that could have a nonzero trace.

Onefactor of 1/N is implicit in the definition of trace. Integration over U can be performedseparately on every link.

The integral over the link ⟨xy⟩depends only on the eigenvalues ofΦ(x) and Φ(y). Here it is easier to work with the traces, trΦi, i = 1, 2, · · ·, rather than theeigenvalues.

Only N of them are independent, but it is convenient to keep all of them. The1

effective action for the scalars arising from the U−integrations is of the formSeff= N2 XxtrV (Φ(x)) −N2 X⟨xy⟩J(Φ(x), Φ(y)),(2)where J comes from the link integral,exphN2J(Φ(x), Φ(y))i=ZdU exphN2trΦ(x)UΦ(y)U†i. (3)An explicit expression for J is not needed for our purpose.

Note that J can be regarded as anindependent function of all the traces of Φ(x)’s and Φ(y)’s. This follows from expanding theexponential inside the integral and integrating over U[6].

An expansion of J in its argumentsis of the formJ(Φ(x), Φ(y)) =XnαJnα(Φ(x), Φ(y)),(4)where Jnα is a product of the various traces that together involve n of the Φ(x)’s and n ofthe Φ(y)’s, α labeling the different possibilities. For instance, at the x end or the y end,J1α ∝trΦ,J2α ∝trΦ2, (trΦ)2,J3α(Φ) ∝trΦ3, trΦ2trΦ, (trΦ)3.

(5)For large N, J has no explicit dependence on N except for that absorbed into the definitionof trace. Expansion of the partition function by perturbing J generates various graphs, treesas well as loops.

For every choice of n and α, J provides us with an edge for constructingthe graphs. Graphs arise when these edges are joined together by contracting the Φ’s.

As itturns out, only the trees contribute to the large N limit.Let us first discuss a primary edge of order n = 2 having trΦ2 attached to the ends. Tomake the counting of N’s easier, let us scale Φ as Φ →Φ/N leading to a factor 1/N2 infront of this edge term.

Each such edge now supplies 1/N2. A vertex of order p, formedwhen p edges are glued at a site by contracting the Φ’s, involves a summation over the N2components of Φ and hence contributes a factor N2.

This is true for the open ends (p = 1)and the joints (p = 2) as well. The Φ propagator involved in the Φ contractions has onefactor of N after scaling that compensates the 1/N factor implicit in the definition of tracein trΦ2.

If the coupling constants of the potential also take part in the gluing process, anexercise in the zero dimensional matrix field theory of Φ defined at a site shows that N2 isthe leading contribution from a vertex. Note that the coupling constants could carry 1/N’safter scaling.

The product of N2’s is thus (N2)V −E at most, where V is the total number2

of vertices and E is the total number of edges. This equals (N2)1−L for a connected graphhaving L loops.

Clearly, only the trees are responsible for the leading, order N2, behavior.The potential is not required to be even in Φ for this discussion. Φ may have a nonzeroexpectation value that is proportional to the identity with a factor N due to scaling.

Theseconclusions hold good for all the primary edges, that is, edges of order n having one traceof the form N2−ntrΦn (N’s coming from scaling) attached to the ends.The remaining edges have more than one traces attached to the ends. After scaling, anorder n edge receives a factor 1/N2 for itself and a factor N2−n at the ends that carry thetraces.

One may view this as a primary edge having some more traces attached. If theadditional traces that are of the form N−mtrΦm are self-contracted, they simply contributea factor unity.

Then the arguments are the same as that for the primary edges and only treescontribute at large N. In general, a whole branch of a tree can grow from the additionaltraces and hence from all the traces. The branch attached to a trace N2−mtrΦm contributesat most N2 like an open end of a primary edge.

This suggests that trΦm supplies a factorNm and hence the product of traces a factor Nn. This along with N2−n coming from scalingcombines to give N2.

Each end of an edge thus carries a N2 and 1/N2 is supplied by the edgeitself, giving together N2, the leading order for the whole tree. This argument is applicableto the branches that grow from the traces as well.

Any other way of constructing the graphsis found to be of higher order in 1/N. Thus, in general, only trees contribute at large N.There is one more possible source of N. This is due to the presence of N independenttraces.

But as we have seen, all the traces are on a similar footing in defining the edgesand generating graphs, and it is unlikely for N of them to enhance the loops. This alsofollows from the fact that the sum of all the trees constructed above gives a result thatagrees with that of the saddle point method.

To see this, let us derive an expression thatsums up the leading trees. In the process, we obtain an equation that is analog of the saddlepoint equation of the lattice model.

First, introduce a λ for each of the traces, that is λi fortrΦi. As noted earlier, it is possible to regard J as an independent function of all the traces.Expanding J around λ givesJ (Φ(x), Φ(y)) = J(λ) +XitrΦi(x) + trΦi(y) −2λi∂iJ(λ)/2 + eJ,(6)where ∂iJ(λ) = ∂J(λ)/∂λi and eJ is the remainder.

For the trees constructed earlier, the open3

ends carry a weight given by the product of the expectation values (in the zero dimensionalmatrix field theory at a site) of all the traces attached. Let us choose λi to be the expectationvalue of trΦi when the above expansion of J is used in Eq.

(2) without eJ. Including eJ as aperturbation, note that the graphs generated exclude the trees constructed earlier.

This isbecause eJ depends on the traces as trΦi −λi and our choice of λ makes it not possible fortrΦi −λi to create open ends of leading order. This suggests that the lowest order (that is,without eJ) should be summing up the trees.

This sum isW (D∂J(λ)) + DJ(λ) −DXiλi∂iJ(λ)(7)per site, where W is defined byexphN2W(j)i=ZdΦ exp"−N2trV (Φ) + N2 XijitrΦi#. (8)Derivative of W(j + D∂J(λ)) −P jλ with respect to j at j = 0 gives the leading weightassociated with the open ends.

This should vanish by our choice of λ, giving∂iW (D∂J(λ)) −λi = 0,(9)where ∂iW(j) = ∂W(j)/∂ji. This equation determining λ is analogous to the saddle pointequation of the lattice model and is equivalent to the latter at large N. W at large N, whenused in (7), gives the free energy per site at the saddle point.

The integral determining W isdominated at a saddle point for large N leading to a saddle point equation of its own, whichagrees with that of the lattice model when j = D∂J(λ). The equation derived above simplytells us to identify j with D∂J(λ).

It also extremizes (7). Second and higher derivativesof W evaluated at D∂J(λ) help us sum the branches that could grow from the vertices ofloops.These conclusions reinforce ones conviction that the saddle point method is closely relatedto summing the tree graphs.

This can be translated to the statement that, at large N,only trivial Wilson loops bounding zero area contribute to the gauge action obtained byintegrating away the scalars. The analysis so far involved the effective scalar action.

Thegauge action can be studied by perturbing the link term of (1). The link terms when joinedby contracting the Φ’s also generate graphs.

To relate the two types of graphs, expand theexponential inside the U−integral in Eq. (3) as well as the exponential of N2J.

This leads4

to a series of relations of the kind,N2ZdU trΦ(x)UΦ(y)U†=N2J1,12N4ZdUhtrΦ(x)UΦ(y)U†i2=N2J2 + 12N4J21,(10)where Jn =Pα Jnα. One may view these as decomposing the U−integrals into connectedpieces[6].

Nontrivial Wilson loops do not contribute at large N simply because there areno such Φ contractions to do the job. The first relation lets us replace a J1 edge, whenpresent alone on a link, byR dU tr(Φ(x)UΦ(y)U†).

Trees made of J1 edges (one per link)are thus the trivial Wilson loops bounding zero area. To understand the other relations, itis convenient label the Φ’s such that those connected by U have the same label.

This definesa labeling of the Φ’s for the J’s. For instance, a labeling in [tr(Φ(x)UΦ(y)U†)]2 relates totwo J1 and one J2 edges.

Integrating over U and collecting the connected pieces, one findsthat a J edge carries the same set of labels at its x and y ends. A nontrivial Wilson loopdefines a sequence of labels along the loop corresponding to a sequence of Φ contractions.Such a sequence can not arise from the trees made of J edges.

It is thus not possible tofeel the presence of nontrivial Wilson loops at large N. For the trivial ones, the productof U’s multiplies to unity and the gauge variables decouple. To see the effects of plaquettelike terms, one needs to go to subleading orders.

This is true for a fundamental scalar aswell and perhaps is also true for other models where the angular variables decouple from theeffective scalar action. A model with N fundamental scalars is free of these problems, but itis very difficult to solve.The large N result could still be useful for discovering a double scaling limit which is quitea generic phenomenon of scalar field theories[7].

The present analysis helps us in identifyingthe 1/N corrections. The tree graphs themselves receive corrections at the vertices, the loopgraphs become significant and the link integral gets modified.

1/N expansion in these modelsis possibly related to a strong coupling expansion of the induced gauge theory. The largeN result is just a constant part of the induced gauge action, but it does have implicationsfor the subleading terms.

The tree graphs modify the vertices of all the loops and the gaugecoupling will be strongly dependent on them. The lattice model involving a fundamentalscalar will provide a simpler framework to study these matters.To summarize, the Kazakov-Migdal model at large N retains only a constant part of the5

effective action for the gauge variables obtained by integrating the scalars. This arises fromthe zero area Wilson loops that are effectively trees.

To feel the presence of plaquette liketerms, one needs to go beyond the leading order. The large N result could still be useful asit modifies indirectly the strength of the subleading terms and is expected to contribute tothe gauge coupling constant as well as critical behavior.This work is supported by NSF Grant PHY-9023257.References[1] V. A. Kazakov and A.

A. Migdal, preprint PUPT-1322, LPTENS-92/15, to be pub-lished in Nucl. Phys.

B . [2] D. Gross, Phys.

Lett. B 293, 181 (1992).

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

Lett. 69, 2747 (1992).

[4] A. A. Migdal, preprint PUPT-1323.

[5] I. I. Kogan, G. W. Semenoffand N. Weiss, Phys. Rev.

Lett. 69, 3435 (1992); A. A.Migdal, preprint PUPT-1332; M. Caselle, A. D’Adda and S. Panzeri, Phys.

Lett. B 293,161 (1992); I. I. Kogan, A. Morozov, G. W. Semenoffand N. Weiss, preprint UBCTP92-26, ITEP M6/92; S. Khokhlachev and Yu.

Makeenko, preprint ITEP-YM-5-92; A.A. Migdal, preprint LPTENS-92/22; I. I. Kogan, A. Morozov, G. W. SemenoffandN. Weiss, preprint UBCTP 92-27, ITEP M7/92; M.I.

Dobroliubov, I.I. Kogan, G.W.Semenoffand N. Weiss, preprint PUPT 1358, UBCTP92-032.

[6] Integration over U’s and U†’s is given by a sum of products of Kronecker delta symbols.For details see, M. Creutz, J. Math.

Phys. 19, 2043 (1978).

[7] B. S. Balakrishna, preprint AZPH-TH/92-43, COLO-HEP/295.6

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