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An Exact Solution of Induced Large-N

arXiv:hep-th/9303138v3 27 Apr 1993ITEP-YM-2-93March, 1993An Exact Solution of Induced Large-NLattice Gauge Theory at Strong CouplingYu. Makeenko1Institute of Theoretical and Experimental PhysicsB.Cheremuskinskaya 25, 117259 Moscow, RFAbstractI show that the strong coupling solution of the Kazakov–Migdal model with ageneral interaction potential V (Φ) in D dimensions coincides at large N with thatof the hermitean one-matrix model with the potential ˜V (Φ):(2D −1) ˜V ′ = (D −1)V ′ + Dq(V ′)2 + 4(1 −2D)Φ2,whose solution is known.

The proof is given for an even potential V (Φ) = V (−Φ)by solving loop equations.Submitted to Modern Physics Letters A1E–mail: makeenko@nbivax.nbi.dk / makeenko@desyvax.bitnet / makeenko@vxitep.itep.msk.su

1IntroductionSolvable matrix models are usually associated with D ≤1 dimensional theories. Kazakovand Migdal [1] have recently proposed that the model (KMM) defined by the partitionfunctionZKMM =Z Yx,µdUµ(x)YxdΦx ePx Nc tr−V (Φx)+PDµ=1 ΦxUµ(x)Φx+µU†µ(x),(1.1)where the scalar field Φx is in the adjoint representation of the gauge group SU(Nc) andthe link variable Uµ(x) belongs to the gauge group, is solvable in the limit of large numberof colors, Nc, for D > 1.

Since stringy phenomena are associated with the strong couplingphase (see [2] for a review), one is interested in the solution of KMM at strong coupling.The original idea to solve KMM is based on the fact that the scalar field can bediagonalized by a (local) gauge transformation so that only O(Nc) degrees of freedomare left and the saddle-point method is applicable as Nc →∞. Migdal [3] proposed tosolve the saddle point equation by the Riemann–Hilbert method and derived a masterfield equation to determine the Nc = ∞solution in the strong coupling phase.Anexplicit solution of this equation for the quadratic potential was found by Gross [4].

Asurprising property of the master field equation (not yet completely understood) is thatit admits [3, 5] self-consistent scaling solutions with non-trivial critical indices.There are two other approaches to solving KMM at strong coupling. The first oneis based on loop equations which were derived and solved for KMM with the quadraticpotential by the author [6] recovering the solution of Ref.

[4]. The second one, proposedby Boulatov [7], relies on the relation of KMM to a matrix model on the Bethe lattice,whose “naive” continuum limit is equivalent for D > 1 to a one-matrix model with theupside-down potential.In the present paper I extend the approach based on loop equations to the case ofKMM with an arbitrary potential.

On the one hand, this provides a way to solve themodel at strong coupling which is an alternative to the Riemann–Hilbert method. Onthe other hand, this method is along the line of modern studies of matrix models of 2Dquantum gravity by means of loop equations (for a review, see Ref.

[8]).The main results of this paper are as follows.The loop equations are drasticallysimplified in the strong coupling phase at Nc = ∞due to the fact that the averagesof closed Wilson loops vanish except for the loops with vanishing minimal area. Theresulting equations are satisfied for an even potential V (Φ) = V (−Φ) by the ansatz whichreduces KMM in D dimensions to a hermitean one-matrix model with the potential ˜V (Φ):˜V ′(Φ) = D −12D −1V ′(Φ) +D2D −1q(V ′(Φ))2 + 4(1 −2D)Φ2 .

(1.2)The solution of this model with an arbitrary potential is well-known.2

In Section 2 I derive the exact loop equations for KMM with an arbitrary potentialand any Nc. In Section 3 I show how these equations are simplified in the strong couplingphase at Nc = ∞.

In Section 4 I define the ansatz and obtain the exact solution of theNc = ∞loop equations in the case of the even potential. Section 5 is devoted to theexplanation of the form of the solution from the viewpoint of the large mass expansion.In Section 6 I discuss some properties of the solution and show how it can be comparedto the one obtained by the Riemann–Hilbert method.

Appendix A contains the detailsof derivation of the loop equations. In Appendix B I analyze the large mass expansion ofthe one-link correlator of the gauge fields at Nc = ∞.2Loop equations for arbitrary potentialThe loop equations of KMM relate the closed adjoint Wilson loopsWA(C) =* 1N2c| tr U(C)|2 −1+(2.1)to the open ones with the matter field attached at the ends:Gλ(Cxy) =* 1NctrΦxU(Cxy)1λ −ΦyU†(Cxy)+.

(2.2)The loop equations result from the invariance of the measure in Eq. (1.1) under anarbitrary shift of Φ and read* 1NctrV ′(Φx)U(Cxy)1λ −ΦyU†(Cxy)+−DXµ=−Dµ̸=0Gλ(C(x+µ)xCxy)= δxy* 1NctrU(Cxy)1λ −Φy 1Nctr1λ −ΦyU†(Cxy)+(2.3)where the path C(x+µ)xCxy on the l.h.s.

is obtained by attaching the link (x, µ) to the pathCxy at the end point x as is depicted in Fig. 1.

The details of derivation are presented inAppendix A. I have omitted here and below additional contact terms which arise at finiteNc due to the fact that Φ belongs to the adjoint representation, so that Eq. (2.3) is writtenfor the hermitean matrices.

This difference should disappear, however, as Nc →∞whichcan be easily proven for the even potential when ⟨1N tr V ′(Φx)⟩vanishes.The path Cxy on the r.h.s. of Eq.

(2.3) is always closed due to the presence of thedelta-function. The explicit equation for the vanishing contour Cxx = 0 at large Nc, whenthe factorization holds, readsZC1dω2πiV ′(ω)λ −ωEω −2DGλ(1) = E2λ(2.4)3

xyxy✧✧✧x + µa)b)✻✻❞t❄❞t❄✧✧✧CyxCxyCyxCx(x+µ)C(x+µ)xCxyFig. 1:The graphic representation for Gλ(Cxy) (a) and Gλ(C(x+µ)xCxy) (b) enter-ing Eq.

(2.3). The empty circles represent Φx or Φx+µ while the filled onesrepresent1λ−Φy .

The oriented solid lines represent the path-ordered productsU(Cxy) and U(C(x+µ)xCxy). The color indices are contracted according to thearrows.whereEλ ≡ 1Nctr1λ −Φx= 1λ(Gλ(0) + 1)(2.5)with Gλ defined by Eq.

(2.2). I have denoted the one-link average byGλ(1) = Gλ(C(x±µ)x)(2.6)since the r.h.s.

does not depend on x and µ due to the invariance under translations bya multiple of the lattice spacing and/or rotations by a multiple of π/2 on the lattice.The contour C1 encircles singularities of Eω so that the integration over ω on the l.h.s. ofEq.

(2.4) plays the role of a projector picking up negative powers of λ.3Loop equations at large NcThe loop equations for non-vanishing contours Cxy ̸= 0 are drastically simplified atNc = ∞in the strong coupling region where the closed adjoint Wilson loops (2.1) vanishexcept the contractable loops (i.e. those with vanishing minimal area Amin(C) which areequivalent to Cxx = 0 due to the unitarity of U’s):WA(C) = δ0Amin(C) + O 1N2c!.

(3.1)4

While the averages of a new kind arise on the r.h.s. of Eq.

(2.3) for Cxx ̸= 0, they obeyat Nc = ∞the following analogue of Eq. (3.1) 1NctrU(Cxx)1λ −Φx 1NctrU†(Cxx)1λ −Φx= δ0,Amin(C)E2λ + O 1N2c!(3.2)i.e.

vanish for Cxx ̸= 0.Hence, the strong coupling loop equation for Cxy ̸= 0 at Nc = ∞reads* 1NctrV ′(Φx)U(Cxy)1λ −ΦyU†(Cxy)+−DXµ=−Dµ̸=0Gλ(C(x+µ)xCxy) = 0(3.3)independently of whether Cxy is closed or open.Therefore, the r.h.s. of the loop equation in nonvanishing at Nc = ∞only for Cxy = 0(modulo backtrackings) when the proper equation is given by Eq.

(2.4). This property ofthe strong coupling loop equations at Nc = ∞allows to find a simple solution.4The strong coupling solutionLet us solve the set (2.4), (3.3) of the Nc = ∞loop equations at strong coupling by thefollowing ansatz in the case of the even potential 1NctrF(Φx)Uµ(x)Φx+µU†µ(x)= 1NctrF(Φx)ΦxΛ(Φx)(4.1)where F(Φ) is arbitrary.

The function Λ(ω) is analytic at ω = 0:Λ(ω) =∞Xk=0Λkωk . (4.2)For Gλ(1) which is defined by Eq.

(2.6), Eq. (4.1) can be written asGλ(1) =ZC1dω2πiωΛ(ω)λ −ω Eω(4.3)where the contour C1 encircles singularities of Eω, i.e.

the same as in Eq. (2.4).The formula (4.3) extends to the general potential the oneGλ(1) = Λ0λEλ(quadratic potential)(4.4)for the quadratic potential which is associated with [6]Λ(ω) = Λ0(quadratic potential) ,(4.5)i.e.

Λk = 0 for k ≥1.5

The substitution of the ansatz (4.3) into Eq. (2.4) yieldsZC1dω2πi˜V ′(ω)λ −ω Eω = E2λ(4.6)where˜V ′(ω) = V ′(ω) −2DωΛ(ω)(4.7)which coincides with the loop equation for the hermitean one-matrix model with the po-tential ˜V .

2The dependence of Λ(ω) on the potential V can be determined from Eq. (3.3).

Thesimplest way to do this is to consider Eq. (3.3) in the case when Cxy is just one link (x, µ0)and to take the 1/λ2 term of the 1/λ expansion.

The resulting equation reads explicitly 1NctrV ′(Φx)Uµ0(x)Φx+µ0U†µ0(x)− 1Nctr Φ2x+µ0−DXµ=−Dµ̸=0µ̸=µ0 1NctrΦx−µUµ(x −µ)Uµ0(x)Φx+µ0U†µ0(x)U†µ(x −µ)= 0(4.8)which reduces after the substitution of the ansatz (4.3) toZC1dω2πi V ′(ω)ωΛ(ω) −1 −(2D −1)Λ2(ω)!ω2Eω = 0 . (4.9)This equation is satisfied provided that Λ(ω) obeys the quadratic equationV ′(ω)ω−1Λ(ω) −(2D −1)Λ(ω) = 0(4.10)for the generic potentialV (ω) =∞Xn=1t2nω2n(4.11)and t2 ≡m20.The solution to Eq.

(4.10) readsΛ(ω) =2V ′(ω)ω+rV ′(ω)ω2 + 4(1 −2D)(4.12)which recovers the one [6] for the quadratic potential whenV ′(ω) = 2m20ω(quadratic potential)(4.13)and Eq. (4.5) holds.

The fact that the ansatz (4.3) with Λ(ω) given by Eq. (4.12) satisfiesEq.

(3.3) means that it is indeed a solution providing Eq. (4.6) with˜V ′(ω)ω= D −12D −1V ′(ω)ω+D2D −1vuut V ′(ω)ω!2+ 4(1 −2D)(4.14)2For a review, see Ref.

[8].6

is satisfied.The one-cut solution to this equation for an arbitrary potential is well-known [9]Eλ =ZC1dω4πi˜V ′(ω)λ −ωvuut (λ −x)(λ −y)(ω −x)(ω −y)(4.15)where x and y are expressed via ˜V byZC1dω2πi˜V ′(ω)q(ω −x)(ω −y)= 0 ,ZC1dω2πiω ˜V ′(ω)q(ω −x)(ω −y)= 2 . (4.16)One gets x = −y for the even potential.The formulas (4.3), (4.14), (4.15) and (4.16) completes the solution of the strongcoupling loop equations of KMM at Nc = ∞.

The saddle point value of Φx is totallydetermined to be x-independent and is described, modulo a gauge transformation, by thespectral densityρ(λ) =12π2Z xy dt˜V ′(t) −˜V ′(λ)t −λvuut(x −λ)(λ −y)(x −t)(t −y)fory < λ < x(4.17)with support y < λ < x. Properties of this solution are discussed in the next section.5Relation to the large mass expansionThe peculiar form (4.3) of the strong coupling solution at Nc = ∞can be understoodin the framework of the large mass expansion.

To this aim let us consider the one-linkcorrelator of the gauge fields 1NctrtaUΦx+µU†U≡R dU eNc trΦxUΦx+µU†1Nc trtaUΦx+µU†R dU eNc trΦxUΦx+µU†(5.1)where the averaging is only w.r.t. U while Φx and Φx+µ play the role of external fields.

ta(a = 1, . .

. , N2c −1) stand for generators of SU(Nc) which are normalized by1Nctr tatb = δab .

(5.2)As was proposed in Refs. [3, 10], the following formula holds at Nc = ∞: 1NctrtaUΦx+µU†U=∞Xm=0Λm1NctrtaΦm+1x.

(5.3)It is shown in Appendix B how this formula can be obtained in the large mass expansion.7

Eq. (5.3) allows to explain the solution of the previous section as follows.Let usmultiply both sides of Eq.

(5.3) bytr(taF(Φx)) which gives, using the completenesscondition (A.4),1NctrF(Φx)UΦx+µU†=1NctrF(Φx)ΦxΛ(Φx)+ 1NctrF(Φx) 1NctrΦx+µ −ΦxΛ(Φx)(5.4)where Λ(Φ) is defined by Eq. (4.2).

The second term on the r.h.s. which is due to thedifference between the adjoint representation and the hermitean matrices vanishes for theeven potential at Nc = ∞.

Hence, Eq. (5.4) for Φx and Φx+µ given by the saddle pointmatrix ΦS recovers Eq.

(4.1). On the other hand, the solution of the previous sectionallows to calculate Λm in Eq.

(5.3) as the coefficients on the expansion of (4.12) in ω.6DiscussionThe above strong coupling solution is realized at given D only in some region of thecouplings t2k’s entering the potential (4.11). At D = 0 it coincides with the well-knownsolution of the hermitean one-matrix model.

At any D but t2 ̸= 0, t2k = 0 for k ≥1, thesolution coincides with the one for the quadratic potential [4]. For this reason I expectthat it is realized in some region around this point.

The condition is that Eq. (4.16)should yield real x and y and the spectral density (4.17), which describes the distributionof eigenvalues of the saddle point matrix ΦS, should be positive.

This is a restriction onthe one-cut solution which is satisfied in some region of values of the couplings t2k’s. It iswell known that for the simple quartic potential˜V (ω) = ˜t2ω2 + ˜t4ω4(6.1)the one-cut spectral density is positive for |˜t4| ≤˜t22/12.

This is, however, a nontrivialrestriction on the potential V sinceV ′(ω) = Dq( ˜V ′(ω))2 + 4ω2 −(D −1) ˜V ′(ω) . (6.2)One more restriction on the one-cut solution is given by the requirement that theexpession under the square root in Eq.

(4.14) must be positive for any ω which belongs tothe support of the spectral density. If this expression becomes negative for some values oft2k’s this simply means that the one-cut solution is not realized and one should look fora more sofisticated support (multi-cut solutions).

This is quite standard for the large-Nphase transitions which occur at the values of couplings where the behavior of the spectraldensity changes.It is interesting to compare our solution with that obtained by the Riemann–Hilbertmethod. It is easy to identify the function Tλ(z) which determines the solution of Ref.

[3]8

with the one-link correlatorTλ(z) −1 = 1Nctr1z −ΦSU1λ −ΦSU†U(6.3)which is defined by the same average as in Eq. (5.1).

The proof is based on the followingextension of the Migdal procedure. Let us define [3]Gλ(Φx) ≡1I(Φx, Φx+µ)1λ −1Nc∂∂ΦxI(Φx, Φx+µ)(6.4)where I stands for the Itzykson–Zuber integralI(Φx, Φx+µ) =ZdU e Nc tr(ΦxUΦx+µU†) .

(6.5)By a direct differentiation of Eq. (6.5) one gets1Nctr1z −ΦxGλ(Φx)=* 1Nctr1z −ΦxU1λ −Φx+µU†+U.

(6.6)Rewriting the l.h.s. via the spectral density, substituting for Φx and Φx+µ the saddle pointvalue ΦS and remembering the definition of Tλ(z) [3]Tλ(z) = 1 +Zdµρ(µ)Gλ(µ)µ −λ,(6.7)one proves Eq.

(6.3).From Eq. (6.6) it is easy to see alternatively that Nc = ∞Tλ(z) −1 =* 1Nctr1z −ΦxUµ(x)1λ −Φx+µU†µ(x)+(6.8)where the average is w.r.t.

the same measure as in Eq. (1.1).

Therefore, we get asymp-toticallyTλ(z) = 1 + Eλz + Gλ(1)z2+ . .

.as z →∞(6.9)so that our Gλ(1) is to be compared with the O(z−2) term in the expansion of Tλ(z).It would be very interesting to calculate exactly the correlator on the r.h.s. of Eq.

(6.8),which should be manifestly symmetric w.r.t. λ and z, by solving the loop equations inorder to compare with Tλ(z) of Ref.

[3].It is worth mentioning that in finding the solution I did not use the fact that Φx canbe diagonalized. For this reason a solution analogous to that of this paper exists forthe adjoint fermion model [11] which reduces to the complex one-matrix model [8].

Thisresult, as well as an analysis of the D ≤1 case, will be published elsewhere. The mostinteresting question is whether the solution of this paper admits a continuum limit forD > 1.AcknowledgementsI thank the theoretical physics department of UAM for the hospitality in Madrid wherea part of the work was done.

This paper was not supported by grands of APS or IFS.9

Appendix ADerivation of the loop equationsLet us consider an equation which results from the invariance of the measure over Φ inthe open-loop average (2.2) under an infinitesimal shiftΦx →Φx + ξx(A.1)of Φx at the given site x with ξx being an infinitesimal hermitean matrix. For KMMone should impose tr ξx = 0 in order for the shifted matrix to belong to the adjointrepresentation of SU(Nc).

Since this condition should be inessential as Nc →∞, I deriveloop equations for the hermitean model, defined by the partition function (1.1) with theintegration going over arbitrary hermitean matrices Φx. ξx in Eq.

(A.1) is then arbitraryhermitean.It is convenient to introduce N2c generators[tA]ij =δij, [ta]ij(A.2)with ta (a = 1, . .

. , N2c −1) being the standard generators of SU(Nc).

The generators (A.2)obey the following normalization1Nctr tAtB = δAB(A.3)and completeness condition[tA]ij[tA]kl = Ncδilδkj . (A.4)An arbitrary Nc × Nc hermitean matrix Φ can be represented asΦ = tAΦAwhereΦA = 1Nctr Φ, 1Nctr taΦ(A.5)with Φ0 =1Nc tr Φ vanishing if Φ is taken in the adjoint representation of SU(Nc).To derive the loop equation I apply a trick similar to that used in deriving loopequations of QCD [9].

Let us consider the loop average* 1NctrtAU(Cxy)1λ −ΦyU†(Cxy)+= 0 ,(A.6)where the averaging is taken with the same measure as in Eq. (1.1), which vanishes dueto the gauge invariance.

Performing the shift (A.1) of Φx, using the invariance of themeasure and calculating ∂/∂ΦB(x), one gets* 1NctrtBV ′(Φx) 1NctrtAU(Cxy)1λ −ΦyU†(Cxy)+−DXµ=−Dµ̸=0* 1NctrtBUµ(x)Φx+µU†µ(x) 1Nctr tAU(Cxy)1λ −ΦyU†(Cxy)+= δxy* 1N3ctrtAU(Cxy)1λ −ΦytB1λ −ΦyU†(Cxy)+. (A.7)10

The l.h.s. of this equation results from the variation of the action while the r.h.s.

representsthe commutator term resulting from the variation of the integrand.The averaging over the gauge group picks up two nonvanising invariant equations forthe hermitean matrices.The first one can be obtained contracting Eq. (A.7) by δAB(A, B = 0, .

. .

, N2c −1) while the second one is given by the A, B = 0 component.The first equation reads* 1NctrV ′(Φx)U(Cxy)1λ −ΦyU†(Cxy)+−DXµ=−Dµ̸=0* 1NctrΦx+µU(C(x+µ)xCxy)1λ −ΦyU†(C(x+µ)xCxy)+= δxy* 1NctrU(Cxy)1λ −Φy 1Nctr1λ −ΦyU†(Cxy)+(A.8)where the contour C(x+µ)xCxy is obtained by attaching the link (x, µ) to the path Cxy atthe end point x as is depicted in Fig. 1.

Using the definition (2.2), this equation can bewritten finally in the form (2.3).The second equation which is given by the A, B = 0 component of Eq. (A.7) reads* 1Nctr V ′(Φx) 1Nctr1λ −Φy+−DXµ=−Dµ̸=0* 1Nctr Φx+µ1Nctr1λ −Φy+=* 1N3ctr1λ −Φy2+.

(A.9)In the large-Nc limit when the factorization holds, Eq. (A.9) is automatically satisfied asa consequence of the O(λ−1) term in Eq.

(A.8).Appendix BThe one-link correlator at large NcEq. (5.3) can be derived for Φ given by the saddle point (master field) configurationanalyzing the large mass expansion which allows to calculate the one-link correlator (5.1)in the strong coupling phase (i.e.

before an expected large-Nc phase transition).Tocalculate it, let us expand the numerator in powers of Φ as is depicted in Fig. 2.The idea is now not to calculate the complicated integral over Uµ(x) in Eq.

(5.1)but rather substitute for Φx and Φx+µ the saddle point value ΦS which is determinedby the future integration over Φ according to Eq. (1.1).

Notice, that the integrals overΦx and Φx+µ are independent to each order of the large mass expansion. Therefore, onesubstitutes(Φax)S(Φbx)S = K 1N2cδab(B.1)11

✛❡✲✉✲✉✉✛✲✉✉✛✲✉✉✛taΦxΦx+µ......Fig. 2:The graphic representation of the large mass expansion of the one-link corre-lator (5.1).

The right filled circles represent Φx+µ while the left ones representΦx. The empty circle represents ta.and(Φax)S(Φbx+µ)S = 0(B.2)whereK = 1Nctr Φ2S(B.3)and (B.2) vanishes due to the gauge invariance.After the use of Eqs.

(B.1), (B.2) and the completeness condition (A.4) the colorindices are contracted in such a way that all the matrices Uµ(x) disappear due to theunitarity. The simplest contractions are depicted in Fig.

3. Only the connected diagramsshould be taken into account due the presence of the denominator.

The diagrams of thetype of Fig. 3a are the only ones which emerge for the quadratic potential.

They alwaysresult in Λ0. The diagrams of the type Fig.

3b appear when the interaction is present.They result in Λm with m ≥1.Finally, let us notice that this structure of Λm is not spoiled by the fact that Φx enters2D links emanating from the point x. This affect only combinatorics making K to beD-dependent.12

✘✙✥✦✛✛✲✲❡✉=K1Nc tr ta Φxa)taΦx✘✙✔✕✘✙✔✕✛✛✲✲✲✛✉❡✉=t3 K31Nc tr ta Φ2xb)taΦxFig. 3:The contraction of color indices for diagrams of Fig.

2.a) The diagram results in the contribution to Λ0.b) The diagram results in the contribution to Λ1.References[1] V.A. Kazakov and A.A. Migdal, Induced QCD at large N, LPTENS-92/15 / PUPT-1322(June, 1992).

[2] V.A. Kazakov, D-dimensional induced gauge theory as a solvable matrix model, CERN-TH-6754/92 (December, 1992);G.J.

Gross, Some new/old approaches to QCD, LBL 33232 / PUPT 1355 (November,1992);G. Semenoffand N. Weiss, Symmetry and observables in induced QCD, UBC-TP-93-4(March, 1993);Yu. Makeenko, Matrix models of induced large-N QCD, ITEP-YM-11-92 (December, 1992).

[3] A.A. Migdal, Mod. Phys.

Lett. A8 (1993) 359.

[4] D. Gross, Phys. Lett.

293B (1992) 181. [5] A.A. Migdal, Mod.

Phys. Lett A8 (1993) 153.

[6] Yu. Makeenko, Mod.

Phys. Lett.

A8 (1993) 209. [7] D.V.

Boulatov, Infinite-tension strings at d > 1, NBI-HE-92-78 (November, 1992). [8] Yu.

Makeenko, Mod. Phys.

Lett. A6 (1991) 1901, and references therein.

[9] A.A. Migdal, Phys. Rep. 102 (1983) 199.

[10] A.A. Migdal, Mod. Phys.

Lett. A8 (1993) 245.

[11] S. Khokhlachev and Yu. Makeenko, Mod.

Phys. Lett.

A7 (1992) 3653.13

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