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Matrix Models of Induced Large-N QCD

arXiv:hep-th/9303079v1 14 Mar 1993ITEP-YM-11-92December, 1992Matrix Models of Induced Large-N QCDYu. Makeenko1Institute of Theoretical and Experimental PhysicsB.Cheremuskinskaya 25, 117259 MoscowAbstractI review recent works on the problem of inducing large-N QCD by matrix fields.In the first part of the talk I describe the matrix models which induce large-N QCDand present the results of studies of their phase structure by the standard latticetechnology (in particular, by the mean field method).

The second part is devotedto the exact solution of these models in the strong coupling region by means of theloop equations.Talk at the Seminar on Strings, Matrix Models and all that, Rakhov, FSU, October 1992.1E–mail: makeenko@vxitep.itep.msk.su / makeenko@desyvax.bitnet / makeenko@nbivax.nbi.dk

1IntroductionRecently there has renewed interest in the problem of inducing QCD by means of somepre-theory. As was proposed by Kazakov and Migdal [1], such a theory can be potentiallysolvable in the limit of large number of colors, Nc, providing the inducing model is thatof the (self-interacting) matrix scalar field in the adjoint representation of the gaugegroup SU(Nc) on the lattice.

The gauge field is attached in the usual way to make themodel gauge invariant except no kinetic term for the gauge field. The latter circumstancediffers the Kazakov–Migdal model from the standard lattice Higgs–gauge models.

It wasconjectured in Ref. [1] that the model undergoes, with decreasing the bare mass of thescalar field, a second order phase transition which is associated with continuum QCDwhen the critical point is approached from the strong coupling region.To solve the Kazakov–Migdal model in the strong coupling region, Migdal [2] appliedthe Riemann–Hilbert method and derived the master field equation to determine theNc = ∞solution.

An explicit solution of this equation for the quadratic potential isfound by Gross [3]. A surprising property of the master field equation (not yet completelyunderstood) is that it admits [2, 4] self-consistent scaling solutions with non-trivial criticalindices for the non-quadratic potentials.

Moreover, the very Riemann–Hilbert method ofRef. [2] was developed, in fact, to find such a scaling solution.These nice features of the Kazakov–Migdal model are due to the fact that the scalarfield is in the adjoint representation of the gauge group.

For this reason it can be di-agonalized by a (local) gauge transformation so that only O(Nc) degrees of freedom areleft and the saddle-point method is applicable as Nc →∞. However, a price for havingthe adjoint-representation field is an extra local ZN symmetry which leads [5] at Nc = ∞to local confinement (i.e.

the infinite string tension) rather than area law for the Wilsonloops.In the present talk I review the papers [6, 7, 8] where the questions of which modelsinduce large-Nc QCD with normal area and how to solve these models in the strongcoupling region were answered2. In Sect.

2 I describe the models, both scalar and fermionones, which induce large-Nc QCD. The mechanism exploited is based on the first orderphase transition which occurs with decreasing bare mass of the inducing field and isassociated with freezing the gauge field and the restoration of area law.In Sect.

3 Idiscuss the exact strong coupling solution for the quadratic potential, both in scalar andin fermion cases. It is obtained by solving loop equations which turn out to be a usefultool for studies of the matrix models.2For a review of other approaches, see recent surveys [9, 10, 11] and references therein2

2The models of induced large-N QCDSince the inducing matter field is in the adjoint representation of SU(Nc), Wilson loopsvanish to each order of the large mass expansion at Nc = ∞. This situation is associatedwith local confinement.

However, the area law is restored with decreasing bare mass at thepoint of the first order large-Nc phase transition. Its existence can be rigorously provenfor the single-plaquette adjoint action and is expected for more complicated models on thebasis of the mean field method.

Once the first order phase transition occurs, the propermodel will induce large-Nc QCD in the continuum.2.1Adjoint scalar modelA simplest way to induce large-Nc QCD is by adjoint representation scalars at Nf flavors(i.e. the number of different species).

The adjoint scalar model ASM is defined by thepartition functionZASM =Z Yx,µdUµ(x)YxNfYf=1dΦf(x) ePNff=1Px Nc tr−V [Φf (x)]+PDµ=1 Φf(x)Uµ(x)Φf (x+µ)U†µ(x)(2.1)where the fields Φf(x) (f = 1, . .

. , Nf) take values in the adjoint representation of thegauge group SU(Nc) and the link variable Uµ(x) belongs to the gauge group.

The potentialV [Φ] is given byV [Φ] = m2 Φ2 + . .

. (2.2)where m is the (square of the) bare mass of the scalar field.

The original Kazakov–Migdalmodel [1] corresponds to Nf = 1. Notice that the action in Eq.

(2.1) is diagonal w.r.t.the flavor indices.The case of Nf = 1 is a unique one when the matrix Φ(x) can be reduced to a diagonalform at each site of the lattice by means of a gauge transformation. Only at Nf = 1 theRiemann-Hilbert method of Ref.

[2] is therefore applicable. The alternative method ofsolving ASM with the quadratic potential at strong coupling is based on loop equations [7]and can be used at any Nf while gives at Nf = 1 the same result as the Riemann–Hilbertmethod.It is worth noting that one can integrate in (2.1) over arbitrary hermitean matricesrather than over those in the adjoint representation which gives at finite Nc a differentmodel for the case of a non-quadratic potential.These two modeld should coincide,however, as Nc →∞.3

2.2Adjoint fermion modelAn alternative to ASM is the adjoint fermion model (AFM) which is defined by thepartition functionZAF M =Z Yx,µdUµ(x)YxNfYf=1dΨf(x)d¯Ψf(x) e −SF [Ψ, ¯Ψ,U]. (2.3)Here SF[Ψ, ¯Ψ, U] is the lattice fermion actionSF[Ψ, ¯Ψ, U] =NfXf=1XxNc trVeven(¯Ψf(x)Ψf(x))−DXµ=1[¯Ψf(x)P −µ Uµ(x)Ψf(x + µ)U†µ(x) + ¯Ψf(x + µ)P +µ U†µ(x)Ψf(x)Uµ(x)](2.4)whereVeven(¯ΨΨ) = m¯ΨΨ + .

. .

(2.5)is a fermionic analogue of the potential (2.2) and m is the bare mass of the fermion field.In Eqs. (2.3), (2.4) Ψf(x) is the Grassmann anticommuting Nc ×Nc matrix field whileP ±µ = r ± γµ(2.6)stand for the projectors.

The case r = 0 corresponds to chiral fermions while r = 1 isassociated with Wilson fermions. As is well known, the chiral fermions describe 2DNfflavors in the naive continuum limit while Wilson fermions are associated with Nf flavors.2.3The induced actionThe above matrix models can be represented in the form of a gauge theory given by thepartition functionZ =Z Yx,µdUµ(x) e −Sind[Uµ(x)] ,(2.7)where the induced action for the gauge field Uµ(x) is defined by the integral over Φ(x) inEq.

(2.1) or over Ψ(x) and ¯Ψ(x) in Eq. (2.3):e −Sind[Uµ(x)] =Z YxNfYf=1dΦf(x) ePNff=1Px N tr−V [Φf(x)]+PDµ=1 Φf (x)Uµ(x)Φf (x+µ)U†µ(x)(2.8)ore −Sind[Uµ(x)] =Z YxNfYf=1dΨf(x)d¯Ψf(x) e −SF [Ψ, ¯Ψ,U] ,(2.9)with SF[Ψ, ¯Ψ, U] given by Eq.

(2.4).4

For the quadratic potential the result of integrating over Φ(x) or over Ψ(x) and ¯Ψ(x)is given by the large mass expansion:Sind[U] = −Nf2XΓ| tr U(Γ) |2l(Γ) 2ml(Γ)(2.10)for scalars orSind[U] = −NfXΓ| tr U(Γ) |2l(Γ)ml(Γ) SpYl∈ΓP ±µ(2.11)for fermions. In Eq.

(2.11) Sp stands for the trace over the spinor indices of the path-ordered product of the projectors (2.6) (plus or minus depends on the orientation of thelink l) along the loop Γ.One easily sees that only single plaquettes survive in the sum over path on the r.h.s.’sof Eqs. (2.10) and (2.11) if m ∼N1/4fas Nf →∞, so that the single plaquette adjointaction arises in the large-Nf limit:SA = −βA2Xp| tr U(∂p) |2(2.12)withβA = 4Nfm4for scalars;βA = 2D2 −1Nf(1 + 2r2 −r4)m4for fermions .

(2.13)This shows of how ASM and AFM induce the single plaquette lattice gauge theory withadjoint action.2.4The large-N phase transitionThe inducing of large-Nc QCD relies on the fact [12] that the lattice gauge theory definedby the partition function (2.7) with the action (2.12) undergoes the first order large-Ncphase transition at βA ≈2 after which the gauge field Uµ(x) becomes frozen near somemean-field value η (η →1 as βA →∞).The proof of the existence of the phase transition is based solely on the factorizationat large-Nc which says that the adjoint action (2.12) is equivalent at Nc = ∞to thesingle-plaquette fundamental actionSF[U] = Nc ¯βXpℜtr U(∂p)(2.14)providing the coupling ¯β is determined by¯β = βAWF(∂p; ¯β)(2.15)5

where WF(∂p; ¯β) stands for the plaquette averageWF(∂p; ¯β) ≡R Qx,µ dUµ(x) e −SF [U] 1Nc tr U(∂p)R Qx,µ dUµ(x) e −SF [U].(2.16)Eq. (2.15) can be naively obtained substituting one of two traces in the action (2.12) bythe average.

A rigorous proof [12] is based on the loop equations.The existence of the first order phase transition for the action (2.12) with decreasingβA can be seen as follows. Let us solve Eq.

(2.15) for ¯β(βA) substituting for (2.16) thestrong coupling expansionWF(∂p; ¯β) =¯β2 +¯β58at strong coupling . (2.17)Eq.

(2.15) possesses at any ¯β a trivial solution ¯β = 0. However, one more solution appearsfor ¯β ≈2:¯β ∝12 −1βA 14 ,(2.18)which matches the weak coupling solution¯β →βA −14asβA →∞.

(2.19)Notice that ¯β ≪1 for the solution (2.18) when βA ≈2 so that the strong couplingexpansion is applicable.The adjoint plaquette averageWA(∂p; βA) ≡R Qx,µ dUµ(x) e −SA[U] 1N2c| tr U(∂p) |2 −1R Qx,µ dUµ(x) e −SA[U](2.20)which is given due to the factorization byWA(∂p; βA) =WF(∂p; ¯β)2 = ¯ββA!2(2.21)is depicted in Fig. 1.

Since the slope is negative for the solution (2.18) near βA = 2,a first order phase transition must occur with increasing βA.This negative slop is aconsequence solely of the positive sign of the second term on the r.h.s. of Eq.

(2.17). Thepredicted value of the critical coupling β∗A, at which the phase transition occurs, obeysβ∗A < 2, as is seen from Fig.

1, to be compared with the result of Monte–Carlo simulationsβ∗A = 1.7 −1.8.2.5The mean field phase diagramAt finite Nf the induced actions (2.10) or (2.11) can not be exactly analyzed even atNc = ∞. An approximate mean field method, which usually works very well in the casesof first order phase transitions, was applied to obtain the phase diagram in Refs.

[6, 8].6

Naively, the mean field approximation consists in substituting the link variable Uµ(x)by the mean field value[Uµ(x)]ij = η δij(2.22)everywhere but one link and writing a self-consistency condition at this link. The self-consistency condition is given by the one-link problemη2 =R dU ebA2 | tr U |2 1N2 | tr U |2R dU ebA2 | tr U |2(2.23)wherebA =R Qx dΦ(x) ePx N tr−V [Φ(x)]+η2 Pµ Φ(x)Φ(x+µ)1N tr Φ(0) Φ(0 + µ)R Qx dΦ(x) ePx N tr−V [Φ(x)]+η2 Pµ Φ(x)Φ(x+µ).

(2.24)These naive mean field formulas can be obtained [6] in the framework of the variationalmethod.The mean field phase diagram which was obtained by an analysis of Eqs. (2.23) and(2.24) is depicted in Fig.

2. At Nf = 1 there is no first order phase transition for thequadratic potential in the stability region m > 2D.

For m < 2D the model is unstableand were in the Higgs phase if the stabilizing higher order in Φ terms would be added tothe potential (2.2). The desired large-Nc phase transition appears for Nf > N∗f ≈30.3ASM looks in this region exactly like the single-plaquette adjoint model discussed in theprevious subsection.A similar phase diagram for AFM is depicted in Fig.

3. Now there is no Higgs phase (oran unstability region for the quadratic potential) due to the fermionic nature of inducingfields.

For the cases of chiral and Kogut–Susskind fermions the first order phase transitionis present already for Nf = 1 while the result for Wilson fermions is less certain.2.6Area law versus local confinementAt the point of the first order large-Nc phase transition, the area law behavior of the(adjoint) Wilson loops which is associated with normal confinement is restored in ASMor AFM analogously to the single-plaquette adjoint action [12].In order to see this, let us consider the adjoint Wilson loop which is defined byWA(C) =* 1N2c| tr U(C) |2 −1+(2.25)where the average is understood w.r.t. the same measure as in Eq.

(2.1) or in Eq. (2.3).Alternatively, one can average w.r.t.

the induced actions (2.10) (2.11) which recovers at3Such a phase diagram is compatible with the Monte–Carlo studies of Ref. [13].7

Nf = ∞the single plaquette adjoint action (2.12). In this limiting case the followingextension of the factorization formula (2.21) holds at Nc = ∞:WA(C; βA) =WF(C; ¯β)2 ,(2.26)where WF(C; ¯β) is defined by the same formula as (2.16) with ∂p replaced by an arbitrarycontour C and ¯β versus βA given by Eq.

(2.15).Since ¯β = 0 for βA < β∗A, WA(C) vanishes in this region due to Eq. (2.26) except theloops with vanishing minimal area Amin(C):WA(C) = δ0Amin(C) + O 1N2c!.

(2.27)On the contrary, the area law with the string tensionKA(βA) = 2 KF(¯β(βA))(2.28)holds for βA > β∗A when Eq. (2.15) possesses the non-trivial solution.

An extension ofthese formulas to finite Nf is given in Ref. [7].While the first order phase transition associated with the restoration of area law lookssimilar for ASM and AFM, the continuum limits should be approached in different ways.For ASM the continuum QCD is reached at the line of second order phase transitionswhich separates the area law and Higgs phases provided that one approaches it from thearea law phase.

For AFM there is no Higgs phase and continuum QCD is reached asm →0.3Loop equations at strong couplingThe loop equations of ASM or AFM relate the closed adjoint Wilson loop (2.25) tothe open ones with scalars or fermions at the ends. The loop equations are drasticallysimplified at Nc = ∞in the strong coupling region where the closed loops obey Eq.

(2.27).The exact solution can be obtained in both cases for the quadratic potential when the loopequations turns out to be equivalent to those for the hermitean and complex one-matrixmodels, respectively.3.1Loop equations for arbitrary potentialThe generic object which appear in the loop equations are open Wilson loopsδff′Gλ(Cxy) =* 1Nctr Φf(x)U(Cxy)1λ −Φf′(y)U†(Cxy)! +.

(3.1)8

The appearance of the δ-symbol w.r.t. the flavor indices f and f ′ is due to the fact thatthe interaction terms in the action entering Eq.

(2.1) are diagonal over the flavor indices.The loop equations of ASM result from the invariance of the measure in Eq. (2.1)under an arbitrary shift of Φf(x) and reads* 1NctrV ′(Φf(x))U(Cxy)1λ −Φf′(y)U†(Cxy)+−DXµ=−Dµ̸=0* 1NctrΦf(x + µ)U(C(x+µ)xCxy)1λ −Φf′(y)U†(C(x+µ)xCxy)+=·δff′δxy* 1NctrU(Cxy)1λ −Φf(y) 1Nctr1λ −Φf(y)U†(Cxy)+(3.2)where the path C(x+µ)xCxy on the l.h.s.

is obtained by attaching the link (x, µ) to the pathCxx at the end point x as is depicted in Fig. 4.

I have omitted additional contact termswhich arise at finite Nc due to the fact that Φ belongs to the adjoint representation, sothat Eq. (3.2) is written for the hermitean matrices.

This difference disappears, however,as Nc →∞.The analogues of Eqs. (3.1) and (3.2) for AFM readδff′Gijλ (Cxy) =* 1NctrΨif(x)U(Cxy)λ¯Ψjf′(y)λ2 −¯Ψf′(y)Ψf′(y)U†(Cxy)+(3.3)where i and j are spinor indices, andD 1NctrΨf(x)V ′even(¯Ψf(x)Ψf(x))U(Cxy)¯Ψf(x)λ −¯Ψf(x)Ψf′(y)U†(Cxy) E−DXµ=1D 1NctrP +µ Ψf(x + µ)U(C(x+µ)xCxy)λ¯Ψf′(y)λ −¯Ψf′(y)Ψf′(y)U†(C(x+µ)xCxy)+P −µ Ψf(x −µ)U(C(x−µ)xCxy)λ¯Ψf′(y)λ −¯Ψf′(y)Ψf′(y)U†(C(x−µ)xCxy)E= δff′δxyD 1NctrU(Cxy)λλ2 −¯Ψf(y)Ψf(y) 1Nctrλλ2 −¯Ψf(y)Ψf(y)U†(Cxy) E.(3.4)The matrix multiplication over the spinor indices is implied in this equation.3.2Loop equations at large NcThe path Cxy on the r.h.s.

of Eq. (3.2) (or Eq.

(3.4)) is always closed due to the presenceof the delta-function. The explicit equation for the case of vanishing (or contractable)contour Cxx = 0 at large Nc, when the factorization holds, readsZC1dω2πiV ′(ω)(λ −ω)Eω −2DGλ(1) = E2λ(3.5)9

whereEλ ≡* 1Nctr1λ −Φf(x)+= 1λ(Gλ(0) + 1)(3.6)with Gλ is defined by Eq. (3.1).

I have denoted the one-link average byGλ(1) = Gλ(C(x±µ)x)(3.7)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. (3.5) plays the role of a projector picking up negative powers of λ.For Cxx ̸= 0, the averages of a new kind arise on the r.h.s.

of Eq. (3.2) (or Eq.

(3.4)).However, these averages obey at Nc = ∞the following analogue of Eq. (2.27)* 1NctrU(Cxx)1λ −Φf(x) 1NctrU†(Cxx)1λ −Φf(x)+= δ0,Amin(C)E2λ + O 1N2c!(3.8)i.e.

vanish for Cxx ̸= 0.Hence, the loop equation for Cxy ̸= 0 at Nc = ∞reads* 1NctrV ′(Φf(x))U(Cxy)1λ −Φf′(y)U†(Cxy)+−DXµ=−Dµ̸=0Gλ(C(x+µ)xCxy) = 0(3.9)independently of whether Cxy is closed or open. Therefore, the r.h.s.

of the loop equationin nonvanishing at Nc = ∞only for Cxy = 0 (modulo backtrackings) when the properequation is given by Eq. (3.5).Finally, the fermionic analogues of Eqs.

(3.8) and (3.6) read* 1Nctr U(Cxx)λλ2 −¯Ψf(x)Ψf(x)! 1Nctr U†(Cxx)λλ2 −¯Ψf(x)Ψf(x)!+= δ0,Amin(C)E2λ + O 1N2(3.10)andEλ =* 1Nctrλλ2 −¯Ψf(x)Ψf(x)+= 1λ(Giiλ(0) + 1)(3.11)with Gλ defined by Eq.

(3.3). Therefore, the r.h.s.

of Eq. (3.4) involves at Nc = ∞onlyEλ similarly to the scalar case.3.3The quadratic potentialThe quadratic potential is always solvable, even in the non-diagonizable cases, for thefollowing reasons.

Let us consider the one-link correlatorD 1Nctr taUχf′U†EU ≡R dU e Nc Pf ϕfUχf U† 1Nc tr taUχf′U†R dU e Nc Pf ϕfUχf U†(3.12)10

where ta (a = 1, . .

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

(3.13)At Nc = ∞the formulaD 1Nctr taUχf′U†EU = Λ 1Nctr taϕf′ ,(3.14)where Λ is a constant to be determined below, can be proven for ϕ and χ given by themaster field for the quadratic potential analyzing the large mass expansion. The point isthat terms like tr taϕkf′ with k > 1 never appear for the quadratic potential.

Analogouslyit can be proven thatD 1Nctr taU1λ −χf′ U†EU = Λ 1Nctr ta1λ −ϕf′ . (3.15)For Nf = 1 Eqs.

(3.14) and (3.15) recovers the ones of Ref. [14].For Gλ(Cxy) defined by Eq.

(3.1), Eq. (3.15) impliesG(Cxy) = ΛLEλ(3.16)where L is the algebraic length (i.e.

the one after contracting the backtrackings) of Cxy.The fermionic analogue of this formula readsGijλ (Cxy) = ΛLEλYl∈CxyP ±µij(3.17)where the plus or minus signs correspond to the direction of the link l which belongs tothe contour Cxy. The spin factor will provide below the cancellation of the projectors inEq.

(3.4).The constant Λ can be determined by substituting the ansatz (3.16) into the Cxy ̸= 0loop equation (3.9) which simplifies for the quadratic potential as [7]mGλ(Cxy) −DXµ=−Dµ̸=0Gλ(C(x+µ)xCxy) = 0. (3.18)The ansatz (3.16) satisfies this equation for any Cxy ̸= 0 providingΛ =2m +qm2 + 4(1 −2D)σ(3.19)where σ = 1 for scalars orσ = P +µ P −µ = r2 −1(3.20)11

for fermions (σ = −1 for chiral fermions and σ = 0 for Wilson fermions).The remaining function Eλ can now be determined from Eq. (3.5) which for thequadratic potential reads˜mEλ = E2λ ,˜m = m −2DΛ(3.21)and coincides with the loop equation for the Gaussian hermitean one-matrix model (fora review, see Ref.

[15] and references therein). The solution of Eq.

(3.21) which satisfiesEλ →1λas λ →∞,(3.22)as it should be due to the definition (3.6), is unambiguous:2Eλ = ˜mλ −˜mqλ2 −4/ ˜m . (3.23)The imaginary partℑEλ ≡ρ(λ) = 14π ˜mq4/ ˜m −λ2for−2/√˜m ≤λ ≤2/√˜m(3.24)recovers the solution [3].Analogously, the Cxx = 0 loop equation for AFM with thequadratic potential is reduced to the loop equation for the complex one-matrix model [15].One should not be surprised that the exact strong coupling solution for the quadraticpotential does not depend on Nf which is a consequence of the peculiar behavior of theWilson loops (2.27).

This independence does not contradict to the fact that the firstorder phase transition discussed in Sect. 2 occurs only for Nf > N∗f .

The point is thatthe strong coupling solution is not sensitive to the phase transition which occurs due toanother (thermodynamic) reason.3.4Interpretation as the 1D tree problemA question arises what combinatorial problem are the exact solutions of the previoussubsection associated with? To answer, let us consider the open loop correlatorδff′G(Cxy) = 1NctrΦf(x)U(Cxy)Φf′(y)U†(Cxy) .

(3.25)which is nothing but the λ−2 term of the expansion of (3.1) in λ−1. At Nc = ∞thestandard sum-over-path representation of G(Cxy) readsG(Cxy) =XΓyx 2ml(Γ)+1WA(CxyΓyx)(3.26)where the contour Γyx forms together with Ccy a closed loop passing x and y.12

Since in our case the formula (2.27) associated with the infinite string tension holds,Γyx must coincide with (Cxy)−1 (i.e. passed in opposite direction) modulo backtrackingsof Γ. Eq.

(3.26) then yieldsG(Cxy) =XΓyx 2ml(Γ)+1(3.27)where the sum goes over contours of the type depicted in Fig. 5.4The fermonic analogues of Eqs.

(3.25) to (3.27) readGij(Cxy) =XΓyx 1ml(Γ)+1WA(CxyΓyx)Yl∈CxyP ±µij(3.28)andGii(Cxy) =XΓyx 1ml(Γ)+1SpYl∈CxyP ±µ(3.29)with summing again over contours depicted in Fig. 5.The proper combinatorial problem is, therefore, that of summing over 1-dimensionaltrees embedded in a D-dimensional space.

The exact solution of loop equations for thequadratic potential represents the solution to this problem:G(Cxy) = ΛL2D −1m(D −1) + Dqm2 + 4(1 −2D)σ,(3.30)where L is the algebraic length of Cxy and Λ is defined by Eq. (3.19).

Such a dependenceon Λ is evident from the representation (3.27) (or (3.29)) since the trees are uniformlydistributed along Cxy.As is already mentioned in the previous section, this solution coincides for scalarswith that of Ref. [3].Eq.

(3.30) takes an especially simple form for Wilson fermionswhen the backtracking parameter σ, given by Eq. (3.20), vanishes so that there are nobacktrackings.

For chiral fermions when σ = −1 the solution (3.30) coincides with thatof Ref. [16] for the case of the fundamental representation and vanishing constant in frontof the plaquette term in the action.

The point is that Wilson loops vanish in this case aswell (except for those with vanishing minimal area) and exactly the same combinatorialproblem of summing the diagrams of the type depicted in Fig. 5 emerges.AcknowledgementI am grateful to the theoretical physics department of the University of Zaragoza for thehospitality last December when the manuscript was being prepared for publication.4One should not confuse the double lines in Fig.

5 which are due to backtrackings with the doublelines in Fig. 4 which represent the adjoint representation.13

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Figures✚✛✭✭✭✭✭✭✭✭✭✭✭✭✭✭12β∗AWA(∂p; βA)βA0∞0Fig. 1The two solutions of Eq.

(2.15). The line which starts at βA = 2 is associated with thesolution (2.18).

Since the slope is negative for this solution near βA = 2 (and WA ≪1),the first order phase transition occurs at some β∗A < 2 so that the actual behavior ofWA(∂p; βA) is depicted by the bold line.15

❏❏❏❏❏❏❏❏❏✡✡✡✡✡✡✡✡✡N∗f∞Nf1/2D1/m0∞1ArealawLocalconfinementHiggsphaseFig. 2The mean field prediction for the phase diagram of ASM.

The bold lines whichbounds the local confinement phase is that of first order phase transitions. The line whichseparates the area law and Higgs phases is that of second order phase transitions.16

∞Nf1/m∗1/m0∞1LocalconfinementArealawFig. 3The mean field prediction for the phase diagram of AFM.

The bold line which separatesthe local confinement and the area law phases is that of first order phase transitions.17

xyxy✧✧x + µa)b)✻✻❝s❄❝s❄✧✧Fig. 4The graphic representation for Gλ(Cxy) (a) and Gλ(C(x+µ)xCxy) (b) entering Eq.

(3.2).The empty circles represent Φf(x) or Φf(x + µ) while the filled ones represent1λ−Φf′(y).The oriented solid lines represent the path-ordered products U(Cxy) and U(C(x+µ)xCxy).The color indices are contracted according to the arrows.✁✄✂✄✂✁✁✄✂✁yxCxyΓyxFig. 5The typical paths Γyx which contribute the sum on the r.h.s.

of Eq. (3.27) (andEq.

(3.29)). These Γyx coincide with Cxy passed backward modulo backtrackings whichform a 1D tree.

The sum over Γyx is reduced to summing over the backtrackings.18

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