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Induced QCD Without Local Confinement

arXiv:hep-th/9212011v1 2 Dec 1992PUPT 1358, UBCTP92-032December 2 1992Induced QCD Without Local ConfinementM.I.Dobroliubov1,a, I.I.Kogan2,b, G.W.Semenoff1 andN.Weiss11. Department of Physics, University of British Columbia,Vancouver, British Columba, Canada V6T 1Z12.

Department of Physics, Princeton University,Princeton, New Jersey 08544 U.S.A.We examine some properties of the filled Wilson loop observables in the Kazakov-Migdal model of induced QCD. We show that they have a natural interpretation ina modification of the original model in which the ZN gauge symmetry is brokenexplicitly by a Wilson kinetic term for the gauge fields.

We argue that there aretwo large N limits of this theory, one leads to ordinary Wilson lattice gauge theorycoupled to a dynamical scalar field and the other leads to a version of the Kazakov-Migdal model in which the large N solution found by Migdal can still be used. Wediscuss the properties of the string theory which emerges.

(a) Permanent Address: Institute for Nuclear Research, Academy of Sciences ofRussia, 60TH October Anniversary Prospekt 7A, Moscow 117312, Russia. (b) Permanent Address: Institute for Theoretical and Experimental Physics, Bol-shaya Cheremushkinskaya, 117259 Moscow, Russia.1

The dynamics of QCD is known to simplify somewhat in the limit where the numberof colors, N, is large [1]. Only planar graphs contribute to scattering amplitudes and theresulting perturbation series exhibits some of the qualitative features of the strong interac-tions.

However so far, no explicit solution is available in the large N limit and it has thusled to very few quantitative results.Recently, Kazakov and Migdal [2] have proposed a novel approach which has the hopeof providing an exact solution of the large N limit of QCD. They consider induced QCDwhich is obtained by integrating over the scalar fields in the lattice gauge theory with thepartition functionZKM =Zdφ[dU] exp−NXxTrV [φ(x)] + NXTrφ(x)U(xy)φ(y)U†(xy)(1)where φ(x) are N×N Hermitean matrices which reside on lattice sites x, U(xy) are unitaryN×N matrices which reside on links < xy > between neighboring sites x and y, dφ is theEuclidean integration measure for Hermitean matrices, [dU] is the invariant Haar measure forintegration over the unitary group U(N) and V [φ] is a potential for the scalars.

This model isinvariant under the gauge transformations φ(x) →ω(x)φ(x)ω†(x), U(xy) →ω(x)U(xy)ω†(y)where ω(x) is an element of U(N). By restricting the trace of φ to zero and the determinantof U to one in (1) we could also consider a model with SU(N) gauge symmetry.This model is soluble in the large N limit.

By explicitly integrating over the unitarymatrices in (1) (see Appendix A) one obtains a field theory for the eigenvalues of φZKM ∝Z Yx,idφi(x)∆2[φ(x)]e−N Px V [φi(x)] Ydetij eNφi(x)φj(y)∆[φ(x)]∆[φ(y)](2)where ∆[φ] = detij(φi)j−1 = Qi

Corrections tothe classical behavior and the spectrum of elementary excitations can also be computed [4].This model has been considered further in [5] - [21].If the model (1) has a second order phase transition and if the fluctuations in the vicinityof the critical point are non-Gaussian, one might expect that the critical behavior shouldbe represented by QCD, the only known nontrivial four dimensional field theory with non-Abelian gauge symmetry. However it was pointed out in [5] that, like all adjoint latticemodels, the Kazakov-Migdal model has an extra gauge symmetry which is not a symmetryof continuum QCD.

The action in (1) is invariant under redefining any of the gauge matricesby an element of the center of the gauge group, U(xy) →z(xy)U(xy), φ(x) →φ(x), wherez(xy) ∈U(1) if the gauge group is U(N) and z(xy) ∈ZN if the gauge group is SU(N). (We shall call the symmetry a ZN gauge symmetry in either case.

This symmetry is themaximal subgroup of the transformations discussed by Gross [10] and Boulatov [18] whichcan be implemented with field independent elements z(xy).) Because of this symmetry theconventional Wilson loop observables of lattice gauge theory have vanishing average unlessthey have either equal numbers of U and U† operators on each link or else, in the case ofSU(N), unless they have an integer multiple of N U’s or N U†’s.2

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

The parameter α is the string tension. On the other hand if the expectationvalue of the Wilson loop goes like the exponential its perimeter then the potential is notconfining.

In the Kazakov-Migdal model, due to the ZN symmetry, the expectation value ofthe Wilson Loop is identically zero for all loops with non-zero area. We can interpret thisas giving an area law with α = ∞, and no propagation of colored objects is allowed at all.

(An exception is the baryon (UN) loops in the case of SU(N) where the correct statementis that N-ality cannot propagate.) It is for this reason that the original Kazakov-Migdalmodel has difficulty describing pure gluo-dynamics.There are currently several points of view on how to cure this difficulty.

In [5] it wassuggested that if there is a phase transition so that the ZN symmetry is represented in a Higgsphase, the resulting large distance theory would resemble conventional QCD. This approachhas been pursued in [8], [11], [12].

An alternative, which was advocated in [5,6], is to useunconventional observables such as filled Wilson loops which reduce to the usual Wilsonloop in the naive continuum limit but which are invariant under ZN. The third possibility isto break the ZN symmetry explicitly.

This was suggested by Migdal [14] in his mixed modelin which he breaks the ZN symmetry by introducing into the model heavy quarks in thefundamental representation of the gauge group. In this Letter we shall examine a differentexplicit ZN symmetry breaking and the use of filled Wilson loop observables.

We shall showthat the filled Wilson loops arise naturally from ordinary Wilson loops in a modified versionof the Kazakov-Migdal model which has additional explicit symmetry breaking terms. Weshall argue that one version of this modified model should be solvable in the large N limit.It thus provides a link between the approach which uses unconventional observables to solvethe ZN problem and the approach in which the symmetry is explicitly broken.We begin with a brief description of the filled Wilson loop operators which were intro-duced in [5] and discussed in detail in [6].

These are a special class of correlation functionswhich survive the ZN symmetry. They are defined by considering an oriented closed curveΓ made of links of the lattice.

The ordinary Wilson loop operator on Γ is given byW[Γ] = TrY∈ΓU(xy). (3)For any surface S which is made of plaquettes such that the boundary of S is the curve Γwe defineWF[Γ, S] = W[Γ]Y✷∈SW †[✷](4)where ✷denotes an elementary plaquette in the surface S. The filled Wilson loop for Γ isnow defined asWF[Γ] =XSµ(S)WF[Γ, S](5)3

where the sum is over all surfaces S whose boundary is the loop Γ with some (yet to bespecified) weight function µ(S). Notice that for each plaquette ✷∈S we have inserted thenegatively oriented Wilson loop W †[✷].

Thus for arbitrary weight functional µ(S) the filledWilson loop operator is invariant under the local ZN gauge symmetry since it has equalnumbers of U and U† operators on each link. Although we have assumed that the loop isfilled with elementary plaquettes this can be easily generalized to other fillings (the otherextreme case being the adjoint loop W[Γ]W †[Γ]).

We can also define the ‘filled correlator”of more than one loop by summing over all surfaces whose boundary is given by those loops.We now review some of the properties of the filled Wilson loop. In ref.

[6] it was shownthat computing the expectation value of WF[Γ] is equivalent to computing the partition func-tion of a certain statistical model on a random two–dimensional lattice. When computingZN gauge invariant correlation functions of U–matrices in the master field approximationthe φ–integral is evaluated by substituting the master field ¯φ = diag(¯φ1, .

. .

, ¯φN) for theeigenvalues of φ.< Ui1j1 . .

. U†k1l1 .

. .

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

. U†k1l1 .

. .R dφ[dU]e−Tr(PV [φ]−PφUφU†)(6)≈R d¯φ[dU]eTr(P ¯φU ¯φU†)Ui1j1 .

. .

U†k1l1 . .

.R d¯φ[dU]eTr(P ¯φU ¯φU†)If we consider for the moment surfaces which are not self-intersecting so that the filledWilson loop correlator has at most one UU† pair on any link we need to consider only thetwo field correlator < UijU†kl >. Gauge invariance implies that [6]< UijU†kl >= CijδilδjkwithCij =R [dU]eN PTr(¯φU ¯φU†)|Uij|2R [dU]eN PTr(¯φU ¯φU†)(7)Thus, in the master field approximation, the expectation value of the filled Wilson loop isgiven by< WF[Γ] >=XSµ(S)Ysitesx∈SNXi(x)=1Ylinks∈SCi(x) j(x)(8)This is a generalized Potts model on a random surface in which N–component spins resideat each site and the Boltzmann weights Cij for the bonds are correctly normalized to beconditional probabilities; Pi Cij=1, Pj Cij=1.Techniques for evaluating Cij for general N and for arbitrary ¯φ are presented in [16] and[17].

An explicit formula for SU(2) is given in [6]. Although the general formula for Cij inSU(N) is quite difficult to deal with, it is still possible to estimate the surface dependenceof the statistical model partition function in (8) when ¯φ is homogeneous by considering twodifferent limits.

(The details of the computation of Cij in these limits is described in theAppendix.) First is the limit in which ¯φ is small.

We call this the “High Temperature” limitsince in this limitCHTij= 1N + . .

. (9)4

is independent of i and j. It thus represents the Bolzman weights for a highly disorderedsystem.

We also consider the limit in which ¯φ is large. We call this the “Low Temperature”limit since in this limitCLTij = δij + .

. .

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

These two cases lead to profoundly different behaviorfor the filled Wilson loop. We shall assume that, by choosing the potential for the scalarfield in (2) appropriately, either of these limits could be obtained (the eigenvalue repulsiondue to the Vandermonde determinants in (2) and the possibility of adding repulsive centralpotentials makes the low temperature limit more natural).We begin by estimating the value of the filled Wilson loop for a fixed surface S. In the“high temperature” case the statistical model is disordered.

The sums over configurationsat the various sites are independent and they contribute an overall factor NV (where V isthe number of vertices on the surface) to the expectation value of the filled Wilson loop.Furthermore each link contributes a factor Cij=1/N so that the links contribute a totalfactor of N−L where L is the total number of links. It follows that the expectation value ofthe filled Wilson loop goes like< WF[Γ, S] >HT∼NV −L = N2−2g(S)N−A(S)(11)where A(S) is the area and g(S) is the genus of the surface S (i.e.

the number of plaquettescomprising S) and we have used Euler’s theorem, χ ≡2 −2g = V −L + A. We thus get therenormalization of the string tension δαHT = log N. Notice also that higher genus surfacesare suppressed and that the loop (genus) expansion parameter is 1/N2.

This is preciselywhat is obtained in the conventional strong coupling expansion of Wilson’s lattice gaugetheory which is known to describe a string theory with extra degrees of freedom associatedwith self-intersections of the string [22].In the “low temperature” case, the statistical system is ordered. The spins on all thesites are frozen at a uniform value.

In this case the partition function is proportional to thedegeneracy of the ground state,< WF[Γ, S] >LT= N(12)Note that in this case the statistical model gives no contribution to the string tension(δαLT ≈0) and there is no suppression of higher genus surfaces.In order to proceed to the evaluation of the filled Wilson loop we need to choose a weightfunction µ(S) in order to perform the sum over surfaces. The most reasonable criterion forchoosing such a weight function is our desire to get a finite physical string tension in thecontinuum limit.

In order to accomplish this goal we must choose a weight function µ(S)which depend on the area of the surface differently in the low and in the high temperaturecases. It is known that the number of closed surfaces with a given area grows exponentiallyasn(A) ∼Aκ(g)eµ0A(13)5

where κ(g) is a universal constant which depends only on the genus of the surface and µ0 is anon-universal, regulator dependent constant [23] which will lead to a renormalization of thestring tension. In our case, although the surfaces are open, the above formula should still bevalid for surfaces whose area is much larger than the area of the minimal surface boundedby Γ.

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

To accomplish the same goal in the “low temperature” phase weshould use µLT(S) ∼e−µ0A(S). Although these choices of µ(S) give the desired result, it israther unnatural to have to choose µ(S) in such an ad hoc fashion.Fortunately there is a very natural way to obtain the sum over surfaces in (8).

Considerthe following expectation value< WF[Γ] >= < W[Γ]eλP✷(W [✷]+W †[✷]) >< eλ P✷(W (✷)+W †[✷]) >(14)where W[Γ] is the conventional Wilson loop. Remember that the average is weighted by theKazakov–Migdal action as in (6): In the master field limit it is computed by integrating onlyover U–matrices with φ = ¯φ and with the Kazakov–Migdal action.

Note that the exponentin (14) is simply the conventional Wilson kinetic term for the gauge fields in lattice gaugetheory. If we expand the right hand side of (14) in λ the non–vanishing terms are all ofthose surfaces which fill the Wilson loop.

The result is thus a filled Wilson loop with asurface weight µ(S) = λA(S). It is clear that we could obtain exactly the same expression (inthe master field approximation) by evaluating the expectation value of the ordinary Wilsonloop operator in a modified version of the Kazakov–Migdal model in which a conventionalWilson term (λP✷W(✷) + W †[✷]) is added to the action.This term breaks the ZNgauge symmetry explicitly and allows Wilson loop operators with non–zero area to havenon–zero expectation values.

We would expect that it is necessary to keep λ small if oneis to maintain the successes of the Kazakov–Migdal model. We shall now argue that in the“low temperature” limit this picture is self–consistent in the sense that the physical stringtension is finite when λ is small and consequently the saddle point solution of the originalmodel is unchanged.

We shall also see that this is not the case in the “high temperature”phase.Let us begin by determining how λ should behave in the continuum limit if we are tohave a finite physical string tension. As discussed above a necessary condition for havinga finite physical string tension is that the string tension in lattice units should vanish.

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

Noticethat this λ is proportional to N in the “high temperature” phase and thus cannot be assumedsmall in large N.In the large N limit of conventional lattice gauge theory the coefficient of the Wilsonterm must be proportional to N if one is to obtain a consistent large N expansion. In ourcase we see that this is true for the “high temperature” phase in which case the Wilson termis of the same order as the Kazakov–Migdal term and it thus plays an important role in the6

infinite N limit. One can say that in this phase we have ordinary QCD.

Unfortunately it isimpossible to preserve the master field solution of the Kazakov-Migdal model in this limitsince the Wilson term, being of order N, would modify the large N solution, ruining theself-consistency of the mean-field approximation as described here.The situation is much more appealing in the “low temperature” phase. In this case therequired coefficient of the Wilson term is of order one.

It is subdominant and thereforenegligible in the large N limit. Thus, Migdal’s solution [3] of the Kazakov-Migdal modelin the large N limit should still apply to our proposed modification of the action.

In factthe only reason that the Wilson term is important at all in the large N limit of the “lowtemperature” phase is related to the collective phenomenon which orders the statisticalsystem on the surfaces. It effectively makes the statistical model’s contribution to the stringtension much smaller than would be expected from naive counting of powers of 1/N and atruly infinitesimal breaking of the ZN gauge symmetry (λLT/N →0 as N →∞) is sufficientto make the averages of Wilson loop operators non-vanishing.

The self-consistency of thispicture can also be demonstrated by computing the contribution of the Wilson term to thefree energy. This can be computed in a small λ expansion.

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

. !

(15)is of order N (where V is the volume, D is the dimension). This should be compared withthe free energy in the pure Kazakov-Migdal model which is proportional to N2.

Here, thefirst term in the free energy is the contribution of the doubled elementary plaquette andthe second term is due to the two orientations of the elementary cube. It is interestingthat, to order 6, there is no energy of interaction of doubled elementary plaquettes witheach other.

We conjecture that the interaction energy of surfaces is absent to all orders andthe free energy obtains contributions from all possible topologically distinct surfaces whichcan be built from elementary plaquettes. This suggests a free string picture of the “lowtemperature” limit of the Kazakov-Migdal model at lattice scales.We have thus far neglected the self-intersecting surfaces in the sum (8) which are gener-ated by the expansion of (14) in λ.

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

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

. .

UinjnU†k1l1 . .

. U†knln >= δi1j1 .

. .

δknlnSi1...ink1...kn(16)where Si1...ink1...kn is the tensor which is one if i1 . .

. in is a permutation of k1 .

. .

kn and is zerootherwise. It is now evident that in this limit the U–matrices are replaced by unit matriceswhich freeze together the spin degrees of freedom on the various intersecting surfaces.

As aspecial case we can consider a single, connected, self–intersecting surface. In this case all thespin indices on the surface are equal and since Si1...k1... = 1 when all arguments are equal thepartition function of the statistical model corresponding to that surface is simply N just asit was for a non-intersecting surface.

Thus just as the statistical model does not contributeto the string tension it also does not contribute to the interaction energy of self–intersectingsurfaces. This implies that in the “low temperature” limit, the sum over connected surfaces7

which have a common boundary behaves like a Nambu-Goto string theory with no internaldegrees of freedom.In summary, the self-consistency of the “low temperature” limit leads us to a new largeN limit of the conventional lattice gauge theory coupled to scalars:Z =Zdφ[dU] exp−NXxTrV [φ(x)] + NXTrφ(x)U(xy)φ(y)U†(xy)++λX✷(W(✷) + W †(✷))! (17)The conventional large N limit occurs when λ is of order N and describes scalar QCD.The other limit occurs when N →∞with λ of order one.

This model is soluble using theKazakov–Migdal approach.It is the latter case in which λ remains constant that is of special interest to us. In thiscase we saw that the large N expansion corresponds to a string theory with some unusualfeatures.

The partition function and the Wilson loop expectation value can be described asa sum over surfaces. What is unusual is that the genus of the surfaces is not suppressed inthe large N limit, as it is in continuum QCD.

(We do of course expect the higher correctionsin 1/N to suppress higher genus terms.) For a continuum string theory this sum over thegenus is badly divergent.

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

It is still a mystery to us how the sum over all surfaces at the lattice scaleshould turn into the sum over planar diagrams in the continuum theory of QCD.An alternative to the model presented here is the mixed model which was inventedby Migdal [14] to solve the problem of ZN symmetry.It contains heavy quarks in thefundamental representation of the gauge group. Despite the obvious differences between ourmodel and Migdal’s mixed model they have many features in common.

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

there wouldexist a size scale which is far enough into the infrared region that the quark potential islinear but the interaction energy is not yet large enough that it is screened by producingquark-antiquark pairs. Thus, in QCD we expect that adding heavy quarks would not ruinthe area law for Wilson loops smaller then some scale.The mixed model has just the opposite scenario, it is possible to get an area law only whenthe heavy quarks are light enough.

This is a result of the fact that, in the Kazakov-Migdalmodel, no Wilson loops are allowed at all unless the ZN symmetry is explicitly broken. Inthe mixed model, the ZN charge of links in a Wilson loop must be screened by the heavyquarks.

This can happen in two ways. First, the Wilson loop can just bind a heavy quarkto form an adjoint loop - giving a perimeter law for the free energy of the loop.

This isthe leading behavior if the fermion mass, M, is large. The free energy would go like 1/MPwhere P is the perimeter.

The only way an area law might arise is when the fermions arelight enough that their propagators could from a filled Wilson loop with free energy 1/M2Lwhere L ≈2A is the number of links. Then, since the entropy for filled loops is much larger8

than that for adjoint loops, these configurations would be important if M4 < eµ0. Then, theasymptotics behavior of the Wilson loop would still have a perimeter law but there wouldbe loops with 4A −P < µ0/ ln M where there would be an approximate area law.Acknowledgement This work is supported in part by the Natural Sciences and En-gineering Research Council of Canada and by the National Science Foundation grant #NSF PHY90-21984.

We thank D. Gross, Yu. Makeenko, A. Migdal, A. Morozov and S.Shatashvili for discussions.APPENDIX:Asymptotics of CorrelatorsThe key to the solvability of the Kazakov–Migdal model is the fact that the single–linkItzyksen–Zuber integral can be done analytically [24,25]IIZ =Z[dU]eN Pφiχj|Uij|2 = det(ij) eφiχj∆[φ]∆[χ](18)In this Appendix we consider two limiting cases of this integral and its correlators.

Thesehave already been solved in [16,17]. Here we present a quick and simple derivation in twolimiting cases.1.Low Temperature:First consider the case where ¯φi are large.

We also assumethat the eigenvalues ¯φi are not too close to each other in the sense thatPi̸=j1(¯φi−¯φj)2 << N(these two conditions are satisfied in the case of the semicircle distribution of the eigenvalues).The integral in (18) is known to be exact in the semi-classical approximation (see [6] for adiscussion). The classical equation of motion is [U ¯φU†, ¯φ] = 0 which, since ¯φ is diagonal,is solved by any U of the form U0 = DP where D is a diagonal unitary matrix and P isa matrix which permutes the eigenvalues, (P ¯φP)ij = δij ¯φP (i).

Also, when N is large and ¯φis not too small the identity permutation gives the smallest contribution to the action in(18) and therefore is the dominant classical solution. In this case we use this minimum toevaluate the correlators,I−1IZZ[dU]eN P ¯φi ¯φj|Uij|2Ui1j1 .

. .

UinjnU†k1l1 . .

. U†knln = δi1j1 .

. .

δknlnSi1···ink1...kn(19)where we have written the normalized integral over diagonal matricesSi1···ink1...kn =R QℓdθℓQp

. in is a permutation of k1 .

. .

kn0otherwise(20)We have decomposed the integration over unitary matrices into an integration over thediagonals and an integration over the unitary group modulo diagonals [26]. The diagonals arethe ‘zero modes’ for the semiclassical integral and must be integrated exactly.

The unitarymodulo diagonal integral is damped by the integrand and is performed by substituting theclassical configuration. Of course, to get the next to leading order the latter integration9

must be done in a Gaussian approximation. It can be done for the first few correlators, Theresult isCii = 1 −1NXk̸=i1¯φi −¯φk2when i ̸= jCij = 1N1¯φi −¯φj2Similar calculations can be easily done for correlators of more than two U’sCij,kl = I−1IZZ[dU]eN P ¯φi ¯φj|Uij|2|Uij|2|Ukℓ|2(21)i = j ,k = ℓCii,kk = 1 −1NXn̸=i"1(¯φi −¯φn)2 +1(¯φk −¯φn)2#i ̸= j ,k = ℓCij,kk = 1N1(¯φi −¯φj)2(22)i ̸= j ,k ̸= ℓCij,kl = 1N21(¯φi −¯φj)21(¯φk −¯φℓ)2 (1 + δikδjℓ) ,where the next corrections will be of the form1N21(¯φi−¯φj)2Pm̸=k1(¯φk−¯φm)2 ..

Note also thatfor the semicircle distribution of the eigenvalues one hasXj̸=i1¯φj −¯φin = 0 ,for n > 22. High Temperature: This corresponds to small ¯φ.

We obtain the correlators by Taylorexpansion:Cij =Z[dU]1 + NTr ¯φU ¯φU† + . .

.|Uij|2 = 1N +¯φi ¯φjN+ . .

.10

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[26] Despite the simple form of S its tensor structure is quite complicated. For exampleSi1i2k1k2 = δi1k1δi2k2 + δi1k2δi2k1 −δi1i2δk1k212

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