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Quark confinement, topological susceptibility and all

arXiv:hep-ph/9207238v1 14 Jul 1992CERN-TH.6564/92Quark confinement, topological susceptibility and allthat in 4 dimensional gluodynamics.A.R.Zhitnitsky 1CERN , Geneva, SwitzerlandandInstitute of Nuclear Physics, Academy of Sciences of the USSR, 630090 Novosibirsk,USSR.AbstractWe discuss a few tightly connected problems, such as the U(1) problem, confine-ment, the θ -dependence within a framework of the dynamical toron approach. Wecalculate two fundamental characteristics of the theory: the vacuum expectationvalue (vev) of the Wilson loop and the topological susceptibility.

The analogy withwell known 2+1 dimensional QED which exhibits confinement phenomenon is alsodiscussed.CERN-TH.6564/92July 19921e-mail address is zhitnita@vxcern.cern.ch1

1.Introduction.It is generally believed that the most profound features of the QCD- confinement andnontrivial θ dependence , the chiral symmetry breaking and the absence of the ninthGoldstone boson are tightly connected. Let me briefly sketch these old arguments.We use the standard notations and the θ vacuum is introduced to the theory in thefollowing way:∆L = θQ(1)where Q is the topological charge which can be written for the 4d gluodynamics as follows:Q =132π2Zd4xGaµν ˜Gaµν =132π2Zd4x∂µKµ,a = 1, 2, 3. µ, ν = 1, 2, 3, 4.

(2)Here Gaµν is the field strength tensor. It is known , the θ term preserves renormalizabilityof the theory but is P and T odd.

As it can be seen from (2) the θ term is a full divergenceand therefore it is equivalent to certain boundary conditions . However as is known, theθ is the physical parameter of the theory and the θ- dependence of physics is linked tothe U(1) problem.

[1],[2] . Indeed, if we believe that the resolution of the U(1) problemappears within the framework of these papers ,we must assume that the correlatorK = iZd4x < 0|T132π2Gaµν ˜Gaµν(x),132π2Gaµν ˜Gaµν(0)|0 >(3)is nonzero in pure Yang Mills theory (YM).

It means that the vacuum expectation value(vev) of the topological density< |132π2Gaµν ˜Gaµν| >= 12Kθ,θ ≪1. (4)is also nonzero .

Therefore, the θ is observable parameter. The nonzero vev < G ˜G ≯= 0does imply the CP -violation in physical transition and leads , for example ,to the mixingof the heavy quarkonium levels with JP = 0+ and JP = 0−.

Only dispersion relationsare used to translate < G ˜G ≯= 0 into a proof of CP- violation in physical effects. [3].So, if we believe that U(1) problem is solved in the framework of Witten-Venezianoapproach [1],[2] we automatically get K ̸= 0 (3) and therefore, the nontrivial θ dependence.With introduction of the light quarks (u,d,s) the value of correlator K(3) changes 2,but θparameter is still an experimentally observed quantity.The next link comes from the analysis of the Ward Identities (WI) which imply [4]that the U(1) problem is linked to the chiral symmetry breaking phenomenon.

Indeed,the relevant WI ( in QCD with Nf flavors) takes the form:KQCD = mNf< ¯ψψ > −m2N2fZdx ¯ψγ5ψ(x), ¯ψγ5ψ(0)(5)and if we have the dynamical solution of the U(1) problem (by other words,we are in po-sition to calculate the singlet correlator in (5) to show its smallness in the chiral limit) we2It is obviously that K ∼m1q irrespectively to the number of light flavors [2].1

should explain the relation between < ¯ψψ > condensate ,responsible for the chiral sym-metry breaking and topological susceptibility , responsible for the solution U(1) problem.Besides that , in a similar way one can check that nontrivial θ dependence in pure YMtheory is through θ/N at large N ( in particular,< ˜GG >∼sin( θN ) )[2]. Such a functioncan be periodic in θ with period 2π only if there are many vacuum states for given valuesof θ.

These vacua should not be degenerated due to the vacua transitions , however thetrace of the enlargement number of the vacuum states have to be seen in the course ofa dynamical solution of the U(1) problem. Therefore , we have a link between the U(1)problem and an additional (discrete) classification of the vacuum states in the theoryapart from the standard θ classification.Another link , between U(1) problem and confinement phenomenon , can be under-stood from the analysis of the effective lagrangian describing the low-energy spectrumand dynamics of the pseudoscalar nonet in the large N limit [5],[6].

The most impor-tant assumption which has been made in the deriviation of the corresponding effectivelagrangian was confinement. As it is known the obtaining lagrangian perfectly describesall properties mentioned above .

In particular this lagrangian reproduces the correct θ/Ndependence in YM theory.Indeed, gluodynamics can be understood as a QCD with very large quark’s mass. Inthis limit effective lagrangian was founded in [5] and it turns out that the number ofvacua is of order N at N →∞.

This fact actually is coded in the effective lagrangiancontaining the multi-branched logarithm log det(U). In the Veneziano approach [2] thesame fact can be seen from the formula for multiple derivation of the topological densityQ with respect to θ at θ = 0.∂2n−1∂θ2n−1 < Q(x) >∼( 1N )2n−1 , n = 1, 2...(6)Therefore,the main idea of this sketch is as follows.

All problems under considerationare tightly connected.Thus , any selfconsistent dynamical solution of one of them shouldbe necessarily accompanied by the resolution of the rest problems within same approach.The purpose of this letter is to demonstrate that such links indeed take place withinframework of the dynamical toron 3 approach which was discussed early in context ofdifferent field theories, see ref. [7] and references therein.

I would like to recall that inall known cases, the toron calculations give, at least, the selfconsistent results. Thus,one may expect that the analogous situation should takes place in the theory underconsideration,i.e.

in 4 dimensional gluodynamics with SU(2) gauge group.We discuss the statistical ensemble of quasiparticles which, presumably [8], describesthe grand partition function of the 4d YM theory and which possesses strong quasipar-ticle interaction. It was shown that this ensemble describes the system with nontrivial3 We keep the term ”toron”, introduced in ref.[9].

By this means we emphasize the fact that theconsidering solution minimizes the action and carries the topological charge Q = 1/2,i.e. it possesses allthe characteristics ascribed to the standard toron [9].

However I should note from the very beginning ofthis paper, that our solution has nothing to do with the standard toron and it is formulated in principlein another way than in ref.[9]. The keeping of this term has a historical origin.2

θ- dependence irrespectively to the strength of quasiparticle interaction . It should benoted,that the coexistence of the strong quasiparticle interaction and the nontrivial θdependence is the nice feature of YM theory .

3d QED also possesses the long rangequasiparticle interaction and confinement phenomenon [10]. In this simple model physicsbecomes θ independent just because of the strong quasiparticle interaction [11].

Crucialpoint is a very nontrivial algebraic structure of the quasiparticle interaction in YM theory,in comparision with analogous calculation of ref. [11] in Polyakov’s model.We will discuss the main assumptions which have been made in the description [8] ofthis statistical ensemble a bit later.Now I would like to cite the result of calculation ofthe topological density as a simplest application of the formulae obtained in ref.

[8]:< ˜GG >= i2Λ4 sin(θ2),−π ≤θ ≤π,(7)where the Λ is some renormalization invariant calculable combination 4The nonzero value for topological density and its θ dependence are in agreement withmain assumption of Witten-Veneziano approach [1],[2]. Indeed, the formula (7) impliesthat the topological susceptibility which is nothing but derivative of the topological densitywith respect to θ, is nonzero.

Besides that, this formula is in agreement with Venezianoexpression (6) for the multiple deriviation with respect to θ.As it was argued above, the dynamical calculation of the topological density shouldbe accompanied by the resolution of the rest problems. One such link was establishedalready in the previous papers [8], [12].

Namely,it was demonstrated that the calculationof < ˜GG > is accompanied by appearing of the additional quantum number (apart fromthe standard parameter θ) classifying a vacuum states.In this letter we will consider the less trivial applications of the formulae describingthe grand partition function of YM theory , ref.[8]. Namely, we will calculate the vev ofthe Wilson loop to demonstrate the confinement in this theory.

This is the main result ofthe work.Besides that, we find the topological susceptibility by explicit calculation ( and notby differentiating < ˜GG > with respect to θ) in order to check the selfconsistency ofthe approach. In such calculation we are able to keep the nonleading term (as k →0)in the correlator for the susceptibility.

It turns out ,that this term is of order (k4) andnot of order (k2) as it could be expected naively. Such behavior for the correlator is thedirect result of the strong quasiparicle interaction and might have a phenomenologicalconsequences.For example, the proton ”spin” problem can be reduced to the problem ofcalculation of the first moment of the topological susceptibility [13].Before we proceed to the detail consideration of the Wilson loop and topologicalsusceptibility let me briefly formulate the basic assumptions of the toron approach.i) I allow the configurations with fractional topological charge (one half for SU(2)group) in the definition of the functional integral.

It means that a multivalued functionsappear in the functional integral. However, the main physical requirement is - all gauge4Formula (7) is written in Euclidean space because its calculation based on the self dual solutiondefined in this space.

In contrary, the formulae (3,4) is in Minkowski space.3

invariant values must be singlevalued . Thus , the different cuts accompany the multi-valued functions should be unobservable, i.e.

the gauge invariant values coincide on theupper and on the lover edges of the cut.The direct consequence of the such definition of the functional integral is the appearingof the new quantum number classifying a vacuum states. Indeed, as soon as we allowedone half topological charge,the number of the classical vacuum states is increased bythe same factor two in comparision with a standard classification, counting only integerwinding numbers |n >.Of course,vacuum transitions eliminate this degeneracy.

However the trace of enlarge-ment number of the classical vacuum states does not disappear.Vacuum states nowclassified by two numbers : 0 ≤θ < 2π and k = 0, 1. Let me repeat that origin for thisis our main assumption that fractional charge is admitted and therefore the number ofclassical vacuum states is multiplied by a factor two.I have to note that the same situation takes place in the supersymmetric YM theory,but in this case the vacuum states are still degenerate after vacuum transitions.

Thenumber k in this case just numerates different vacua at the same θ. However, the gaugeclassification for the winding vacua is the same for both models (supersymmetric andnonsupersymmetric one) before vacuum transitions.ii) The next main point of the toron approach may be formulated as follows.

Wehope that in the functional integral of the gluodynamics , when the bare charge tends tozero and when we are calculating some long range correlation function, only certain fieldconfigurations ( the toron of all types) are important. In this case the hopeless problem ofintegration over all possible fields is reduced to the problem of summation over classicaltoron configurations.I have no proof that the system of solutions which have been takeninto account is a complete system.

But I would like to stress that a lot of problems ( likethe θ dependence , the U(1) problem , the counting of the discrete number of vacuumstates , the confinement , the nonzero value for the vacuum energy and so on...) can bedescribed in a very simple manner from this uniform point of view.Both these points are quite nontrivial ones. However , I would like to convince thereader in the consistency of these assumptions by considering a more simple models , wherethe answers are well known beforehand.

By this reason let me recall in passing that allcalculations, based on the toron solution [7] demonstrate its very nontrivial role in differentfield theories. Most glaringly these effects appear in supersymmetric variants of a theory.In particular, in the supersymmetric CP N−1-theories, the torons (point defects) can ensurea nonvanishing value for the < ¯ψψ >∼exp(2iπk/N+iθ/N) with right θ-dependence.

Suchbehavior is in agreement with the value of the Witten index which equals N[14] and inagreement with the large N-expansion [15]. In analogous way, the chiral condensates canbe obtained for 4d theories: supersymmetric YM (SYM), supersymmetric QCD (SQCD)(see also calculations [16],[17], based on the standard ’t Hooft solution).

In these cases alot of various results are known from independent consideration (such as the dependenceof condensates on parameters m, g; the Konishi anomaly equation and so on...). [18].Toron approach is in agreement with these general results.

The same approach can beused for physically interesting theory of QCD with Nf = Nc. In this case an analogous4

calculation of < ¯ψψ > does possible because of cancellation of nonzero modes, like insupersymmetric theories. For this theory the contribution of the toron configurations tothe chiral condensate has been calculated and is equal to: < ¯ψψ >= −π2 exp(5/12)24Λ3[7].

As is well known in any consistent mechanism for chiral breaking a lot of problems,such as: the U(1)-problem, the number of discrete vacuum states, the θ-puzzle, low energytheorems and so on, must be solved in an automatic way. We have checked that all theseproperties [7] are consistent with the toron calculation.2.The calculation of the topological susceptibility.Let me start by giving a few formulae from ref.[8].

The grand partition function isgiven byZ =∞Xk=0Λ4(k1+k2)(k1)!(k2)!XIα,qαk1+k2Yi=1d4xiexp(−ǫint.),(8)ǫint. = −43Xi>jqiIiln(xi −xj)2qjIj + 23 ln L2(XiqiIi)2,Λ4−1/3 = cM4−1/30g2(M0) exp(−4π2g2(M0)).where two different kinds of torons classified by the weight Ii of fundamental represen-tation of the SU(2) group and qi is the sign of the topological charge.

Besides that, informula (8) the value g2(M0) is the bare coupling constant and M0 is ultraviolet regular-ization, so that eq. (8) depends on the renormalization invariant combination Λ.

As it canbe seen from (8) the only configurations satisfying the neutrality conditionXiqiIi = 0(9)are essential in thermodynamic limit L →∞. In obtaining (8) we took into account thatthe classical contribution to Z from k torons is equal toZ ∼exp(−4π2g2 k).

(10)Besides that the factor d4xi in eq. (8) is due to the 4 translation coordinates accompanyan each toron 5 and combinatorial factor k1!k2!

is necessary for avoiding double countingfor k1 torons and k2 antitorons; lastly, the average overall configurations q, I is an averageover all isotopical directions and topological charge signs of torons.The constant c in thedefinition of Λ is the calculable constantc = 25π2 exp(−α(1)/2)(11)where the coefficient α(1) is tabulated in ref. [19].To compute some vacuum expectation values it is convenient to use the correspondencebetween the grand partition function for the gas (8) and field theory with Sine-Gordon5 Let us recall that the one toron has exactly four zero modes in contrary with instanton possessingby eight zero modes.5

interaction , as it was done by Polyakov in ref. [10] for 3d QED.

Let us rewrite (8) in theform:Zθ =ZD⃗φexp(−Zd4xLeff. ),✷≡∂µ∂µ, (12)Leff = 1/2(✷⃗φ)2 −X⃗IαΛ4 exp(i8π/√3⃗Iα⃗φ + iθ/2) −X⃗IαΛ4 exp(−i8π/√3⃗Iα⃗φ −iθ/2).In this deriviation it was used the fact that the logarithm function which appears in theformula for the interaction (8) is the Green function for the operator ✷✷.

After that wecan use the method [10] to express the generating functional in terms of effective fieldtheory (12).In this effective field theory the sum over ⃗Iα runs over the 2 weights of the fundamentalrepresentation of SU(2) group. Note, that the first interaction term is related to toronsand the second one to antitorons.

Besides that, since we wish to discuss the θ dependence, we also include a term proportional to the topological charge densityθ32π2Gµν ˜Gµν to thestarting lagrangian and corresponding track from this to the effective lagrangian(12).The most important result from ref. [8] is the nontrivial dependence on θ of the topo-logical density and susceptibility.These quantaties are relevant for the solution of the U(1)problem :⟨132π2Gµν ˜Gµν⟩≡iF(θ) = i2Λ4 sin(θ/2), −π ≤θ ≤π.

(13)Zd4x exp(ikx)⟨132π2Gµν ˜Gµν(x),132π2Gµν ˜Gµν(0)⟩k→0 ∼dF(θ)dθ∼12 cos(θ2). (14)As was discussed in ref.

[12], the reason to have the nontrivial θ- dependence (13,14) aswell as the strong quasiparticle interaction ∼ln(xi−xj)2 (8) in YM theory, is the presenceof the nontrivial algebraic structure ∼qi ⃗µiqj ⃗µj in the expression for ǫint (8). Just thisfact was crucial in the analysis of the 2 dimensional CP N−1 model also [8].Such structurefor ǫint is in the striking contrast with 2+1 Polyakov’s model [10], where the interactionenergy proportional to the topological (magnetic) charges of quasiparticles only:ǫint ∼qiqj|xi −xj|.

(15)Just this difference leads to the existence of the nontrivial θ dependence in 4d YM theory(in spite of the fact of the strong quasiparticle interaction ∼ln(xi −xj)2) in contrary withPolyakov’s model where physics does not depend on θ.We proceed now to the direct calculation of the topological susceptibility(14).From the eq. (12) it is easy to reproduce the formula (13) for the vev of the topologicaldensity.To this aim we differentiate the Zθ with respect to θ and put ⃗φ = 0 in thevacuum state.

To compute a correlation functions depending on x2 we have to introduce2 different fields χα, α = 1, 2 in place of the ⃗φ:χα ≡8π√3⃗Iα⃗φ. (16)6

These χα fields should be considered as independent in the course of calculation , becausethe constraintXχα = 0(mod2π)(17)should be set to zero only at the end of calculation. The reason for this is as follows.The constraint (17) is appeared as a manifestation of the neutrality condition (9)in theinfinite volume limit of the functional description (12) for our ensemble (8).

We shouldbe very careful in taking such limit in the course of calculation of a different topologicalcharacteristics. It can be argued that this limit should be set at the last stage of calculationonly ( otherwise, we would get some senseless results).

Thus , the constraints (17) as wellas (9) will be fixed at the end of calculation.With the above consideration taken into account , the topological susceptibility canbe expressed in terms of the new variable χα:χt(k) ≡Zd4x exp(ikx)⟨132π2Gµν ˜Gµν(x),132π2Gµν ˜Gµν(0)⟩=(18)−Λ8Zd4x exp(ikx)⟨sin χ1(x), sin χ1(0) + sin χ2(x), sin χ2(0)⟩Thus, the problem of calculation of the topological susceptibility reduces to the problemof calculation of the following correlation function:⟨M+α (x1), Mα(x2)⟩,Mα(x) ≡exp(iχα). (19)where the operator M is so called disorder operator.The importance of disorder variables in gauge theories has been emphasized by ’tHooft [20], who has argued that rather than instantons it is the field configurations withnontrivial ZN topological charge that should be considered responsible for the long rangeconfinement of quarks.

Analogous disorder variables have been used in different fieldsof physics [21].In the context under consideration this variable was introduced in 2+1Polyakov’s model in ref. [22] and in two dimensional CP N−1 model in ref.

[8].A set of operators is now defined as follows. M is an operator that acts on originalAaµ fields by gauge transforming them by Ux0(x); this gauge transformation is singular atx0 and has the property that as x encircles x0,U does not return to its original value ( asit happens in the instanton case), but acquires a ZN phase (N = 2 for SU(2) group):Ux0(φ = 2π) = exp(−i2π/N)Ux0(φ = 0),U = exp(2iIφ)(20)where φ is an angle variable in the x −y plane and the point x0 lies at the same plane.From its definition it must be clear that M(x) absorbs one half topological unit, so wesay that M(x) is the annihilation operator for one point toron at x with weight I andM+(x) is the creation operator for one toron.6 It should be clear , that U depends on all6 Let me note that the small size instanton (ρ →0) at x0 is also can be created by the singular gaugetransormation U(x).

However, in the instanton case U does return to its original value ( in contrast with(20)) as x encircles x0 in the xy plane.7

xµ variables , so that M is the annihilation operator for the point defect. However, at thex −y plane, U depends only on angle variable φ.

The singularity of Aµ = iU+∂µU mustbe smeared over an infinitesimal region around x as it was done for the separate toronsolution7. We will not consider the regularization problem in this paper.Now we would like to demonstrate that the definition of M as an operator of largegauge transformation in terms of original fields exactly leads to the expression (19)interms of the effective field theory (12).

To this aim, let us consider the ǫint in the formula(8) after the action of the gauge transformation (20) at point x0. Because this gaugetransformation creates an additional toron at point x0 with isospin I0 in the system ofthe other torons placed at xi with isospins Ii we will obtain an additional contribution tothe ǫint.

Namely, after action of the operator M we have an additional interaction termbetween created toron I0 and torons Ii from the system∆ǫint ∼XiI0Ii ln(x0 −xi)2. (21)It is easy to understand that this interaction after simply repeating the deriviation ofeq.

(12),reduces to the following expression in the effective field theory:⟨M(x0)⟩=ZD⃗φexp(−Zd4xLeff.) exp(i8π/√3⃗I0⃗φ(x0) + iθ/2), ✷≡∂µ∂µ,(22)Leff = 1/2(✷⃗φ)2 −X⃗Iα=± 12Λ4 exp(i8π/√3⃗Iα⃗φ + iθ/2) −X⃗IαΛ4 exp(−i8π/√3⃗Iα⃗φ −iθ/2).Thus, the operator M under consideration in the effective theory looks like thatMα(x) = exp(i8π/√3⃗Iα ⃗φ(x) + iθ/2) = exp(iχα(x) + iθ/2),(23)where we rewrite eq.

(23) in terms of the χ fields (16).I would like to emphasize , that the expression for the operator of the large gaugetransformation in terms of the effective variables (23) has the same form like in anotherfield theories, 2+1 dimensional QED [22] and 2 dimensional CP N−1 models [8]. As it wasargued in the last reference, it is not an accidental coincidence, but this fact has a verygeneral origin.Now we are in position to calculate the correlation functionZd4x exp(ik(x1 −x2))⟨M(x1), M+(x2)⟩(24)As it was explained in refs.

[10], [22] in the analogous calculations in 3 dimensional QEDwe can not neglect terms related to insertions of M(x). Rather, this functional integralcan be estimated by the steepest descent method.

Thus, if we assume that this functionalintegral is dominated by the classical field, χ satisfies✷✷χ + 2Λ4( 4π√3)2 sin χ = i( 4π√3)2(δ4(x −x1) −δ4(x −x2)). (25)7 Usually we have in mind the special kind of regularization which preserves the self-duality equation,so that the classical action for this configuration equals 8π2g2N .8

The physics suggests linearization is legitimate so we can substitute χ instead of sin(χ) inthe equation (25) and find χcl(x) by means of Fourier transformation. However, for ourpurpose an explicit form of χcl(x) is not needed , because we are interesting in the Fouriertransformed form of the correlator (24).

Simple calculation gives the following expressionfor this correlation function at k →0:Zd4x exp(ik(x1 −x2))⟨M(x1), M+(x2)⟩∼(k4364π2 + Λ4)−1. (26)Now several comments are in order.

First, in comparision with the well known expres-sion for topological susceptibility at k = 0 (14) we were able to find the k dependence atsmall k. It turns out that the k dependence comes in this formula through k4. Such be-havior, from the one hand, is the direct consequence of the strong logarithmic interactionof the pseudoparticles (8) and from the other hand it implies thatχ′t(k = 0) ≡∂∂k2Zd4x exp(ikx)⟨132π2Gµν ˜Gµν(x),132π2Gµν ˜Gµν(0)⟩k=0 →0.

(27)If such behavior were in QCD theory , this would mean the zero magnitude for the singletaxial form factor, GA(0) [13] in agreement with EMC experiment. However, this is notthe case, because we have dealt with the pure YM theory without quarks and one couldexpect that the dynamical light quarks will change this situation.

But we may expect thatsuch k4 behavior can not disappears without leaving a trace in the full theory. However ,we can only speculate on this topic now.As a second remark, we note that the link between the θ dependence in the the-ory, discrete number of vacuum states there and confinement phenomenon mentioned inIntroduction can be understood by the following way.From the one hand, we have nonzero value for the vev of the disorder operator M (23)⟨M⟩∼exp(2iπk/N + iθ/N),k = 0, 1, 2..., N −1.

(28)In obtaining eq. (28) we took into account that in different vacua the fields χα take Ndifferent vacuum values χα = 2πk/N , as was discussed in ref.

[8]8.On the classicallevel we have ZN degeneracy , but on the quantum level, taking into account the toronvacuum transitions, these vacua have different energy as was the case in the standard|θ >= P exp(inθ)|n > vacuum consideration, but now there is only a finite number ofvacua.The operator M plays the role analogous to the chiral condensate in QCD. Differentworlds discussed above are labeled by the phase of this disorder parameter M.From the other hand, one could expect that < M ≯= 0 just corresponds to confine-ment phase.

I have no rigorous proof for this, however , an analogous considerations in2 dimensional CP (N −1) model [8] and in 3 dimensional Polyakov’s model [22], wherevev of the disorder operator have been calculated ,verify this conjecture. Moreover, the8It is easy to understand that this is the direct consequence of the periodicity of the effective lagrangian(12).Although each of the separate solutions has a θ period of 2πN,the overall minimum has a period 2πbecause of jumping from one value to another at θ = π.9

explicit calculation of the vev of the Wilson line in 4 dimensional gluodynamics (see thenext section), also demonstrates the nontrivial relation between nonzero value for < M >and area law for Wilson loop < W >.As a last remark, I would like to note,that the correlator (18), describing a gluoniumstates, can be expressed in terms of the operator M.The corresponding correlationfunction in the x space⟨M(x1), M+(x2)⟩(x1−x2)→∞−⟨M⟩2 ∼exp −|x1 −x2|(29)shows an exponential fall offat large distances. Thus, one could expect that the oper-ator M can be identified with the creation operator of the lowest gluonium state.

It isinteresting to note that a solitonic operator in the different field theories as well as thevertex operator in the conformal field theories have the same exponential form. Thus,the creation operator M which should be highly nonlocal and nontrivial in terms of theoriginal fields (Aaµ-gluons) has a very simple form in the terms of the auxiliary (dual)variables χα.3.

Wilson loop operatorIn order to show directly ( without references to the analogy with 2 and 3 dimensionalsystems) that there are electric strings in the ensemble of our pseudoparticles (8) let uscalculate the vev of the Wilson loop. As was explained above, the torons at large distanceslook like singular pure gauge field with definite isotopical direction and so, the Aaµ fieldis abelian at large distances (in a more detail see [8]).

Thus, the standard quasiclassicalapproximation, when we substitute for Aaµ the corresponding classical solution, leads (after simply repeating the deriviation of (12)) to the following expression for < W > atθ = 0⟨W⟩= ⟨Tr exp(Il iqAµdxµ)⟩=(30)ZD⃗φexp(−Zd4xLeff. ),Leff = 1/2(✷⃗φ)2 −X⃗Iα2Λ4 cos(8π/√3⃗Iα⃗φ + ⃗Iα⃗Φ).where term proportional to Φ is related to Wilson loop insertion and has the followingproperty : Φ(x) is equal to the external charge 2πq if x ∈S, Wilson plane, and Φ = 0otherwise.

In this deriviation we took into account that if the toron is in the S plane,then the integral over Gµνdσµν is non-zero, and it is equal to zero otherwise (see for amore detail about toron properties the ref . [8] ).Since the field Φ(x) is strong enough , we can not neglect non-linearities in (30) just likein the calculation of the topological susceptibility (see previous section).

The estimationof this functional integral can be done as before by the steepest descent method withrespect to φ. The corresponding field φcl is determined from the classical field equation,✷✷χ′ + 2Λ4( 4π√3)2 sin(χ′) = 2πθS(z, t)δ′(x)δ′(y),(31)where χ′ ≡χ + ⃗Φ⃗I, δ′(x) ≡dδ(x)dx,θS(z, t) = 1 if z, t ∈S and θS(z, t) = 0 otherwise.

Theright- hand side of this equation is related to the Wilson loop insertion,i.e. with function10

Φ(x). Indeed, for the Wilson loop placed in the t −z plane we have Φ ∼δx,0δy,0 and thus, acting by the operator ✷✷on Φ we exactly reproduce the right hand side of the eq.

(31)( without loss of generality we consider x > 0, y > 0):✷✷δx,0δy,0 ∼✷✷(1 −ǫ(x))(1 −ǫ(y)) ∼✷[δ′(x)(1 −ǫ(y)) + δ′(y)(1 −ǫ(x))],(32)✷≡∂2∂x2 + ∂2∂y2,x > 0, y > 0.where we have substituted the expression 1 −ǫ(x) in place of δx,0 and used the followingfeature of the ǫ(x) function :✷ǫ(x) ∼δ′(x). The next step of calculation of the r.h.s.

(32)is the using of the Fourier representation for the functions ǫ(x) ∼R ∞0sin kxkdk and δ(x) ∼R ∞−∞exp(ikx)dk which leads to the desired result:r.h.s. (32) ∼✷Zdkxkx[1 −2πZdkysin(kyy)ky] exp(ikxx) + (x →y)(33)∼Z ∞−∞dkxkx exp(ikxx)Z ∞0dkyky sin(kyy) + (x →y) ∼δ′(x)δ′(y).Now we are ready to estimate the functional integral (30) by the steepest descentmethod.

For this purpose , we substitute χ instead of sin χ in the eq. (31) and find χclby means of Fourier transformation just as it was done before in the previous sectionfor the calculation of the topological susceptibility.

We expect that the accuracy for suchprocedure is not very high and the numerical coefficient given bellow should be consideredas an estimation of the order of value. With these remarks in mind we obtain⟨W⟩∼exp(−√3π64π Λ2S)(34)The result (34) implies that between two fixed charges there exists an electric string withenergy density ∼Λ2.Now several comments are in order.First of all, I have to stress that the unusual kineticterm (✷χ)2 in the effective action (12)plays a crucial role in this deriviation.

From the onehand ,such kinetic term is the direct consequence of the strong pseudoparticle interaction(8) ,and from the other hand ,just this term gives a correct expression ∼δ′(x)δ′(y) for theright hand side of eq. (31), leading to the existence of the solitonic shape solution in the x, ydirections.9 This solitonic shape guarantee the convergence of the corresponding integralR dxdy over x, y directions, perpendicular to Wilson plane z, t and leads to confinementformula (34).

The thickness of the sheet with Wilson loop as boundary is of order Λ−1,plasma screening length, and not ∆→0, toron size. This is just the reason to get thefinite answer for the string tension (34) in comparision with our very rough previous9 Let us note that the analogous problem in 3 dimensional QED looks as follow .

If Wilson loop is placedin the xy plane, the equation , analogous to (31) is essentially one dimensional [10] and (χcl)depends onlyon z . The solitonic shape for the corresponding solution is due to the non-linearity (∼δ′(z)) related tothe Wilson loop insertion.

This solitonic shape guarantee the convergence of the corresponding integralRdz, over perpendicular to Wilson loop direction and leads to the confinement formula.11

estimation [8],where back reaction of the Wilson loop insertion has not been taken intoaccount.As a second remark ,we note that our explicit calculation of the < W >∼exp(−S)is in the consistency with our conjecture that the confinement phase just corresponds tothe nonzero value of disorder operator < M > (28). As it was mentioned above, it isalso correct in the 2 dimensional CP N−1 model [8] and 3 dimensional QED [22] where anexplicit calculations do possible.4.Final remarks.The main point of this Letter is that the dynamical solution of the one from thefollowing problems : U(1) problem,confinement problem, multiplicity of the vacuum statesN, θ/N dependence, should be necessarily accompanied by the resolution of the restproblems within the same approach.

We have checked this conjecture in the frameworkof the dynamical toron calculation. I expect that two main assumptions of this approachI am usingi)the multivalued functions are admissible in the definition of the functional integral;ii) the only certain field configurations ( torons) are important and the problem ofintegration over all possible fields is reduced to the problem of summation over classicaltoron configurations, are consistent assumptions.

I have no proof for this, but I would liketo stress that all problems mentioned above can be considered from this uniform point ofview.The main quantitative results, which have been obtained can be formulated as follow:χt(k →0) ∼( 3k464π2 + Λ4)−1(35)< M >∼exp(iθ/2 + iπk), k = 0, 1. (36)< W >∼exp(−σS)(37)The first eq.

(35) shows that the k dependence of the topological susceptibility comesthrough k4. Such behavior might be important for explanation of EMC experiment.The second eq.

(36) explicitly demonstrates that vacua in YM theory are classified bythe phase of the disorder operator M. Apart from standard parameter θ, 0 ≤θ ≤2π,there is a new label k = 0, 1. Although each of the separate solutions has a θ period4π, the overall solution has a θ period of 2π because of the jumping from the one value(k = 0) to another (k = 1) at θ = π.

Such behavior is in agreement with large N results,where the number of vacuum states is equal N and θ dependence appears in the formθ/N. Besides that,< M ≯= 0 is in consistency with the confinement property of thetheory.The explicit calculation of the vev of the Wilson line (37) also confirms this conjecture.AcknowledgmentsI am very grateful to G. Veneziano for stimulating discussions, H.Leutwyler andA.Smilga for organization of Moscow - Bern seminar , where this work was presentedand also to the CERN Theory Division where this work was done.12

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[16] E. Cohen and C. Gomez, Phys.Rev.Lett.52(1984)237[17] C. Gomez, Phys.Lett.B 141 (1984),366. [18] D.Amati et al., Phys.Rep.162,169(1988).

[19] G ’t Hooft, Phys.Rev.D14,3432 (1976). [20] G ’t Hooft, Nucl.Phys.B138,1(1978).13

[21] L.P.Kadanoffand H.Geva, Phys, Rev.B3,3918(1971).E.Fradkin and L.P.kadanoff, Nucl.Phys.B170,1(1980).V.Fateev and A.B.Zamolodchikov, Sov.Phys.JETP 82,215 (1985). [22] N.J.Snyderman , Nucl.Phys.B 218,381 (1983).14

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