# A gauge invariant order parameter for monopole condensation in QCD vacuum

AA gauge invariant order parameter for monopole condensation in

QCD vacuum.

Adriano Di Giacomo Pisa University and I.N.F.N. Sezione di Pisa ∗ (Dated: January 1, 2021)In this paper we improve the existing order parameter for monopole condensation in gauge theoryvacuum, making it gauge-invariant from scratch and free of the spurious infrared problems whichplagued the old one. Computing the new parameter on the lattice will unambiguously detect weatherdual superconductivity is the mechanism for color conﬁnement.As a byproduct we relate conﬁnement to the existence of a ﬁnite correlation length in the gauge-invariant correlator of chromo-electric ﬁeld strengths. I. INTRODUCTION

Conﬁnement of color is a fundamental problem in par-ticle physics: hadrons are made of quarks and gluonsbeyond any reasonable doubt, but no free quark or gluonhas ever been observed neither in nature nor as a prod-uct of a reaction. This phenomenon is known as con-ﬁnement of color.

QCD as the theory of strong inter-actions should have the explanation of conﬁnement builtin. However at large distances

QCD becomes stronglyinteracting and impossible do deal with the known tech-niques of ﬁeld theory, except for numerical simulations ona lattice. The strategy is then to look for ”mechanisms”,i.e. to explore whether conﬁnement can structurally besimilar to some other known physical phenomenon andcheck this possibility by use of numerical simulations.A theoretically attractive mechanism for color conﬁne-ment is dual superconductivity of the vacuum [1] [2].Conﬁnement of chromo-electric charges is produced bycondensation of magnetic charges in the vacuum via dualMeissner eﬀect, in the same way as magnetic charges areconﬁned in ordinary superconductors by condensation ofCooper pairs.What makes this mechanism particularly attractiveis the fact that it is based on symmetry. The de-conﬁnement phase transition is a change of symmetryand this provides a ”natural” explanation of the fact thatthe measured upper limits to the existence of free quarksare very small (typically 10 − [3]): by this mechanismthe number of free quarks in nature is strictly zero bysymmetry.An alternative mechanism suggested in Ref [4] is per-colation of center vortices through space-time.Both mechanisms have been widely studied by use nu-merical simulations on the lattice. For reviews se e.g. [5]for monopoles and [6] for vortices. No conclusive evidencefor either mechanism has yet been found.The most popular line of investigation of the monopolemechanism has been to select a gauge such that the corre-sponding lattice monopoles would dominate the dynam-ics of the system (monopole dominance) [7]. This gauge ∗ Electronic address: [email protected] proves empirically to be the so called maximal abeliangauge[8]. The technique was to show that the U (1) sys-tem selected by that gauge, and speciﬁcally the monopolecomponent of it, would reproduce with good approxima-tion physical observables.The line of the competing approach based on vorticeswas very similar: identify a gauge such that the corre-sponding central vortices would dominate the dynamics(center dominance). This gauge here is called the maxi-mal central gauge [9].Such kind of approaches will never be conclusive: dom-inance is neither a necessary nor a suﬃcient condition formonopole condensation or vortex percolation. Indeed itis not obvious at all that one can describe e.g. a super-conducting system by Cooper pair dominance.A diﬀerent approach is to directly look for symmetry[10]: the vacuum expectation value of an operator µ creating a monopole can be the order parameter, likethe creator of a Cooper pair in an ordinary supercon-ductor. In the superconducting phase the system is asuperposition of states with diﬀerent magnetic chargesand (cid:104) µ (cid:105) (cid:54) = 0, in the deconﬁned phase (cid:104) µ (cid:105) = 0.In the U (1) pure gauge theory a construction of theorder parameter exists which is rigorous at the level ofa theorem [11] [12][13]. Lattice simulations show thatthe system has a conﬁned phase at low β ’s [ in the usualnotation β ≡ Ng for gauge group SU ( N )] and adeconﬁned phase for β ≥ β c , some critical value. Theoperator which creates a monopole is uniquely identiﬁedas the shift of the vector potential by the classical ﬁeldof the monopole [11], and the shift is generated by thecanonically conjugate momentum, the transverse electricﬁeld operator. In formulae the operator µ ( (cid:126)x, t ) whichcreates a monopole in the point (cid:126)x at the time t is µ ( (cid:126)x, t ) = exp( (cid:90) d y (cid:126)E ( (cid:126)y, t ) 1 g (cid:126)A ⊥ ( (cid:126)y − (cid:126)x )) (1) g (cid:126)A ⊥ ( (cid:126)y − (cid:126)x ) is the vector potential at the point (cid:126)y producedby a monopole sitting at (cid:126)x , in the transverse gauge. Thefactor g comes from the magnetic charge. An additionalfactor g appears in the lattice formulation, where theelectric ﬁeld is g times the canonical electric ﬁeld, so thatthe operator µ has the form µ = exp( − β ∆ S ) (2) a r X i v : . [ h e p - l a t ] D ec Note that only the transverse electric ﬁeld survivesthe convolution in Eq(1), which is the conjugate momen-tum to the transverse vector potential in whatever gauge.Lattice simulations [11] show that (cid:104) µ (cid:105) is an order parame-ter for conﬁnement: (cid:104) µ (cid:105) (cid:54) = 0 in the conﬁned phase β ≤ β c and (cid:104) µ (cid:105) = 0 in the deconﬁned phase β > β c showing thatin the U (1) system the mechanism of conﬁnement is dualsuperconductivity of the vacuum.The extension of this construction to a generic gaugegroup was attempted in subsequent steps by the Pisa lat-tice group. The basic diﬃculty is that the monopole is a U (1) conﬁguration [14] [15]: its existence requires a Higgsbreaking of the gauge symmetry to the U (1) subgroup inwhich the Higgs scalar has a deﬁnite direction in colorspace. In QCD there is no Higgs ﬁeld. Any operator inthe adjoint representation can in principle act as an ef-fective Higgs ﬁeld [8], selecting what is called an ”abelianprojection”. The general attitude of the lattice commu-nity was to assume that monopoles belonging to diﬀerentabelian projections are diﬀerent objects and to look foran abelian projection in which monopole dominance weremost eﬀective [5].From the point of view of symmetry instead the orderparameter is the creation operator of a monopole, andthe creation of a monopole should be a gauge indepen-dent process since a monopole has a non trivial topology.Creating a monopole in any abelian projection amountsto create it in all projections. See on this point Ref [16]and [17]. With this idea in mind the order parameter wasthen tentatively constructed for SU (2) gauge group asthe creation operator of a monopole in a generic abelianprojection [ U (1) subgroup ] speciﬁcally along the nomi-nal 3-axis used in the numerical simulations [18][19]. (cid:104) µ (cid:105) was expected to go to a ﬁnite non zero value in the ther-modynamic limit V → ∞ below β c and to vanish in thesame limit for β > β c . The approach looked successfulwithin the lattice sizes available at that time. Instead anattempt to extend the construction to the gauge group G µ would tend to zero in the ther-modynamic limit also in the conﬁned phase and thus isno order parameter. Also the determinations for SU (2)and SU (3) gauge groups at larger volumes showed thesame problem [20]. The origin of the problem was identi-ﬁed and an improved version of the order parameter wasproposed, infrared subtracted and numerically tested for SU (2) pure gauge theory in Ref.[21].In this paper we elaborate more on the constructionof the operator µ by expanding ρ ≈ log( µ ) , deﬁned inEq(14) below, in a power series of ∆ S i.e. of the magneticcharge of the created monopole [Section II] . The coef-ﬁcients of the expansion are integrals on 3d space or on4d space of connected correlators of ﬁeld strengths or ofﬁeld strengths squared, which only depend on diﬀerencesof positions by translation invariance, times functions ofthe classical ﬁeld of the monopole which also depend onthe sum of the position vectors. The correlation func-tions are exponentially vanishing at large distances inthe conﬁned phase and therefore the integrals are ﬁnite. In the de-conﬁned phase instead there is no length scale,the dependence on the diﬀerence of coordinate is dic-tated by scale dimensions and can give logarithmicallydivergent integrals which correspond to zero’s of µ , sig-naling de-conﬁnement. The advantage of this expansionis a transparent bookkeeping of the terms which can besensitive to the de-conﬁning transition. [ Section III].Moreover the expansion allows to study the divergen-cies which originate from the integration on the sum ofthe positions : the ﬁeld correlators are translation invari-ant, i.e. they do not depend on the sum of the positionvectors. Were not for the monopole classical conﬁgura-tion which breaks translation invariance ρ would divergeas V at large volumes V . In fact we show that thereis only a linear divergence ∝ V and only in the ﬁrstfew terms of the expansion, up to ∆ S . These divergen-cies which we will call ”kinematic” can be isolated and aregularized ρ can thus be deﬁned to which the analysispresented above applies.In Section IV we compute the kinematically divergentpart of ρ deﬁned in Section 3 by strong coupling expan-sion. We show that it vanishes to all orders for gaugegroup U (1), thus conﬁrming the validity of the order pa-rameter, already proved with diﬀerent arguments in Ref’s[11][12][13]. In [21] it was shown that the strong couplingexpansion of ρ is ﬁnite to ﬁfth order.Divergencies exist instead at that order for SU (2)gauge group, they are not even gauge invariant, they arepresent both in the conﬁned and in the deconﬁned phaseand are the origin of the problems of the order parame-ter observed on the lattice [20]. A possible way out is tosubtract them by hand [21]. A better way is to improvethe deﬁnition of the order parameter.In Section V, on the basis of the results presented inSection 4, we trace the origin of our problems and mod-ify the order parameter making it gauge invariant fromscratch, with no ”kinematic” divergence, and conceptu-ally correct. The analyses of sections 2, 3 ,4 are immedi-ately extended to the new operator.The new operator needs no infrared subtraction. Inaddition the analysis allows to connect conﬁnement tothe two point gauge invariant correlator of electric ﬁelds:its contribution to ρ is ﬁnite in the conﬁned phase whereit decreases exponentially at large distances x , but candiverge logarithmically in the deconﬁned phase where itbehaves as x − . We are able to prove in our analysisthat this contribution is negative deﬁnite, and thus thedivergence corresponds to a zero of the order parameter (cid:104) µ (cid:105) .An overall discussion of the results is presented in Sec-tion VI. II. THE ORDER PARAMETER.

We start analyzing the order parameter as deﬁned inRef’ s [11][19]. As a consequence of the proportionalityof the monopole ﬁeld to the inverse of the gauge coupling g Eq(1) the order parameter µ ≡ (cid:104) µ ( (cid:126)x, t ) (cid:105) has the formof the ratio of two partition functions [11] [18] µ = Z ( S + ∆ S ) Z ( S ) (3)where Z ( S ) = (cid:82) dA exp( − βS ) is the QCD partitionfunction.In the ( Wilson) lattice formulation, for pure gaugetheory S = Σ n,µ,ν (cid:60) [ (cid:0) P µν ( n ) − (cid:1) ] (4) (cid:60) denotes real part, β = Ng for group SU ( N ) and n = ( (cid:126)n, t ) denotes the lattice site. P µν ( n ) is the plaquette P µν ( n ) = 1 N T r [ U µ ( n ) U ν ( n + ˆ µ ) U † µ ( n + ˆ ν ) U † ν ( n ) (cid:3) (5) S + ∆ S has instead the form S + ∆ S = Σ n,µ,ν (cid:60) (cid:0) − P (cid:48) µν ( n ) (cid:1) (6)with P (cid:48) µν ( n ) = P µν ( n ) everywhere and f or all µ, ν exceptP (cid:48) i ( (cid:126)n, t ) = 1 N T r [ U i ( (cid:126)n, t ) U ( (cid:126)n + ˆ i, t ) M i ( (cid:126)n + ˆ i ) U † i ( (cid:126)n, t + 1) U † ( (cid:126)n, t ) (cid:3) (7)Here t is the time at which the monopole is created, whichwe could ﬁx at any value by use of translation invariance. M i ( (cid:126)m ) = exp (cid:0) ig σ (cid:126) ¯ A i ( (cid:126)m − (cid:126)x ) (cid:1) (8) (cid:126) ¯ A i ( (cid:126)y − (cid:126)x ) is the classical vector potential produced at (cid:126)y by a monopole sitting at (cid:126)x in the transverse gauge (cid:126) ∇ (cid:126) ¯ A i ( (cid:126)y − (cid:126)x ) = 0. It is easily shown by successive changesof variables in the Feynman path integral that replacing S by S +∆ S is equivalent to add a monopole at the point (cid:126)x in the subgroup U (1) generated by T at all times t > P (cid:48) i ( (cid:126)n,

0) = P i ( (cid:126)n,

0) cos (cid:0) g (cid:126) ¯ A i ( (cid:126)n + ˆ i − (cid:126)x ) (cid:1) + i sin (cid:0) g (cid:126) ¯ A i ( (cid:126)n + ˆ i − (cid:126)x ) (cid:1) Q i ( (cid:126)n,

0) (9)where P i ( (cid:126)n,

0) is deﬁned by Eq(5) and Q i ( (cid:126)n, ≡ N T r (cid:2) U i ( (cid:126)n, U ( (cid:126)n +ˆ i, σ U † i ( (cid:126)n, U † ( (cid:126)n, (cid:3) (10) We have assumed gauge group SU (2) for the sakeof simplicity. The extension to generic gauge group isstraightforward.From Eq.’s (4), (6) and (7) ∆ S is non zero only in thehyperplane n = t . In the following we shall omit thedependence on time, if not specially needed, as well asthe dependence on (cid:126)x , the position of the monopole beingﬁxed once and for all. From the deﬁnition of S and of S + ∆ S it follows∆ S = Σ (cid:126)n Σ i [( C i ( (cid:126)n ) − (cid:60) P i ( (cid:126)n, − S i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n, C i ( (cid:126)n ) and S i ( (cid:126)n ) the cosine and the sine appearing in the expressionEq.(9), leaving the only relevant dependences, on (cid:126)n andon i .Note that the presence of a monopole at (cid:126)x breaks theinvariance under spatial translations of S + ∆ S .Since | (cid:126) ¯ A i ( (cid:126)n ) | ≈ n at large distances(1 − C i ( (cid:126)n )) ≈ n S i ( (cid:126)n ) ≈ n (12)In the special case N = 1 the gauge group is abelian,everything commutes, Q i = P i , P i = exp iθ i . Theelementary links have the form U µ ( n ) = exp iθ µ ( n ), θ µν ≡ θ µ ( n ) + θ ν ( n + ˆ µ ) − θ µ ( n + ˆ ν ) − θ ν ( n ), P (cid:48) i ( (cid:126)n, t ) = P i ( (cid:126)n, t ) exp igA i ( (cid:126)n, t ) and∆ S = Σ (cid:126)n Σ i [( C i ( (cid:126)n ) − (cid:60) P i ( (cid:126)n, − S i ( (cid:126)n ) (cid:61) P i ( (cid:126)n, ρ ≡ ∂ log( µ ) ∂β = (cid:104) S (cid:105) S − (cid:104) ( S + ∆ S ) (cid:105) ( S +∆ S ) (14)The brackets denote average, the subscript on the rightbrackets denotes the action used to weight the average. (cid:104) S (cid:105) S ≡ (cid:82) dA exp( − βS ) S (cid:82) dA exp( − βS ) (cid:104) ( S + ∆ S ) (cid:105) ( S +∆ S ) ≡ (cid:82) dA exp (cid:0) − β ( S + ∆ S ) (cid:1) ( S + ∆ S ) (cid:82) dA exp (cid:0) − β ( S + ∆ S ) (cid:1) (15)Since µ ( β = 0) = 1 µ ( β ) = exp (cid:0) (cid:90) β ρ ( β (cid:48) ) dβ (cid:48) (cid:1) (16)If superconductivity of the vacuum is the correct mech-anism for conﬁnement ρ is expected to be ﬁnite in theconﬁned phase β < β c in the inﬁnite volume limit, sothat µ (cid:54) = 0. In the deconﬁned phase β > β c ρ must di-verge negative in the thermodynamic limit, so that µ = 0.This we want to investigate using the deﬁnitions Eq(4)and (6) of S and S + ∆ S .We shall expand (cid:104) ( S + ∆ S ) (cid:105) ( S +∆ S ) in powers of ∆ S .In our notation (cid:104) O (cid:105) ≡ (cid:82) dA exp( − βS ) O (cid:82) dA exp( − βS ) for any operator O . The expression for ρ Eq(14) is ρ = (cid:104) S (cid:105) − (cid:104) ( S + ∆ S ) (cid:0) (cid:80) ∞ n =0 ( − β ∆ S ) n n ! (cid:1) (cid:105) (cid:80) ∞ n =0 (cid:104) ( − β ∆ S ) n (cid:105) n ! (17)∆ S is the integral on 3-d space of an electric plaque-tte Π i ( (cid:126)n,

0) and of a Q i ( (cid:126)n,

0) multiplied by numericalcoeﬃcients, and S is in the same way the integral on 4-d space time of a plaquette times numerical coeﬃcients.Therefore ρ Eq(14) is expressed in terms of integrals ofcorrelators of plaquettes and Q i (cid:48) s .We shall ﬁrst construct an expansion of ρ in powers of∆ S i.e. in powers of the charge of the external monopole.The eﬀect of the series in the denominator of ρ Eq(17)is simply to cancel disconnected parts of the correlatorsand ρ = − ∞ (cid:88) ( − β ) n n ! (cid:104)(cid:104) ∆ s n +1 (cid:105)(cid:105) − (cid:104)(cid:104) S ∞ (cid:88) ( − β ) n n ! ∆ S n (cid:105)(cid:105) (18)The notation (cid:104)(cid:104) .. (cid:105)(cid:105) indicates connected correlators, i.e.correlators with all the disconnected parts subtracted.We prove Eq(18) in Appendix 1.As a second basic point of our strategy we note that atthe leading infrared order a plaquette is proportional tothe square of a component of the ﬁeld strength, and Q i to the i (cid:104) µ (cid:105) Eq(3). Higher terms will prove to be irrelevant.The interesting part of ρ will ﬁnally be a sum ofgauge-invariant connected correlators of two electric ﬁeldstrengths. Correlators of gauge invariant ﬁelds are knownin the literature [23] [24] [22] and have been numericallystudied on the lattice [25]. We shall come back to thispoint below.The physical idea is that in the conﬁned phase there isa ﬁnite correlation length and the correlators are expo-nentially decreasing at large distances thus making thethe integrals infrared convergent, ρ ﬁnite and (cid:104) µ (cid:105) (cid:54) = 0 .In the deconﬁned phase instead there is no length scalein the game, so that the behavior at large distance is dic-tated by the dimension in length, is power-like and theintegral can be negative infrared divergent at large vol-umes thus making (cid:104) µ (cid:105) = 0 in the thermodynamic limit V → ∞ . This aspect we analyze in the next section. III. COMPUTING ρ We now analyze in detail Eq(18) both for SU (2) and U (1) gauge groups. U (1) will provide a safe test since in that case the orderparameter is well deﬁned at a rigorous level [11] [12][13]. The generic term (cid:104)(cid:104) ∆ S n (cid:105)(cid:105) is an n-fold 3-d inte-gral of products of factors (cid:60) P i k ( (cid:126)n k )[ C i ( (cid:126)n ) −

1] and (cid:61) Q i ( (cid:126)n ) S i ( (cid:126)n ).Notice that only terms with an even number of factors Q i ( (cid:126)n ) survive the average.This is easily seen in the case U (1) where (cid:61) Q i ∝ sin( θ i ) is odd under the change θ µ → − θ µ which is asymmetry both of the action (cid:80) (1 − cos( θ i ) and of themeasure Π (cid:82) + π − π dθ µ ( (cid:126)n ).For SU (2) gauge group the change of variables U µ ( n ) → Π † U µ ( n )Π (19)with Π ≡ exp( i π σ ) (20)inverts the sign of σ i.e. of (cid:104) Q i (cid:105) but leaves the actionand the measure invariant.For a generic group σ is replaced by the third compo-nent of the SU (2) subgroup in which the monopole livesand the construction is the same.The generic term of the expansion Eq(18) say (cid:104)(cid:104) ∆ S n (cid:105)(cid:105) will have the form (cid:104)(cid:104) ∆ S n (cid:105)(cid:105) ∝ (cid:88) j s (cid:126)n s Π ks =1 (cid:88) i r (cid:126)n r Π n − kr =1 S j s ( (cid:126)n s )[ C i r ( (cid:126)n r ) − F i r j s ( (cid:126)n r , (cid:126)n s )(21)The number of factors (cid:61) Q i is k and must be even.At the lowest order in the lattice spacing a a plaquette P µν has the form P µν ≈ − a (cid:126)G µν (cid:126)G µν + .... (22)The term 1 does not contribute to connected correla-tors, the operator in the second term has dimension -4 inlength. (cid:61) Q io instead has the form (cid:61) Q i ≈ a G (3) i + .... (23)and has dimension -2 in length. The upper index (3)denotes direction in color space.Higher terms in the expansion of P i ( (cid:126)n ) and Q i ( (cid:126)n )have higher dimension in inverse length.The function F i r j s ( (cid:126)n r , (cid:126)n s ) in Eq(21) only depends ondiﬀerences of (cid:126)n ’s by translation invariance and is ex-pected to be cut-oﬀ exponentially at large distances inthe conﬁned phase, making all the integrals convergentin the inﬁnite volume limit. In the deconﬁned phaseinstead there is no intrinsic scale and F i r j s ( (cid:126)n r , (cid:126)n s ) willdepend on inverse powers of the distances with expo-nent dictated by the scale dimension. Each factor ∆ S in the correlator contributes the dimension in length l of the function as d n [1 − C i ( (cid:126)n )] (cid:126)G i (cid:126)G i , i.e. by − P i . The insertion of a factor Q i instead as d nS i ( (cid:126)n )) (cid:126)G i changes the dimension by 0.Therefore terms in Eq(21) containing factors (cid:60) P i tendto stay ﬁnite both in the conﬁned and in the deconﬁnedphase. Terms containing only Q i ’s have dimension 0 andcan produce a logarithmic divergence in the deconﬁnedphase.Higher terms in the expansions Eq(22) and Eq(23) inpowers of the lattice spacing a have higher inverse dimen-sion in length and as a consequence they are irrelevant.As for the terms in Eq(18) (cid:104)(cid:104) S ∆ S n (cid:105)(cid:105) the result is sim-ilar.The insertion of a factor S means, at the leadinginfrared order, (cid:82) d n a (cid:126)G µν (cid:126)G µν which has dimension 0.We are tacitly assuming that everything in Eq(21) iswell deﬁned and ﬁnite.We immediately realize that this is not the case bylooking at the ﬁrst few terms of the expansion Eq(18) upto n = 2. The reason is that the correlators are transla-tion invariant, and only depend on relative distances, butintegration on the coordinates also includes an integralon their sum. Translation invariance is broken by thefactors S i ( (cid:126)n ) and ( C i ( (cid:126)n ) −

1) containing the ﬁeld of themonopole. If there are enough of them the integral is con-vergent [Eq(12)]. By dimensional argument two factorsof type (cid:60) P i and four factors of type (cid:61) Q i are suﬃcient.In order to understand these ”kinematic” infrared diver-gencies and get rid of them it will be then suﬃcient tostudy the expansion Eq(18) up to second order in ∆ S ,namely ρ ≈ −(cid:104) ∆ S (cid:105) + (cid:104)(cid:104) βS ∆ S (cid:105)(cid:105) + β (cid:104)(cid:104) ∆ S (cid:105)(cid:105) − β (cid:104)(cid:104) S ∆ S (cid:105)(cid:105) + .... (24)All the other terms are ﬁnite.From Eq(11) one easily gets (cid:104) ∆ S (cid:105) = (cid:104)(cid:60) P i (cid:105) (cid:88) i =1 , (cid:88) (cid:126)n ( C i ( (cid:126)n ) − ∝ (cid:88) (cid:126)n n (25)The term in (cid:104)(cid:61) Q i (cid:105) Eq(11) vanishes by symmetry. (cid:104)(cid:60) P i ( (cid:126)n ) (cid:105) is independent on (cid:126)n due to translation in-variance and on the index i due to invariance under 90degree rotations around the coordinate axes of the lat-tice. (cid:104) ∆ S (cid:105) Eq(25) diverges linearly with the spatial linearsize L of the lattice, both in the conﬁned and in thedeconﬁned phase independent of the gauge group.The term β (cid:104)(cid:104) S ∆ S (cid:105)(cid:105) in Eq(24) contains a term pro-portional to P i and a term proportional to Q i . Thelatter vanishes by symmetry and what is left is linearlydivergent in the infrared β (cid:104)(cid:104) S ∆ S (cid:105)(cid:105) = K (cid:88) (cid:126)n (cid:88) i =1 − ( C i ( (cid:126)n ) −

1) (26)with K = (cid:88) nµν (cid:104)(cid:104) P µν ( n ) P i ( (cid:126)m, t ) (cid:105)(cid:105) (27) K is independent on (cid:126)m and t , and is ﬁnite in the conﬁnedphase but also in the deconﬁned phase having dimension in length -4. Note that the integral on the time axis iscut-oﬀ by T the inverse of the temperature.More interesting is the term β (cid:104)(cid:104) ∆ S (cid:105)(cid:105) in the expansionEq(24). By use of Eq(11) we get (cid:104)(cid:104) ∆ S (cid:105)(cid:105) = (cid:88) (cid:126)n (cid:126)n i i (cid:2) (cid:104)(cid:104)(cid:60) P i ( (cid:126)n ) (cid:60) P i ( (cid:126)n ) (cid:105)(cid:105) ( C i ( (cid:126)n ) − C i ( (cid:126)n ) −

1) + (cid:104)(cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) (cid:105)(cid:105) S i ( (cid:126)n ) S i ( (cid:126)n ) (cid:3) (28)The term (cid:104)(cid:104) P i Q j (cid:105)(cid:105) vanishes by symmetry. Theﬁrst term in Eq(28) is convergent both in the con-ﬁned and in the deconﬁned phase: indeed the correlator (cid:104)(cid:104)(cid:60) P i ( (cid:126)n ) (cid:60) P i ( (cid:126)n ) (cid:105)(cid:105) has dimension l − thus makingthe sum on (cid:126)n − (cid:126)n convergent in both phases and thefactor ( C i ( (cid:126)n ) − C i ( (cid:126)n ) −

1) behaves as | (cid:126)n + (cid:126)n | − at large distances, thus making the sum on (cid:126)n ≡ (cid:126)n + (cid:126)n ﬁnite. We shall not consider that term any more and weshall concentrate on the second term. Indeed adding aﬁnite constant to ρ which is the logarithm of the orderparameter (cid:104) µ (cid:105) is equivalent to change the order parame-ter by a non zero factor and hence it is irrelevant: whatmatters is that the order parameter be zero or non zero.We ﬁrst notice that (cid:104)(cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) (cid:105)(cid:105) = (cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) (cid:105) since (cid:104) Q i ( (cid:126)n ) (cid:105) = 0 by symmetry.Moreover at the leading infrared order (cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) (cid:105) ∝ (cid:104) E (3) i ( (cid:126)n ) E (3) i ( (cid:126)n ) (cid:105) (29) (cid:126)E (3) i ( (cid:126)n ) is the i component of the electric ﬁeld in colordirection 3. For U (1) gauge group the color index canbe disregarded. Invariance under parity implies that thecorrelator is an even function of (cid:126)n − (cid:126)n . Moreover thecorrelator is non zero only if i = i . Indeed if one ofthe electric ﬁelds is parallel to (cid:126)n − (cid:126)n and the other per-pendicular the correlator vanishes due to the invarianceunder rotations around the direction of (cid:126)n − (cid:126)n . If theyare both transverse to it and perpendicular to each othera rotation of angle π around any of them changes thesign of the other but does not aﬀect the dependence onthe distance since the dependence on it is even. There-fore we have for the relevant part of (cid:104)(cid:104) ∆ S (cid:105)(cid:105) , neglectingterms which are ﬁnite both in the conﬁned and in thedeconﬁned phase which are irrelevant to our argument (cid:104)(cid:104) ∆ S (cid:105)(cid:105) ≈ (cid:88) i,(cid:126)n ,(cid:126)n (cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) (cid:105) S i ( (cid:126)n ) S i ( (cid:126)n )(30)This expression diverges linearly when summed on (cid:126)n = (cid:126)n + (cid:126)n : indeed the correlator does not depend on (cid:126)n andthe two factors S i ( (cid:126)n ) S i ( (cid:126)n ) behave as n each at largedistances. We regularize it and isolate the diverging partby adding and subtracting the contact termΣ = (cid:88) i,(cid:126)n,(cid:126)n − (cid:126)n (cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) (cid:105) S i ( (cid:126)n ) S i ( (cid:126)n ) ∝ (cid:88) (cid:126)n n (31)We deﬁne a quantity (cid:104)(cid:104) ∆ S (cid:105)(cid:105) subtracted as (cid:104)(cid:104) ∆ S (cid:105)(cid:105) subtracted = (cid:104)(cid:104) ∆ S (cid:105)(cid:105) − Σ (32)We ﬁnally consider the last term of Eq(24) , namely − β (cid:104)(cid:104) S ∆ S (cid:105)(cid:105) . As for was for the term β (cid:104)(cid:104) ∆ S (cid:105)(cid:105) ofEq(28) there is a term proportional to P i ( (cid:126)n ) P i ( (cid:126)n )which is convergent both kinematically i.e. in the sumover (cid:126)n and by dimension i.e. in the sum over (cid:126)n − (cid:126)n . Wecan then disregard it. As was for Eq(28) the cross term P i Q i vanishes by symmetry. We are then left with − β (cid:104)(cid:104) S ∆ S (cid:105)(cid:105) ≈ β (cid:88) nµν (cid:88) i i (cid:126)n (cid:126)n S i ( (cid:126)n ) S i ( (cid:126)n ) (cid:104)(cid:104)(cid:60) P µν ( n ) (cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) (cid:105)(cid:105) (33)This expression is ”kinematically ” divergent both in theconﬁned and in the deconﬁned phase. As for the otherterms of Eq(28) we shall isolate and subtract the diver-gent part. By dimensional arguments also this term couldbe candidate to produce a logarithmic divergence in thedeconﬁned phase: in fact this is not true because therange of the sum on temporal coordinates is cut oﬀ at T at non zero temperature in the deconﬁned phase.In conclusion the only term of the series deﬁning ρ which can diverge logarithmically in the deconﬁnedphase after removal of the kinematic divergences is β (cid:104)(cid:104) ∆ S (cid:105)(cid:105) subtracted of Eq(32).We notice that it is negative deﬁnite. In-deed starting from the obvious inequality | (cid:80) i S i ( (cid:126)n ) S i ( (cid:126)n ) | ≤ S i ( (cid:126)n )+ S i ( (cid:126)n )2 and calling f ( (cid:126)n − (cid:126)n ) ≡ (cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) (cid:105) to simplify thenotation we have the chain of inequalities |(cid:104)(cid:104) ∆ S (cid:105)(cid:105)| = | (cid:88) i,(cid:126)n ,(cid:126)n f i ( (cid:126)n − (cid:126)n ) S i ( (cid:126)n ) S i ( (cid:126)n ) |≤ (cid:88) i,(cid:126)n ,(cid:126)n S i ( (cid:126)n ) + S i ( (cid:126)n )2 f i ( (cid:126)n − (cid:126)n ) ≡ Σ (34)The last inequality is only true if (cid:80) (cid:126)n f i ( (cid:126)n ) ≥

0. Indeed (cid:88) (cid:126)n (cid:104)(cid:61) Q i ( (cid:126)n (cid:61) Q i ( − (cid:126)n )2 ) (cid:105) = (cid:80) k |(cid:104) |(cid:61) Qi (cid:126) | k (cid:105)| (cid:104) k | k (cid:105) (35)The sum is extended to the states | k (cid:105) of zero momentumand is certainly positive if the operator Q i is gauge in-variant so that only states of positive metric contribute.We shall come back to this point below.To summarize we have shown that up to order ∆ S ρ is ﬁnite, except for a possible logarithmic divergence inthe two point correlator of the chromo-electric ﬁeld. Aﬁnite contribution to ρ reﬂects in a non-zero multiplica-tive factor in µ which is irrelevant to symmetry. Higherorder terms in the expansion of the plaquettes in termsof lattice spacing Eq’s (22) and (23) are then irrelevant,as well as all the terms of order > (cid:60) P µν . All the terms which only contain factors (cid:61) Q i can diverge log-arithmically and be relevant to symmetry. We have noidea about the sign of these terms, which could inﬂuencethe critical index of (cid:104) µ (cid:105) at the transition. In the spiritof Stochastic Vacuum [23] [24] the two point functionshould dominate. In any case our analysis shows thatthe order parameter (cid:104) µ (cid:105) can be traded with the inverseof the correlation length of the theory. In Section 6 wediscuss the connection to lattice results on the subject[26] [27]. We close this section by rewriting the sum ofthe kinematically divergent parts of ρ , ρ div ρ div = −(cid:104) ∆ S (cid:105) + (cid:104)(cid:104) βS ∆ S (cid:105)(cid:105) + (cid:88) i,(cid:126)n ,(cid:126)n [ β (cid:104)(cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) (cid:105)(cid:105)− β (cid:104)(cid:104) S (cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) (cid:105)(cid:105) ] S i ( (cid:126)n ) (36)with (cid:126)n ≡ (cid:126)n + (cid:126)n IV. COMPUTING THE KINEMATICDIVERGENCES.

In this section we compute the divergent part of ρ div byuse of a strong coupling expansion. It is already knownthat the term O ( β ) is convergent for U (1) gauge group,and infrared divergent for SU (2) [21].Here we show that for U (1) gauge group ρ div vanishesto all orders of the strong coupling expansion, as expected[11] [12][13].For any gauge group and for any operator O the strongcoupling expansion of (cid:104) O (cid:105) is [30] (cid:104) O (cid:105) ≡ (cid:82) Π dU µ ( n ) exp( − βS ) O (cid:82) Π dU µ ( n ) exp( − βS ) = (cid:80) ∞ n =0 ( − β ) n n ! (cid:104)(cid:104) OS n (cid:105)(cid:105) . The integral is a group integral, the link U µ ( n ) be-ing an element of the group. Here again the doublebracket means connected graph, the disconnected partsbeing canceled by the denominator.We compute in the strong coupling expansion the fourterms in Eq(36): For the ﬁrst term we get −(cid:104)(cid:104) ∆ S (cid:105)(cid:105) ≡ D (cid:80) i(cid:126)n ( C i ( (cid:126)n ) − D = −(cid:104)(cid:60) P i (cid:105) SC = − ∞ (cid:88) n =0 ( − β ) n n ! (cid:104)(cid:104)(cid:60) P i S n (cid:105)(cid:105) (37)Only the term of the action (cid:80) nµν P µν ( n ) Eq(4) con-tributes to the connected part and not the term 1.Each link must appear an even number of times in thegraphs to give a non zero result when integrated over. Itfollows that only odd values of n are non zero in the sumEq(37). We rewrite it as D = ∞ (cid:88) n =0 β n +1 (2 n + 1)! (cid:104)(cid:104)(cid:60) P i S n +1 (cid:105)(cid:105) (38)In the same way we get for the second term β (cid:104)(cid:104) S ∆ S (cid:105)(cid:105) ≡ D (cid:80) i(cid:126)n ( C i ( (cid:126)n ) −

1) with D = β (cid:80) ∞ n =0 ( − β ) n n ! (cid:104)(cid:104)(cid:60) P i S n +1 (cid:105)(cid:105) Here only the even values of n contribute, but there isan extra β with respect to D and after some algebra D = ∞ (cid:88) n =0 β n +1 (2 n )! (cid:104)(cid:104)(cid:60) P i S n +1 (cid:105)(cid:105) (39)The sum of the two terms gives ﬁnally − ∆ S + β (cid:104)(cid:104) S ∆ S (cid:105)(cid:105) = (cid:80) i(cid:126)n ( C i ( (cid:126)n ) − D + D ) with D + D = ∞ (cid:88) n =0 β n +1 (2 n + 1)! 2( n + 1) (cid:104)(cid:104)(cid:60) P i S n +1 (cid:105)(cid:105) (40)We now move to the remaining two terms ofEq(36). We get for the third one [See Eq(31)] Σ = D (cid:80) i(cid:126)n S i ( (cid:126)n ) , and for the fourth term − β (cid:104)(cid:104) S ∆ S (cid:105)(cid:105) = D (cid:80) i(cid:126)n S i ( (cid:126)n ) . D = − ∞ (cid:88) n =0 ( − β ) n +1 n ! (cid:88) (cid:126)n (cid:104)(cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) S n (cid:105)(cid:105) (41)Here only even values of n contribute ad therefore D = ∞ (cid:88) n =0 β n +1 (2 n )! (cid:88) (cid:126)n (cid:104)(cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) S n (cid:105)(cid:105) (42)For the last term we get D = − β ∞ (cid:88) n =0 ( − β ) n n ! (cid:88) (cid:126)n (cid:104)(cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) S n +1 (cid:105)(cid:105) (43)Here only odd values of n contribute, so that D = 12 ∞ (cid:88) n =1 β n +1 (2 n − (cid:88) (cid:126)n (cid:104)(cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) S n (cid:105)(cid:105) (44)Finally for we can write for the sum of the two terms D + D = ∞ (cid:88) n =0 β n +1 (2 n + 1)! ( n + 1)(2 n + 1) (cid:88) (cid:126)n (cid:104)(cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) S n (cid:105)(cid:105) (45)Putting everything together the coeﬃcient of the diverg-ing part , D + D − ( D + D ) is ( the diverging factorof D , D is C i ( (cid:126)n ) −

1, that of D and D is S i ( (cid:126)n ) ) (cid:88) n β n +1 (2 n + 1)! ( n + 1) (cid:2) (cid:104)(cid:104)(cid:60) P i S n +1 (cid:105)(cid:105)− (2 n + 1) (cid:88) (cid:126)n (cid:104)(cid:104)(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) S n (cid:105)(cid:105) (cid:3) (46) The analysis above is valid for any gauge group. Forthe group U (1) Q i = P i (cid:60) P i and (cid:61) Q i are the realpart and the imaginary part of the plaquette and areboth gauge invariant. We now show that the divergencecancels to all orders as expected [11]. For higher groupsthe problem is not even well deﬁned, since (cid:61) Q i is notgauge invariant. In addition there is a divergence at order β in the strong coupling expansion, which is absent inthe case of U (1), in agreement with Ref([21]).To show that we rewrite the expression in Eq(46) in amore convenient form: (cid:104)(cid:104)(cid:60) P i S n +1 (cid:105)(cid:105) = (2 n + 1) (cid:88) (cid:126)n (cid:104)(cid:104)(cid:60) P i ( (cid:126)n ) (cid:60) P i ( (cid:126)n ) S n (cid:105)(cid:105) The choice (cid:126)n can be done by use of translation invari-ance and the factor (2 n + 1) is purely combinatorial andcomes from the exponent of S n +1 .For any value of (cid:126)n , (cid:126)n and n the contribution toEq(46) is proportional to D n ( (cid:126)n , (cid:126)n ) ≡ (cid:104)(cid:104) (cid:2) (cid:60) P i ( (cid:126)n ) (cid:60) P i ( (cid:126)n ) −(cid:61) Q i ( (cid:126)n ) (cid:61) Q i ( (cid:126)n ) (cid:3) S n (cid:105)(cid:105) (47)In the case of U (1) gauge group Q i = P i and D n ( (cid:126)n , (cid:126)n ) = 14 (cid:104)(cid:104) (cid:2) ( P i ( (cid:126)n ) + P ∗ i ( (cid:126)n ))( P i ( (cid:126)n ) + P ∗ i ( (cid:126)n ))+( P i ( (cid:126)n ) − P ∗ i ( (cid:126)n ))( P i ( (cid:126)n ) − P ∗ i ( (cid:126)n )) (cid:3) S n (cid:105)(cid:105) (48)The only non zero contributions are those proportionalto P i ( (cid:126)n ) P ∗ i ( (cid:126)n ) and P ∗ i ( (cid:126)n ) P i ( (cid:126)n ), and they cancelbetween the two terms in Eq(48) so that D n ( (cid:126)n , (cid:126)n ) = 0and there is no divergence. The physical reason is that tohave non zero correlation function of two operators theymust have total electric ﬂux zero, since the electric ﬂuxis odd under charge conjugation.Notice that the conﬁgurations which survive the groupintegration (cid:82) π − π dθ i π are those in which each link U i ( (cid:126)n ) =exp( iθ i ( (cid:126)n ) appears the same number of times as its com-plex conjugate and therefore contributes 1.The proof can be made more detailed.We only pre-sented the main point of the argument.We already knowby independent arguments [11] [12] [13]that no kinematicdivergence is present in the order parameter of the U (1)gauge theory.To illustrate the argument and to compare to the nonabelian case consider the contribution wit n = 2 i.e. O ( β ) depicted in Fig.1 Here (cid:126)n = (cid:126)n + ˆ k with k a spatialdirection orthogonal to i . The two external plaquettes P i ( (cid:126)n ) and P i ( (cid:126)n ) are depicted by full oriented links,the plaquettes coming from S by dotted oriented links.The two cubes correspond to the two terms in Eq(48) andthe black dots in the right upper vertices of the externalplaquettes represent the σ insertion in Q i : for the U (1)case they are trivially 1.From the graphs it is easy to see that if the links inthe external plaquette in (cid:126)n rotate clockwise those in the − FIG. 1: A contribution O ( β ) to ρ div . The black circles onthe right cube are insertions of σ . external plaquette in (cid:126)n rotate anti-clockwise and vice-versa if we want in each link two lines with opposite di-rection. This is an example of the general argument givenabove. For U (1) gauge group there is no σ insertion, theblack dots are equal to 1 and the two graphs in ﬁgure areequal and cancel each other.For SU ( N ) gauge group the graphs can be directlycomputed by use of the group integration formulae inAppendix 2 [30]. The result is N for the ﬁrst cube andzero for the second one. The two contributions do notcancel and the kinematic divergence with them. This weknew already from Ref. [21]. The second cube is noteven gauge invariant: a generic gauge transformation ro-tates diﬀerently the two σ ’s in the plaquette at (cid:126)n andin that at (cid:126)n One way out is to operate a subtractionwhich eliminates the kinematic divergence but preservesthe possibility of connecting a logarithmic divergence ofthe two-point function of the electric ﬁeld to the vanish-ing of the order parameter. This was done in Ref. [21].A more satisfactory solution in all respects suggestedby our analysis is a better deﬁnition of the order param-eter which is gauge invariant from scratch. This we willdiscuss in the next section. We show there that in thisway the kinematic divergences sum to zero.

V. A GAUGE INVARIANT ORDERPARAMETER

The analysis of Section 4 naturally leads to a fun-damental improvement of the order parameter, whichmakes it gauge invariant and free of ”kinematic ” diver-gences and sheds light on the meaning of abelian projec-tions.To cancel kinematic divergences, i.e. to have D n as de-ﬁned by Eq(48) equal to zero at any order n , Q i ( (cid:126)n ) hasto be gauge invariant. This looks impossible since Q i ( (cid:126)n )transforms as the third component of a vector under lo-cal gauge transformations and generically by a diﬀerentangle in (cid:126)n and (cid:126)n . A change of abelian projection by alocal gauge transformation does not help, as well as a nonlocal transformation like a parallel transport to points atﬁnite distance.A gauge-invariant σ can however be deﬁned in anypoint ( (cid:126)m, t ) of space-time by parallel transport to inﬁnity,along any path C by a unitary operator V C ( (cid:126)m, t ) which depends on the point ( (cid:126)m, t ) and on the path C . We deﬁne¯ σ ( (cid:126)m, t ) = V † C ( (cid:126)m, t ) σ V C ( (cid:126)m, t ) (49)Any path C gives a gauge invariant Q i ( (cid:126)n ).Physically this is procedure is related to the fact thata monopole breaks some SU (2) symmetry , (genericallya subgroup of the gauge group), to U (1), the little groupof the Higgs ﬁeld [14][15]. This breaking can not be abreaking of the local gauge symmetry, which is forbidden[28], but of a global symmetry. The global symmetrylives on the hypersphere at inﬁnity, where the direction σ is deﬁned, and is gauge invariant.Indeed any action of the gauge group on a ﬁeld systemhas the form [29] U G ( x ) = U ( x ) U B (50)where U ( (cid:126)x ) is the usual gauge transformation at the point x in the bulk of the system and U ( x ) = 1 on the borderat inﬁnity where the ﬁelds vanish. Instead U B = 1 atﬁnite distances, is non trivial at inﬁnity and is a globaltransformation: it is relevant whenever there are ﬁeldswhich are non zero at inﬁnity, like the Higgs ﬁeld in thebroken phase of a Higgs system.We have shown that replacing σ by a parallel trans-port of it to inﬁnity is a necessary condition to satisfyEq(48). However is not generally suﬃcient as is e.g. inthe case for the axial gauge, which is deﬁned by a paralleltransport along a line parallel say to the z axis at (cid:126)x and (cid:126)y ﬁxed. It is easily seen that the product of two Q i ’s cor-responding to two diﬀerent values of ( (cid:126)x, (cid:126)y ) is zero to allorders in the strong coupling expansion and thus the sec-ond term in Eq(48) is zero for all values of (cid:126)n (cid:126)n exceptfor a set of zero measure ( (cid:126)n ) x = ( (cid:126)n ) x , ( (cid:126)n ) y = ( (cid:126)n ) y .The second term in Eq(48) thus vanishes and can notcancel the ﬁrst term. It is easy to see that the only wayto have the second term in Eq(48) non zero is that thepaths C from diﬀerent points (cid:126)n to inﬁnity coincide aftersome point P in their way to inﬁnity: we shall assumethat for all the paths C independent of the point.We discuss in detail below the cancellation of D n i.e. ofthe kinematic divergence. Before discussing the cancella-tion of the kinematic divergence of the new parameter, weshow in detail that it is an order parameter for monopolecondensation. We have redeﬁned the order parameter byreplacing σ by ¯ σ ( (cid:126)m, t ) in the expression Eq(8). We geta modiﬁed M i ( (cid:126)m ) which we call M i ( (cid:126)m, t ) M i ( (cid:126)m, t ) = exp (cid:0) ig ¯ σ ( (cid:126)m, t )2 (cid:126) ¯ A i ( (cid:126)m − (cid:126)x ) (cid:1) = V † C ( (cid:126)m, t ) M i ( (cid:126)m ) V C ( (cid:126)m, t ) (51)We ﬁrst show that the new operator creates a monopoleas did the old one. Replacing the action S by S + ∆ S at any time t is equivalent to create a monopole at alltimes > t , in a similar way as for the old deﬁnition. Thechange of variables in the Feynman path integral [11] U i ( (cid:126)n, t + 1) → U i ( (cid:126)n, t + 1) M i ( (cid:126)m, t )leaves the measure invariant and sends the quantity P (cid:48) i ( (cid:126)n, t ) of Eq(7) to P i ( (cid:126)n, t ), the same as with the olddeﬁnition. The spatial links appearing in the magneticplaquettes at time t + 1 get modiﬁed as U i ( (cid:126)n, t + 1) → M i ( (cid:126)m, t ) U i ( (cid:126)n, t + 1)which means that a monopole has been added in the colordirection ¯ σ ( (cid:126)m, t ) or in the gauge invariant direction σ on the sphere at ∞ . The old deﬁnition would add it in thecolor direction of the σ axis. Finally the link aﬀected bythe change of variables appears in the plaquette P (cid:48) i ( (cid:126)n, t +1) ≡ P i ( (cid:126)n, t + 1) which is changed to P (cid:48) i ( (cid:126)n, t + 1) = 1 N T r (cid:2) U i ( (cid:126)n, t + 1) U ( (cid:126)n + ˆ i, t + 1) U † ( (cid:126)n + ˆ i, t + 1) M i ( (cid:126)n + ˆ i, t ) U ( (cid:126)n + ˆ i, t + 1) U † ( (cid:126)n, t + 2) U † ( (cid:126)n, t + 1) (cid:3) (52)We can deﬁne M i ( (cid:126)n +ˆ i, t +1) = U † ( (cid:126)n +ˆ i, t ) M i ( (cid:126)n +ˆ i, t ) U ( (cid:126)n +ˆ i, t ) (53)or V C ( (cid:126)n, t + 1) = V C ( (cid:126)n, t ) U ( (cid:126)n, t ) (54)The net eﬀect of the change of variables has been to ex-pose the monopole at time t + 1 and to reproduce at time t + 1 the same situation that existed originally at time t with the new path of the form Eq(54).Iterating the change of variables proves our statement.Comparing to the old deﬁnition the expression for ∆ S Eq(11) stays unchanged with the same (cid:60) P i ( (cid:126)n ) in theﬁrst term but a modiﬁed Q i ( (cid:126)n ) with respect to the onedeﬁned in Eq(10): in the new deﬁnition σ is replaced by¯ σ ( (cid:126)m, t ) Q i ( (cid:126)n, t ) = 1 N T r (cid:2) U i ( (cid:126)n, t ) U ( (cid:126)n +ˆ i, t )¯ σ U † i ( (cid:126)n, t +1) U † ( (cid:126)n, t ) (cid:3) (55)The new Q i is a gauge invariant electric ﬁeld strength.As for the old one the vacuum correlator of an oddnumber of Q i ’s is zero. Indeed if we replace eachlink U µ ( n ) in the Feynman integral by Π U µ ( n )Π † withΠ = exp( iσ π ), the action and the measure stay invari-ant but ¯ σ ( (cid:126)m, t ) changes sign.The analysis of Section 3 based on dimension inlength of the correlation functions stays unchanged. Butnow the two point function (cid:104)(cid:104)(cid:61) Q i ( (cid:126)n , (cid:61) Q i ( (cid:126)n , (cid:105)(cid:105) isthe gauge invariant connected correlator of two electricﬁelds. Such quantities have been studied in the liter-ature [23][24] [22],in particular on the lattice [25]. Itssubtracted version Eq(32) is the candidate part to signalde-conﬁnement by diverging logarithmically at the phasetransition, by pure dimensional arguments. With thenew deﬁnition that quantity is gauge invariant and welldeﬁned and the proof that it is negative deﬁnite Eq(35)is valid, being Q i gauge invariant. ( ) N a + N b = FIG. 2: The product of two gauge-invariant ﬁelds. a = N − , b = − a N . The proof in Appendix B Finally we argue that the new parameter has no kine-matic divergence. The terms D and D of Section 4 arenow well deﬁned as in U (1) gauge theory and not gaugedependent. We show that they cancel with D and D like in the U (1) theory order by order in the strong cou-pling expansion.To be deﬁnite consider in the case of the two pointfunction (cid:104)(cid:104) Q i ( (cid:126)n , Q i ( (cid:126)n , (cid:105)(cid:105) for the path C a straightline from ( (cid:126)n ,

0) ( (cid:126)n ,

0) to ( (cid:126)n + (cid:126)n ,

0) and then a com-mon path along some axis to inﬁnity [Fig(2)]. It is eas-ily seen[Appendix 2] that the result is a gauge invariantconnected two point function of electric ﬁeld strengthsas deﬁned in Ref [25]: See Fig 2. The meaning of theequality in Fig.2 is that the graphs represented can beparts of a generic conﬁguration to be integrated over thelinks in the strong coupling expansion of any correla-tion function: only the integral on the four overlappinglinks along the time axis has been performed. The re-sult is independent of the line transporting to ∞ : theproduct of any number of tensors of the form Eq(65)is atensor of the same form, i.e. that tensor is a projector.If we choose a diﬀerent path for the parallel transportthe result is a connected two point correlator with theconnecting line of diﬀerent form: the only condition isthat the two paths originating from Q i ( (cid:126)n ) and Q i ( (cid:126)n )overlap at some point on their way to ∞ . All this gamecan be repeated for an n-point function of gauge invari-ant chromo-electric ﬁelds but we will not do that sincehigher correlators are not relevant to the diverging partof ρ and we shall neglect their contribution to ρ itselfin the spirit of the stochastic vacuum model [23] [24].To compare to the old approach, whenever we have two Q i ’s in a term of the strong coupling expansion, whichwere represented as plaquettes with a black dot we haveto replace them by the expression in Fig.2, and then per-form the integrations on the links. In Appendix 2 we dothat explicitly for the right cube in Fig.1 with the resultthat it exactly cancels with the left cube. The generalprocedure is to integrate ﬁrst on all the links diﬀerentfrom those appearing in Fig.2. If after that the paralleltransport between the two ﬁelds acquires no overlappinglinks the result is zero because of Eq(64) and of the factthat b = − N a . If instead two extra links are left over-lapping with the parallel transport in Fig.2 it is easily0shown by use of the result in Fig.2 and of Eq(65) thatthe term cancels exactly the corresponding term comingfrom two P i ’s. In principle one could expect that alsoterms exist for which more than one pair of extra linksappear, and one should extend the proof to them. Weshall not do that here and assume that cancellation as anatural conjecture for the time being. VI. DISCUSSION

The order parameter for monopole condensation is (cid:104) µ (cid:105) the vev of the creation operator of a monopole. µ is theshift by the classical ﬁeld of a monopole of the trans-verse vector potential operated by use of the conjugatemomentum, which is the transverse electric ﬁeld. This isbasic quantum mechanics. If dual superconductivity isthe mechanism for conﬁnement we expect (cid:104) µ (cid:105) (cid:54) = 0 in theconﬁned phase and (cid:104) µ (cid:105) = 0 in the deconﬁned one. Thisis exactly what happens in the U (1) gauge theory on thelattice [11] [12] [13].In the non abelian case, say SU ( N ) one would naivelyexpect that the order parameter is the vev of the operatorwhich creates a monopole in some U (1) subgroup of thegauge group. The argument is that creating a monopoleis a gauge invariant operation, since the monopole is aconﬁguration with non trivial topology [17]: as a con-sequence the speciﬁc choice of a U (1) subgroup shouldbe irrelevant [19]. The result of this procedure, however,proves to be a nonsense: the resulting order parametervanishes in the thermodynamic limit V → ∞ both in theconﬁned and in the deconﬁned phase and therefore is noorder parameter [21].In this paper we have analyzed in detail the structureof the order parameter for generic gauge group, by ex-panding the quantity ρ Eq(14) in powers of ∆ S [Eq(6)and (11)]. We found that the ﬁrst few terms of the ex-pansion are divergent at large volumes independent ofthe gauge group and of the dynamics of the gauge the-ory. We have isolated these divergences which we callkinematic divergences. We show that for gauge group U (1) they cancel among themselves, so that the orderparameter is well deﬁned. For non abelian theories in-stead we have tracked the origin of our problems in thefact that the kinematic divergences do not cancel, ρ di-verges in the thermodynamic limit both in the conﬁnedand in the deconﬁned phase thus spoiling the possibilityof (cid:104) µ (cid:105) of being an order parameter. In addition the di-vergent part is not even gauge invariant. This indicatesthat there is something deeply wrong in the procedure.Indeed a monopole breaks some SU (2) symmetry to U (1)and that SU

2) can not be a gauge symmetry but onlya global symmetry, e.g. a group SU (2) at inﬁnity. Theﬁeld strengths are to be replaced by gauge invariant ﬁeldstrengths [23] [24] [22]. As a byproduct this makes thekinematic divergence zero and the order parameter welldeﬁned and gauge invariant.What is left of the order parameter after the cancella- tion of the kinematic divergences is ﬁnite in the conﬁnedphase if the correlation functions of the ﬁeld strengths areexponentially cut-oﬀ at large distances as is in presenceof a mass gap. The only way to have (cid:104) µ (cid:105) → ρ ≈ V →∞ K ln( V ) with K < K is related to the criticalindex δ by which (cid:104) µ (cid:105) → T c . We identify all the terms in our expansion which canin principle have such a behavior: they are all the cor-relators which do not contain P i ’s (plaquettes or ﬁeldstrengths squared) but only Q i ’s ( ﬁeld strengths). Inparticular we show that the two point function of gaugeinvariant electric ﬁelds is negative deﬁnite and can pro-duce a zero of (cid:104) µ (cid:105) . In the spirit of the stochastic vacuummodel [23] [24] this term should dominate. Attempts ex-ist in the literature to determine numerically its behavioras a function of the temperature [26] [27]. A precise de-termination of K and a comparison to the measured valueof δ could in principle say something on the contributionof higher correlators i.e. on the validity of the stochasticvacuum model. In any case our analysis relates conﬁne-ment to the existence of a ﬁnite length at least in thecorrelation of chromo-electric ﬁelds. This subject is alsostudied in diﬀerent approaches [ see e.g. Ref. [31]].Finally a comment about the uniqueness of the orderparameter (cid:104) µ (cid:105) . The direction of the common path toinﬁnity is irrelevant by symmetry reasons, as well as theposition of the point P on it. Diﬀerent choices for thepath before the point P lead to a diﬀerent line of paralleltransport between the two points. As long as for all ofthem there is a ﬁnite correlation length in the conﬁnedphase and not in the deconﬁned phase, they all producethe same order parameter. Indeed adding a ﬁnite numberto ρ which is roughly the logarithm of (cid:104) µ (cid:105) , results in anon zero multiplicative constant for the order parameterwhich we have considered irrelevant in the whole analysisof this paper, since it does not aﬀect the fact that it iszero or non zero.The possibility should also be studied of computing thenew gauge-invariant ρ on a lattice. A choice for the im-plementation could be to have all the parallel transportsgo to a point, say the origin of spatial coordinates andthen to ∞ as in Fig.2. This research would clarify in anunambiguous way whether dual superconductivity of thevacuum is the correct mechanism for conﬁnement. VII. APPENDIX 1

We want to prove Eq(18) .To do that we ﬁrst compute the series expansion of thequantity D with D the denominator in Eq(17). D = ∞ (cid:88) n =0 (cid:104) ( − ∆ S ) n (cid:105) n ! (56)1The result is D = 1 + (cid:80) ∞ n =1 d n with d n = ( − ) n +1 n ! [ (cid:104) ∆ S n (cid:105) − n − (cid:88) k =1 n ! k !( n − k )! (cid:104) ∆ S (cid:105) k (cid:104) ∆ S (cid:105) n − k + (cid:88) k ≥ ,k ≥ n ! k ! k !( n − k − k )! (cid:104) ∆ S k (cid:105)(cid:104) ∆ S k (cid:105)(cid:104) ∆ S n − k − k (cid:105)− ..... + ( − ) n +1 n !1! n (cid:104) ∆ S (cid:105) n ] (57)The sums over the k i ’s in all terms run on positiveintegers with the condition n − (cid:80) k i ≥

1. The last termis the on in which all of the k i = 1, ( i = 1 ...n −

1) and n − (cid:80) k i = 1.According to the deﬁnition of connected correlator d n = ( − ) n +1 n ! (cid:104)(cid:104) ∆ S n (cid:105)(cid:105) (58)Indeed the expression in Eq(57) subtracts all the discon-nected parts from the correlator (cid:104) ∆ S n (cid:105) .We then compute the series expansion of the quantity T ≡ D exp( − ∆ S ) which appears in Eq(18) .We get T = (cid:80) ∞ n =0 T ( n ) . T (0) = 1 and for n ≥ T ( n ) = n (cid:88) k =0 ( − ∆ S ) n − k ( n − k )! d k (59)or, isolating the term with k = n , T ( n ) = ( − ) n n ! ¯∆ S n + d n (60)where ¯∆ S n ≡ ∆ S n − n − (cid:88) k =1 n ! k !( n − k )! ∆ S k (cid:104) ∆ S (cid:105) n − k + (cid:88) k ≥ ,k ≥ n ! k ! k !( n − k − k )! ∆ S k (cid:104) ∆ S k (cid:105)(cid:104) ∆ S n − k − k (cid:105)− ..... + ( − ) n +1 n !1! n ∆ S (cid:104) ∆ S (cid:105) n − (61)It is immediately seen that (cid:104) ¯∆ S n (cid:105) = (cid:104)(cid:104) ∆ S n (cid:105)(cid:105) Moreover by use of Eq(58) and Eq(60) (cid:104) T ( n ) (cid:105) = 0 sothat (cid:104) T (cid:105) = T = 1 as it should be.¯∆ S n is the connected part of ∆ S n . If O is any localoperator the quantity (cid:104) OT ( n ) (cid:105) = ( − ) n n ! (cid:2) (cid:104) O ¯∆ S n (cid:105) − (cid:104) O (cid:105)(cid:104)(cid:104) ∆ S n (cid:105)(cid:105) (cid:3) = (cid:104)(cid:104) O ∆ S n (cid:105)(cid:105) (62)is fully connected being the connected part of the correla-tor of O with a connected correlator. Taking O = S +∆ S proves Eq(18). The result for the term proportional to ∆ S , T isknown in the literature [32].Indeed T = (cid:80) ∞ n =0 (cid:104) ( − ∆ S ) n +1 (cid:105) n ! (cid:80) ∞ n =0 (cid:104) ( − ∆ S ) n (cid:105) n ! = ∂ λ ln( (cid:104) exp( − λ ∆ S ) (cid:105) ) λ =1 or [32] T = ∞ (cid:88) m =0 ( − ) m m ! (cid:104)(cid:104) ∆ S m +1 (cid:105)(cid:105) (63)The logarithm of a generating functional is the gener-ator of the connected correlators, a well known fact. VIII. APPENDIX 2

We make use in our strong coupling computations oftwo basic formulae which we take from Ref.[30]. The ﬁrstone is (cid:90) dU U α β U † β α = 1 N δ β β δ α α (64)The group is SU ( N ), U is an N × N matrix in the fun-damental representation, and the integration ranges onthe group.The second basic formula is (cid:90) dU U α β U † β α U α β U † β α = a [ δ α α δ α α δ β β δ β β + δ α α δ α α δ β β δ β β ] + b [ δ α α δ α α δ β β δ β β + δ α α δ α α δ β β δ β β ] (65)with a = 1 N − b = − N a (66)A ﬁrst consequence of Eq(64) is that the average valueof any closed path covered by two lines circulating in op-posite direction is equal to 1. This allows to immediatelycompute the left cube in Fig.1: the integral on the hor-izontal pairs of links connecting the front and the rearplaquette gives by Eq(64) N times the product of thetwo plaquettes covered each by two lines circulating inopposite directions, which is = 1. In conclusion the vev of the left cube is N .As for the cube on the right in Fig.1 we can repeatthe procedure, but now of the two overlapping lines bothin the front and in the rear plaquette one contains a σ inserted and when the average is taken by use of Eq(64)the result is proportional to ( T rσ ) and thus is zero. Thediﬀerence of the two cubes is non zero and with it thekinematic divergence.We now prove the equality in Fig.2 . We integrate onthe four overlapping links in the central vertical line byuse of Eq(65). Of the four terms two are proportional2to [ T rσ ] and vanish, the other two are proportional to T r ( σ ) = N which is the factor in front of the result.The coeﬃcients a and b are computed in Ref [30]. Wecould have extended arbitrarily the length of the centralline down to ∞ , with the same result. It is indeed easy toshow that the product of two tensors of the form Eq(65)has the same form with the same coeﬃcients a and b .We could also have chosen a diﬀerent form of the pathsmerging in a single line after some point on the way toinﬁnity, as well as a diﬀerent direction to inﬁnity.Theresult would only be a diﬀerent path for parallel transportconnecting the two plaquettes in Fig.2. Notice that theequality in Fig.2 means that the ﬁelds there can be partof a generic conﬁgurations to integrate over. The onlyintegral which has already been performed is that on thelink in which the lines merge. To have the equality inFig.2 an exact equality it is necessary to extend the lineto inﬁnity. At any ﬁnite order of the strong couplingexpansion the connected contributions to any correlationfunction extend to a ﬁnite distance, which, however cantend to inﬁnity with increasing order.Finally we discuss the cancellation of the kinematicdivergence. We start computing the right cube in Fig.1.With the gauge-invariant order parameter the two exter-nal Q i ’s are replaced by the two terms in the right handside in Fig.2. Integrating on the double links between thefront and the rear plaquette by use of Eq(64) and then on the fourfold link by use of Eq(65) gives after somealgebra N multiplied by the quantity aN [ a ( N + 1) + 2 bN ] + bN = a ( N −

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