A non Abelian effective model for ensembles of magnetic defects in QCD(3)
AA non Abelian effective model for ensemblesof magnetic defects in
QC D C. D. Fosco a and L. E. Oxman b a Centro Atómico Bariloche and Instituto BalseiroComisión Nacional de Energía AtómicaR8402AGP Bariloche, Argentina. b Instituto de FísicaUniversidade Federal FluminenseCampus da Praia VermelhaNiterói, 24210-340, RJ, Brazil.
Abstract
We construct a non Abelian model for SU (2) QCD in Euclideanthree-dimensional spacetime and study its different phases. The modelcontains a center vortex sector coupled to a dual effective field encodinginformation about how the vortices are paired in the ensemble. Thepossible phases in parameter space are interpreted in terms of theproliferation of either closed center vortices or closed chains, wherethe endpoints of open vortices are attached in pairs to monopole-likedefects.
The vortex model introduced by t’ Hooft [1] is a low energy effective theorythat successfully describes some aspects of the confinement mechanism in dimensional SU ( N ) Yang-Mills theories. It is defined in terms of adynamical variable which is a complex scalar field V equipped with a discrete Z ( N ) symmetry, realized with a Lagrangian, L = ∂ µ ¯ V ∂ µ V + µ ¯ V V + α ( ¯ V V ) + β ( V N + ¯ V N ) . (1)Its form is based on a study of the possible nontrivial vortex correlationfunctions in the original theory. In particular, the confining phase is described1 a r X i v : . [ h e p - t h ] M a r s one where the discrete Z ( N ) symmetry is spontaneously broken, due tothe presence of a vortex condensate. The possibility of an effective fieldrepresentation for D center vortices relies on the fact that an ensembleof stringlike objects can be thought of as a sum over different numbers ofparticle worldlines, which corresponds to a second quantized field theory [2]-[7]. Based on this observation, one of us proposed, in refs. [8, 9], a generalizedvortex model where ∂ µ in eq. (1) is substituted by the covariant derivative D µ , that depends on a dual vector field λ µ describing the off-diagonal sector.The dynamics is completed with a Proca action term for λ µ . In ref. [10], wederived this model by considering an ensemble of chains, where the vortexendpoints are attached in pairs to monopole-like defects, and following recentpolymer techniques to compute the vortex end-to-end probability.Effective field models can also be obtained in scenarios based just on themonopole (instanton) component. In this case, the assumption of Abeliandominance and the associated monopole ensemble is encoded in a sine-Gordontype model for a scalar dual field (see [11, 12] and references therein), as oc-curs in the case of compact QED (3) , discussed by Polyakov in ref. [13].In spite of the fact that the initial theory is a non Abelian one, theseeffective models are
Abelian , that is, additional information regarding thistransition is already incorporated, while it would be desirable to see it ap-pearing as a phase transition in a previous non Abelian model.In addition, the different ideas regarding the magnetic sector have beenexplored in the lattice, relying only on monopoles [14]-[17], only on centervortices [18]-[22], or on chains [23]-[25]. Therefore, it would be interesting toconstruct a model where the possible phases in parameter space correspondto the different ensembles.In this article, we construct a non Abelian effective model which en-compasses a description of interacting effective gluons and center vortices.Depending on the choice of parameters, the vortices can be found in differ-ent states, including a phase where they are closed, and a phase where theirendpoints become paired to form closed chains.To that aim, we use a parametrization that treats the different colorcomponents in a symmetric way [26], and describes correlated monopolesand center vortices as defects of a local color frame ˆ n a , a = 1 , , . Thisparametrization is based on the usual manner to introduce thin center vor-tices in Yang-Mills theories [27, 28], and corresponds to a symmetric formof the Cho-Faddeev-Niemi (CFN) decomposition [29]-[31], used to representmonopoles as defects of the third component ˆ n = ˆ n . For a descriptionof center vortices in the CFN framework, and related consequences in thecontinuum, see refs. [8, 9].Since center vortices can be joined in pairs to pointlike monopoles, the2atural non Abelian field content of the model is given by a scalar fieldwith one (magnetic) color index, generalizing the vortex field V in eq. (1),and a scalar field with two color indices, generalizing the scalar dual field inscenarios only involving the monopole component . The order parameterspresent in the effective model bear a relation to the nature of the phasetransition one may describe. In this respect, the interesting point has beenraised [12] about whether the confining/deconfining phase transition is ofthe KT or Ising model type. The former involves the monopole sector: athigh temperatures, the instanton magnetic flux is distributed along the twospatial directions, thus leading to effective logarithmic interactions. Then,because of dimensional reduction, instantons and anti-instantons tend to besuppressed by forming pairs. On the other hand, the latter naturally involvesthe vortex degrees of freedom, as they are the objects where the discretesymmetry transformations act.In the model we construct and study below, since it does contain orderparameters for both the center vortices and the distribution of monopole-like defects they can concatenate to form chains, an interesting frameworkto discuss the competition between different phases shall emerge, originatinga phase diagram with a rich structure. This paper is organized as follows:in section 2, we deal with the topological defects included in the model, inparticular, their parametrization, and the functional and ensemble integra-tion over them. In section 3, based on the previous section results, and afterdiscussing the possible symmetries, we construct an action for the effectivemodel in terms of the fields introduced therein. Finally, in section 4, wepresent a study of the phase structure of the model, based on some assump-tions about the relative strength of its different terms. We shall start from the SU (2) Yang-Mills action, S Y M , which may be writtenas follows: S Y M = 14 (cid:90) d x (cid:126)F µν · (cid:126)F µν , (2)where (cid:126)F µν is the non Abelian field-strength tensor. We use an arrow ontop of any object to denote the -component vector formed by its compo-nents on the su (2) Lie algebra basis, whose elements are the (Hermitian)generators ( T a ) a =1 . In the concrete case we are considering, they can beconveniently realized as T a = τ a / , where τ a denotes a Pauli matrix; theysatisfy (cid:2) T a , T b (cid:3) = i (cid:15) abc T c , and tr( T a T b ) = δ ab . Interestingly, isospin two order parameters appear in models for liquid crystals [32]. (cid:126)F µν , as follows: (cid:126)F µν · (cid:126)T = ig [ D µ , D ν ] , D µ = ∂ µ − ig (cid:126)A µ · (cid:126)T , (3)where D µ has been used to denote the covariant derivative operator, whenacting on fields in the fundamental representation.It goes without saying that a ‘canonical color basis’ (cid:0) ˆ e a (cid:1) a =1 (with colorcomponents ˆ e ab = δ ab ) can be introduced, so that (cid:126)A µ = (cid:126)A aµ ˆ e a . This seeminglytrivial remark is made in order to highlight the next step; namely, that onecould have used a different basis. Indeed, in order to describe configurationswith defects, in a symmetric way that admits its extension to finite sizeobjects, we introduce a space-dependent color basis (ˆ n a ) a =1 , related to theoriginal one by: ST a S − = ˆ n a · (cid:126)T (with S ∈ SU (2) ). Thus the new basisis connected to the canonical one by an orthogonal space-dependent matrix R ( S ) : ˆ n a = R ( S )ˆ e a , which belongs to the adjoint representation. In thisrepresentation, the corresponding infinitesimal generators shall be denoted by M a , with ( M a ) bc ≡ − i(cid:15) abc . They satisfy (cid:2) M a , M b (cid:3) = i(cid:15) abc M c , tr ( M a M b ) =2 δ ab . At this point, and equipped with the local basis, we consider theparametrization of the gauge field [26]: (cid:126)A µ = ( A aµ − C aµ ) ˆ n a , (4)where the frame dependent fields, C aµ = − g (cid:15) abc ˆ n b · ∂ µ ˆ n c , (5)satisfy the properties: ˆ n b · ∂ µ ˆ n c = − g(cid:15) abc C aµ , C aµ M a = ig R − ∂ µ R . (6)This corresponds to a symmetric form of the Cho-Faddeev-Niemi (CFN)decomposition [29]-[31]. In terms of the parametrization (4) of the gaugefield, we note that the field-strength tensor becomes: (cid:126)F µν = G aµν ˆ n a , G aµν = F aµν ( A ) − F aµν ( C ) , (7)with F aµν ( A ) ≡ ∂ µ A aν − ∂ ν A aµ + g(cid:15) abc A bµ A cν (and an analogous expression for F aµν ( C ) ), while the Yang-Mills action is given by: S Y M = (cid:90) d x G aµν G aµν . (8)4egarding the color components of the frame-dependent tensor F aµν ( C ) , theycan also be obtained by commuting covariant derivatives in the adjoint rep-resentation: F aµν ( C ) M a = ig (cid:2) D µ , D ν (cid:3) , D µ ≡ ∂ µ − igC aµ M a , (9)so that the second equality in (6) implies the alternative expression for F aµν ( C ) : F aµν ( C ) = i g tr ( M a R − [ ∂ µ , ∂ ν ] R ) . (10)This equation highlights the meaning of F aµν ( C ) , by showing that it can onlybe different from zero where R has defects; these, are characterized hereby the noncommutativity of the mixed partial derivatives. These defectsare zero measure objects; in other words, the partial derivatives will failto commute on zero measure regions. Being this an effective theory, thisshould be interpreted as the assumption that the model describes physics atdistances much larger than the size of the defects.Of course, there are infinitely many different local frames, and corre-sponding fields A aµ , that can be used to describe one and the same gaugefield configuration, A aµ . One can use that large amount of freedom in orderto split it into its ‘regular’ and ‘singular’ parts. Indeed, the A aµ measurewill represent topologically trivial fluctuations. The singular configurations,described by the frames, will have a measure representing an ensemble inte-gration over defects.In ref. [26], one of us has shown that the configuration in (4) is tantamountto the usual way [27, 28] to introduce thin center vortices on top of a trivialfield configuration A aµ ˆ e a , namely, (cid:126)A µ · (cid:126)T = S (cid:126) A µ · (cid:126)T S − + ig S∂ µ S − − (cid:126)I µ ( S ) · (cid:126)T . (11)Because of the presence of the last term, this is not just a gauge trans-formation of the topologically trivial gauge field. Indeed, the (cid:126)I µ ( S ) fieldcorresponds to the so called ideal center vortex, and is localized on a hyper-surface Σ . This is the region which, when traversed, makes S change by acenter element. It is designed to cancel the contribution in the second termoriginated from the discontinuity of S − , only leaving the effect of the borderof Σ where the thin center vortices are located. That is, we can write, ig S∂ µ S − | Σ = (cid:126)I µ ( S ) · (cid:126)T , (12)5here the subscript in the left-hand side amounts to just keeping in thecalculation the term originated from the derivative of the discontinuity in S − . Considering two regular mappings U , ˜ U , the ideal vortex satisfies, (cid:126)I µ ( U S ˜ U − ) · (cid:126)T = U (cid:126)I µ ( S ) · (cid:126)T U − , (13)obtained from ∂ µ ( ˜ U − S − U − ) | Σ = ˜ U − ∂ µ S − | Σ U − , as the term localizedon Σ is only generated when ∂ µ acts on S − . The gauge field (cid:126)A µ = (cid:126)A µ ( (cid:126) A , S ) in (11) enjoys the following properties, (cid:126)A U ( (cid:126) A , S ) = (cid:126)A ( (cid:126) A , U S ) , (cid:126)A ( (cid:126) A , S ) = (cid:126)A ( (cid:126) A ˜ U , S ˜ U − ) . (14)Then, in terms of the (cid:126) A , S variables we have a double redundancy, theusual one associated with invariance of the Yang-Mills action under gaugetransformations, (cid:126)A Uµ · (cid:126)T = U (cid:126)A µ · (cid:126)T U − + ig U ∂ µ U − , represented by S → U S ,and other originated from the different ways to express the same vector field,combining the transformation (cid:126) A ˜ Uµ · (cid:126)T = ˜ U (cid:126) A µ · (cid:126)T ˜ U − + ig ˜ U ∂ µ ˜ U − , togetherwith a right multiplication of S .At this point, we would like to emphasize that a nonperturbative defi-nition of the path integral in Yang-Mills theory is still lacking. This comesabout as a gauge fixing procedure generally leads to Gribov copies [33] in thatregime, so that it is difficult to define an appropriate object where each phys-ical situation is counted only once. The restriction to the modular region hasbeen usually implemented by means of the Zwanzinger action [34]. In thisframework, in the infrared regime, the path integral has been shown to bedominated by configurations near the Gribov horizon. On the other hand, asis well-known, configurations containing magnetic objects proliferate at thehorizon [35]-[37]. From this perspective, it is natural to fix the redundancyby introducing the identity F P [ A ] (cid:82) [ d ˜ U ] δ [ f ( A ˜ U )] , in the perturbativesector where the Faddeev-Popov procedure is well defined. In addition, as the S sector parametrizes correlated monopoles and center vortices, it representsconfigurations at the horizon, relevant to describe the large distance physics.Giving a configuration S , gauge fixing amounts to choose a representative ofthe orbit U S . Any condition imposed on (cid:126)A ( (cid:126) A , S ) will be invariant under the ˜ U -transformations in eq. (14). This is also the case for conditions dependingon (cid:126)I ( S ) , as it is invariant under right multiplication (cf. eq. (13)).In this article we shall not attempt to derive a precise construction ofthe integration measure. Rather, having the previous remarks, notation, andconventions in mind, we argue that it is quite natural to propose the followingpath integral, Z Y M = (cid:90) [ d A ][ dS ] ∆ F P [ A ] δ [ f ( A )] e − S Y M [ A ] , (15)6here dS , represents the ensemble integration over monopoles and thin cen-ter vortices, that is supposed to include its own appropriate gauge fixingcondition. Note that, in the trivial sector, where S = S r is regular, we have (cid:126)A ( (cid:126) A , S r ) = (cid:126) A S r and the associated contribution to (15) is the usual, pertur-bative one. Only in that sector (cid:126)A ( (cid:126) A ˜ U , S ) = (cid:126)A ( (cid:126) A , S ˜ U ) may be identified witha gauge transformation.As a final step, and as a guide to the construction of the effective model,we rewrite the partition function in the equivalent form: Z Y M = (cid:90) [ d A ][ dS ][ dλ ] ∆ F P [ A ] δ [ f ( A )] e − (cid:82) d x [ λ aµ λ aµ + iλ aµ ( F aµ ( A ) −F aµ ( C )) ] , (16)where F aµ = (cid:15) µνρ F aνρ , and we have introduced a color-valued auxiliary field λ aµ to deal with a first-order version of (8). Let us now derive a non Abelian effective field theory for the sector of de-fects. The derivation will become possible by relying on the symmetriesexhibited by the ensemble integration. This effective theory shall containmass parameters, which we assume are originated from those present in a(phenomenological) ansatz for the action of the defects. In this regard, wenote that up to now we have considered thin center vortices, parametrized asin (4). However, lattice simulations [18] point to the idea that they becomethick objects, characterized by some finite radius of the order of 1fm. More-over, as discussed in [26], the stable objects in the continuum could in factcorrespond to some deformation of the thin objects given in (4), where the“thin” quantities C aµ are replaced by some smooth finite radius profiles C aµ .If this is assumed to be the case, rather than eqs. (7), (8), the Yang-Millsaction would have the form, S Y M = (cid:90) d x
14 ( F aµν ( A ) − F aµν ( C )) + R , (17)where R vanishes for thin center vortices. Note that the first term can belinearized, as we did before, by introducing the fields λ aµ . Besides, at largedistances, approximating C aµ by C aµ , this term shall originate the terms ap-pearing in the exponent of eq. (16), when the center vortices were consideredto be thin. On the other hand, the second term ( R ), will be concentratedon the center vortices and at large distances will produce instead an addi-tional action S d for the defects. Therefore, in the general case, the ensemble7ntegration must be written in the form, e − S v,m [ λ ] = (cid:90) [ dS ] e − S d + i (cid:82) d x λ aµ F aµ ( C ) . (18)The second term in the exponent above has a local SO (3) symmetry underright multiplication: S → S ˜ U , changing the color basis from ˆ n a · (cid:126)T = ST a S − to ˆ n (cid:48) a · (cid:126)T = S ˜ U T a ˜ U − S − , that is, ˆ n (cid:48) a = R ( S ) R ( ˜ U )ˆ e a . Note that, using (10),and that R ( ˜ U ) contains no defects, we have, F aµν ( C (cid:48) ) = i g tr ( R ( ˜ U ) M a R − ( ˜ U ) R − ( S )[ ∂ µ , ∂ ν ] R ( S )) . (19)In other words, a regular local rotation of λ aµ can be translated to a regularlocal transformation of S . Then, if S d were nullified, that is, if we weredealing with strictly thin center vortices, S v,m [ λ ] would be invariant underlocal SO (3) rotations, as the transformation S → S ˜ U could be absorbed bythe integration measure dS . In this regard, we would like to underline thatthis measure is to be accompanied by an appropriate gauge fixing conditionthat is invariant under right multiplication (see the discussion at the end ofthe previous section). However, in S v,m [ λ ] , that symmetry will be broken toa global one because of the thick character expected for center vortices. Tohave a simple picture about this statement we note that an action for thickcenter vortices will typically contain a Nambu-Goto term plus other termsdescribing the center vortex rigidity [27, 38]. These pieces can be generated,for instance, from a large distance approximation of the more symmetric term(in color space), (cid:90) d x d y F aµν ( C ) | x G M ( x − y ) F aµν ( C ) | y , where G M is a kernel localized on a scale /M . Now, as we have seen ineq. (19), the field strength F aµν ( C ) will rotate under local ˜ U transforma-tions. Therefore, as for any finite M the integrand above depends on thefield strength at different spacetime points, it will change under the localtransformations, only leaving a symmetry under the global ones.Based on purely geometrical/mathematical grounds, the possible kindsof defects can be straightforwardly classified as follows:i) Closed center vortices.ii) Monopoles and antimonopoles, joined by center vortices (each pointlikeobject is joined by a pair of center vortices).8ii) A particular limit of ii): A coincident pair of center vortices, whichshould correspond to an unobservable Dirac string.To proceed, let us consider a type ii) configuration (correlated monopolesand center vortices). To that end we recall that, in refs. [8]-[10], we haveconsidered a particular case of that situation, namely, when the center vor-tex color points along the (locally) diagonal direction ˆ n . In that case, theeffective field describing these objects corresponds to a complex vortex field V . In particular, in ref. [10], we have shown how the ensemble integrationover open center vortices, whose endpoints are joined in pairs to form closedchains, leads to an Abelian Z (2) effective theory that can be written in termsof V , thus making contact between the initial representation and the finaleffective field theory. For this aim, we applied recent polymer techniques[39, 40] to deal with the end-to-end probability associated with center vor-tices interacting with a general vector field λ µ , and a scalar field needed torepresent vortex-vortex interactions. However, it is far from straightforwardto extend this type of derivation to the non Abelian context. Therefore, inour case, we will propose a model relying on the symmetries displayed bythe initial representation, that strongly constraints the possible associatedeffective theories.In our case, the candidate for a vortex field has to be a real -componentfield φ a ( a = 1 , , ), because of the global SO (3) symmetry of the action S [ λ ] . We shall also introduce an isospin- field Q , where Q is a tracelesssymmetric × real matrix, encoding information about how the monopole-like defects that center vortices can concatenate are distributed. We maythen consider in the effective theory, an invariant term V I that couples themonopole and vortex sectors: V I = ζ φ T Q φ , ζ ≡ constant (20)which is invariant under the local SO (3) transformations: φ ( x ) → R ( x ) φ ( x ) , Q ( x ) → R ( x ) Q ( x ) R T ( x ) . (21)There are also invariant terms involving just either the vortex or themonopole field. Regarding the former, we may include a ‘potential’ term V φ ,with the general structure: V φ = µ φ T φ + λ φ T φ ) , (22)where µ and λ are arbitrary constants. On the other hand, for the case of themonopole field, we recall that an order parameter Q , with a similar structure,9s well-known in the context of liquid crystals. Thus, we expect the relevantterms in the effective theory to be of the same kind, namely, we may includea potential V Q [32]: V Q = A δ + B C δ + D δ ∆ + E + F δ , (23)where A, . . . , F are constants, and we have introduced two independent SO (3) invariants that can be built in terms of Q : δ = T r Q , ∆ = T r Q . (24)Thus, the three terms V φ , V Q and V I have the local symmetry (21). Thislocal symmetry will be broken to its global counterpart by the kinetic terms;however, these terms shall be constructed in such a way that they are compat-ible with a local discrete gauge symmetry. This symmetry must be present,at least in a phase where the vacuum is symmetric (no spontaneous symmetrybreaking).In this regard, the field strength tensor F aµ ( C ) can be written as, F aµ ( C ) = 12 (cid:15) µνρ F aνρ ( C )= (cid:15) µνρ ∂ ν C aρ + g (cid:15) µνρ (cid:15) abd C bν C dρ , (25)where, C µ = − g ˆ n · ∂ ν ˆ n C µ = − g ˆ n · ∂ ν ˆ n C µ = − g ˆ n · ∂ ν ˆ n . (26)We will show that F aµ ( C ) can be rewritten as, F aµ ( C ) = ˜ h aµ − h aµ , (27) ˜ h aµ = (cid:15) µνρ ∂ ν C Aρ , h aµ = − g (cid:15) µνρ ˆ n a · ( ∂ ν ˆ n a × ∂ ρ ˆ n a ) , (28)where, in the second tensor, no summation over a is understood. Being a traceless real symmetric matrix, the invariant content of Q can be generatedby two real invariants. For example, two of its eigenvalues. F µ = ˜ h µ − h µ , (29) ˜ h µ = (cid:15) µνρ ∂ ν C ρ , h µ = − g(cid:15) µνρ C ν C ρ . (30)In order to show that h µ = − g (cid:15) µνρ ˆ n · ( ∂ ν ˆ n × ∂ ρ ˆ n ) , (31)we can simply note that, ∂ ν ˆ n = (ˆ n · ∂ ν ˆ n )ˆ n + (ˆ n · ∂ ν ˆ n )ˆ n + (ˆ n · ∂ ν ˆ n )ˆ n = (ˆ n · ∂ ν ˆ n )ˆ n + (ˆ n · ∂ ν ˆ n )ˆ n = g ( C ν ˆ n − C ν ˆ n ) . (32)Then, replacing in the second member of (31), and using ˆ n × ˆ n = ˆ n , etc.,it is straightforward to make contact with (30).The important point is that (27) and (28) imply that for a fixed monopolebackground correlated with center vortices, the integral of each componentover a closed surface ∂ϑ (given as the border of a three-volume ϑ ), (cid:73) ∂θ dS µ F aµ ( C ) = (cid:73) ∂θ dS µ (˜ h aµ − h aµ ) = 12 g (cid:73) ∂θ dS µ (cid:15) µνρ ˆ n a · ( ∂ ν ˆ n a × ∂ ρ ˆ n a ) , (33)gives the Π topological charge for the mapping ∂ϑ → ˆ n a . More precisely, (cid:73) ∂θ dS µ F aµ ( C ) = 4 πg ( n + ( ϑ ) − n − ( ϑ )) , (34)where n + ( ϑ ) ( n − ( ϑ ) ) is the number of monopole (antimonopole) defects in-side ϑ , for the component ˆ n a .It may appear that the previous expression sets a preferred direction incolor space. This impression can be dispelled by considering the effect thata space independent change of color basis has on the expression (34). Tothat end, we consider a new basis (cid:0) ˆ n (cid:48) a (cid:1) a =1 , related to the original one by a(constant) matrix R (cid:48) , namely: ˆ n (cid:48) a = R (cid:48) ˆ n a , a = 1 , , . In components, andusing an obvious notation, the last relation means: (ˆ n (cid:48) a ) b = ( R (cid:48) ) bc (ˆ n a ) c . (35)Thus, we see that: (cid:73) ∂θ dS µ (cid:15) µνρ ˆ n (cid:48) a · ( ∂ ν ˆ n (cid:48) a × ∂ ρ ˆ n (cid:48) a ) = (cid:73) ∂θ dS µ (cid:15) µνρ (cid:15) b b b (ˆ n (cid:48) a ) b ∂ ν (ˆ n (cid:48) a ) b ∂ ρ (ˆ n (cid:48) a ) b (cid:73) ∂θ dS µ (cid:15) µνρ (cid:15) b b b ( R (cid:48) ) b c ( R (cid:48) ) b c ( R (cid:48) ) b c (ˆ n a ) c ∂ ν (ˆ n a ) c ∂ ρ (ˆ n a ) c = (det R (cid:48) ) (cid:73) ∂θ dS µ (cid:15) µνρ ˆ n a · ( ∂ ν ˆ n a × ∂ ρ ˆ n a ) , (36)where we have used the property: (cid:15) b b b ( R (cid:48) ) b c ( R (cid:48) ) b c ( R (cid:48) ) b c = (det R (cid:48) ) (cid:15) c c c . (37)A similar relation holds if one changes the original canonical basis by a con-stant rotation matrix. What this proves is that it is possible to generate amonopole charge along any color direction as long as one needs how to dothat for, say, the third one.Thus, coming back to the discussion on the possible form of the kineticterms, they must be -at least in the symmetric phase- compatible with thelocal discrete gauge symmetry: λ aµ → λ aµ + ∂ µ ω a , (38)where ω a is a discontinuous function taking values ± g inside a three-volume ϑ a , and zero outside. Note also that in a phase only containing closed centervortices, a larger symmetry, λ aµ → λ aµ + ∂ µ ϕ a , (39)for any smooth ϕ a , is expected: in this case, the absence of monopoles wouldimply ∂ µ F aµ ( C ) = 0 .Then, the kinetic terms must have the global SO (3) symmetry plus thelocal Abelian one. The simplest choice, which, in the effective theory ap-proach spirit we shall consistently adopt, is to minimally couple φ and Q to λ aµ (note that these couplings do not have the local (cid:126)λ ( x ) → R ( x ) (cid:126)λ ( x ) symmetry). Thus, the structure of the kinetic term K is as follows: K = K φ + K Q , (40)where: K φ = 12 ( ∇ µ φ ) a ( ∇ µ φ ) a K Q = 12 ( ∇ µ Q ) ab ( ∇ µ Q ) ab (41)where ∇ µ denotes the covariant derivative operator (consistent with the sym-metries mentioned above), which shall adopt a different expression whenacting on each one of the fields. Explicitly: ( ∇ µ φ ) a = ∂ µ φ a − ig φ λ bµ (cid:15) abc φ c ( ∇ µ Q ) ab = ∂ µ Q ab − ig Q λ cµ (cid:15) acd Q db + ig Q Q ad (cid:15) dcb λ cµ , (42)12here g φ and g Q are constants.The global SO (3) symmetry is evident, while by imposing g φ = g Q , theeffective action will display a non Abelian gauge symmetry, and the differentphases for the ensemble of monopoles and center vortices will arise as differentpossible vacua when the system undergoes SSB. Note also that as the field φ represents center vortices that in the case of Abelian configurations posses amagnetic charge π/g , the natural choice is g φ = 2 π/g , which also matchesthe correct dimensions in eq. (42) as [ λ ] = 3 / , [ g ] = 1 / .Then, joining the different pieces and taking into account eq. (16), thefollowing model, encoding a general ensemble of magnetic defects, can beproposed, L eff = L v,m + 12 λ aµ λ aµ + iλ aµ F aµ ( A ) , (43) L v,m = K Q + K φ + V φ + V Q + V I . (44)We would like to underline that according to the discussion at the end of §2,and beginning of §3, the symmetry displayed by the second and third termsin eq. (43), namely a transformation (cid:126) A ˜ U , accompanied by the local SO (3) rotation of λ aµ , is not the gauge symmetry that operates on (cid:126)A µ . Therefore,the noninvariance of L v,m under local SO (3) rotations of λ aµ is not an explicitbreaking of the gauge symmetry in our effective model. Only in the trivialsector (cid:126) A ˜ U may be associated with a gauge transformation, in other words,our model refers to the interaction of effective fields, parametrizing a generalensemble, with effective gluons represented by (cid:126) A µ . In order to analyze the possible phases of the model, it is necessary to studyall the possible scenarios regarding both the φ and Q dependent potentials, V φ and V Q , as well as the interaction V I . This will yield information aboutthe possible translation invariant configurations that will determine the prop-erties of each phase. Non translation invariant configurations, on the otherhand, are important to understand the mechanism driving the phase transi-tions between them. Of course, that will require the inclusion of the derivativeterms into the game.Thus we consider the minima of V T = V φ + V Q + V I . (45)This analysis is greatly simplified if we note that Q can always be diagonalized13y a similarity transformation Q = R T DR , with D = − q − η − q + η
00 0 q . (46)Defining Rφ = ψ , the potential V T adopts the form, V T = A δ + B C δ + D δ ∆ + E + µ ψ T ψ + λ ψ T ψ ) + ζψ T D ψ, (47) δ = (3 q + η ) /
2∆ = 3 q ( q − η ) / ψ T D ψ = − q ψ + ψ ) + η ψ − ψ ) + qψ . (48)Here, the term δ that was present in eq. (23) has been discarded, as it doesnot modify the qualitative structure of the minima [32].Now, to find the minima of the potential, we will suppose that the chainof spontaneous breaking of the symmetries is dominated by the monopolesector. Concretely, this approximation amounts to finding the minima of V Q ,and using the configurations q , η that Q adopts in those minima as a fixedbackground where we look for the vortex field configuration that minimizesthe remaining potential. Then, the whole space of minima is generated bymeans of R -rotations of the former.The minima of V Q are determined by: ∂ q V Q (cid:12)(cid:12) q ,η = 0 , ∂ η V Q (cid:12)(cid:12) q ,η = 0 , (49)plus the usual conditions on the second derivatives. We will consider CE > D / , and will follow the discussion in [32], where the different kinds ofminima are obtained by varying A and B . Changing the independent vari-ables q and η , the region δ ≥ is mapped, and the strict inequalityoccurs for η (cid:54) = 0 . Then, the points obtained by simply minimizing with re-spect to δ , ∆ as independent variables (in this case the potential contains apositive definite quadratic form) can only correspond to η (cid:54) = 0 . Otherwise,the potential must be minimized with the constraint δ = 6∆ , in which casetwo different situations are obtained, q = 0 , η = 0 or q (cid:54) = 0 , η = 0 (when CE < D / , only the two latter possibilities can be realized).Then, the ψ -field minima follow from the study of the ‘effective potential’ V ( ψ ) , defined by: V ( ψ ) = V φ + V I ( q , η ; ψ ) , (50)14 N + N − N b IA B
Figure 1: A-B phase diagram for the monopole sector, when
CE > D / , D < .and which explicit form is: V ( ψ ) = 12 (cid:2) µ − ζ ( q + η ) (cid:3) ψ + 12 (cid:2) µ − ζ ( q − η ) (cid:3) ψ + 12 ( µ + 2 ζq ) ψ + λ ψ + ψ + ψ ) . (51)It is now clear what kind of vacua may emerge, depending on the relativevalues of the parameters. We first note that stability requires λ ≥ . For D < , the monopole phase diagram is that of fig. 1 [32]. If the parameters A , B are initially in region I, we have Q = 0 ( q = η = 0 ), and the effectivevortex potential results, V I ( ψ ) = µ ψ + λ ( ψ ) . Then, if µ ≥ , the mini-mization gives ψ = 0 . With this vacuum, S v,m displays a non Abelian gaugesymmetry, much larger than the Abelian symmetries in eqs. (38), (39), typ-ically obtained when monopoles and center vortices are present. Therefore,this phase represents a situation where monopoles and center vortices do notproliferate (deconfining phase). Still in the Q = 0 phase, but with µ < ,the system undergoes SSB leaving an Abelian symmetry. If the mass scalegenerated for the off-diagonal fields is large, they will be suppressed andthen the effective theory will essentially be invariant under Abelian gauge15ransformations of the form, (cid:126)λ µ · ˆ φ → (cid:126)λ µ · ˆ φ + ∂ µ ϕ. Thus, recalling eq. (39), this phase describes an ensemble of closed centervortices.Now, in order to continue the analysis, it is convenient to define a complexfield V = √ ( ψ + iψ ) , and rewrite eq. (51) in the form (we consider ζ < ), V ( φ ) = ( µ + | ζ | q ) ¯ V V + 12 | ζ | η ( V + ¯ V )+ 12 ( µ − | ζ | q ) ψ + λ (cid:18) ¯ V V + 12 ψ (cid:19) . (52)When A is lowered from positive to negative values, after a first order tran-sition, we will enter the uniaxial nematic phase N + ( q > ) or the N − ( q < ), depending on whether B < or B > . These phases are charac-terized by η = 0 . In what follows, to simplify the analysis, we will suppose µ > . Then, if we enter the N − phase, after a discontinuous transition,the effective potential V − ( ψ ) will be minimized by ψ = 0 . In the monopolesector, the vacuum will be invariant under rotations around the third axis,while in the V -sector this symmetry will undergo a U(1) SSB or not, de-pending on the sign of ( µ + | ζ | q ) . In addition, the N − phase will inducea mass of order q for the charged dual vector fields λ µ and λ µ , originatedfrom the covariant derivative of Q in eq. (42). If we assume this mass to belarge when compared with the other mass scales in the problem, these dualvector fields will become suppressed.If we further diminish A , after a second order phase transition, we willeventually reach the biaxial phase N b where η (cid:54) = 0 . As this transition iscontinuous, and we are approaching from the N − phase, we will start with ψ and η small. In the N b phase, the U (1) symmetry of the effective action inthe former N − phase will be broken to a discrete one under π -rotations alongthe third axis. Again, in the monopole sector the vacuum is invariant, whilein the vortex sector it will display SSB of the discrete π -rotations dependingon the sign of ( µ + | ζ | q ) . When this quantity is negative, at the minima,the V field can take a pair of values V , − V connected by a Z (2) symmetry.That is, the obtained effective potential coincides with the confining phaseof the vortex model introduced by t’ Hooft, relying on the possible nontrivialvortex correlators in the initial theory. In this phase, the spontaneous Z (2) symmetry breaking leads to domain walls attached to Wilson loops, thusproviding an area law. Still in the ( µ + | ζ | q ) < case, in the intermediate N − phase, the vacuum no longer displays the Abelian symmetry present16n the initial phase, where center vortices are only closed objects, nor thediscrete symmetry of the last phase, typical of open center vortices whoseendpoints are joined in pairs to monopole-like objects that proliferate. Fromthis perspective, we speculate that the N − phase might be associated withone where monopoles and antimonopoles are still bound in pairs. We have constructed a novel non Abelian effective model for SU (2) QCD inEuclidean three-dimensional spacetime that allows for the description of aphase diagram with a rich structure. The construction is based on a specialparametrization of the gauge field configurations (cid:126)A µ in terms of a vectorfield (cid:126) A µ , representing a topologically trivial sector of smooth fluctuations,and a local color frame ˆ n a containing defects, the nontrivial sector describingmonopoles and thin center vortices. The frame can be written as a local SO (3) rotation R of the canonical basis ˆ e a , which can be also expressed inthe form R = R ( S ) , where S is in the fundamental representation.This parametrization is used to write the Yang-Mills action, what definesthe weight assigned to each configuration. On the other hand, as in anynon-perturbative definition of the functional integration measure in a nonAbelian gauge theory, one is faced with the usual stumbling blocks, related tothe Gribov problem. We do not attempt to tackle this problem; rather, sincewe use the functional integral just as a guide for the subsequent derivationof the effective model, we use instead a definition of the measure which: (a)reduces to the proper one for topologically trivial configurations and (b) isconsistent with (although not uniquely determined by) the properties of thegauge field parametrization used.The next step in the construction of the effective model proceeds with theintroduction of an auxiliary field (cid:126)λ µ that linearizes the Yang-Mills action, andthe incorporation of a phenomenological weight S d that senses the geometryof the defects. It is at this point where the real reduction to an effectivetheory is implemented. Indeed, the symmetries are identified here, for agiven classification of defects, what allows us to construct an effective model.If S d were nullified (thin objects), the partition function for the sector ofdefects should be invariant under local SO (3) rotations of (cid:126)λ , as they could beabsorved by a frame redefinition, transforming S under right multiplicationby an appropriate regular SU (2) matrix ˜ U − . In the Yang-Mills partitionfunction, the symmetry should also be accompanied by the transformation (cid:126) A ˜ Uµ . However, this symmetry is the one associated with the many differentways a given gauge field (cid:126)A µ containing thin defects can be decomposed, so17hat it is expected to be broken as soon as center vortices become thick.Alternatively, this could be seen as the noninvariance of the effective phe-nomenological action S d under local frame rotations, only leaving a global SO (3) .An interesting point is that in order to guide the construction of theeffective model for the ensemble integration, not only the global SO (3) sym-metry is important but also a new symmetry comes into play. At least in thesymmetric phase, due to the topological structure of monopoles, the modelshould be invariant under a local discrete gauge symmetry. This led us topropose a non Abelian model describing the interaction of the natural or-der parameters for monopoles and center vortices with the effective gluonfield (cid:126) A µ . As center vortices can be attached in pairs to the non Abelianmonopoles, the corresponding order parameters are given by fields φ and Q ,carrying isospin one and two, respectively. The effective character of thegluons is due to the fact that gauge transformations of the Yang-Mills fields (cid:126)A µ act as a left multiplication of the S sector, leaving (cid:126) A µ invariant.The effective model we introduced exhibits a rich phase diagram. Forinstance, the monopole sector of the effective potential depends on two in-variants, δ = T r Q , ∆ = T r Q . If this sector is supposed to dominatethe transitions, the phase diagram inherits, by construction, some of theproperties found in liquid crystals. In this case, if the quadratic form inthe quantities δ and ∆ is positive definite, and the coefficient of the linear ∆ -term is positive, an interesting chain of phase transitions is obtained.Initially, when the coefficient of the linear δ -term ( A ) is varied from posi-tive to negative values, a first order transition from the isotropic deconfiningphase to a uniaxial monopole condensate takes place. In this process, in thevortex sector, the “third” component becomes suppressed, while the other twocomponents can be arranged as an Abelian complex vortex field V display-ing U (1) SSB. In this example, the vortex mass scales have been supposed tobe negligible when compared with those generated in the monopole sector.The further reduction of A produces a second order phase transition, andthe monopole condensate becomes biaxial. Here, center vortices are left in aglobal Z (2) SSB phase, thus making contact with the ’t Hooft vortex model,and arriving to the confining phase expected in D Yang-Mills theories.
Acknowledgements
The Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) and PROPPi-UFF are acknowledged for the financial support.C. D. F. acknowledges financial support from CONICET and UNCuyo.18 eferences [1] G. ’t Hooft, Nucl. Phys. B138 (1978) 1.[2] J. Ambjorn, Quantization of geometry, in Les Houches (eds. F. David,P. Ginsparg, J. Zinn-Justin, 1994), hep-th/9411179.[3] K. Bardakci and S. Samuel, Phys. Rev.
D18 (1978) 2849.[4] M. Kiometzis, H. Kleinert, A. M. J. Schakel, Fortschr. Phys. (1995)697.[5] M. B. Halpern, A. Jevicki and P. Senjanovic, Phys. Rev. D16 (1977)2476.[6] M. B. Halpern and W. Siegel, Phys. Rev.
D16 (1977) 2486.[7] H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, PolymerPhysics, and Financial Markets (World Scientific, Singapore, 2006).[8] L. E. Oxman, JHEP (2008) 089.[9] L. E. Oxman, Phys. Rev. D82 (2010) 105020.[10] A. L. L. de Lemos, L. E. Oxman, and B. F. I. Teixeira, to be published.[11] D. Antonov, Surveys High Energ. Phys. (2000) 265.[12] I. I. Kogan and A. Kovner, “Monopoles, Vortices and Strings: Con-finement and Deconfinement in 2+1 Dimensions at Weak Coupling”,hep-th/0205026.[13] A. M. Polyakov, Phys. Lett. B59 (1975) 82; Nucl. Phys.
B120 (1977)429.[14] M. N. Chernodub, M. I. Polikarpov, Lectures given at the Workshop“Confinement, Duality and Non-Perturbative Aspects of QCD”, Cam-bridge, England, 1997, arXiv:hep-th/9710205.[15] M. N. Chernodub, F. V. Gubarev, M. I. Polikarpov, A. I. Veselov, Prog.Theor. Phys. Suppl. (1998) 309.[16] A. Di Giacomo, B. Lucini, L. Montesi, and G. Paffuti, Phys. Rev.
D61 (2000) 034503. 1917] A. S. Kronfeld, M. L. Laursen, G. Schierholz, and U. J. Wiese, Phys.Lett.
B198 (1987) 516.[18] L. Del Debbio, M. Faber, J. Greensite, S. Olejnik, Phys. Rev.
D55 (1997) 2298.[19] L. Del Debbio, M. Faber, J. Giedt, J. Greensite, and S. Olejnik, Phys.Rev.
D58 (1998) 094501.[20] P. de Forcrand and M. D’Elia, Phys. Rev. Lett. (1999) 4582.[21] J. Greensite, Prog. Part. Nucl. Phys. (2003) 1.[22] M. Engelhardt, M. Quandt, H. Reinhardt, Nucl. Phys. B685 (2004)227.[23] J. Ambjorn, J. Giedt, and J. Greensite, Nucl. Phys. Proc. Suppl. (2000) 467.[24] Ph. de Forcrand and M. Pepe, Nucl. Phys. B598 (2001) 557-577.[25] F. V. Gubarev, A. V. Kovalenko, M. I. Polikarpov, S. N. Syritsyn, V. I.Zakharov, Phys. Lett.
B574 (2003) 136.[26] L. E. Oxman, JHEP (2011) 078.[27] M. Engelhardt, H. Reinhardt, Nucl.Phys. B567 (2000) 249.[28] H. Reinhardt, Topology of Center Vortices, Nucl. Phys.
B628 (2002)133.[29] Y. M. Cho, Phys. Rev.
D21 (1980) 1080; Phys. Rev. Lett. (1981)302; Phys. Rev. D23 (1981) 2415.[30] L. Faddeev and A. J. Niemi, Phys. Rev. Lett. (1999) 1624.[31] S. V. Shabanov, Phys. Lett. B458 (1999) 322.[32] P. G. de Gennes and J. Prost, The Physics of Liquid Crystals (ClarendonPress, Oxford, 1993).[33] N. Gribov, Nucl. Phys.
B139 (1978) 1.[34] D. Zwanziger, Nucl. Phys.
B323 (1989) 513, Nucl. Phys.
B399 (1993)477. 2035] F. Bruckmann, T. Heinzl, A. Wipf, and T. Tok, Nucl. Phys.
B584 (2000)589.[36] J. Greensite, S. Olejnik, and D. Zwanziger, JHEP (2005) 070.[37] A. Maas, Nucl. Phys. A790 (2007) 566.[38] P. V. Buividovich, M. I. Polikarpov, V. I. Zakharov, PoS(LATTICE2007) 324.[39] D. C. Morse and G. H. Fredrickson, Phys. Rev. Lett.73