Effective confinement theory from Abelian variables in SU(3) gauge theory
aa r X i v : . [ h e p - t h ] M a r Effective confinement theory from Abelian variables in SU (3) gauge theory L. S. Grigorio,
1, 2, ∗ M. S. Guimaraes, † R. Rougemont, ‡ and C. Wotzasek § Instituto de F´ısica, Universidade Federal do Rio de Janeiro,21941-972, Rio de Janeiro, Brazil Centro Federal de Educa¸c˜ao Tecnol´ogica Celso Suckow da Fonseca,28635-000, Nova Friburgo, Brazil Departamento de F´ısica Te´orica, Instituto de F´ısica,UERJ - Universidade do Estado do Rio de Janeiro,Rua S˜ao Francisco Xavier 524, 20550-013 Maracan˜a, Rio de Janeiro, Brazil (Dated: June 19, 2018)In this Letter we use the Julia-Toulouse approach for condensation of defects in order to obtainan effective confinement theory for external chromoelectric probe charges in SU (3) gauge theory inthe regime with condensed chromomagnetic monopoles. We use the Cho decomposition of the non-Abelian connection in order to reveal the Abelian sector of the non-Abelian gauge theory and theassociated topological defects (monopoles) without resorting to any gauge fixing procedure. Usingonly the Abelian sector of the theory, we construct a hydrodynamic effective theory for the regimewith condensed defects in such a way that it is compatible with the Elitzur’s theorem. The resultingeffective theory describes the interaction between external chromoelectric probe charges displayinga short-range Yukawa interaction plus a linear confining term that governs the long distance physics. Keywords: Topological defects, monopoles, confinement.
I. INTRODUCTION
There is a very popular proposal regarding the issueof static quark confinement that conjectures that theQCD vacuum should contain a condensate of chromo-magnetic monopoles, constituting a dual superconduc-tor which should generate an asymptotic linear chromo-electric confining potential through the dual Meissnereffect [3]. Since monopoles are absent in pure Yang-Mills theories as classical solutions with finite energy,the dual superconductor scenario of color confinementis usually approached by using an Abelian projection[12], which corresponds to some partial gauge fixing con-dition implementing the following explicit breaking pat-tern: SU ( N ) → U (1) N − , followed by the discarding ofthe off-diagonal sector of the theory. It can be shownthat under an Abelian projection the Abelian (diagonal)sector of the gluon field behaves like if chromomagneticmonopoles (whose chromomagnetic charges are definedwith respect to the residual unfixed maximal Abeliansubgroup U (1) N − ) were sitting in points of the spacewhere the Abelian projection becomes singular. It isthen expected that if somehow these chromomagnetic de-fects proliferate (condense), the chromoelectric sourcesbecome confined; in particular, for the SU (2) case, thelattice data [15] show the very interesting result that ifone fixes the maximal Abelian gauge (MAG) and fur-ther discards the off-diagonal sector of the theory, thenthe Abelian confining string tension σ U (1) obtained re- ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] produces ∼
92% of the full string tension σ SU (2) , a phe-nomenon that is known as the Abelian dominance (forthe confining string tension). For a review, see [22] (seealso the discussions in [7]).Despite the success of the MAG calculations, there isan apparent problem of gauge dependence in such reason-ing, since in the Abelian Polyakov gauge the monopolecondensation does not lead to static quark confinement[16]. Indeed, within the Abelian projection approach itcould appear in principle that monopoles are merely agauge artifact. Since no physical phenomenon can de-pend on an arbitrary gauge choice, there are certainlysome missing pieces in this puzzle. In particular, it seemsthat an important step to be taken is to develop a gaugeindependent way of revealing the Abelian sector of theYang-Mills theory and the associated topological struc-tures (monopoles). Regarding this point, a very inter-esting proposal was made by Cho [4, 5], consisting in areparameterization of the Yang-Mills connection, calledthe Cho decomposition, which has the feature of explic-itly exposing its Abelian component and the associatedtopological structures without resorting to any gauge fix-ing procedure.In this Letter we make use of a generalization of theJulia-Toulouse Approach (JTA) for condensation of de-fects [8, 9], as put forward recently in [20], in order toobtain, out of the Abelian sector of the SU (3) Yang-Mills theory singled out in a gauge independent wayvia the Cho decomposition of the SU (3) connection [5],an effective confinement theory for static chromoelectricprobe charges due to the condensation of chromomag-netic monopoles.The main idea in the JTA is to allow one to studygauge theories in the presence of defects that eventuallycondense. If we are not interested in the details of thecondensation process we can ask ourselves if, having theknowledge of the model that describes the system beforethe condensation, we are able to determine the effectivemodel describing the system in the condensed regime.The condensate of topological defects establishes a newmedium in which the defects constitute a continuous dis-tribution in space. The low energy excitations of sucha medium represent the new degrees of freedom of thecondensed regime. Julia and Toulouse [8] specified a pre-scription to identify these new degrees of freedom, know-ing the model that describes the regime with diluted de-fects. This prescription does not deal with the dynamicalreasons responsible for the condensation process: this isconsidered a separate issue, beyond the scope of the pre-scription that is concerned only with the properties of thenew degrees of freedom once the condensation of topo-logical defects has taken place. However, the work ofJulia and Toulouse has taken place in the context of or-dered systems in condensed matter and due to the possi-ble non-linearity of the topological currents, the absenceof relativistic symmetry and the need for the introduc-tion of dissipative terms in this scenario, the constructionof effective actions is a very complicated issue.Latter, the JTA was extended by Quevedo and Trugen-berger [9] who showed that in theories involving p -forms,which are very common in effective descriptions of stringtheories, these difficulties do not show up. They showedthat in this context the prescription can be defined intoa more precise form, which leads to the determinationof the effective action describing the system in the con-densed regime. They have also shown that this leads nat-urally to the interpretation of the Abelian Higgs Mecha-nism as dual to the JTA. Based mainly on the ideas devel-oped in [8, 9], we recently developed a general procedure[20] to address the condensation of these defects, whosemain features are a careful treatment of a local symmetrycalled as brane symmetry (which is independent of theusual gauge symmetry, as discussed by Kleinert in [10]),which consists in the freedom of deforming the unphysicalDirac strings without any observable consequences, andthe development of the JTA to be completely compatiblewith the Elitzur’s theorem [2].In [20], we also have shown that the JTA may be re-alized both in the direct space of the potentials (as inthe proposal of [9]) or in its dual space, which is therealization chosen in the present work, as we shall dis-cuss in section IV. The resulting effective theory here ob-tained with the joint use of the Cho decomposition andthe JTA is compatible with the Elitzur’s theorem in thecondensed regime and describes the interaction betweenexternal chromoelectric charges displaying a short-rangeYukawa interaction plus a linear confining term that gov-erns the long distance physics. II. SU (2) CHO DECOMPOSITION
In this section we review some relevant points of theCho decomposition of the SU (2) connection. The start- ing point is the introduction of a unitary color tripletand Lorentz scalar ˆ n and the definition of the so-calledrestricted connection ˆ A µ which leaves ˆ n invariant underparallel transport on the principal bundle [4],ˆ D µ ˆ n := ∂ µ ˆ n + g ˆ A µ × ˆ n ≡ ~ ⇒ ˆ A µ = A µ ˆ n − g ˆ n × ∂ µ ˆ n, (1)where g is the Yang-Mills coupling constant.Due to the fact that the space of connections is anaffine space, a general SU (2) connection ~A µ can be ob-tained from the restricted connection ˆ A µ by adding afield ~X µ that is orthogonal to ˆ n [4]. Thus, the generalform of the SU (2) Cho decomposition is given by: ~A µ = ˆ A µ + ~X µ = A µ ˆ n − g ˆ n × ∂ µ ˆ n + ~X µ , ˆ n = 1 and ˆ n · ~X µ = 0 . (2)From the infinitesimal SU (2) gauge transformation de-fined by: δ ~A µ = 1 g D µ ~ω := 1 g ( ∂ µ ~ω + g ~A µ × ~ω ) ,δ ˆ n = − ~ω × ˆ n, (3)it follows that: δA µ = 1 g ˆ n · ∂ µ ~ω,δ ˆ A µ = 1 g ˆ D µ ~ω,δ ~X µ = − ~ω × ~X µ . (4)We see from (4) that A µ transforms like an U (1) con-nection, being the abelian component of the SU (2) con-nection explicitly revealed by the Cho decompositionwithout any gauge fixing procedure. Thus, we say thatthe unitary triplet field ˆ n selects the Abelian direction inthe internal color space for each spacetime point. Fur-thermore, we also see from (3) and (4) that the restrictedconnection ˆ A µ transforms like the general SU (2) connec-tion ~A µ since the restricted covariant derivative is ex-pressed in the adjoint representation, like the general co-variant derivative, in terms of the SU (2) structure con-stants defining the cross product. Hence, the restrictedconnection is already an SU (2) connection carrying allthe gauge degrees of freedom (but not all the dynami-cal degrees of freedom) of the non-Abelian gauge theory,being ~X µ a vector-colored source term called the valencepotential which carries the remaining dynamical degreesof freedom of the theory. Notice also from (4) that ~X µ transforms covariantly as a matter field in the adjointrepresentation.Using the covariant gauge fixing condition for the re-stricted connection, ∂ µ ˆ A µ = ~ ⇒ ∂ µ A µ = 0 and ˆ n × ∂ ˆ n − gA µ ∂ µ ˆ n = ~ , (5)we see that the 2 independent components of the fieldˆ n are completely fixed by a gauge condition and henceit is not a dynamical variable of the theory. The twodynamical variables of the theory are the fields A µ and ~X µ .The curvature tensor ~F µν associated to the gauge con-nection ~A µ is given by: ~F µν = ∂ µ ~A ν − ∂ ν ~A µ + g ~A µ × ~A ν = ˆ F µν + ˆ D µ ~X ν − ˆ D ν ~X µ + g ~X µ × ~X ν , (6)where the restricted curvature tensor ˆ F µν is given by:ˆ F µν = ∂ µ ˆ A ν − ∂ ν ˆ A µ + g ˆ A µ × ˆ A ν = ( F µν + H µν )ˆ n, (7) F µν = ∂ µ A ν − ∂ ν A µ , (8) H µν = − g ˆ n · ( ∂ µ ˆ n × ∂ ν ˆ n ) . (9)The Lagrangian density of the theory reads: L = − ~F µν = −
14 ˆ F µν −
14 ( ˆ D µ ~X ν − ˆ D ν ~X µ ) + − g F µν · ( ~X µ × ~X ν ) − g ~X µ × ~X ν ) . (10)The Euler-Lagrange equations of motion for A µ aregiven by: ∂ µ ( F µν + H µν + X µν ) = − g ˆ n · [ ~X µ × ( ˆ D µ ~X ν − ˆ D ν ~X µ )] , (11)where we defined: X µν := g ˆ n · [ ~X µ × ~X ν ] , (12)while the Euler-Lagrange equations of motion for ~X µ read:ˆ D µ ( ˆ D µ ~X ν − ˆ D ν ~X µ ) = g ( F µν + H µν + X µν )ˆ n × ~X µ . (13)Notice that the constraints of the theory imply that ~X µ ⊥ ˆ n and ˆ D µ ~X ν ⊥ ˆ n , such that ~X ν × ( ˆ D µ ~X ν − ˆ D ν ~X µ ) k ˆ n , and hence g ~X ν × ( ˆ D µ ~X ν − ˆ D ν ~X µ ) = g (ˆ n · [ ~X ν × ( ˆ D µ ~X ν − ˆ D ν ~X µ )])ˆ n . Thus, combining the equations ofmotion for the fields A µ and ~X µ , ˆ n (11)+(13), we get theusual Yang-Mills equations of motion: D µ ~F µν = ~ . (14)We can obtain an explicit form for the unitary tripletˆ n through an Euler rotation of the internal global Carte-sian basis { ˆ e a , a = 1 , , } . Parameterizing the ele-ments of the adjoint representation by the three Eu-ler angles, S = exp( − γJ ) exp( − θJ ) exp( − ϕJ ) ∈ SO (3), we can define the local internal basis by { ˆ n a := S − ˆ e a , a = 1 , , } , where ˆ n := ˆ n = S − ˆ e = (sin( θ ) cos( ϕ ) , sin( θ ) sin( ϕ ) , cos( θ )). With this parame-terization for ˆ n it is easy to show that the monopolecurvature tensor given by (9) reproduces the magneticfield generated by an antimonopole at the origin: H µν = − g sin( θ )( ∂ µ θ∂ ν ϕ − ∂ ν θ∂ µ ϕ )= − ( δ µθ δ νϕ − δ νθ δ µϕ ) 1 gr , (15)where in the last line we identified the internal and thephysical polar and azimuthal angles and used spheri-cal coordinates to write the components of the gradi-ent operator: ∂ := ∂ t ≡ ∂/∂t , ∂ := ∂ r ≡ ∂/∂r , ∂ := ∂ θ ≡ (1 /r ) ∂/∂θ and ∂ := ∂ ϕ ≡ (1 /r sin( θ )) ∂/∂ϕ .In fact, eq. (15) gives us the first homotopy class of themapping π ( SU (2) /U (1) ≃ S ) = Z defined by ˆ n (thesecond homotopy group is associated with the 2-sphere S phys at the spatial infinity of the physical space foreach fixed time). The complete set of homotopy classesof this mapping defining the chromomagnetic chargesof the theory is obtained by making the substitution ϕ mϕ, m ∈ Z in the above parameterization for ˆ n [4], from which we obtain the chromomagnetic charge ofan antimonopole in the m -th homotopy class of π ( S ):˜ g ( m ) := I S phys d~S · ~H ( m ) = I S phys dS i ǫ iµν H µν ( m ) = Z π r dθ sin( θ ) Z π dϕ (cid:18) − mgr (cid:19) = − πmg , m ∈ Z , (16)which is the non-Abelian version of the Dirac quantiza-tion condition [1, 22].The so-called magnetic gauge [4] is defined by fixingthe local color vector field ˆ n in the ˆ e -direction in theinternal space by means of the S -rotation. In this gauge,the restricted curvature tensor is written as ˆ F µν = ( F µν + H µν )ˆ e . Defining the so-called magnetic potential by theexpression: ˜ C µ := 1 g (cos( θ ) ∂ µ ϕ + ∂ µ γ ) , (17)we see that we can rewrite the monopole curvature tensor H µν as [7]: H µν = ∂ µ ˜ C ν − ∂ ν ˜ C µ − χ µν , (18)where: χ µν := 1 g (cos( θ )[ ∂ µ , ∂ ν ] ϕ + [ ∂ µ , ∂ ν ] γ ) , (19)is the chromomagnetic Dirac string associated to themonopole [1]. It is easy to see now that the restrictedconnection transforms like ˆ A µ ( A µ + ˜ C µ )ˆ e underthe gauge transformation that leads us to the magneticgauge.The magnetic potential ˜ C µ describes the potential of amonopole, being singular over its associated Dirac string,as we can easily see following the example discussed in[21] - if we consider γ = − ϕ , we have from (17) that:˜ C µ = 1 g (cos( θ ) − ∂ µ ϕ = 1 g (cos( θ ) − r sin( θ ) δ µϕ , (20)which is the monopole potential singular over the Diracstring arbitrarily placed (by the choice made for γ ) inthe negative ˆ e -axis ( θ = π ). Notice, however, that themonopole curvature tensor (18) is regular, as it shouldbe.If we now discard the valence potential ~X µ , that is, ifwe discard the off-diagonal sector of the theory, we endup with the restricted Lagrangian density defining theAbelian sector of the gauge theory: L ( R ) = −
14 ˆ F µν = −
14 [ ∂ µ ( A ν + ˜ C ν ) − ∂ ν ( A µ + ˜ C µ ) − χ µν ] . (21)Notice that, being orthogonal to the internal unitarytriplet ˆ n := ˆ n , the valence potential ~X µ features 8 de-grees of freedom. In fact, we can write ~X µ = X µ ˆ n + X µ ˆ n , where each vector field X iµ , i = 1 ,
2, has 4 degreesof freedom. The gauge transformation law (4) for thevalence potential corresponds to transformations of thelocal internal base vectors ˆ n i , i = 1 ,
2, while the compo-nents X iµ , i = 1 ,
2, of ~X µ do not change at all. Hence, thediscarding of the off-diagonal degrees of freedom of thetheory described by the vector fields X iµ , i = 1 ,
2, whichimplies in the discarding of the valence potential ~X µ ,constitutes a color gauge invariant approximation whichdistinguishes our approach from the Abelian projectionmethod.If we further absorb the singular magnetic potential˜ C µ into the regular Abelian gluon field A µ by making theshift ( A µ + ˜ C µ ) A µ , we rewrite (21) in the followingremarkable simple form: L ( R ) = −
14 ( F µν − χ µν ) , (22)where now the field A µ is singular over the chromomag-netic Dirac strings. Equation (22) describes the Maxwelltheory with the vector potential A µ non-minimally cou-pled to monopoles. In [19] we added a minimal couplingof the gauge field with external chromoelectric probecharges to this Lagrangian density and applied a gener-alization [17–20] of the Julia-Toulouse approach for con-densation of defects, compatible with the Elitzur’s theo-rem [2], to obtain a confining low energy effective theoryfor these charges in the static limit assuming the estab-lishment of a stable chromomagnetic condensate (in [6]it is proposed a demonstration of the establishment ofsuch a stable chromomagnetic condensation in the SU (2)gauge theory). We shall take these steps in details in sec-tion IV for the SU (3) case. III. SU ( N ) AND SU (3) CHO DECOMPOSITIONS
The generalization of the SU (2) Cho decomposition(2) for the SU ( N ) case proposed by Shabanov is givenby [7]: ~A µ = ˆ A µ + ~X µ = X ( k ) (cid:20) A ( k ) µ ˆ n ( k ) − g ˆ n ( k ) × ∂ µ ˆ n ( k ) (cid:21) + ~X µ , ˆ n ( i ) · ˆ n ( k ) = δ ( i )( k ) and ˆ n ( k ) · ~X µ = 0 , (23)where the referred “cross product” is defined by the SU ( N ) structure constants and the indices betweenparentheses refer to the ( N −
1) elements of the Cartansubalgebra of SU ( N ).Since our main objective in the present Letter is toinvestigate the contribution of the Abelian sector of the SU (3) gauge theory to the issue of static chromoelectricconfinement, from now on we focus only on the SU (3)restricted connection, given by [5]:ˆ A µ = X ( k )=3 , (cid:20) A ( k ) µ ˆ n ( k ) − g ˆ n ( k ) × ∂ µ ˆ n ( k ) (cid:21) , ˆ n = ˆ n = 1 and ˆ n (3) · ˆ n (8) = 0 . (24)The restricted connection (24) is the connection thatleaves the unitary octets ˆ n (3) and ˆ n (8) invariant underparallel transport on the principal bundle. Actually, themagnetic condition ˆ D µ ˆ n (3) = ~ n (8) through the definition ˆ n (8) := √ n (3) ⋆ ˆ n (3) ,where the star product is defined by the symmetric SU (3) d -constants. The chromomagnetic charges of the theoryare labeled by two integers according to the mapping π ( SU (3) /U (1) × U (1)) = Z × Z defined by ˆ n (3) , where U (1) × U (1) is the maximal Abelian subgroup of SU (3)selected in each spacetime point in a gauge independentway by ˆ n (3) and ˆ n (8) :˜ ~g ( N, N ′ ) = ˜ g (cid:18) N − N ′ , √ N ′ (cid:19) , N, N ′ ∈ Z , ˜ g = 4 πg . (25)Analogously to the discussion made around equation(16), the components (3) and (8) of ˜ ~g ( N, N ′ ) are definedas being the chromomagnetic fluxes of the types (3) and(8) through S phys by using explicit parameterizations forˆ n (3) and ˆ n (8) obtained by means of a rotation of the in-ternal global base elements ˆ e (3) and ˆ e (8) realized by anarbitrary element of the adjoint representation of SU (3)parameterized by eight “Euler angles”. As we can seefrom (25), this is equivalent to take ˜ ~g ( N, N ′ ) as a linearcombination with integer coefficients of the simple rootsof the SU (3) algebra. Also, as before, the magnetic po-tentials ˜ C (3) µ and ˜ C (8) µ are revealed in the restricted con-nection by fixing the magnetic gauge which is defined byan internal rotation that sends the local internal vectorsˆ n (3) and ˆ n (8) to the ˆ e (3) and ˆ e (8) directions, respectively(see [5] and references therein for details).The chromoelectric charge operator in the fundamentalrepresentation of SU (3) is given by (see appendix D.2 of[22]): ~ G := g~ T = g (cid:0) T (3) , T (8) (cid:1) , T ( k ) = 12 λ ( k ) , (26)where λ ( k ) are Gell-Mann matrices.The restricted curvature tensor in the magnetic gaugeis given by (absorbing, as before, the singular mag-netic potentials ˜ C ( k ) µ into the regular Abelian gluon fields A ( k ) µ ): ˆ F µν = X ( k )=3 , ( F ( k ) µν − χ ( k ) µν )ˆ e ( k ) , (27)where, as before, the chromomagnetic string terms χ ( k ) µν appear due to the angular (multivalued) nature of thecomponents of the magnetic potentials ˜ C ( k ) µ . The associ-ated Lagrangian density is given by: L ( R ) = − X ( k )=3 , ( F ( k ) µν − χ ( k ) µν ) . (28)If we now minimally couple external chromoelectricprobe charges to the restricted connection in the mag-netic gauge, we get the following Lagrangian density:¯ L ( R ) = X ( k )=3 , (cid:20) −
14 ( F ( k ) µν − χ ( k ) µν ) − j ( k ) µ A µ ( k ) (cid:21) . (29)We must now specify the chromoelectric charge struc-ture of the probe current ~j µ = ( j (3) µ , j (8) µ ). This can bedone by comparison with the quarkionic current ~j ψµ =¯ ψγ µ ~ G ψ that couples minimally to the gauge connectionˆ A µ = ( A (3) µ + ˜ C (3) µ )ˆ e (3) + ( A (8) µ + ˜ C (8) µ )ˆ e (8) : ~j ψµ · ˆ A µ = ¯ rγ µ (cid:20) g A µ (3) + ˜ C µ (3) ) + g √ A µ (8) + ˜ C µ (8) ) (cid:21) r ++ ¯ bγ µ (cid:20) − g A µ (3) + ˜ C µ (3) ) + g √ A µ (8) + ˜ C µ (8) ) (cid:21) b ++ ¯ yγ µ (cid:20) − g √ A µ (8) + ˜ C µ (8) ) (cid:21) y, (30)where ¯ ψ = (¯ r, ¯ b, ¯ y ) is the internal antitriplet, being ¯ r ,¯ b and ¯ y the red, blue and yellow antiquark spinors, re-spectively. From (30) we see that the red, blue and yel-low quarks have chromoelectric charges ~Q = ( Q (3) , Q (8) )given by ( g , g √ ), ( − g , g √ ) and (0 , − g √ ), respectively(see also table (D.10) of [22] and [5]).We now construct the chromoelectric probe current ~j µ = ( j (3) µ , j (8) µ ) in such a way that it is compatible withthe above charge structure: j (3) µ = g δ µ ( x ; L r ) − g δ µ ( x ; L b )= 12 ǫ µναβ ∂ ν Λ αβ (3) , (31) Λ αβ (3) = g δ αβ ( x ; S r ) − g δ αβ ( x ; S b ) , (32) j (8) µ = g √ δ µ ( x ; L r ) + g √ δ µ ( x ; L b ) − g √ δ µ ( x ; L y )= 12 ǫ µναβ ∂ ν Λ αβ (8) , (33)Λ αβ (8) = g √ δ αβ ( x ; S r ) + g √ δ αβ ( x ; S b ) − g √ δ αβ ( x ; S y ) , (34)where Λ ( k ) µν are the chromoelectric Dirac string terms, be-ing L r = ∂S r , L b = ∂S b and L y = ∂S y the worldlinesof the red, blue and yellow probe charges, boundaries ofthe worldsurfaces S r , S b and S y of the chromoelectricDirac strings of the red, blue and yellow probe charges,respectively. IV. THE EFFECTIVE THEORY OF COLORCONFINEMENT
If we now assume that in a certain regime of the theoryit is established a stable chromomagnetic monopole con-densate, we can ask ourselves what should be the form ofthe effective theory describing the low energy excitationsof such a condensate.One interesting approach to the general issue of thedetermination of the form of the effective field theory de-scribing a regime with condensed defects, having previousknowledge of the form of the theory in the regime wherethese defects are diluted, was introduced by Julia andToulouse within the context of ordered solid-state media[8] and further developed by Quevedo and Trugenbergerwithin the relativistic field theory context [9], constitut-ing the so-called Julia-Toulouse approach (JTA) for con-densation of defects. Another interesting approach tothis same issue was developed by Banks, Myerson andKogut within the context of relativistic lattice field theo-ries [11] and also by Kleinert within the condensed mat-ter context [10], constituting what we called in [18] theAbelian lattice based approach (ALBA). In a recent work[20], we unified and generalized these two approaches. Byconvention, we keep calling this generalized approach forthe determination of the effective theory describing theregime with condensed defects, simply as JTA.As already mentioned, the JTA can be worked out ei-ther in the direct space of the potentials or in its dualspace. Since we are interested in analyzing some of theconsequences of a monopole condensation, it is interest-ing to go to the dual picture, where we avoid the problemof working with the vector potentials A ( k ) µ in a scenariowhere they are ill-defined in almost the whole spacetimedue to the proliferarion of the chromomagnetic Diracstrings. Thus, our first step consists in dualizing the ac-tion associated to the restricted Lagrangian density (29)[18]: ∗ ¯ S ( R ) = Z d x X ( k )=3 , (cid:20) −
14 ( ˜ F ( k ) µν − Λ ( k ) µν ) + ˜ j ( k ) µ ˜ A µ ( k ) (cid:21) , (35)where the couplings are inverted relatively to the onespresent in (29) - here the dual vector potentials ˜ A ( k ) µ couple minimally to the monopoles and non-minimallyto the chromoelectric charges. Hence, in the dual picturethe chromoelectric charges are seen as defects by the dualvector potentials, which are singular over the associatedchromoelectric strings Λ ( k ) µν given by (32) and (34).Notice that the minimal coupling has support only overthe chromomagnetic worldlines and, since the dual poten-tials are singular over the worldsurfaces of the chromo-electric Dirac strings (which we call as “chromoelectricDirac branes”, the term “brane” here meaning a generichypersurface embedded in spacetime), the minimal cou-pling would be singular in events of the spacetime wherethe chromoelectric Dirac branes cross the chromomag-netic worldlines. Hence, in order to the action (35) tobe regular everywhere in spacetime, the chromoelectricDirac branes must not cross the chromomagnetic world-lines , which is the dual version of the famous Dirac’s veto [1]. Due to the Dirac’s veto, the so-called chromoelectricDirac brane symmetry corresponds to the freedom of mov-ing the unphysical chromoelectric Dirac branes throughthe geometric place of the spacetime not occupied by thechromomagnetic monopoles .We must now specify the chromomagnetic chargestructure that shall undergo a condensation process. No-tice from (25) that there are three monopoles of min-imal chromomagnetic charge (in modulus) in the non-trivial mapping π ( SU (3) /U (1) × U (1)), namely: ˜ ~g :=˜ ~g (1 ,
0) = πg ~ω , ˜ ~g := ˜ ~g ( − , −
1) = πg ~ω and ˜ ~g :=˜ ~g (0 ,
1) = πg ~ω , where ~ω = (1 , ~ω = (cid:16) − , − √ (cid:17) and ~ω = (cid:16) − , √ (cid:17) are the positive roots of the SU (3) alge-bra. The corresponding antimonopole charges (negativeroots) are obtained by inverting the signs of N and N ′ in the previous configurations. It is energetically favor-able that the lowest chromomagnetic charges (in mod-ulus) allowed by the mapping π ( SU (3) /U (1) × U (1))undergo a condensation process, thus, since we are go-ing to deal with three condensing monopole currents ofminimal chromomagnetic charge, we write:˜ j iµ := ˜ gδ µ ( x ; ˜ L i ) = 12 ǫ µναβ ∂ ν χ αβi , (36) χ αβi = ˜ g ˜ δ αβ ( x ; ˜ S i ) , (37)where ˜ L i = ∂ ˜ S i , i = 1 , , ~ω i , the physi-cal boundaries of the worldsurfaces ˜ S i of the unphysi-cal chromomagnetic Dirac strings. The roots appear ex-plicitly in the minimal coupling ˜ j iµ ˜ A µi , where we defined˜ A µi := ~ω i · ˜ ~A µ = ~ω i · ( ˜ A (3) µ , ˜ A (8) µ ). In order to rewrite (35) in an explicitly Weyl-symmetric form (the Weyl symmetry corresponds to theinvariance of the theory under permutations of the in-dices i = 1 , , SU (3)), we shall make use of the following identities: ~V µ = ( V (3) µ , V (8) µ ) = 23 X i =1 ~ω i V iµ ,V iµ := ~ω i · ~V µ ⇒ X i =1 V iµ = 0 (constraint); ~V µ · ~R µ = V (3) µ R µ (3) + V (8) µ R µ (8) = 23 X i =1 V iµ R µi . (38)Thus, the Weyl-symmetric representation of (35) withthe monopole current given by (36) is: ∗ ¯ S ( R ) = Z d x X i =1 (cid:20) −
16 ( ˜ F iµν − Λ iµν ) + ˜ j iµ ˜ A µi (cid:21) , (39)where the chromoelectric branes acquire the following re-markable symmetric form:Λ µν = ~ω · ~ Λ µν = Λ (3) µν = g δ µν ( x ; S r ) − g δ µν ( x ; S b ) , (40)Λ µν = ~ω · ~ Λ µν = −
12 Λ (3) µν − √
32 Λ (8) µν = − g δ µν ( x ; S r ) + g δ µν ( x ; S y ) , (41)Λ µν = ~ω · ~ Λ µν = −
12 Λ (3) µν + √
32 Λ (8) µν = g δ µν ( x ; S b ) − g δ µν ( x ; S y ) . (42)In order to allow the monopoles to proliferate we mustgive dynamics to their associated chromomagnetic Diracbranes χ iµν , since the proliferation of them is directlyrelated to the proliferation of the monopoles and theirworldlines. Thus, our second step consists in supplement-ing the dual action (39) with a kinetic term for the chro-momagnetic Dirac branes of the form − m ˜ j µi , which isthe term in a derivative expansion with the lowest orderin derivatives of χ iµν (that is, the dominant contributionfor the hydrodynamic limit of the theory) satisfying therelevant (Lorentz, gauge and brane) symmetries of thesystem. Such a contribution corresponds to an activa-tion term for the chromomagnetic loops [10]. Hence, thepartition function associated to the extended dual actiondescribing the regime with condensed chromomagneticmonopoles reads: Z c := Y i =1 Z D ˜ A iµ δ [ ∂ µ ˜ A µi ] e i R d x [ − ( ˜ F iµν − Λ iµν ) ] Z c [ ˜ A iµ ] , (43)where the Lorentz gauge was adopted for the dual poten-tials ˜ A iµ and where the partition functions for the branesectors are given by: Z c [ ˜ A iµ ] = X { ˜ L i } δ [ ∂ µ ˜ j µi ] exp (cid:26) i Z d x (cid:20) − m ˜ j µi + ˜ j iµ ˜ A µi (cid:21)(cid:27) , (44)(without sum in i ) where the functional δ -distributionenforces the closeness of the monopole worldlines (thechromomagnetic loops) giving the current conservationlaws ∂ µ ˜ j µi = 0, which are identically satisfied due to (36).An observation is in order at this point. Notice that theactivation term for the chromomagnetic loops is highlysingular. This singularity is associated to the hydrody-namic limit, where we consider the coherence length ofthe condensate to be zero. As discussed in [10], this ac-tivation term can be regularized be smoothing out the δ -distributions over the real coherence length of the chro-momagnetic condensate, that is non-zero (this, in fact,gives the thickness of the confining chromoelectric fluxtube, when external charges are present in this medium),such that the regularized activation term gives an en-ergy contribution proportional to the total length of thechromomagnetic loops ˜ L i . Such a regularization can alsobe done in the explicit evaluation of the confining poten-tial through the introduction of an ultraviolet cutoff scalecorresponding to the inverse of the coherence length ofthe condensate [20].The third step in our approach consists in the use ofthe Generalized Poisson’s Identity (GPI) (see appendix Aof [18] for a detailed discussion on the subject) in d = 4: X { ˜ L i } δ [ η iµ − δ µ ( x ; ˜ L i )] = X { ˜ V i } e πi R d x ˜ δ µ ( x ; ˜ V i ) η µi , (45)(without sum in i ) where ˜ V i is the 3-brane Poisson-dualto the 1-brane ˜ L i . The GPI works as a brane analogueof the Fourier transform: when the line configurations˜ L i in the left-hand side of (45) proliferate, the volumeconfigurations ˜ V i in the right-hand side become diluted and vice-versa. Using (45) we can rewrite (44) as: Z c [ ˜ A iµ ] = Z D η iµ X { ˜ L i } δ " ˜ g η iµ ˜ g − δ µ ( x ; ˜ L i ) ! δ " ˜ g ∂ µ η iµ ˜ g ! exp (cid:26) i Z d x (cid:20) − m η µi + η iµ ˜ A µi (cid:21)(cid:27) = N Z D η iµ X { ˜ V i } e πi R d x ˜ δ µ ( x ; ˜ V i ) ηµi ˜ g Z D ˜ θ i e i R d x ˜ θ i ∂ µ ηµi ˜ g exp (cid:26) i Z d x (cid:20) − m η µi + η iµ ˜ A µi (cid:21)(cid:27) = N N ′ X { ˜ V i } Φ[˜ θ V iµ ] Z D ˜ θ i Z D η iµ exp (cid:26) i Z d x (cid:20) − m η µi − η µi ˜ g ( ∂ µ ˜ θ i − ˜ θ V iµ − ˜ g ˜ A iµ ) (cid:21)(cid:27) , (46)(without sum in i ) where we defined the Poisson-dualcurrent (to the chromomagnetic current ˜ j iµ ), ˜ θ V iµ :=2 π ˜ δ µ ( x ; ˜ V i ), being N a constant associated to the useof the functional generalization of the identity δ ( ax ) = δ ( x ) / | a | and N ′ a constant associated to the fact thatthere is an overcounting of physically equivalent config-urations in the partition function for the brane sectorwithout the brane fixing functional Φ[˜ θ V iµ ] (see [10] fora discussion on the subject). Since the constant prod-uct
N N ′ is canceled out in the calculation of correlationfunctions and VEV’s, we shall effectively neglect themfrom now on (the partition function is only defined up toglobal constant factors).Notice from the geometric interpretation of the GPIgiven above that the proliferation (dilution) of the en-semble of chromomagnetic worldlines n ˜ L i o is associatedto the dilution (proliferation) of the Poisson-dual ensem-ble of volumes n ˜ V i o , what tells us that the Poisson-dualcurrents ˜ θ V iµ must be interpreted as closed chromoelec-tric vortices describing regions of the spacetime wherethe chromomagnetic condensate has not been established[10, 20].Integrating out the auxiliary fields η iµ in the partialpartition functions (46) and substituting the result backin the complete partition function (43) we obtain, as thelow energy effective theory for the chromomagnetic con-densed regime in the dual picture, the hydrodynamic (orLondon) limit of a U (1) × U (1) dual Abelian Higgs model(DAHM): Z c = Y i =1 X { ˜ V i } Φ[˜ θ V iµ ] Z D ˜ B iµ exp (cid:26) i Z d x " −
16 ( ˜ G iµν − Λ iµν ) + m (cid:18) ˜ B iµ + 1˜ g ˜ θ V iµ (cid:19) = Y i =1 X { ˜ V i } Φ[˜ θ V iµ ] Z D ˜ B iµ exp (cid:26) i Z d x (cid:20) −
16 ( ˜ G iµν − L iµν ) + m B µi (cid:21)(cid:27) , (47)(without sum in i ) where in the last line we madethe shift ˜ B iµ := ˜ A iµ − g ∂ µ ˜ θ i ˜ B iµ − g ˜ θ V iµ . Noticethat m is the mass acquired by the gauge invariantfield ˜ B iµ due to the chromomagnetic condensate. Also,˜ G iµν := ∂ µ ˜ B iν − ∂ ν ˜ B iµ is the strength tensor field and L iµν := Λ iµν + g ( ∂ µ ˜ θ V iν − ∂ ν ˜ θ V iµ ) is the so-called chro-moelectric brane invariant [19, 20]: as discussed before,the chromoeletric brane symmetry corresponds to thefreedom of moving the unphysical chromoelectric Diracbranes through the geometric place of the spacetimenot occupied by the chromomagnetic monopoles. Butsince in the chromomagnetic condensed regime the onlyplace not occupied by the chromomagnetic monopolesis the interior of the closed chromoelectric vortices, thechromoelectric Dirac strings are necessarily placed overthe closed vortices connected to a pair of probe quark-antiquark. In such a setup, which can be read off fromthe expression for L iµν with non-trivial Λ iµν (in regionsof the spacetime where Λ iµν = 0 and ˜ θ V iµ = 0, we havefrom the expression for L iµν the closed chromoelectricvortices), the flux inside the chromoelectric Dirac stringsis canceled out by part of the flux inside the closed vor-tices, leaving as result only open chromoelectric vorticeswith a pair of probe quark-antiquark in their ends. Theseopen vortices correspond to the confining chromoelectricflux tubes [20].Notice from (47) that it is impossible to realize a com-plete chromomagnetic condensation (meaning that themonopoles proliferate in such a way that they occupythe whole space) when there are external chromoelec-tric sources embedded into the system: such a completechromomagnetic condensation would imply in the com-plete dilution of the ensemble of internal defects n ˜ V i o ,what would destroy the brane invariants L iµν and spoilthe local chromoelectric Dirac brane symmetry and theElitzur’s theorem by “making the unphysical chromoelec-tric Dirac strings become real, constituting the confiningchromoelectric flux tubes”, what is clearly an absurd.This restriction over the realization of the chromomag-netic condensation in the system in the presence of ex-ternal chromoelectric sources is easily comprehensible inphysical terms: the dual Meissner effect, generated by the mass m of the gauge invariant field ˜ B iµ in the condensedregime, expels the chromoelectric fields generated by theexternal charges of almost the whole space constitutedby the dual superconductor, however, these fields cannotsimply vanish - they become confined in regions of thespace with minimal volume corresponding to the chro-moelectric confining flux tubes described by the braneinvariants L iµν [19, 20].Before returning to the direct picture, we can rewritethe effective action present in the partition function (47)in the Cartan representation as: S = Z d x X ( k )=3 , (cid:20) −
14 ( ˜ G ( k ) µν − L ( k ) µν ) + ˜ m B µ ( k ) (cid:21) , (48)where we defined ˜ m := q m . Its dual action is given by[18]: ∗ S = Z d x X ( k )=3 , (cid:20) −
12 ( ∂ µ ˜ Y µν ( k ) ) + ˜ m Y µν ( k ) ++ ˜ m Y ( k ) µν L µν ( k ) (cid:21) , (49)where the Kalb-Ramond fields ˜ Y ( k ) µν describe themonopole condensate in the direct picture: notice the rank-jumping observed in the direct picture due to themonopole condensation - in the diluted regime describedby (29) the system is characterized by the massless 1-forms A ( k ) µ , while in the condensed regime described by(49) the system is characterized by the massive 2-forms˜ Y ( k ) µν . The rank-jumping is a signature of the defects con-densation and the mass generation in the JTA [17–20].Notice also that (49) is the generalization of [9] compati-ble with the Elitzur’s theorem and the local chromoelec-tric brane symmetry: in [9] the last term in (49) featuresa minimal coupling of the Kalb-Ramond field directlywith the chromoelectric Dirac strings instead of the chro-moelectric brane invariants, thus violating the Elitzur’stheorem and the local chromoelectric brane symmetry by“making the unphysical chromoelectric Dirac strings be-come real, constituting the confining chromoelectric fluxtubes”.The partition function describing the chromomagneticcondensed regime in the direct picture is, then, given by: Z c = Y ( k )=3 , X { ˜ V ( k ) } Φ[˜ θ V ( k ) µ ] Z D ˜ Y ( k ) µν exp (cid:26) i Z d x (cid:20) −
12 ( ∂ µ ˜ Y µν ( k ) ) + ˜ m Y µν ( k ) + ˜ m Y ( k ) µν L µν ( k ) (cid:21)(cid:27) , (50)(without sum in ( k )). As discussed before, it is impos-sible to realize a complete chromomagnetic condensationin the presence of external chromoelectric sources: thebest the dual Meissner effect can do is to completely di-lute the closed chromoelectric vortices disconnected fromthe chromoelectric Dirac strings. In such a case, by inte-grating out the fields ˜ Y ( k ) µν in (50), we obtain [20]: Z c = Y ( k )=3 , exp (cid:26) i Z d x (cid:20) − j ( k ) µ ∂ + ˜ m j µ ( k ) (cid:21)(cid:27)X { ¯ S ( k ) } exp (cid:26) i Z d x (cid:20) − ˜ m L ( k ) µν ∂ + ˜ m L µν ( k ) (cid:21)(cid:27) , (51)(without sum in ( k )) where the geometric sum is nowtaken over all the possible shapes of the confining fluxtubes. This sum is very difficult to realize in general.However, if we consider a static probe quark-antiquarkconfiguration with a spacial separation L and the asymp-totic time regime T → ∞ , then it is reasonable to approx-imate the sum over brane invariants in (51) by taking intoaccount only its dominant contribution, which is given bya linear flux tube of lenght L corresponding to the sta-ble asymptotic configuration which minimizes the energyof the system [20]. In this limit, one obtains from (51)the following static interquarks potential (see [20] for thedetailed evaluation): V static ( L ; g, ˜ m, ˜ M ) = σ ( g, ˜ m, ˜ M ) L − ( Q + Q )4 π e − ˜ mL L , (52)where the string tension is given by: σ ( g, ˜ m, ˜ M ) = ( Q + Q ) ˜ m π ln ˜ m + ˜ M ˜ m ! = g ˜ m π ln ˜ m + ˜ M ˜ m ! , (53)where ˜ M is an ultraviolet cutoff corresponding to theHiggs mass (the mass of the monopoles), whose inversegives the coherence length of the monopole condensateand where we used the fact that for any mesonic config-uration ( r − ¯ r , b − ¯ b or y − ¯ y ), we have Q + Q = g .The static interquarks potential (52) was originally ob-tained in [14], where the free parameters ( g, ˜ m, ˜ M ) werefixed by fitting the profile of the phenomenological Cor-nell potential as being (5 . , . GeV, . GeV ). This setof values reproduces the experimental value of the stringtension, σ ≈ (440 M eV ) , obtained from the slope of theRegge trajectories, and gives the prediction that the dualsuperconductor realizing the static chromoelectric con-finement in the QCD vacuum should be of the type II.Notice also that taking ˜ m = 0 leads us back to the dilutedregime characterized by the long-range Coulomb interac-tion, eliminating the monopole condensate and destroy-ing the chromoelectric confinement.It is important to stress here the two main differencesbetween our approach and the approach of [14]:a) Our work has as the starting point the Abelianaction with chromomagnetic monopoles (35), obtained using the SU (3) Cho decomposition and the discardingof the off-diagonal sector of the theory parametrized bythe valence potential, while the starting point in [14] isthe Abelian action with chromomagnetic monopoles ob-tained in [13] using the SU (3) Abelian projection im-plemented specifically in the MAG. Since the Abelianprojection method involves a partial gauge fixing condi-tion, there is an ambiguity involved in the choice of aparticular Abelian gauge and in the corresponding def-inition of the monopoles as discussed in [16]: the dif-ferent Abelian gauges that can be fixed in the Abelianprojection method lead, in general, to different resultsfor the string tension. In particular, the Abelian stringtension obtained in the MAG in the SU (2) case givessupport to the Abelian dominance hypothesis. However,this result is obscure from a physical point of view, sincethe value of a physical observable like the string tensionshould not depend on an arbitrary gauge choice. Onthe other hand, the Cho decomposition allows one to re-veal the monopoles in the non-Abelian theory withoutresorting to any gauge fixing procedure, what representsan apparent advantage over the usual Abelian projectionmethod, since we do not have an ambiguity in the choiceof a particular Abelian gauge and in the definition of themonopoles via the Cho decomposition. In fact, if the off-diagonal sector of the theory can be discarded at all insome regime of the theory, then the result obtained forthe string tension using the Cho decomposition shouldbe unique in principle, corresponding to the result foundin our work, which agrees with the result of the Abelianprojection method implemented specifically in the MAG.Although the effective potential is the same in both cases,via the Abelian projection method the choice of the MAGis in principle only one between many different possibil-ities, while via Cho decomposition the result we haveobtained is in principle unique. In this sense, we see ourresult as being gauge independent;b) In what concerns specifically to the evaluation ofthe confining potential, there is another important con-ceptual difference between the procedure of [14] and ours.In the calculation of [14], the chromoelectric Dirac stringis taken as being the linear confining chromoelectric fluxtube. This cannot be correct as a matter of principle,since the Dirac string is non-physical. As discussed here,the confining flux tubes correspond actually to open chro-moelectric vortices with a pair of probe quark-antiquarkin their ends, which are described in our formalism bybrane invariants. These open vortices emerge in our for-malism due to the mutual cancellation between part ofthe chromoelectric flux inside the closed chromoelectricvortices connected to the Dirac strings and the chromo-electric flux inside the strings. This mutual cancellationnecessarily happens due to the Dirac’s veto. This is a for-mal advance featured in our approach. Notice also thatin the partition function (51), all the possible shapes ofthe confining flux tubes contribute in the sum over thebrane invariants. Hence, in general, our result is differ-ent from the result of [14], which takes into account only0the straight shape for the string. However, for a staticprobe quark-antiquark configuration in the asymptotictime regime T → ∞ , as discussed in details in [20], thedominant contribution in the sum over configurations ofthe brane invariants is given by the tube with the minimalvolume (stable configuration that minimizes the energyof the system), which corresponds to a straight flux tube.In this limit, it is reasonable to approximate the effectivestatic interquarks potential by taking into account onlythe contribution of the straight tube, which is the basicconsideration that makes the form of our effective poten-tial equivalent to the one obtained in [14]. V. CONCLUDING DISCUSSION
In this Letter we used a generalization of the Julia-Toulouse approach for condensation of defects to studyhow the confinement of static external chromoelectricprobe charges could emerge from the Abelian sector ofthe pure SU (3) gauge theory due to the condensation ofchromomagnetic monopoles.We took as the starting point to the novel approachhere presented, regarding the monopole condensation,the expression for the restricted SU (3) gauge theory de-fined by means of the Cho decomposition of the non-Abelian connection. This decomposition consists in areparameterization of the non-Abelian connection whichreveals its Abelian sector and the associated topological structures (monopoles) without resorting to any gaugefixing procedure, hence providing a gauge invariant defi-nition of these defects in the Yang-Mills theory.With the discarding of the off-diagonal sector ofthe theory (assuming the validity of the hypothesis ofAbelian dominance), we showed that the action in theregime with diluted defects can be put in the form ofa Maxwellian theory non-minimally coupled to chromo-magnetic monopoles and minimally coupled to externalchromoelectric probe charges. This was the crucial pointthat allowed us to apply the generalized form of theJulia-Toulouse approach for condensation of defects andobtain a hydrodynamic effective theory for the regimewhere the monopoles are condensed. The effective the-ory that we derived with such an approach gives aninteraction potential between two static chromoelectricprobe charges of opposite signs embedded in the chro-momagnetic monopole condensate consisting in a sum ofa Yukawa and a linear confining term in the asymptotictime regime T → ∞ . Our result is in principle gaugeindependent. VI. ACKNOWLEDGEMENTS
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