arX

iv:1

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v1 [

hep-

th]

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Aug

201

1

Effective confinement theory from Abelian variables in SU(3) gauge theory

L. S. Grigorio,1, 2, ∗ M. S. Guimaraes,3, † R. Rougemont,1, ‡ and C. Wotzasek1, §

1Instituto de Fısica, Universidade Federal do Rio de Janeiro,21941-972, Rio de Janeiro, Brazil

2Centro Federal de Educacao Tecnologica Celso Suckow da Fonseca,28635-000, Nova Friburgo, Brazil

3Departamento de Fısica Teorica, Instituto de Fısica,UERJ - Universidade do Estado do Rio de Janeiro,

Rua Sao Francisco Xavier 524, 20550-013 Maracana, Rio de Janeiro, Brazil(Dated: August 15, 2011)

We extend to the SU(3) gauge group a previous approach that we applied to the SU(2) gaugegroup in order to obtain an effective confinement theory for static chromoelectric probe chargesdue to the condensation of chromomagnetic monopoles. We use the so-called Cho decomposition ofthe non-Abelian connection in order to reveal the Abelian sector of the non-Abelian gauge theoryand the associated topological structures (monopoles) without resorting to singular gauge fixingprocedures. Using only the Abelian sector of the theory, we proceed to apply a generalization ofthe Julia-Toulouse approach for condensation of defects, compatible with the Elitzur’s theorem, andconstruct the effective theory for the regime where the topological defects proliferate in such a waythat their collective behavior becomes describable by a (continuous) field. The resulting effectivetheory describes the interaction between external chromoelectric charges displaying a short-rangeYukawa interaction plus a linear confinement 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−1. It can be shown that un-der an Abelian projection the Abelian (diagonal) sec-tor of the gluon field behaves like if chromomagneticmonopoles (whose chromomagnetic charges are definedwith respect to the residual unfixed maximal Abeliansubgroup U(1)N−1) 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 [14] show the very interesting result that ifone fixes the maximal Abelian gauge (MAG) and fur-ther discards the off-diagonal sector of the theory, then

∗ leogrigorio@if.ufrj.br† msguimaraes@uerj.br‡ romulo@if.ufrj.br§ clovis@if.ufrj.br

the Abelian confining string tension σU(1) obtained re-produces ∼ 92% of the full string tension σSU(2), a phe-nomenon that is known as the Abelian dominance (forthe confining string tension). For a review, see [21] (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[15]. 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 ina reparametrization 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 extend to the SU(3) gauge group aprevious approach that we applied to the SU(2) gaugegroup [18] in order to obtain an effective confinementtheory for static chromoelectric probe charges due to thecondensation of chromomagnetic monopoles.

We begin in section II with a review of the SU(2)Cho decomposition [4]. In section III we briefly pointout the general form of the SU(N) Cho decompositionproposed by Shabanov [7] and discuss with some moredetails the SU(3) case [5], which constitutes the focus

2

of the present Letter. In section IV we discard the off-diagonal sector of the theory and use only its Abeliansector in the so-called magnetic gauge, where the theorycan be put into a Maxwellian form with chromomagneticdefects, applying in the sequel a generalization [16–19] ofthe Julia-Toulouse approach for condensation of defects[8, 9], mainly based on the works of Kleinert [10] andBanks, Myerson and Kogut [11], which is compatible withthe Elitzur’s theorem [2], and obtain an effective theoryfor the regime where the monopoles condense. The re-sulting effective theory describes the interaction betweenexternal chromoelectric charges displaying a short-rangeYukawa interaction plus a linear confinement term thatgoverns 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 n and the definition of the so-calledrestricted connection Aµ which leaves n invariant underparallel transport on the principal bundle [4],

Dµn := ∂µn+gAµ×n ≡ ~0 ⇒ Aµ = Aµn−1

gn×∂µn, (1)

where g is the Yang-Mills coupling constant.Due to the fact that the space of connections is an

affine space, a general SU(2) connection ~Aµ can be ob-

tained from the restricted connection Aµ by adding a

field ~Xµ that is orthogonal to n [4]. Thus, the generalform of the SU(2) Cho decomposition is given by:

~Aµ = Aµ + ~Xµ = Aµn− 1

gn× ∂µn+ ~Xµ,

n2 = 1 and n · ~Xµ = 0. (2)

From the infinitesimal SU(2) gauge transformation de-fined by:

δ ~Aµ =1

gDµ~ω :=

1

g(∂µ~ω + g ~Aµ × ~ω),

δn = −~ω × n, (3)

it follows that:

δAµ =1

gn · ∂µ~ω,

δAµ =1

gDµ~ω,

δ ~Xµ = −~ω × ~Xµ. (4)

We see from (4) that Aµ 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 that

the unitary triplet field n selects the Abelian direction inthe internal color space for each spacetime point. Fur-thermore, we also see from (3) and (4) that the restricted

connection Aµ transforms like the general SU(2) connec-

tion ~Aµ 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 ~Xµ a vector-colored source term called the valencepotential which carries the remaining dynamical degrees

of freedom of the theory. Notice also from (4) that ~Xµ

transforms covariantly as a matter field in the adjointrepresentation.Using the covariant gauge fixing condition for the re-

stricted connection,

∂µAµ = ~0 ⇒ ∂µA

µ = 0 and n× ∂2n− gAµ∂µn = ~0, (5)

we see that the 2 independent components of the fieldn 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 Aµ and~Xµ.

The curvature tensor ~Fµν associated to the gauge con-

nection ~Aµ is given by:

~Fµν = ∂µ ~Aν − ∂ν ~Aµ + g ~Aµ × ~Aν

= Fµν + Dµ~Xν − Dν

~Xµ + g ~Xµ × ~Xν , (6)

where the restricted curvature tensor Fµν is given by:

Fµν = ∂µAν − ∂νAµ + gAµ × Aν = (Fµν +Hµν)n, (7)

Fµν = ∂µAν − ∂νAµ, (8)

Hµν = −1

gn · (∂µn× ∂ν n). (9)

The Lagrangian density of the theory reads:

L = −1

4~F 2µν

= −1

4F 2µν −

1

4(Dµ

~Xν − Dν~Xµ)

2+

− g

2Fµν · ( ~Xµ × ~Xν)−

g2

4( ~Xµ × ~Xν)

2. (10)

The Euler-Lagrange equations of motion for Aµ aregiven by:

∂µ(Fµν +Hµν +Xµν) = −gn · [ ~Xµ × (Dµ~Xν − Dν

~Xµ)],(11)

where we defined:

Xµν := gn · [ ~Xµ × ~Xν ], (12)

3

while the Euler-Lagrange equations of motion for ~Xµ

read:

Dµ(Dµ~Xν− Dν

~Xµ) = g(Fµν+Hµν+Xµν)n× ~Xµ. (13)

Notice that the constraints of the theory imply that~Xµ ⊥ n and Dµ

~Xν ⊥ n, such that ~Xν×(Dµ~Xν−Dν

~Xµ) ‖n, and hence g ~Xν × (Dµ

~Xν − Dν~Xµ) = g(n · [ ~Xν ×

(Dµ~Xν − Dν

~Xµ)])n. Thus, combining the equations of

motion for the fields Aµ and ~Xµ, n(11)+(13), we get theusual Yang-Mills equations of motion:

Dµ ~Fµν = ~0. (14)

We can obtain an explicit form for the unitary tripletn through an Euler rotation of the internal global Carte-sian basis {ea, a = 1, 2, 3}. Parametrizing the ele-ments of the adjoint representation by the three Eu-ler angles, S = exp(−γJ3) exp(−θJ2) exp(−ϕJ3) ∈SO(3), we can define the local internal basis by{na := S−1ea, a = 1, 2, 3}, where n := n3 =S−1e3 = (sin(θ) cos(ϕ), sin(θ) sin(ϕ), cos(θ)). With thisparametrization for n it is easy to show that themonopole curvature tensor given by (9) reproduces themagnetic field generated by an antimonopole at the ori-gin:

Hµν = −1

gsin(θ)(∂µθ∂νϕ− ∂νθ∂µϕ)

= −(δµθδνϕ − δνθδµϕ)1

gr2, (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: ∂0 := ∂t ≡ ∂/∂t, ∂1 := ∂r ≡ ∂/∂r,∂2 := ∂θ ≡ (1/r)∂/∂θ and ∂3 := ∂ϕ ≡ (1/r sin(θ))∂/∂ϕ.In fact, eq. (15) gives us the first homotopy class of themapping π2(SU(2)/U(1) ≃ S2) = Z defined by n (thesecond homotopy group is associated with the 2-sphereS2phys at the spatial infinity of the physical space for

each fixed time). The complete set of homotopy classesof this mapping defining the chromomagnetic chargesof the theory is obtained by making the substitutionϕ 7→ mϕ, m ∈ Z in the above parametrization for n[4], from which we obtain the chromomagnetic charge ofan antimonopole in the m-th homotopy class of π2(S

2):

g(m) :=

∮

S2

phys

d~S · ~H(m) =

∮

S2

phys

dSi1

2ǫ0iµνH

µν

(m)

=

∫ π

0

r2dθ sin(θ)

∫ 2π

0

dϕ

(

− m

gr2

)

= −4πm

g, m ∈ Z, (16)

which is the non-Abelian version of the Dirac quantiza-tion condition [1, 21].

The so-called magnetic gauge [4] is defined by fixingthe local color vector field n in the e3-direction in theinternal space by means of the S-rotation. In this gauge,the restricted curvature tensor is written as Fµν = (Fµν+Hµν)e3. Defining the so-called magnetic potential by theexpression:

Cµ :=1

g(cos(θ)∂µϕ+ ∂µγ), (17)

we see that we can rewrite the monopole curvature tensorHµν as [7]:

Hµν = ∂µCν − ∂νCµ − χµν , (18)

where:

χµν :=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 restricted

connection transforms like Aµ 7→ (Aµ + Cµ)e3 underthe gauge transformation that leads us to the magneticgauge.The magnetic potential Cµ describes the potential of a

monopole, being singular over its associated Dirac string,as we can easily see following the example discussed in[20] - if we consider γ = −ϕ, we have from (17) that:

Cµ =1

g(cos(θ) − 1)∂µϕ =

1

g

(cos(θ)− 1)

r sin(θ)δµϕ, (20)

which is the monopole potential singular over the Diracstring arbitrarily placed (by the choice made for γ) inthe negative e3-axis (θ = π). Notice, however, that themonopole curvature tensor (18) is regular, as it shouldbe.If we now discard the valence potential ~Xµ, that is,

if we discard the off-diagonal sector of the theory, weend up with the restricted Lagrangian density definingthe Abelian sector of the gauge theory in the magneticgauge:

L(R) = −1

4F 2µν

= −1

4[∂µ(Aν + Cν)− ∂ν(Aµ + Cµ)− χµν ]

2. (21)

If we further absorb the singular magnetic potentialCµ into the regular Abelian gluon field Aµ by making the

shift (Aµ + Cµ) 7→ Aµ, we rewrite (21) in the followingremarkable simple form:

L(R) = −1

4(Fµν − χµν)

2, (22)

where now the field Aµ is singular over the chromomag-netic Dirac strings. Equation (22) describes the Maxwelltheory with the vector potential Aµ non-minimally cou-pled to monopoles. In [18] we added a minimal cou-pling of the gauge field to external chromoelectric probe

4

charges to this Lagrangian density and applied a gener-alization [16–19] of the Julia-Toulouse approach for con-densation of defects [8, 9], compatible with the Elitzur’stheorem [2], to obtain a confining low energy effectivetheory for these charges in the static limit assuming theestablishment 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]:

~Aµ = Aµ + ~Xµ =∑

(k)

[

A(k)µ n(k) − 1

gn(k) × ∂µn

(k)

]

+ ~Xµ,

n(i) · n(k) = δ(i)(k) and n(k) · ~Xµ = 0, (23)

where the referred “cross product” is defined by theSU(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 to

investigate the contribution of the Abelian sector of theSU(3) gauge theory to the issue of static chromoelectricconfinement, from now on we focus only on the SU(3)restricted connection, given by [5]:

Aµ =∑

(k)=3,8

[

A(k)µ n(k) − 1

gn(k) × ∂µn

(k)

]

,

n2(3) = n2

(8) = 1 and n(3) · n(8) = 0. (24)

The restricted connection (24) is the connection thatleaves the unitary octets n(3) and n(8) invariant underparallel transport on the principal bundle. Actually, themagnetic condition Dµn

(3) = ~0 automatically generates

the field n(8) through the definition n(8) :=√3n(3) ⋆ 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 mappingπ2(SU(3)/U(1)× U(1)) = Z × Z defined by n(3), whereU(1)× U(1) is the maximal Abelian subgroup of SU(3)selected in each spacetime point in a gauge independentway by n(3) and n(8) [5]:

~g(N,N ′) = g

(

N − N ′

2,√3N ′

2

)

, N,N ′ ∈ Z. (25)

The chromoelectric charge operator in the fundamentalrepresentation of SU(3) is given by (see appendix D.2 of[21]):

~G := g~T = g(

T(3),T(8)

)

, T(k) =1

2λ(k), (26)

where λ(k) are Gell-Mann matrices.The typical contribution for the partition function due

to deformations of the Dirac strings is of the form [5, 10]:

exp[i~G · ~g] = exp

igg

N/2 0 00 (−N +N ′)/2 00 0 −N ′/2

.

(27)

Since the presence of monopole-defects is associatedwith the local brane symmetry [19] consisting in the free-dom of deforming the unphysical Dirac strings without

any observable effects [1, 10, 21], we see from (27) thatwe have to impose as a consistency condition the non-Abelian version of the Dirac charge quantization:

gg = 4π, (28)

in order to have exp[i~G · ~g] = 13×3. Hence, the chargequantization follows as a consequence of the brane sym-metry (unobservability of the Dirac strings) [10, 18, 19].The term “brane” in this context refers to any hypersur-face (for example, a string is a 1-brane while its world-surface is a 2-brane).As before, explicit parametrizations for n(3) and n(8)

can be obtained by means of a rotation of the inter-nal global base elements e(3) and e(8) realized by an ar-bitrary element of the adjoint representation of SU(3)parametrized by 8 “Euler angles”. Also, the magnetic

potentials C(3)µ and C

(8)µ are revealed in the restricted

connection by fixing the magnetic gauge which is definedby an internal rotation that sends the local internal vec-tors n(3) and n(8) to the e(3) and e(8) directions, respec-tively (see [5] and references therein for details).The restricted curvature tensor in the magnetic gauge

is given by (absorbing, as before, the singular mag-

netic potentials C(k)µ into the regular Abelian gluon fields

A(k)µ ):

Fµν =∑

(k)=3,8

(F (k)µν − χ(k)

µν )e(k), (29)

where, as before, the chromomagnetic string terms χ(k)µν

appear due to the angular (multivalued) nature of the

components of the magnetic potentials C(k)µ . The associ-

ated Lagrangian density is given by:

L(R) = −1

4

∑

(k)=3,8

(F (k)µν − χ(k)

µν )2. (30)

If we now minimally couple external chromoelectricprobe charges to the restricted connection in the mag-netic gauge, −~jµ · Aµ, we get the following Lagrangiandensity:

L(R) =∑

(k)=3,8

[

−1

4(F (k)µν − χ(k)

µν )2 − j(k)µ Aµ(k)

]

. (31)

5

We must now specify the chromoelectric charge struc-

ture of the probe current ~jµ = (j(3)µ , j

(8)µ ). This can be

done by comparison with the quarkionic current ~jψµ =

ψγµ ~Gψ that couples minimally to the gauge connection

Aµ = (A(3)µ + C

(3)µ )e(3) + (A

(8)µ + C

(8)µ )e(8):

~jψµ · Aµ = rγµ

[

g

2(Aµ(3) + Cµ(3)) +

g

2√3(Aµ(8) + Cµ(8))

]

r+

+ bγµ

[

−g2(Aµ(3) + Cµ(3)) +

g

2√3(Aµ(8) + Cµ(8))

]

b+

+ yγµ

[

− g√3(Aµ(8) + Cµ(8))

]

y, (32)

where ψ = (r, b, y) is the internal antitriplet, being r,b and y the red, blue and yellow antiquark spinors, re-spectively. From (32) we see that the red, blue and yel-

low quarks have chromoelectric charges ~Q = (Q(3), Q(8))given by ( g2 ,

g

2√3), (− g

2 ,g

2√3) and (0,− g√

3), respectively

(see also table (D.10) of [21] and [5]).We now construct the chromoelectric probe current

~jµ = (j(3)µ , j

(8)µ ) in such a way that it is compatible with

the above charge structure:

j(3)µ =g

2δµ(x;Lr)−

g

2δµ(x;Lb)

=1

2ǫµναβ∂

νΛαβ(3), (33)

Λαβ(3) =g

2δαβ(x;Sr)−

g

2δαβ(x;Sb), (34)

j(8)µ =g

2√3δµ(x;Lr) +

g

2√3δµ(x;Lb)−

g√3δµ(x;Ly)

=1

2ǫµναβ∂

νΛαβ(8), (35)

Λαβ(8) =g

2√3δαβ(x;Sr) +

g

2√3δαβ(x;Sb)−

g√3δαβ(x;Sy),

(36)

where Λ(k)µν are the chromoelectric Dirac string terms, be-

ing Lr = ∂Sr, Lb = ∂Sb and Ly = ∂Sy the worldlinesof the red, blue and yellow probe charges, boundaries ofthe worldsurfaces Sr, Sb and Sy of the chromoelectricDirac strings of the red, blue and yellow probe charges,respectively.

IV. THE EFFECTIVE THEORY OF COLOR

CONFINEMENT

If we now assume that in a certain regime of the the-ory it is established a stable chromomagnetic monopolecondensation, we can ask ourselves what should be theform of the effective theory describing the low energy ex-citations of 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 [17] theAbelian lattice based approach (ALBA). As we showedin [17], these two approaches are dual to each other in thelimit where a complete defects condensation takes place(meaning that the defects proliferate in such a way thatthey occupy the whole space). However, such a com-plete condensation, as discussed in [18, 19], can only berealized in the absence of external charges, due to a re-striction imposed by the Elitzur’s theorem [2], which pro-hibits the spontaneous breaking of local symmetries (lo-cal symmetries can have different realizations in differentregimes of a theory, but cannot be spontaneously brokenby any physical process, what is indeed a very intuitivefact, since local symmetries are actually redundancies inthe definition of the variables of a theory). As we dis-cussed in [18], when we include external chromoelectricprobe charges in the chromomagnetic condensate, thereis a breaking of the local chromoelectric brane symmetry(which corresponds to the freedom of deforming the chro-moelectric Dirac strings without any observable conse-quences: unobservability of the unphysical Dirac strings)in the original JTA [8, 9], which violates the Elitzur’s the-orem and spoils the charge quantization (since with sucha breaking the strings would become observable). Suchan inconsistency, however, does not happen in the ALBAif we use the concept of brane invariants introduced in[16, 18, 19]. Indeed, this approach generalizes the resultsof the original JTA in such a way to satisfy the Elitzur’stheorem (see the discussion in [18]) and, hence, we shallfollow this route in the sequel.Since we are interested in analyzing some of the con-

sequences of a monopole condensation, it is interestingto go to the dual picture, where we avoid the problem

of working with the vector potentials A(k)µ in a scenario

where they are ill-defined in almost the whole spacetimedue to the proliferarion of the chromomagnetic Diracbranes. Thus, our first step consists in dualizing the ac-tion associated to the restricted Lagrangian density (31)[17]:

∗S(R) =

∫

d4x∑

(k)=3,8

[

−1

4(F (k)µν − Λ(k)

µν )2 + j(k)µ Aµ(k)

]

,

(37)

where the couplings are inverted relatively to the ones

6

present in (31) - 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 associated

chromoelectric strings Λ(k)µν given by (34) and (36).

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 π2(SU(3)/U(1) × U(1)), namely: ~g1 :=

~g(1, 0) = 4πg~ω1, ~g2 := ~g(−1,−1) = 4π

g~ω2 and ~g3 :=

~g(0, 1) = 4πg~ω3, where ~ω1 = (1, 0), ~ω2 =

(

− 12 ,−

√32

)

and

~ω3 =(

− 12 ,

√32

)

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 π2(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:

jiµ := gδµ(x; Li) =1

2ǫµναβ∂

νχαβi , (38)

χαβi = gδαβ(x; Si), (39)

where Li = ∂Si, i = 1, 2, 3 are the worldlines of thethree monopoles associated to the roots ~ωi, the physi-cal boundaries of the worldsurfaces Si of the unphysi-cal chromomagnetic Dirac strings. The roots appear ex-plicitly in the minimal coupling jiµA

µi , where we defined

Aµi := ~ωi · ~Aµ = ~ωi · (A(3)µ , A

(8)µ ).

In order to rewrite (37) in an explicitly Weyl-symmetric form (the Weyl symmetry corresponds to theinvariance of the theory under permutations of the in-dices i = 1, 2, 3 of the fundamental representation ofSU(3)), we shall make use of the following identities:

~Vµ = (V (3)µ , V (8)

µ ) =2

3

3∑

i=1

~ωiViµ,

V iµ := ~ωi · ~Vµ ⇒3∑

i=1

V iµ = 0 (constraint);

~Vµ · ~Rµ = V (3)µ Rµ(3) + V (8)

µ Rµ(8) =2

3

3∑

i=1

V iµRµi . (40)

Thus, the Weyl-symmetric representation of (37) withthe monopole current given by (38) is:

∗S(R) =

∫

d4x

3∑

i=1

[

−1

6(F iµν − Λiµν)

2 + jiµAµi

]

, (41)

where the chromoelectric branes acquire the following re-

markable symmetric form:

Λ1µν = ~ω1 · ~Λµν = Λ(3)

µν

=g

2δµν(x;Sr)−

g

2δµν(x;Sb), (42)

Λ2µν = ~ω2 · ~Λµν = −1

2Λ(3)µν −

√3

2Λ(8)µν

= −g2δµν(x;Sr) +

g

2δµν(x;Sy), (43)

Λ3µν = ~ω3 · ~Λµν = −1

2Λ(3)µν +

√3

2Λ(8)µν

=g

2δµν(x;Sb)−

g

2δµν(x;Sy). (44)

In order to allow the monopoles to proliferate we mustgive dynamics to their associated chromomagnetic Diracbranes χiµν , 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 (41) with a kinetic term for the chro-momagnetic Dirac branes of the form − ǫc

2 j2µi, which is

the term in a derivative expansion with the lowest orderin derivatives of χiµν (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 actionreads:

Zc :=3∏

i=1

∫

DAiµ δ[∂µAµi ]ei∫d4x[− 1

6(F i

µν−Λiµν )

2]Zc[Aiµ],

(45)

where the Lorentz gauge has been adopted for the dualgauge fields Aiµ and the partition functions for the branesectors are given by:

Zc[Aiµ] =

∑

{Li}δ[∂µj

µi ] exp

{

i

∫

d4x[

− ǫc2j2µi + jiµA

µi

]

}

,

(46)

(without sum in i) where the functional δ-distributionenforces the closeness of the monopole worldlines (thechromomagnetic loops) giving the current conservationlaws ∂µj

µi = 0, which are identically satisfied due to (38).

An observation is in order at this point. Notice that theactivation term for the chromomagnetic loops is highlysingular, since it is proportional to the square of a δ-distribution at the same point. This singularity is as-sociated to the hydrodynamic limit, where we considerthe coherence length of the condensate to be zero. Asdiscussed in [10], this activation term can be regularizedbe smoothing out the δ-distributions over the real coher-ence length of the chromomagnetic condensate, that is

7

non-zero (this, in fact, gives the thickness of the confin-ing chromoelectric flux tube, when external charges arepresent in this medium), such that the regularized acti-vation term gives an energy contribution proportional tothe total length of the chromomagnetic loops Li. Such aregularization can also be done in the explicit evaluationof the confining potential through the introduction of anultraviolet cutoff scale corresponding to the inverse of thecoherence length of the condensate [19].The third step in our approach consists in the use of

the Generalized Poisson’s Identity (GPI) (see appendix Aof [17] for a detailed discussion on the subject) in d = 4:

∑

{Li}δ[ηiµ − δµ(x; Li)] =

∑

{Vi}e2πi

∫d4x δµ(x;Vi)η

µi , (47)

(without sum in i) where Vi is the 3-brane Poisson-dual

to the 1-brane Li. The GPI works as a brane analogueof the Fourier transform: when the line configurationsLi in the left-hand side of (47) proliferate, the volume

configurations Vi in the right-hand side become dilutedand vice versa. Using (47) we can rewrite (46) as:

Zc[Aiµ] =

∫

Dηiµ∑

{Li}δ

[

g

(

ηiµg

− δµ(x; Li)

)]

δ

[

g

(

∂µηiµg

)]

exp

{

i

∫

d4x[

− ǫc2η2µi + ηiµA

µi

]

}

= N∫

Dηiµ∑

{Vi}e2πi

∫d4x δµ(x;Vi)

ηµig

∫

Dθi

ei∫d4x θi∂µ

ηµig exp

{

i

∫

d4x[

− ǫc2η2µi + ηiµA

µi

]

}

= NN ′∑

{Vi}Φ[θV iµ ]

∫

Dθi∫

Dηiµ exp

{

i

∫

d4x

[

− ǫc2η2µi −

ηµig(∂µθ

i − θV iµ − gAiµ)

]}

, (48)

(without sum in i) where we defined the Poisson-dual

current (to the chromomagnetic current jiµ), θV iµ :=

2πδµ(x; Vi), 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[θV iµ ] (see [10] fora discussion on the subject). Since the constant prod-uct NN ′ 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).Integrating out the auxiliary fields ηiµ in the partial

partition functions (48) and substituting the result backin the complete partition function (45) we obtain, as the

low 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):

Zc =3∏

i=1

∑

{Vi}Φ[θV iµ ]

∫

DBiµ exp{

i

∫

d4x

[

−1

6(Giµν − Λiµν)

2 +m2

2

(

Biµ +1

gθV iµ

)2]}

=3∏

i=1

∑

{V i}Φ[θV iµ ]

∫

DBiµ exp{

i

∫

d4x

[

−1

6(Giµν − Liµν)

2 +m2

2B2µi

]}

, (49)

(without sum in i) where in the last line we made the

shift Biµ := Aiµ − 1g∂µθ

i 7→ Biµ − 1gθV iµ and where m :=

(√ǫc)

−1 is the mass acquired by the gauge invariant field

(and also brane invariant field, after the above shift) Biµdue to the chromomagnetic condensate. Also, Giµν :=

∂µBiν −∂νBiµ is the gauge (and brane invariant) strength

tensor field and Liµν := Λiµν + 1g(∂µθ

V iν − ∂ν θ

V iµ ) is the

chromoelectric brane invariant which remains fixed underdeformations of the chromoelectric branes (provided theDirac quantization condition (28) is satisfied) [18, 19]:

Λiµν 7→ Λiµν + ∂µΣiν − ∂νΣ

iµ, (50)

Σ1µ =

g

2δµ(x;Vr)−

g

2δµ(x;Vb), (51)

Σ2µ = −g

2δµ(x;Vr) +

g

2δµ(x;Vy), (52)

Σ3µ =

g

2δµ(x;Vb)−

g

2δµ(x;Vy), (53)

where Vr is the 3-volume spanned in spacetime by the de-formation of the worldsurface of the chromoelectric Diracstring of the red probe charge, keeping fixed its (physi-cal) boundary (the worldline of the red probe charge),and similarly for the blue and yellow probe charges.It is important to notice that Liµν is a brane invariant

at the level of the partition function (and not at the levelof the action): any external deformation Σiµ produced in

the chromoelectric Dirac branes Λiµν can be absorbed in

the internal branes 1gθV iµ (provided (28) is satisfied), since

in the presence of external charges all such configurations

are contained in the ensemble of internal defects{

Vi

}

[19]. Since we sum over all the configurations of thisensemble in the partition function, the theory has braneinvariance.

8

The scalar field θi present in Biµ is interpreted as thephase field of the complex Higgs field describing the chro-momagnetic condensate (in the hydrodynamic limit thefluctuations of the modulus of the Higgs field are frozen),

while the current θV iµ codifies the contribution of thevortices at low energies, which constitute defects overthe monopole condensate, localizing regions of the spacewhere the chromomagnetic condensate has not been es-tablished [10, 17–19]. In particular, due to the local chro-moelectric brane symmetry (50), which cannot be bro-ken due to the Elitzur’s theorem [2], in the presence ofexternal chromoelectric charges it is impossible to havea complete chromomagnetic condensation, because thevortices θV iµ connected to the strings Λiµν cannot be di-luted, in order to maintain the structure of the braneinvariant Liµν [18, 19]. In fact, this restriction imposedby the Elitzur’s theorem over the chromomagnetic con-densation when there are external chromoelectric chargesembedded in the system is easy to understand physically:the dual Meissner effect (generated by the mass m of the

gauge invariant field Biµ in the condensed regime) ex-pels the chromoelectric fields generated by the externalcharges of almost the whole space constituted by the dualsuperconductor, however, these fields cannot simply van-ish - they become confined in regions of the space withminimal volume corresponding to the linear chromoelec-tric confining flux tubes described by the brane invariantLiµν [18, 19]. Notice also that in the effective theory (49)both, the local gauge and brane symmetries, are realizedin a hidden fashion in the interior of the gauge and braneinvariants Biµ and Liµν : this hidden realization is com-monly referred in the literature by the inadequate jargonof the “spontaneous breaking” of the gauge and branesymmetries, respectively. This nomenclature loosely re-lates to the fact that there are no gauge and brane sym-metries in terms of the gauge and brane invariants Biµand Liµν in (49). It is important to notice, however, thatthese local symmetries are only defined with respect tothe original variables Aiµ and Λiµν , and that in terms ofthese variables, such symmetries are realized in a hiddenfashion in the interior of the invariants of the theory, notbeing actually broken, in consonance with the Elitzur’stheorem.

Before returning to the direct picture, we can rewritethe effective action present in the partition function (49)in the Cartan representation as:

S =

∫

d4x∑

(k)=3,8

[

−1

4(G(k)

µν − L(k)µν )

2 +m2

2B2µ(k)

]

,

(54)

where we defined m :=√

32m. Its dual action is given by

[17]:

∗S =

∫

d4x∑

(k)=3,8

[

−1

2(∂µY

µν

(k) )2 +

m2

4Y 2µν(k)+

+m

2Y (k)µν L

µν

(k)

]

, (55)

where the Kalb-Ramond fields Y(k)µν describe the

monopole condensate in the direct picture: notice therank-jumping observed in the direct picture due to themonopole condensation - in the diluted regime describedby (31) the system is characterized by the massless 1-

forms A(k)µ , while in the condensed regime described by

(55) 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 [16–19].Notice also that (55) is the generalization of [9] com-

patible with the Elitzur’s theorem and the local chro-moelectric brane symmetry: in [9] the last term in (55)features a minimal coupling of the Kalb-Ramond fielddirectly with the chromoelectric Dirac strings instead ofthe chromoelectric brane invariants, thus violating theElitzur’s theorem through the breaking of the chromo-electric brane symmetry and spoiling the charge quanti-zation as a consequence.

Integrating out the fields Y(k)µν , we obtain the effec-

tive action describing the interaction between the exter-nal chromoelectric probe charges in the chromomagneticcondensed regime:

∗Seff =

∫

d4x∑

(k)=3,8

[

−m2

4L(k)µν

1

∂2 + m2Lµν(k)+

−1

2j(k)µ

1

∂2 + m2jµ(k)

]

. (56)

From (56) we obtain, in the static limit, the follow-ing potential between two static chromoelectric probecharges separated by a distance L embedded in the chro-momagnetic monopole condensate (see [19] for the de-tailed evaluation):

Vstatic(L; g, m, M) = σ(g, m, M)L−(Q2

(3) +Q2(8))

4π

e−mL

L,

(57)

where the string tension is given by:

σ(g, m, M) =(Q2

(3) +Q2(8))m

2

8πln

(

m2 + M2

m2

)

=g2m2

24πln

(

m2 + M2

m2

)

, (58)

where M is an ultraviolet cutoff corresponding to theHiggs mass (the mass of the monopoles), whose inversegives the coherence length of the monopole condensate

9

and where we used the fact that for any mesonic config-

uration (r− r, b− b or y− y), we have Q2(3) +Q2

(8) =g2

3 .

In [13] the free parameters (g, m, M) were fixedby comparison with various lattice data as being(5.5, 0.5GeV, 1.26GeV ). From this set of values, one getsthe string tension σ ≈ (440MeV )2 and the predictionthat the dual superconductor realizing the static chro-moelectric confinement in the QCD vacuum should be ofthe type II.

Notice also that taking m = 0 leads us back to thediluted regime characterized by the long-range Coulombinteraction, eliminating the monopole condensate and de-stroying the chromoelectric confinement.

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 condensationof chromomagnetic monopoles, extending a previous ap-proach that we applied to the SU(2) case [18].

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 areparametrization of the non-Abelian connection whichreveals its Abelian sector and the associated topologicalstructures (monopoles) without resorting to any gaugefixing procedure, hence providing a gauge invariant defi-nition of these defects in the Yang-Mills theory [4, 5, 7].

With the discarding of the off-diagonal sector of thetheory in the magnetic gauge, we showed that the actionin the regime with diluted defects can be put in the formof a Maxwellian theory non-minimally coupled to chro-momagnetic monopoles and minimally coupled to exter-nal chromoelectric probe charges. This was the crucialpoint that allowed us to apply the generalized form of theJulia-Toulouse approach for condensation of defects [19]and obtain an effective theory for the regime where themonopoles are condensed. The effective theory that wederived with such an approach, given by equation (49),

gives an interaction potential between two static chromo-electric probe charges embedded in the chromomagneticmonopole condensate consisting in a sum of a Yukawaand a linear confining term. The monopole condensaterealizing this dual superconductor scenario of color con-finement in the QCD vacuum was found to be of the typeII. The remotion of the monopole condensate destroysthe chromoelectric confinement and leads us back to thediluted regime characterized by the long-range Coulombinteraction.It is important to emphasize the fact that in our ap-

proach the concept of brane invariants [16, 18, 19] was offundamental importance in the construction of the effec-tive theory in a way consistent with the charge quantiza-tion in the condensed regime. Kleinert was the first oneto point out the fact that the brane symmetry is a kind oflocal symmetry independent of the gauge symmetry [10].In [16] (see also [19] for an updated discussion on the sub-ject) some of us showed this point explicitly in studyingthe Maxwell-Chern-Simons theory in the presence of in-stantons. In this system the gauge symmetry is explicitlyrealized while the brane symmetry is realized in a hiddenfashion, thus leading to the confinement of the instan-tons. These two different realizations of the gauge andthe brane symmetries, respectively, in the same phaseof a system, constitutes an unequivocal example of theindependence between these two local symmetries. Thebrane symmetry implies the charge quantization as a con-sistency condition due to the unobservability of the un-physical Dirac strings and it must be maintained also inthe regime with condensed monopoles. Indeed, as it is aredundancy in the definition of the variables of the sys-tem, it cannot be spontaneously broken in any phase ofa system by any physical process, in accordance withthe Elitzur’s theorem [2]. Hence, the effective theorywe obtained (49) features not only the confinement ofstatic chromoelectric probe charges in the fundamentalrepresentation of SU(3), but it also preserves the chargequantization, which is present in the regime with dilutedmonopoles, in the regime where these monopoles are con-densed.

VI. ACKNOWLEDGEMENTS

We thank Conselho Nacional de DesenvolvimentoCientıfico e Tecnologico (CNPq) for financial support.

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