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USM-TH-167

Duality and con?nement in D=3 models driven by condensation of topological defects

Patricio Gaete?

Departamento de F? ?sica, Universidad T? ecnica F. Santa Mar? ?a, Valpara? ?so, Chile and Departamento de F? ?sica, Universidad Tecnol? ogica Metropolitana , Santiago, Chile

Clovis Wotzasek?

Instituto de F? ?sica, Universidade Federal do Rio de Janeiro, Brazil and Departamento de F? ?sica, USACH, Chile We study the interplay of duality and con?nement in Maxwell-like three-dimensional models induced by the condensation of topological defects driven by quantum ?uctuations. To this end we check for the con?nement phenomenon, in both sides of the duality, using the static quantum potential as a testing ground. Our calculations are done within the framework of the gauge-invariant but path-dependent variables formalism which are alternative to the Wilson loop approach. Our results show that the interaction energy contains a linear term leading to the con?nement of static probe charges.

PACS numbers: 11.10.Ef, 11.10.Kk

arXiv:hep-th/0503070v3 25 Aug 2005

I.

INTRODUCTION

This work is aimed at studying the duality symmetry for certain 3D models coupled to sources of di?erent dimensions that eventually condense due to quantum ?uctuations, using the Quevedo- Trugenberger phenomenology [1] to the Julia-Toulouse mechanism [2]. In general studies in Field Theory, the presence of the duality between two models is veri?ed through their equations of motion and the algebra of the observables. However, since quantum ?uctuations may eventually drive the condensation of topological defects destroying the duality, we should be able to look for a distinct property to be used as a point of proof in the condensed phase. In this paper we propose to use the e?ective potential between two static charges as such a testing ground for the existence of duality, after quantum ?uctuations drive the condensation of topological defects. In principle one could check for this proposal studying the interplay between con?nement and duality in an U (1) gauge theory in D spacetime dimensions for Maxwell-like theories of totally anti-symmetric tensors of arbitrary rank. However, due to technical di?culties in the computation of the e?ective static potential in arbitrary D dimensions we shall restrict ourselves to the case D = 3. Based on the common knowledge coming from the continuum abelian gauge theory, the assertion that the such a theory has a con?ning phase may sound strange. In fact the existence of a phase structure for the continuum abelian U(1) gauge theory was obtained by including the e?ects due to compactness of U(1) group that dramatically change the infrared properties of the model. These results, ?rst found by Polyakov [3], have been con?rmed by many distinct techniques basically due to the contribution of the vortices into the partition function of the theory. The condensation of these topological defects then lead to a structural change of the conventional vacuum of the theory into a dual superconductor vacuum. An interesting approach to this problem has recently been proposed by Kondo who derived the e?ective potential from a partition function that includes the contribution of all topologically non-trivial sectors of the theory [4]. In a previous paper [5] we have approached the problem in a phenomenological way using the Julia-Toulouse mechanism [2], as proposed by Quevedo and Trugenberger [1], that considers the condensation of topological defects. This study was undertaken for theories of compact anti-symmetric gauge tensors of arbitrary ranks in D spacetime dimensions that appear as low-energy e?ective ?eld theories of strings. More speci?cally, using the QuevedoTrugenberger phenomenology [1] we studied the low-energy ?eld theory of a pair of compact massless anti-symmetric tensor ?elds, say Ap and Bq with p + q + 2 = D, coupled magnetically and electrically, respectively, to a large set of (q ? 1)-branes, characterized by charge e and a Chern-Kernel Λp+1 [6], that eventually condense. It has been argued that the e?ective theory that results displays the con?nement property. The results of [5] show that the phenomenological action proposed in [1] incorporates automatically the contribution of the condensate of topological defects to the vacuum of the model or, alternatively, the non-trivial topological sectors as in [4].

? Electronic ? Electronic

address: patricio.gaete@usm.cl address: clovis@if.ufrj.br

2 It goes without saying that, on the grounds of the observed electric -magnetic duality, the same result should come through if the dual picture were adopted, i.e., by considering the reversed couplings with a (p ? 1)-brane of charge g and Chern-kernel ?q+1 . Of course this seems to be mandatory if the system displays self- duality, e.g., if the sources have the same dimension and the tensors are of the same rank (p = q ), which only occurs in even dimensions. The Julia-Toulouse mechanism for the condensation of magnetic charges, leading to con?nement of electric charges is dual to the con?nement of monopoles driven by the condensation of electric charges. However, the general results presented in [5] suggest that such a duality between the electric and magnetic views should survive also for systems not presenting self-duality, that is, when p = q . However a clear-cut veri?cation of this possibility is still missing. To explicitly check for this manifestation of the duality phenomenon in the condensed phase using the e?ective static potential is our main motivation in this paper. The way we intend to ?ll up this gap is by studying speci?c examples in D = 3 since they will not display self-duality. Our calculations are done within the framework of the gauge-invariant but path-dependent variables formalism which are alternative to the Wilson loop approach. To consider this simpler situation seems to be our only possibility to check for the proposal of using the e?ective static potential as a point of proof for duality in the condensed phase since for higher dimensions such a computation becomes very messy. There is however an extra technical point of di?culty here. In this dimension one side of duality involves a scalar ?eld which is not a gauge model. In fact it is not even a constrained theory. This poses some di?culties applying the above formalism to compute the e?ective static potential. An extension of the above mentioned method will be presented here that solves this problem. This is a new, although minor contribution of this work. In the next section we discuss the duality both in the dilute and in the condensed phases. In particular we review the main points of the mechanism presented in [5] using the Quevedo-Trugenberger formalism. The phenomenological approach to the condensation of topological defects developed in [1] and the con?nement potential for the e?ective theory of [5] are quickly reviewed in Section 2. In Section 3 we perform explicit computation for the e?ective potential for the examples mentioned above. An alternative derivation of these results together with notation and technical details in the computation of the e?ective potential are included in Appendix-A . A summary of results and future perspective are the subject of our ?nal section.

II. DUALITY AND CONFINEMENT A. Duality in the Dilute Phase

?q , that represent the potentials for Consider for a moment a dual pair of massless antisymmetric tensors, Ap and A a pair of Maxwell-like theories, coupled to closed charged branes of dimension (p ? 1) and (q ? 1). In this dilute phase, duality is manifest in the following sense. From the point of view of the Ap tensor, the (p ? 1)-brane is electric while the (q ? 1)-brane is magnetic. This situation, is illustrated by the following diagram ?> =< 89 :; (q ? 1) ? brane f MM M MMM MM MC MMM & and is formally described by the following action SA = 1 (?1)p 2 [Fp+1 (Ap ) ? g Λp+1 ] + e Ap J p (?) . 2 (p + 1)! (2) ?> =< 89 :; (p ? 1) ? brane 8 q qq qqq q q EC xqqq (1)

Ap

where Λp+1 and ?q+1 are the Chern-Kernel of the (q ? 1)-brane and the (p ? 1)-brane, respectively. For the D = 3 case where p = 1 and q = 0, Eq.(2) represents the action of a vector ?eld coupled minimally to an electric charge and non-minimally to a magnetic instanton. This system displays electric and magnetic symmetries δE A? = ?? ξ ; δE ?? = ?? ξ ; electric δM A? = g χ? ; δM Λ?ν = ?[? χν ] ; magnetic

(3)

with ξ and χ? being the corresponding gauge functions. The new magnetic symmetry [7, 8] appears due to the presence of magnetic sources and is manifest as the invariance of the “generalized” ?eld tensor Hp+1 = [Fp+1 (Ap ) ? g Λp+1 ]. The conventional (electric) gauge symmetry, manifest by F?ν (A) keeps the minimal coupling with the electric pole invariant thanks to the current conservation condition coming from the fact that we are considering closed branes. This

3 second term is also invariant under the magnetic symmetry if the charges of the branes satisfy the Dirac quantization condition, eg = 2πn ; n ∈ Z . (4)

?q , the electric and magnetic couplings with the branes have reversed From the point of view of the dual ?eld A character, as depicted by the diagram, ?q fL 8A LL rr LL MC r r EC r LL r LL r r r LL r x r & ?> ?> =< =< 89 :; 89 :; (q ? 1) ? brane (p ? 1) ? brane whose action reads, SA ? = 1 (?1)q ? ?q ) ? e ?q+1 Fq+1 (A 2 (q + 1)!

2

(5)

?q J q (Λ) . +gA

(6)

This dual action is invariant under a corresponding pair of electric and magnetic gauge transformation. It is known that upon duality transformation from one picture to the other, a brane-brane term like, Sbrane?brane = e g Λp+1 ?p+1,q+1 ?q+1 (7)

will also be induced. Such a term however disappear from the partition function thanks to the charge quantization condition (4) and the fact that the Chern-kernels are integer functions, rendering the above integral as an integer that represents the intersection number of the two branes. It is worth noticing at this juncture that such a duality equivalence (?0 ), illustrated as, ?q 8A rr O r EC rr r rr r xrr ?> =< 89 :; ?0 (q ? 1) ? brane f MM M MMM MM MC MMM & Ap L f L LL LL MC LL LL L& =< ?> 89 :; (p ? 1) ? brane 8 q q q qqq q q EC xqqq (8)

is not the conventional textbook duality since the potential tensors involved are not, in general, of the same rank and, correspondingly, the branes are not of the same dimensions. For the particular D = 3 case one tensor is of rank-1 (vector) and the other is of rank-0 (scalar) while the sources are a monopole and an instanton. The classi?cation of these topological defects as electric or magnetic will depend on which point of view is taken, as discussed above. When both objects are of the same dimension (and the tensors are of the same rank) we have self-duality. This is the usual electric-magnetic duality discussed in the Maxwell theory in D=4, for instance, being straightforward then that the condensation of either the electric and the magnetic sources leads to identical physical results. However, when the topological defects are of di?erent dimensions duality has to be understood in the more general sense discussed above. Therefore without self-duality it seems doubtful that the eventual condensation of either branes lead to the very same physical picture. In fact, as we will show clearly in section III by the computation of the e?ective static potential, the phenomenon of con?nement occurs irrespective of which brane condense. In particular, we will show that for D = 3 two electric instantons are con?ned when immersed in the condensate of magnetic instantons while the electric charges experience the same result when immersed in the condensate of magnetic monopoles.

B. Duality in the Condensed Phase

In this subsection we use the Quevedo and Trugenberger phenomenology to study the interplay between duality and con?nement. To this end we consider the low-energy ?eld theory of a pair of independent massless anti-symmetric tensor ?elds, say Ap and Bq with p + q + 2 = D, coupled electrically (magnetically) and magnetically (electrically) to a large set of (q ? 1)-branes, characterized by charge e and a Chern-Kernel Λp+1 (resp.,(p ? 1)-branes with charge

4 g and Chern-kernel ?q+1 ) [6], that eventually condense. It will be explicitly shown, for the D=3 example of vector and scalar ?elds coupled to instantons and monopoles, that the induced e?ective theories display the con?nement property by providing a clear-cut derivation of the e?ective potential for a pair of static, very massive point probes immersed in the condensate. Firstly consider that the ?elds Bq and Ap are electrically (EC) and magnetically (MC) coupled to a e-charged, (q ? 1)-brane, respectively, according to the following diagram B q8 O q q q EC qq qqq q q q x ?> =< 89 :; D=p+q+2 (q ? 1) ? brane fMMM MMM MM MC MMM & Ap The action corresponding to this situation is given by Se = 1 (?1)q 1 (?1)p 2 2 [Hq+1 (Bq )] + e Bq J q (Λ) + [Fp+1 (Ap ) ? eΛp+1 ] . 2 (q + 1)! 2 (p + 1)! (10) (9)

Upon the condensation phenomenon, the Chern-Kernel Λp+1 will become the new massive mode of the e?ective theory. Next, we consider the dual picture where the Bq and Ap ?elds are magnetically and electrically coupled to a g -charged, (p ? 1)-brane, respectively, according to the following diagram Bq M O f MMM MM MC MMM MM& ?> =< 89 :; D=p+q+2 (p ? 1) ? brane 8 q q qqq qq q EC q xqq Ap described by the following action Sg = 1 (?1)p 1 (?1)q 2 2 [Fp+1 (Ap )] + g Ap J p (?) + [Hq+1 (Bq ) ? g ?q+1 ] 2 (p + 1)! 2 (q + 1)! (12) (11)

in which case the Chern-Kernel ?q+1 will become the new massive mode of the e?ective theory after the condensation. Our compact notation here goes as follows. The ?eld strengths are de?ned as Fp+1 (Ap ) = F?1 ?2 ...?p+1 = ?[?1 A?2 ···?p+1 ] and Hq+1 (Bq ) = H?1 ?2 ...?q+1 = ?[?1 B?2 ···?q+1 ] , while the Chern-kernels Λp+1 = Λ?1 ···?p+1 and ?q+1 = ??1 ···?q+1 are totally anti-symmetric objects of rank (p + 1) and (q + 1), respectively. The conserved currents J q (Λ) and J p (?) are given by delta-functions over the world-volume of the (q ? 1)-brane and (p ? 1)-brane [9]. These conserved currents may be rewritten in terms of the kernel Λp+1 and ?q+1 as 1 ?q,α,p+1 ?α Λp+1 , (p + 1)! 1 J p (?) = ?p,α,q+1 ?α ?q+1 , (q + 1)! J q (Λ) =

(13)

and ?q,α,p+1 = ??1 ...?q ,α,ν1 ...νp+1 . We review next how to constructed the e?ective interacting action, in the condensed phase, between the antisymmetric tensor ?eld Bq (Ap ) and the degrees of freedom of the condensate Λp+1 (?p+1 ). To this end, after the condensate is integrated out, we compute the e?ective quantum potential for a pair of static probe living inside the condensate, within the framework of the gauge-invariant but path-dependent variables formalism. This will disclose the dependence of the con?nement properties with the condensation parameters coming from the Julia–Toulouse mechanism.

5 We are now ready to discuss the consequences of the Julia-Toulouse mechanism over the action (10). Since the manipulations are quite general, the result for the dual action (12) follows analogously. The initial theory, before condensation, displays two independent ?elds coupled to a (q ? 1)–brane. The nature of the two couplings are however di?erent. The Ap tensor, that is magnetically coupled to the brane, will then be absorbed by the condensate after phase transition. On the other hand, the electric coupling, displayed by the Bq tensor, becomes a “B ∧ F (Λ)” topological term after condensation. Indeed, the distinctive feature is that after condensation, the Chern-Kernel Λp+1 is elevated to the condition of propagating ?eld. The new degree of freedom absorbs the degrees of freedom of the tensor Ap this way completing its longitudinal sector. The new mode is therefore explicitly massive. Since Ap → Λp+1 there is a change of rank with dramatic consequences. The last term in (10), displaying the magnetic coupling between the ?eld-tensor Fp+1 (Ap ) and the (q ? 1)-brane, becomes the mass term for the new e?ective theory in terms of the tensor ?eld Λp+1 and a new dynamical term is induced by the condensation. It is consequential that the minimal coupling of the Bq tensor becomes responsible for another contribution for the mass, this time of topological nature. Indeed the second term (10) becomes an interacting “BF –term” between the remaining propagating modes, inducing the appearance of topological mass, in addition to the induced condensate mass. The ?nal result reads Scond = (?1)q (?1)p+1 (?1)p+1 (p + 1)! 2 2 [Hq+1 (Bq )]2 + e Bq ?q,α,p+1 ?α Λp+1 + [Fp+2 (Λp+1 )]2 ? m Λp+1 (14) 2(q + 1)! 2(p + 2)! 2

where m = Θ/e where Θ represents the condensate density. There has been a drastic change in the physical scenario. To see this we need ?rst to obtain an e?ective action for the Bq tensor that includes the e?ects of the condensate. To this end one integrates out the condensate ?eld Λp+1 to obtain, after some algebra [5] Sef f =

(e)

(?1)q+1 Hq+1 (Bq ) 1 + 2 (q + 1)!

e2 + m2

H q+1 (Bq ) .

(15)

Analogously, for the condensation of the g -charged brane, we obtain from (12) the following e?ective action, Sef f =

(g )

(?1)p+1 Fp+1 (Ap ) 1 + 2 (p + 1)!

g2 + ?2

F p+1 (Ap ) ,

(16)

where ? = Θ/g is the mass parameter for this condensate. In the next section we shall examine the duality versus con?nement issue. To this end we shall consider a speci?c example involving a Maxwell ?eld coupled electrically and magnetically to a monopole and an instanton while the scalar ?elds couples electrically and magnetically to the instanton and the monopole. After the condensation of the monopole we end up with two Maxwell ?elds (the massive one being the condensate) coupled topologically to each other. On the other hand after the condensation of the instanton we end up with a massless scalar ?eld coupled to a massive Kalb-Ramond potential carrying the degrees of freedom for the condensate.

III. INTERACTION ENERGY

Our aim in this Section is to calculate the interaction energy for the e?ective theories computed above, Eqs.(15) and (16), between appropriate external probe for each speci?c model. For D = 3 the e?ective actions that naturally incorporate the contents of the dual superconductor e?ects via Julia-Toulouse mechanism, for p = 1 and q = 0, are Sef f and Sef f

(inst) (pole)

=

1 ? F?ν 4

1+

△2

e2 + m2

F ?ν ? A? J ?

(17)

=

1 g2 ? ?? φ 1 + 2 2 △ + ?2

??φ ? φ J

(18)

where the external current will be chosen so as to represent the presence of the two point probes. The technique for computing the e?ective potential, which is distinguished by particular attention to gauge invariance, has been developed in [10]. However due to the absence of gauge symmetry in one end of the duality, an extension of that technique will be proposed to deal with this issue here. Other then that our notation is de?ned in the Appendix-A where the reader will also ?nd an alternative derivation of such results, see Eq.(A25). We start with

6 the analysis of the theory (17) that display the coupling of a vector ?eld to a pole. To this end consider the potential [10], V ≡ q (A0 (0) ? A0 (y)) , where the physical scalar potential is given by A0 x0 , x =

1

(19)

dλxi Ei (λx) ,

0

(20)

and i = 1, 2. This follows from the vector gauge-invariant ?eld expression [11]: ? ?

x

where the line integral is along a space-like path from the point ξ to x, on a ?xed time slice, see Eq.(A6). The gauge-invariant variables (21) commute with the sole ?rst constraint (Gauss’ law), con?rming that these ?elds are physical variables [12]. Note that Gauss’ law for the present theory reads

0 ?i Πi L = J , i where Πi L refers to the longitudinal part of Π ≡ 1 ? the electric ?eld is given by m2 ?2 ?e2

? A? (x) ≡ A? (x) + ?? ??

ξ

? dz ? A? (z )? ,

(21)

(22)

E i , and E i is the electric ?eld. For J 0 (t, x) = qδ (2) (x)

Ei = q 1 ? where G (x) =

m2 ?2

? i G (x) ,

(23)

1 K0 (M |x|) ; M 2 ≡ m2 + e2 , 2π

(24)

is the Green function for the Proca operator in D = 3. As a consequence, Eq.(20) becomes A0 (t, x) = q 1 ? m2 ?2 G (x) , (25)

after subtraction of self-energy terms. Our next task is the computation of the second term on the right-hand side of Eq. (25). We will make use of the Green function, de?ned in Eq.(A17). Using this in (25) we then obtain G 1 =? (I1 + I2 ) , 2 ? 8πM 2 where the I1 and I2 terms are given by

∞

(26)

I1 =

?∞

dt √

1 exp (iM rt) 1 , 1 + t2 t2 1 ? i/M rt

(27)

and

∞

I1 =

?∞

1 exp (?iM rt) 1 dt √ , 2 t 1 + t 2 1 + i/M rt

(28)

here |x| ≡ r. The integrals (27) and (28) have been explicitly computed in Appendix-A. As a consequence, Eq. (26) reduces to r G = . 2 ? 4M (29)

7 Finally, making use of (29) in (25), the potential for a pair of point-like opposite charges q located at 0 and L, becomes V ≡ q (A0 (0) ? A0 (L)) = ? q 2 m2 q2 K0 (M L) + L, 2π 4M (30)

where |L| ≡ L. This potential displays the conventional screening part, encoded in the Bessel function, and the linear con?ning potential. As expected, the con?nement disappears in the dilute phase (m → 0). Next we perform the analysis of the theory (18) that display the coupling of a scalar ?eld to an instanton. From (18) we obtain the following equation of motion + g 2 + ?2 + ?2 φ=J . (31) φ = ??2 φ. It implies that (31) (32)

Next, we restrict ourselves to static scalar ?elds which allowed us to replace becomes, ?2 ? M2 J φ=? 2 2 2 ? ?? ? where now M2 = ?2 + g 2 . From this we see that φ= 1? ?2 ?2 ? ?2 J ? M2 .

(33)

As before we note that for J (t, x) = qδ (2) (x), the scalar ?eld is given by φ=q 1? ?2 ?2 G (x) (34)

where G (x) is the massive Green function de?ned before, Eq.(24), with M → M. It must now be observed that Eq.(34) is identical to Eq.(25), leading to the same e?ective potential also for the scalar ?eld case, con?rming that the claimed duality between the two models persists after the condensation of the topological defects.

IV. FINAL REMARKS

We have used the con?nement as a criterium to study duality for a pair of antisymmetric tensors coupled to topological defects that eventually condense. To this end we have computed the e?ective static potential for a general e?ective theory in the condensed phase using the Quevedo- Trugenberger formalism. This result was successfully used as a testing grounding for duality using the interaction energy between two point-like probes but, for technical reasons, we had to settle for the speci?c case of D = 3 only. According to the Quevedo-Trugenberger phenomenology, the condensation mechanism for a couple of massless antisymmetric tensors is responsible for the appearance of mass and the jump of rank in the magnetic sector while the electric sector becomes a BF coupling with the condensate. The condensate absorbs and replaces one of the tensors and becomes the new massive propagating mode but does not couple directly to the probe charges. The e?ects of the condensation are however felt through the BF coupling with the remaining massless tensor. It is therefore not surprising that they become manifest in the interaction energy for the e?ective theory. Our results show that in both sides of the duality the interaction energy in fact contains a linear con?ning term. This is an important result showing that the e?ective potential is a key tool to corroborate the existence of duality, which, otherwise are only suggested by other, very formal approaches. Extension of this approach to check for duality in higher dimensions using the con?nement criterium is presently under investigation by the authors.

V. ACKNOWLEDGMENTS

One of us (CW) would like to thank the Physics Department of the Universidad T? ecnica F. Santa Mar? ?a for the invitation and hospitality during the earlier stages of this work. This work was supported in part by Fondecyt (Chile) Grant 1050546 (P.G.) and by CNPq/Pronex and CAPES/Procad (Brazil).

8

APPENDIX A: COMPUTATION OF THE CONFINING POTENTIAL

Our aim in this Appendix is to recover the con?ning potential for the e?ective theory (17) computed between external probe sources. To do this, we will compute the expectation value of the energy operator H in the physical state |Φ describing the sources, which we will denote by H Φ . Our starting point is the e?ective Lagrangian Eq. (17): 1 Lef f = ? F?ν 4 1+ m2 + e2 F ?ν ? A0 J 0 , (A1)

where J 0 is an external current. Once this is done, the canonical quantization of this theory from the Hamiltonian point of view follows straightfor2 0? with the only non-vanishing canonical Poisson brackets wardly. The canonical momenta read Π? = ? 1 + m +e2 F being

ν {A? (t, x) , Πν (t, y)} = δ? δ (x ? y ) .

(A2)

m2 +e2

Since Π0 vanishes we have the usual primary constraint Π0 = 0, and Πi = 1 + is thus HC = 1 d3 x ? Πi 1 + 2 m2 + e2

?1

F i0 . The canonical Hamiltonian

1 Πi + Πi ?i A0 + Fij 4

1+

m2 + e2

F ij + A0 J 0

.

(A3)

Time conservation of the primary constraint Π0 leads to the secondary Gauss-law constraint Γ1 (x) ≡ ?i Πi ? J 0 = 0. (A4)

The preservation of Γ1 for all times does not give rise to any further constraints. The theory is thus seen to possess only two constraints, which are ?rst class, therefore the theory described by (A1) is a gauge-invariant one. The extended Hamiltonian that generates translations in time then reads H = HC + d2 x (c0 (x) Π0 (x) + c1 (x) Γ1 (x)), where c0 (x) ˙ 0 (x) = [A0 (x) , H ] = c0 (x), and c1 (x) are the Lagrange multiplier ?elds. Moreover, it is straightforward to see that A 0 0 0 which is an arbitrary function. Since Π = 0 always, neither A nor Π are of interest in describing the system and may be discarded from the theory. Then, the Hamiltonian takes the form H= 1 d2 x ? Πi 1 + 2 m2 + e2

?1

1 Πi + Fij 1 + 4

m2 + e2

F ij + c (x) ?i Πi ? J 0

,

(A5)

where c(x) = c1 (x) ? A0 (x). The quantization of the theory requires the removal of nonphysical variables, which is done by imposing a gauge condition such that the full set of constraints becomes second class. A convenient choice is found to be [10]

1

Γ2 (x) ≡

Cξx

dz ν Aν (z ) ≡

dλxi Ai (λx) = 0,

0

(A6)

where λ (0 ≤ λ ≤ 1) is the parameter describing the spacelike straight path xi = ξ i + λ (x ? ξ )i , and ξ is a ?xed point (reference point). There is no essential loss of generality if we restrict our considerations to ξ i = 0. In this case, the only non-vanishing equal-time Dirac bracket is

1

Ai (x) , Π (y )

j

?

=

(2) δj iδ

(x ? y ) ?

x ?i 0

dλxj δ (2) (λx ? y) .

(A7)

In passing we recall that the transition to quantum theory is made by the replacement of the Dirac brackets by the operator commutation relations according to {A, B } → (?i) [A, B ] .

?

(A8)

9 We now turn to the problem of obtaining the interaction energy between point-like sources in the model under consideration. The state |Φ representing the sources is obtained by operating over the vacuum with creation/annihilation operators. We want to stress that, by construction, such states are gauge invariant. In the case at hand we consider the gauge-invariant stringy Ψ (y) Ψ (y′ ) , where a fermion is localized at y′ and an anti-fermion at y as follows [12], ? y ? |Φ ≡ Ψ (y) Ψ (y′) = ψ (y) exp ?iq

y′

dz i Ai (z )? ψ (y′) |0 ,

(A9)

where |0 is the physical vacuum state and the line integral appearing in the above expression is along a space-like path starting at y′ and ending y, on a ?xed time slice. It is worth noting here that the strings between fermions have been introduced in order to have a gauge-invariant function |Φ . In other terms, each of these states represents a fermion-antifermion pair surrounded by a cloud of gauge ?elds su?cient to maintain gauge invariance. As we have already indicated, the fermions are taken to be in?nitely massive (static). From our above discussion, we see that H Φ reads H

Φ

= Φ|

1 d2 x ? Πi 1 + 2

m2 + e2

?1

1 Πi + Fij 1 + 4

m2 + e2

F ij

|Φ .

(A10)

Consequently, we can write Eq.(A10) as H = Φ| 1 m2 d2 x ? Πi 1 ? 2 2 ? ? e2

?1

Φ

Πi

|Φ ,

(A11)

where, in this static case, = ??2 . Observe that when m = 0 we obtain the pure Maxwell theory, as already mentioned. From now on we will suppose m = 0. Next, from our above Hamiltonian analysis, we note that

y′ ′ ′

Πi (x) Ψ (y) Ψ (y ) = Ψ (y) Ψ (y ) Πi (x) |0 + q

y

dzi δ (2) (z ? x) |Φ .

(A12)

As a consequence, Eq.(A11) becomes H where H

0 Φ

= H

0

+ V (1) + V (2) ,

(A13)

= 0| H |0 . The V (1) and V (2) terms are given by: V (1) = ? q2 2

y′

d2 x

y

′ (2) dzi δ (x ? z ′ )

1 ?2 2 x ?2 ? M x

y′ y

dz i δ (2) (x ? z ) ,

(A14)

and V (2) = q 2 m2 2

y′

d2 x

y

′ (2) dzi δ (x ? z ′ )

1 ?2 ? M2 x

y′ y

dz i δ (2) (x ? z ) ,

(A15)

′ where M 2 ≡ m2 + e2 and the integrals over z i and zi are zero except on the contour of integration. (1) The V term may look peculiar, but it is just the familiar Bessel interaction plus self-energy terms. In e?ect, expression (A14) can also be written as

V where G is the Green function

(1)

q2 = 2

y′ y

′ z′ dzi ?i

y′ i dz i ?z G (z′ , z) , y

(A16)

G(z′ , z) =

1 K0 (M |z′ ? z|) . 2π

(A17)

10

′ Employing Eq.(A17) and remembering that the integrals over z i and zi are zero except on the contour of integration, expression (A16) reduces to the familiar Bessel interaction after subtracting the self-energy terms, that is,

V (1) = ?

q2 K0 (M |y ? y′ |) . 2π

(A18)

We now turn our attention to the calculation of the V (2) term, which is given by V (2) = q 2 m2 2

y′ y′

dz ′i

y y

dz i G(z′ , z).

(A19)

It is appropriate to observe here that the above term is similar to the one found for the system consisting of a gauge ?eld interacting with an external background[13]. Notwithstanding, in order to put our discussion into context it is useful to summarize the relevant aspects of the calculation described previously [13]. In e?ect, as was explained in Ref. [13], by using the integral representation of the Bessel function

∞ ∞

K0 (x) =

0

cos(x sinh t)dt =

0

cos(xt) √ dt, t2 + 1

(A20)

where x > 0, expression (A19) can also be written as V (2) = q 2 m2 2πM 2

∞

dt

0

1 1 √ (1 ? cos (M Lt)) , t2 t2 + 1

(A21)

where L ≡ |y ? y′ |. Now let us calculate integral (A21). For this purpose we introduce a new auxiliary parameter ε by making in the denominator of integral (A21) the substitution t2 → t2 + ε2 . Thus it follows that V

(2)

≡ lim V

ε→0

(2)

q 2 m2 = lim ε→0 2πM 2

∞

0

dt 1 √ (1 ? cos (M Lt)) . t2 + ε 2 t2 + 1

(A22)

A direct computation on the t-complex plane yields V (2) = q 2 m2 4M 2 1 ? e?MLε ε 1 √ . 1 ? ε2 (A23)

Taking the limit ε → 0, expression (A23) then becomes V (2) = q 2 m2 |y ? y′ |. 4M (A24)

From Eqs.(A18) and (A24), the corresponding static potential for two opposite charges located at y and y′ may be written as V (L) = ? where L ≡ |y ? y′ |. q 2 m2 q2 K0 (M L) + L, 2π 4M (A25)

[1] [2] [3] [4] [5] [6]

F. Quevedo, C. A. Trugenberger, Nucl.Phys.B501 143 (1997). B. Julia and G. Toulouse, J. Physique Lett. 40 L395 (1979). A.M. Polyakov, Nucl. Phys. B180, 429 (1977). K.-I. Kondo, Phys. Rev. D58, 085013 (1998). P. Gaete and C. Wotzasek, Phys. Lett. B601, 108 (2004). [7] F.R. Harvey and H.B. Lawson Jr., A theory of characteristic currents associated with a singular connection, Asterisque 213 (1993), Soci` et` e Math` ematique de France. [7] Z. F. Ezawa and A. Iwazaki, Prog. Theor. Phys. 66, 275 (1981).

11

[8] H. Kleinert, Phys. Lett. B 246, 127 (1990). [9] H. Kleinert, “Gauge Fields In Condensed Matter. Vol. 1: Super?ow And Vortex Lines, Disorder Fields, Phase Transitions,” World Scienti?c Publishing Co., Singapore 1989. [10] P. Gaete, Phys. Rev. D59, 127702 (1999), 355 (1997); Phys. Lett. B515, 382 (2001); Phys. Lett. B582, 270 (2004). [11] P. Gaete, Z. Phys. C76, 355 (1997). [12] P. A. M. Dirac, The Principles of Quantum Mechanics (Oxford University Press, Oxford, 1958); Can. J. Phys. 33, 650 (1955). [13] P. Gaete, “On external backgrounds and linear potential in three dimensions”, hep-th/0501136. [14] P. Orland, Nucl. Phys. B205 107 (1982). [15] A.M. Polyakov, “Gauge Fields and Strings”, Harwood Academic Publishers (1987).

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