Confinement and Screening in Tachyonic Matter

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provided by Open Access Repository Eur. Phys. J. C (2014) 74:3202 DOI 10.1140/epjc/s10052-014-3202-y

Regular Article - Theoretical Physics

Conﬁnement and screening in tachyonic matter

F. A. Brito 1,a,M.L.F.Freire2, W. Seraﬁm1,3 1 Departamento de Física, Universidade Federal de Campina Grande, 58109-970 Campina Grande, Paraíba, Brazil 2 Departamento de Física, Universidade Estadual da Paraíba, 58109-753 Campina Grande, Paraíba, Brazil 3 Instituto de Física, Universidade Federal de Alagoas, 57072-970 Maceió, Alagoas, Brazil

Received: 26 August 2014 / Accepted: 19 November 2014 © The Author(s) 2014. This article is published with open access at Springerlink.com

Abstract In this paper we consider confinement and way in which hadronic matter lives. In three spatial dimen- screening of the electric field. We study the behavior of sions this effect is represented by Coulomb and confinement a static electric field coupled to a dielectric function with potentials describing the potential between quark pairs. Nor- v ( ) =−a + the intent of obtaining an electrical confinement similar to mally the potential of Cornell [1], c r r br, is used, what happens with the field of gluons that bind quarks in where a and b are positive constants, and r is the distance hadronic matter. For this we use the phenomenon of ‘anti- between the heavy quarks. In QED (Quantum Electrodynam- screening’ in a medium with exotic dielectric. We show that ics), the effective electrical charge increases when the dis- tachyon matter behaves like in an exotic way whose associ- tance r between a pair of electron–anti-electron decreases. ated dielectric function modifies the Maxwell equations and On the other hand, in QCD there is an effect that creates a affects the fields which results in confining and Coulombian- color charge which decreases as the distance between a pair like potentials in three spatial dimensions. We note that the of quark and anti-quark decreases. Thus, in this sense, QCD confining regime coincides with the tachyon condensation, develops phenomena effectively opposite to QED. So it is which resembles the effect of confinement due to the conden- natural to try to understand the QCD phenomena in terms sation of magnetic monopoles. The Coulombian-like regime of QED in an “exotic” medium such that it creates an anti- is developed at large distance, which is associated with a screening effect at some regime near the QCD scale. In this screening phase. sense we assume that the electric field is now modified by ( ) E(r) ≡ q r ≡ q a “dielectric function”. Consider r2 G(r)r2 where E(r) represents the field due to an electric charge 1 Introduction q immersed in a polarizable medium playing the role of a “color” charge coupled to a dielectric function G(r) where r The confinement arises naturally in Quantum Chromody- is the distance of the screening (or anti-screening). The action namics (QCD) where the fields of gluons and quarks appear of screening in the QED is expressed according to the effec- in a confined state at low energies. Similarly to the electric tive electrical charge i.e., q(r d) ≤ 1orG(r d) ≥ 1 field, we associate the gluon field charges to “color charges”. and q(r d)>1orG(r d)<1, d being the typi- In our studies we show that in general the confining phe- cal size of the screening produced by the polarization of the nomenon can be considered as an effect of a dielectric on the molecules. On the other hand, in QCD the phenomenon of fields and charges with the opposite effect of the screening, anti-screening manifests with color charge q(r R) 1 called “anti-screening”. Throughout the paper we shall mean or G(r R) 1 and q(r < R) → 1orG(r < R) → 1 by anti-screening the way how an electric field behaves in a and R is the radius of the anti-screening which is the typi- medium with asymptotic behavior that is opposite to the well- cal size of hadrons. Since we are interested in confinement known asymptotic behavior in a usual dielectric medium. The we will focus on the latter case where the dielectric func- hadron matter in its confining phase develops this behav- tion G(r) is such that the anti-screening provides the QCD ior. The tachyonic field is presented here as a good way to color confinement in a pair of heavy quark. In spite of this, develop both anti-screening (the confinement regime) and in our present study, a dynamical G(r) comes about in the screening behavior (the Coulomb-potential behavior at suf- tachyonic matter, such that for sufficiently large distance the ficiently large interquark separation). This seems to be the anti-screening breaks down and a screening phase associated with hadronization (or light quarks pair creation) takes place. a e-mail: [email protected] 123 3202 Page 2 of 8 Eur. Phys. J. C (2014) 74:3202

Indeed the main proposal in the present study is to define a 2 Maxwell’s equations modified by a dielectric function potential model that may develop appropriate potentials for the confining and screening regimes. Whereas the Cornell In this section we apply the theory of electromagnetism in potential is good for describing heavy quarks, for light quarks an exotic dielectric medium to describe the phenomenon of a screening regime occurs for a sufficiently large interquark electric confinement. First we write the Maxwell Lagrangian separation and another appropriate potential is necessary. In in the vacuum without sources our setup the tachyon matter properly develops the desirable 1 μν confining and screening behaviors. Note that we will con- L =− Fμν F . (1) tinue using Abelian gauge fields as in QED, but now G(r) is 4 properly obtained in order to provide both confinement and As is well known the equations of motion for the electromag- screening. We will consider only Abelian projections when netic field are given by these fields are embedded in a color dielectric medium [2,3]. Indeed it is already known that as a result of QCD lattice stud- μν ∂μ F = 0. (2) ies it has been shown that the Abelian part of the string tension accounts for 92 % of the confinement part of the static lattice potential. Thus, it suffices to consider only the Abelian (linear One should note that even in the absence of sources the equa- part) of the non-Abelian strength field [4,5]. These facts are tions of motion for the electromagnetic field produce spher- useful to study the confinement of quarks and gluons inside ically symmetric static vacuum solutions (the Coulomb field the hadrons using a phenomenological effective field theory due to a point charge) [14] as we shall see below. for QCD [1,6–8]. There are different ways to confine the For electromagnetic fields immersed in a dielectric (φ) electric field in a dielectric medium with different functions medium characterized by the dielectric function G , where φ( ) G(r) to obtain confinement. In our case the function G(r) is r is a scalar field that governs the dynamics of the associated with tachyon condensation [9,10]. In general, the medium, we have the following Lagrangian: behavior of the dielectric function G with respect to r can be 1 μν governed by a scalar field φ(r) which describes the dynamics L =− G(φ)Fμν F . (3) of the dielectric medium in tachyon matter. For this, we use 4 a Lagrangian to describe both the dynamics of gauge and of The equations of motion (2) are now described by the scalar fields associated with the tachyon dynamics. The motivation to use this approach is twofold. First, as we shall μν ∂μ G(φ)F = 0, (4) see later, this a very natural way to obtain a phenomenological effective field theory for QCD since the aforemen- where μ = 0, 1, 2, 3. Developing the component ν = 0, we tioned dielectric function is automatically identified with the simply have tachyon potential. As a consequence, this approach may also address the issues of phenomenological aspects of confine- ∇·[G(φ)E] = 0. (5) ment and screening in string theory. Second, since in this setup the electric confinement is related to tachyon conden- We shall be neglecting the magnetic field throughout the sation, it may bring about new insights on confining super- paper. This is because the electric field is sufficient for our symmetric gauge theories such as the Seiberg–Witten theory analysis. [11–13], which is based on electric–magnetic duality and Starting from Eq. (5) we find the electric field E coupled to develops magnetic monopole condensation. We shall con- the dielectric function G(φ). Now working on spherical coor- sider examples of confinement of the electric field in one and dinates and assuming that E(r) and φ(r) are only functions three spatial dimensions. of r and as a consequence G(φ) follows the same condition, The paper is organized as follows. In Sect. 2 we briefly we have the following form: review the theory of electromagnetism in a dynamical dielectric medium. In Sect. 3 we introduce the tachyon Lagrangian 1 ∂ and potentials. We discuss the solutions obtained in one and ∇·[G(φ)E] = (r 2G(φ)E ) = 0. (6) r 2 ∂r r three spatial dimensions. In the latter case we find analyti- cal potentials under certain conditions and a numeric solu- By using this equation we can obtain the vacuum solutions tion connecting the confining behavior at small distance to mentioned above. Thus, now we integrate the differential screening behavior at large distance. We explore the physics equation to find given by the confining and screening potentials. We show how the tachyon condensation is directly related to the con- λ E = . (7) fining phase. In Sect. 4 we present our final considerations. r r 2G(φ) 123 Eur. Phys. J. C (2014) 74:3202 Page 3 of 8 3202

Now it is easy to interpret the constant of integration λ = or be self-interacting at higher orders to form the tachyon con- q/4πε0 to write the Coulomb electric field modified by a densation [9,10,15]. In our study we show that the electric dielectric function G(φ), confinement via tachyon condensation can occur in the same way as the confinement of colorful particles, such as quarks q E = , and gluons through condensation of monopoles [11–13]. 2 (8) 4πε0r G(φ) 3.1 Tachyon Lagrangian with electromagnetic fields where E =|E|=Er . Therefore, we observe that the dielectric function coupled to the electric field E changes its mag- nitude as a function of the radial position r. For simplicity, we first consider the fields depending only on Let us consider a dielectric function G(φ) as a function of the spatial component x, i.e. a dynamical field φ, according to the Lagrangian [16,17] φ = φ(x), Aμ = Aμ(x). (16) 1 μν 1 μ L =− G(φ)Fμν F + ∂μφ∂ φ − V (φ). (9) 4 2 Thus, the equations of motion discussed in the previous section are now given by The behavior of the dielectric function G(φ) will be pre- d sented as a consequence of the solutions of the equations of [G(φ)E] = 0, (17) motion obtained by the above Lagrangian. The equations of dx d2φ ∂V 1 ∂G motion for the electromagnetic field Aμ and the scalar field − + − E2 = 0, (18) φ are given explicitly by the following differential equations: dx2 ∂φ 2 ∂φ μν 01 = ∂μ G(φ)F = 0, (10) where we use the fact that F E. Note that Eq. (17)is a one-dimensional version of Eq. (6). Integrating (17), we μ ∂V (φ) 1 ∂G(φ) μν ∂μ∂ φ + + Fμν F = 0. (11) have ∂φ 4 ∂φ q Then the equations of motion for the dielectric medium and G(φ)E = q ⇒ E = , (19) electric field in spherical coordinates are G(φ) 1 d (r 2G(φ)E) = 0, (12) so that substituting it into Eq. (18) we find r 2 dr 1 d dφ ∂V 1 ∂G 2φ ∂ (φ) ∂ (φ) 2 r 2 = − E2 . (13) d V 1 G q 2 ∂φ ∂φ − + − = 0, (20) r dr dr 2 dx2 ∂φ 2 ∂φ G(φ)2 Based on the previous discussion it is easy to show that the solution of the first equation for the electric field is that given or simply in (8). ∂V (φ) 1 q2 ∂G(φ) To find a confining regime everywhere the dielectric func- −φ + − = 0. (21) tionin(8) must have the following asymptotic behavior: ∂φ 2 G(φ)2 ∂φ

G(φ(r)) = 0asr →∞, (14) We note that in the above equation we have, in principle, G(φ(r)) = 1asr → 0. (15) the potential V (φ) and the function G(φ). However, we can restrict these choices considering G(φ) = V (φ). As we will Particularly, for G(φ(∞)) ∼ 1/r 2,fromEq.(8) we find see below this choice is legitimate when we are working E ≡ const. This uniform electric field behavior agrees with with a Lagrangian that describes the dynamics of tachyons confinement. represented by the scalar field φ. As is well known from string theory, the dynamics of a tachyonic field T (x) coupled with the electric field E(x) is 3 Tachyon condensation and electric confinement given by [9,10,15] −1L =− ( ) − 2 + 01 In this section we discuss the relationship between the phe- e V T 1 T F01 F nomena of tachyon condensation and the confinement of the 1 =−V (T ) 1 − (T 2 + F F01) +··· electric field. When we speak of “tachyon” we refer to parti- 2 01 cles that are faster than light and are associated with instabil- 1 2 1 01 ities. As magnetic monopoles they have never been observed =−V (T ) + V (T )(T ) − V (T )F01 F ) +··· 2 2 isolated in nature, although, specially from the superstring 1 1 =−V (φ) + φ 2 − V (φ)F F01 +··· , (22) point of view, they may always be interacting with other fields 2 2 01 123 3202 Page 4 of 8 Eur. Phys. J. C (2014) 74:3202 √ where e = |g| in a general spacetime. The power expansion for q2 = 2.Thesolutionof(24) for this kind of potential is is justified in slow varying tachyon fields, which are suitable given by to describe tachyon matter [9,10]. This derivation remains valid in 3+1 dimensions for a φ dependence with purely radial 2 φ(x) = arcsin [tan(x)] . (28) coordinate r which can be identified with x. α In the last equation of (22) we use the fact that −αφ Substituting this solution into the potential V = e ,we 2 ∂φ 1 find the dielectric function V (T (φ)) = ⇒ V (T )(T 2) ∂T 2 1 2 ( ) = ( ) = . 1 ∂φ ∂T 1 G x V x (29) = = φ 2, (23) (tan(x)+|sec(x)|)2 2 ∂T ∂x 2 Therefore, the model reproduces well the behavior of the where φ = f (T ),orT = f −1(φ). Note that comparing (22) electric field confinement with (9) we find the equality G = V is legitimate, so that we can write q E(x) = . (30) 2 G(x) ∂V q 1 ∂V −φ + − = 0, ∂φ 2 V 2 ∂φ π Note that in the limit x →± the electric field diverges 2 −1 2 ∂V q −∂V −φ + − = 0, (“superconfining” system). On the other hand, in the limit ∂φ 2 ∂φ x → 0, the electric field tends to a constant value E = q 2 −1 ∂V q ∂V (“confining” regime). φ − − = 0, (24) ∂φ 2 ∂φ 2 −1 3.1.2 Confinement and Coulomb potential for the electric ∂ q V φ + −V − = 0, field in three dimensions ∂φ 2 ˜ ∂V φ − = 0. In three dimensions the extension of Eq. (18) (in the absence ∂φ of magnetic fields) to radial symmetry is direct and is given Thus, Eq. (24) is obtained from Eq. (21) using G=V. Note by (13), which we recast into the form that now we are only left with the equation for the scalar field 1 ∂ ∂φ ∂V 1 ∂G − r 2 + − E2 = 0. (31) ˜ r 2 ∂r ∂r ∂φ 2 ∂φ ∂V φ = (25) ∂φ Recalling that the solution for the electric field is given by

with the potential q E(r) = 2 (32) 4π 0G(φ)r q2 1 V˜ = V + . (26) 2 V and substituting into (31) we find 3.1.1 Confinement potential for the electric field in a 1 ∂ ∂φ ∂V 1 ∂G q 2 − r 2 + − = 0. spatial dimension 2 2 r ∂r ∂r ∂φ 2 ∂φ 4π 0G(φ)r (33) For a tachyon Lagrangian in Eq. (22) expanded polynomi- (φ) ally, it is explicitly clear that the dielectric function G can Now considering the fact that G(φ)=V (φ) and be equal to the potential V (φ). Thus, our dielectric function is naturally identified in the context of tachyon theory. Poten- q λ = , (34) tials describing the tachyon condensation in string theory are 4π 0 of the type that are zero in the vacuum i.e., when φ →±φvac such that V (φ →±φvac) = 0. we have Thus, as an example, we shall adopt the most commonly used tachyon potential of the exponential form 1 ∂ ∂φ ∂V (φ) λ2 ∂V (φ) 1 − r 2 + − = 0. r 2 ∂r ∂r ∂φ 2 ∂φ r 4V (φ)2 −αφ V = e , so that V˜ = 2 cosh(αφ) (27) (35) 123 Eur. Phys. J. C (2014) 74:3202 Page 5 of 8 3202

Assuming again the exponential tachyon potential Substituting (43)into(32), with G(φ(r)) ≡ G(r),wehave −αφ(r) V (φ(r)) = e we have g Nc − 1 2 αφ(r) E(r) = (44) 1 ∂ ∂φ(r) ∂ −αφ( ) λ e π ( ) 2 r 2 = e r − . (36) 4 G r r Nc r 2 ∂r ∂r ∂φ(r) 2 r 4 where G(φ(r)) = V (φ(r)) = exp(−αφ(r)); φ(r) is the Since φ depends only on r, we can write our equation in conﬁning solution given in Eq. (40). Now using this solution terms of ordinary derivatives to simply obtain α =−1 and fφ we arrive at

2 2 d φ(r) 2 dφ(r) −αφ( ) λ αφ( ) 1 rφ + =−αe r + αe r . (37) G(φ(r)) = V (φ(r)) = exp 2ln , (45) dr2 r dr 2 r 4 r

For αφ(r) sufficiently large (which will easily be satisfied or simply / for a large rate r rφ—see below), we can compare our result rφ 2 with the results of Refs. [18,19] to the confinement of quarks G(r) = . (46) and gluons with Nc colors, i.e., r 2φ( ) φ( ) d r + 2 d r Substituting Eq. (46) into Eq. (44), we find the electric field 2 dr r dr modified by a “color dielectric function” G developing a g2 1 φ(r) 1 ‘confining medium’ which means a confining phase given =− 1 − exp − , (38) 64π 2 fφ N fφ r 4 by c because the first term on the right side of Eq. (37) becomes g 1 E(r) = 1 − . (47) negligible. This now allows us to identify our electric charge 2 4πrφ Nc q in terms of the color charge of quarks and gluons g as follows: This solution was first obtained in [18,19] in theories with dilatonic solutions [14]. The electric field is constant and 2 λ2 − g − 1 = α . therefore implies that the lines of force keep together all the 2 1 (39) 64π fφ Nc 2 time, which features a flux tube, a confinement due to an “anti-dielectric medium” developed by the hadronic matter Recall that from Eq. (34) that we have a relation between λ that here is represented by ‘tachyonic matter’. See discussion and q. According to [18,19], the confining solution is given below. by • Confining potential

rφ Integrating (47) we find the confining potential φ(r) = 2 fφ ln . (40) r g 1 v (r) = 1 − r + c, (48) Manipulating Eq. (39)wehave c 2 4πrφ Nc 1 λ2 64π 2 N which is a linear confining potential, − = α c . (41) fφ 2 g2 N − 1 c vc(r) = σr + c, (49) 1 where c is an integration constant and σ is the QCD string Now identifying α =− , we find fφ tension (which in general also depends on r), which breaks at some scale of QCD favoring the production of pairs of

mesons (hadronization). As we will see, after this scale g N − 1 λ = c , appears the regime in which a Coulomb-like type potential π (42) 4 Nc approaches a constant. Interestingly, our numerical solution √ to this problem exhibits this behavior. where we have redefined g → g/ 2. Using Eq. (34) we can The electric potential energy between two punctual now write charges is simply given in terms of the potential and the charge, Nc − 1 q = 0g . (43) Nc uc(r) = qvc(r), (50) 123 3202 Page 6 of 8 Eur. Phys. J. C (2014) 74:3202 that is, Expressing the charge q in terms of g as defined above, we found ( ) = 2 r − 1 uc r 0g 2 1 (51) 2 1 1 4πrφ Nc ucl (r) =− 0g 1 − (57) Nc 4π(r + rφ) where for simplicity we have disregarded the integration con- where for simplicity we also have here disregarded the inte- stant and in Eq. (51), Eq. (43) has been used. So we have a gration constant and in Eq. (57), Eq. (43) has been used. The linear confinement type that describes the confinement of Coulomb-type potential energy for the electric field mimics quarks and gluons. Note that our treatment is mainly based the energy of quarks and gluons in the limit of high ener- on an Abelian gauge theory and therefore does not have all gies as r → 0. Note, however, that the potential is regular the degrees of freedom adequate to describe the colors of at r = 0 unlike a usual Coulomb potential. The confining gluons. However, as the very expression shows, in the limit and Coulomb-like potentials are depicted in Fig. 1. Note that →∞ Nc the electric charge is identical to the color charge. the numerical solution vn (depicted in red) smoothly connects This is the ’t Hooft limit (planar limit) where several Feyn- these two regimes in small and large distances. Here we solve man diagrams disappear and make the non-Abelian theory numerically Eq. (37)forα = 0.01 and λ = 1. With this, we 2 −1 simpler. Thus we can understand our approach as an Abelian use g/4π = 1, Nc 1, rφ = (0.05) , and c˜ = 20 in poten- approach to approximately describe the non-Abelian theory tials vc and vcl . As expected, at small distances the potential of QCD—the same was considered in [18,19]. approaches the confining solution vc (consider c = 0)— • Coulombian potential Fig. 1-top–and at large distances the potential approaches a constant given by vcl (i.e., c˜ = 2mq )—Fig. 1-bottom. This Let us find the Coulomb potential starting from the elec- is the regime where hadronization processes take place via tric field of Eq. (44), based on the non-confining solution of the production of light mesons, i.e., for small masses mq . Eq. (38) for the scalar field [18,19] Finally, we shall comment on the relation between the confinement and the tachyon condensation. We will also r + rφ comment on the connection in between the confinement and φ(r) = 2 fφ ln , (52) r deconfinement phases at small and large distances presented numerically. and we have We have found the linear confining by assuming αφ sufficiently large (i.e., r rφ)inEq.(37) that leads to Eq. (38), 2 r + rφ which for the tachyon potential G(φ(r)) = V (φ(r)) = G(r) = . (53) −αφ r V = e (58)

Recall that fφ =−1/α is used in Eq. (53). Substituting this presents the explicit confining solution (40). When we sub- result into Eq. (44) we find stitute this solution into the tachyon potential we find

−αφ( ) rφ 2 V (r) = e r = . (59) g 1 r E(r) = 1 − , (54) 2 We note from (59) that the tachyon condensation, i.e., V → 4π r + rφ Nc 0, happens when r/rφ 1. This is automatically satisfied by the hypothesis that r rφ. which is a Coulomb potential regularized at distances rφ. On the other hand, from the numerical analysis, since the Integrating the field as a function of r we obtain the dielectric function G(r) ≡ V (r),Fig.2 suggests that the ‘Coulomb’ potential as follows: confinement/condensation condition is again easily satisfied 2 −1 at r 100 since we have chosen rφ = (0.05) . We can therefore conclude that the electric confinement is associated v ( ) =− g − 1 1 +˜, cl r 1 c (55) with tachyon condensation. From the ’t Hooft–Mandelstam 4π Nc r + rφ and Seiberg–Witten [11–13] theories, which deal with the duality between the phenomena of confinement and super- ˜ where c is an integration constant. The potential energy is conductivity, we know that the electric confinement can also given by be associated with the condensation of magnetic monopoles. However, both isolated tachyons and magnetic monopoles ucl (r) = qvcl (r). (56) have never been observed. 123 Eur. Phys. J. C (2014) 74:3202 Page 7 of 8 3202

Fig. 2 The numerical solution for the dielectric function: G(r) develops a decreasing behavior until r 100 showing a conﬁning (anti- screening) phase and then it increases with r developing a deconﬁning (screening) phase

Our numerical solution shows that there is a QCD-like string breaking such that the confining is linear only in a certain range, Fig. 1 (top), and then the potential becomes constant, which is typical of a hadronization phase. As a last remark one should note that a large typical hadron size rφ ∼ 1/mq , which means the limit mq → 0 (light mesons), favors a regime in which a screening phenomenon after the confining phase comes into play, because the QCD string tension given in (48) becomes small. Then it is expected to be broken more easily than for heavy mesons. Thus, for light quarks QCD is never truly confining [20].

Fig. 1 The numerical solution vn smoothly connects the two regimes 4 Conclusions in small (vc)andlarge(vcl ) distances. (top) Potentials at small distances: v approaches v at a linear confinement phase. (bottom) Potentials for n c In this study we found a Coulomb-like and confinement large distances: vn approaches vcl at a deconfinement (or hadronization) phase potential to the electric field that resembles the ones obtained for the quarks and gluons. Our Abelian approach can be understood as an approximation of the non-Abelian QCD Now we focus our attention to the deconfining regime. theory. The confining/deconfining regime of the electric field Following a similar analysis and hypothesis using the non- was obtained considering a dielectric medium whose tachyon confining solution (52) we found dielectric function describes an anti-screening/screening behavior at small and large distance. This medium is regarded 2 r + rφ as a tachyon matter described in terms of a tachyon potential V (r) = → 1. (60) r that vanishes at the minima, that is, in the tachyon condensation. Tachyon condensation was shown to be related to the Again, since G(r) ≡ V (r),Fig.2 suggests that the non- electric confinement. Thus the tachyon condensation plays confining regime at sufficiently large distance does not cor- the same role as the condensation of monopoles in the electric respond to tachyon condensation. Instead, the full numeric confinement phenomenon. The latter corresponds to duality solution depicted in Fig. 1 indeed allows us to identify this between the confinement and the phenomenon of supercon- regime with a hadronization phase. ductivity. 123 3202 Page 8 of 8 Eur. Phys. J. C (2014) 74:3202

Acknowledgments Wewould like to thank CNPq, CAPES, PROCAD- 7. W.A. Bardeen, M.S. Chanowitz, S.D. Drell, M. Weinstein, T.-M. CAPES for partial financial support. We also thank Tiago Mariz and Yan, Phys. Rev. D 11, 1094 (1975) Emanuel Cunha for discussions. 8. R. Friedberg, T.D. Lee, Phys. Rev. D 15, 1964 (1997) 9. A. Sen, JHEP 0207, 065 (2002). arXiv:hep-th/0203265 Open Access This article is distributed under the terms of the Creative 10. A. Sen, JHEP 0204, 048 (2002). arXiv:hep-th/0203211 Commons Attribution License which permits any use, distribution, and 11. S. Mandelstam, Phys. Rep. 23C, 145 (1976) reproduction in any medium, provided the original author(s) and the 12. G. ’t Hooft, in Proceed. of Euro. Phys. Soc. 1975, ed. A. Zichichi source are credited. 13. N. Seiberg, E. Witten, Nucl. Phys. B 426, 19 (1994) Funded by SCOAP3 /LicenseVersionCCBY4.0. 14. M. Cvetic, A.A. Tseytlin, Nucl. Phys. B 416, 137 (1994) 15. B. Zwiebach, String Theory (University Press, Cambridge, 2004) 16. D. Bazeia, F.A. Brito, W. Freire, R.F. Ribeiro, Int. J. Mod. Phys. A 18, 5627 (2003) References 17. D. Bazeia, F.A. Brito, W. Freire, R.F. Ribeiro, Eur. Phys. J. C 40, 531 (2005). arXiv:hep-th/0311160 1. E. Eichten et al., Phys. Rev. Lett. 34, 369 (1975) 18. R. Dick, Phys. Lett. B 409, 321 (1997) 2. L. Wilets, Nontopoloeical Solutions (Word Scientific, Singapore, 19. R. Dick, Eur. Phys. J. C 6(4), 701–703 (1999) 1998) 20. M.J. Strassler, On confinement and duality, in Superstrings and 3. T.D. Lee, Particles Physics and Introduction to Field Theory (Har- Related Matters Trieste, 2001, pp. 105–162 wood Academic, New York, 1981) 4. H. Shiba et al., Phys. Lett. B 333, 461 (1994) 5. G.S. Bali et al., Phys. Rev. D 54, 2863 (1996) 6. A. Chodos, R.L. Jaffe, K. Jonhson, C.B. Thorn, V.F. Weisskopf. Phys. Rev. D 9, 3471 (1974)

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