Chern-Simons-Schwinger model of confinement in QCD
aa r X i v : . [ h e p - t h ] A p r Chern-Simons-Schwinger model of confinement in
QCD
Antonio Aurilia, ∗ Patricio Gaete, † and Euro Spallucci ‡ Department of Physics and Astronomy, California State Polytechnic University-Pomona, Pomona, California 91768, USA Departmento de F´ısica and Centro Cient´ıfico-Tecnol´ogico de Valpara´ıso,Universidad T´ecnica Federico Santa Mar´ıa, Valpara´ıso, Chile Dipartimento di Fisica Teorica, Universit`a di Trieste and INFN, Sezione di Trieste, Italy (Dated: September 10, 2018)It has been shown that the mechanism of formation of glue-bags in the strong coupling limit ofYang-Mills theory can be understood in terms of the dynamics of a higher-rank abelian gauge field,namely, the 3-form dual to the Chern-Simons topological current.Building on this result, we show that the field theoretical interpretation of the Chern-Simonsterm, as opposed to its topological interpretation, also leads to the analytic form of the confinementpotential that arises in the large distance limit of
QCD . In fact, for a (3 + 1)-dimensional general-ization of the Schwinger model, we explicitly compute the interaction energy. This generalization isdue to the presence of the topological gauge field A µνρ . Our results show that the static potentialprofile contains a linear term leading to the confinement of static probe charges.Once the quantum effects of the axial vector anomaly in QCD are taken into account, the newgauge field and its matter-current counterpart provide an exact replica of the Schwinger mechanismof “charge-screening” that operates in
QED . PACS numbers: 14.70.-e, 12.60.Cn, 13.40.Gp
I. INTRODUCTION AND SYNOPSIS
An analytical proof of color confinement in
QCD still eludes us in spite of many ingenious model calculations anddefinite hints from computer simulations based on lattice gauge-theory. Computer simulations also provide someevidence for the existence of glueballs as bound states of pure glue that cannot be accounted for by conventionalperturbative techniques.The root of the problem is well known: while asymptotic freedom is a well established property of the perturbativedynamics of
QCD , the transition to infrared slavery is problematic because of non-perturbative effects that dominatein the large distance limit of the theory. Once this ”large distance limit” is defined in terms of some phenomenologicalscale of distance, the immediate problem is that of identifying the dynamical variables that operate in that limit. Ahint about the nature of those hidden dynamical variables comes from the phenomenological bag models of hadrons[1]: the partial success of those models indicate that, in the large distance limit of
QCD , the spatial extension ofhadrons and the bag degrees of freedom must somehow be included among those new dynamical variables. On theother hand, in order to speak meaningfully of a ”
QCD -solution” of the confinement problem, one would expect that such variables should arise from the very dynamics of
QCD and control the mechanisms of color confinement andhadronization through charge screening.A significant step that meets the above expectation was taken by Gabadadze several years ago [2] building on theearly suggestion that there is a hidden long range force [3] in the topological sector of YM-theory defined by the socalled θ -term [4, 5]. The prevalent “instanton interpretation” of that term [6] has prevented an earlier and widerrecognition of that long range force and its relevance not only for the formation of glueball states but also for thefundamental problem of confinement and screening of color charges.The focus on instanton solutions in YM-theory renders the confinement problem cumbersome, if not intractable: onthe one hand, instanton-configurations of non-abelian gauge fields play an essential role in determining the structureof the Yang-Mills theory vacuum and are responsible for tunneling processes among topologically distinct vacuums.On the other hand, they fail to satisfy the widely accepted criterion for color confinement, namely, the so called ”arealaw” for the Wilson loop [7]. As noted in [2], this failure suggests that the formation of non-perturbative bound states,such as glueballs, requires at least a strong correlation among instantons. However, the study of instanton effects ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] in the strong coupling limit of YM-theory is a cumbersome task precisely because of the lack of non-perturbativecomputational techniques. A notable advance was made several years ago through the string theory spin-off, nowuniversally known as
AdS/CF T , or
AdS/QCD [9]. For an authoritative review see [10]. Capitalizing on the fullpower of the string theory machinery, this new approach offers effective non-perturbative techniques to deal withconfinement in super-symmetric Yang-Mills theory in the Large- N limit [11]. However, we believe that a satisfactoryproof of confinement in real QCD with N = 3, where both super-symmetry and scale invariance are broken, is stillmissing and this is the rationale for considering an alternative approach grounded in the more traditional frameworkof quantum field theory.In the following, we shall argue that the key to the resolution of this quandary is to shift from the conventionaltopological interpretation of the Chern-Simons term to a field theoretical interpretation based on the hidden longrange force as originally suggested in Refs.[3, 8]. Indeed, this was the basic recognition made by Gabadadze whosuccessfully applied the field theoretical approach to the problem of formation of non-perturbative glueball states [12].On our part, we will show that the study of that hidden long range force in QCD casts the problem of colorconfinement in a completely new perspective, one in which calculations can be carried out exactly and whose physicalinterpretation is transparent.The implementation of this idea involves three distinct steps that will be discussed in the following sections: first,we will give a precise meaning to the “field theoretical interpretation” of the Chern-Simons term in
QCD and definethe “long range force” that arises from that interpretation; second, we shall derive the form of the static potentialassociated with that force and show that it satisfies the Wilson loop criterion for confinement; third, we will evaluatethe quantum effects of the axial anomaly in QCD on the static potential in order to account for the phenomenon of“ color charge screening ”.The guiding principle of the above research plan is the correspondence, noted long ago in Ref.[8] but never fullyexploited, between the colorless topological sector of QCD and the zero-charge sector of QED in (1+1)-spacetimedimensions. The latter is also known as the “Schwinger model” [13, 14] and the broader objective of this paper isto give a precise meaning to the above correspondence. This extrapolation from two to four dimensions is importantbecause the occurrence of a rising Coulomb potential, the “screening” of it through the generation of mass and thesubtle interplay with gauge invariance represent the essential ingredients of a mechanism by which one hopes tounderstand the phenomenon of quark-binding into physical hadrons. These issues were first analyzed in
QED in[13–16].On the mathematical side, it seems useful, even at this introductory stage, to point to the nature of the long rangeforce that is inherent, but not manifestly present, in the topological formulation of the Chern-Simons term in QCD .In a nutshell, this force is mediated by a higher rank abelian gauge potential A µνρ which is dynamically realized, in QCD , as the well-known composite, colorless, combination of YM potential A µ and field strength F νρ , namely, thedual of the Chern-Simons current A µνρ ≡ QCD
T r (cid:0) A [ µ F νρ ] (cid:1) = 1Λ QCD
T r (cid:18) A [ µ ∂ ν A ρ ] + 2 g A [ µ A ν A ρ ] (cid:19) = 1Λ QCD (cid:18) δ ab A a [ µ ∂ ν A bρ ] + 2 g f abc A a [ µ A bν A cρ ] (cid:19) . (1)In the above expression we have identified Λ QCD as the energy scale at which the dynamics of
QCD cannot betreated perturbatively. The introduction of this distance scale, at this stage, is required in order to give canonicaldimensions to A µνρ . The confining nature of this higher rank potential was anticipated long ago in Refs.[8, 18] and themanifold applications of the 3-index potential in particle physics and cosmology have been investigated extensively inRefs.[17–20], [21, 22], [23–26], [27–30], [31, 32]. Presently, since the implementation of the Schwinger mechanism in QCD depends critically on the properties of the mathematical construct (1), we shall refer to it as the
Chern-Simons-Schwinger potential, or, CSS-potential for short.The next obvious question that arises in the field-theory interpretation of the Chern-Simons term concerns thenature of the “sources” of the force mediated by A µνρ . Here is where we depart substantially from the conventionalinterpretation of the topological charge density. In the next section we show that the CSS-potential (1) is subjectto a gauge transformation that dictates the nature of its coupling to matter just as in the familiar case of ordinaryelectrodynamics. However, unlike the vector electromagnetic potential, the 3-vector A µνρ does not couple to point-like objects; rather, it couples to relativistic closed membranes which, in our interpretation, constitute the physicalboundary of hadronic “bags” [8].Apart from the noteworthy results obtained by Gabadadze in the study of glue-bags, the physical payoff of thenew formulation can be summarized thus: first, unlike the Euclidean topological interpretation of the Chern-Simonscurrent, the dynamics of the 3-vector gauge field (1) is exactly solvable in four-dimensions as is the dynamics of thecorresponding 1-vector gauge field in two dimensions. Indeed, solving this field equations for the CSS-potential, onefinds that the gauge field A µνρ does not transmit physical quanta but gives rise to the same confining potential thatoperates in classical electrodynamics in (1 + 1)- dimensions. Equivalently, the binding potential between two surfaceelements on the boundary of a bag is shown to satisfy the Wilson loop criterion for confinement. Furthermore, atthe quantum level a mechanism sets in, identical to the ”Schwinger mechanism” of QED , leading to the screeningof color charges and ”hadronization” in the form of massive pseudo-scalar particles as a consequence of the chiralanomaly in QCD . It seems noteworthy that this mechanism of mass generation is, a) fundamentally different fromthe conventional Higgs mechanism which is based on spontaneous symmetry breaking, and b) it is also a universal mechanism in the sense that the dynamics of the CSS-potential can be formulated in any number of spacetimedimensions and may have interesting cosmological consequences. Finally, as announced earlier, the variables thatcontrol the dynamics of
QCD in the large distance limit are clearly identified. They are, i) the current associatedwith a relativistic (closed) membrane as the ”matter” constituent of the action, and ii) a Maxwell field of the ”fourthkind” as the field constituent of the action.These points are discussed in a self-contained manner in the following sections. We begin, in Section II, with anoverview of the reinterpretation of the topological term in YM-theory as it constitutes the basis of our approach. InSection III, for a (3 + 1)-dimensional generalization of the Schwinger model, we explicitly compute the interactionenergy between external probe charges. As a result, we obtain a linear term leading to the confinement of static probecharges. As expected, the above potential profile is analogous to that encountered in the usual Schwinger model.Finally, in Section IV, we cast our Final Remarks. Next, in Appendix, we introduce a new way to obtain the anomalyinduced effective action with emphasis on the screening effect and mass generation mechanism by way of the quantumanomaly in QCD.
II. THE CASE FOR AN ABELIAN GAUGE FIELD OF THE FOURTH KIND IN
QCD
A. A new look at the topological charge density in YM-theory
Consider the so called θ -term in the total action for YM-theory, S θ ≡ θ π Z d x T r (cid:0) F ∗ µν F µν (cid:1) . (2)The θ -dependence of the vacuum energy density in the large N -limit has been studied extensively by variousmethods, even in terms of a D -brane construction of YM-theory within the context of string theory [28]. However,the essential point of our discussion is that the above term admits a field theory interpretation in terms of a Lagrangiansystem from which one can extract some physical properties that are far from evident in the conventional topologicalinterpretation.The key idea is to reformulate S θ in terms of the CSS-gauge potential anticipated in the Introduction. This is possiblebecause of the following well known identities in YM-theory: the integrand of the θ -term, namely, ”the topologicalcharge density” Q is identically equal to the divergence of the Chern-Simons current K µ so that Q = ∂ µ K µ = π Tr F µν ˜ F µν ≡ F˜F . On the other hand, the Chern-Simons current is the Hodge dual of A µνρ , K µ = ε µναβ A ναβ .Putting together the above identities, one arrives at the following expression for Q , Q = 14! ε µνρσ F µνρσ , where the differential four-form F ≡ F µνρσ dx µ ∧ dx ν ∧ dx ρ ∧ dx σ , represents the field strength for the three-form potential F = dA , where A ≡ A νρσ dx ν ∧ dx ρ ∧ dx σ .We shall refer to F as a “Maxwell field of the fourth kind” because of its invariance under a generalized gaugetransformation to be specified shortly.As we shall discuss in the following, the action S θ in YM-theory represents the coupling of the F -field to the bulkof a finite vacuum domain, or ”bag” [18], [17] S θ = θ Z V Q d x = − θ Z V F µνρσ dx µ ∧ dx ν ∧ dx ρ ∧ dx σ ≡ − θ Z V F , where V is the region where the topological charge is different from zero.More instructively, using Stokes’ theorem,the same action takes the form S θ = − θ Z V F = − θ Z ∂V A = − θ Z ∂V A νρσ dx ν ∧ dx ρ ∧ dx σ , (3)in which the role of the θ -angle is now that of a ”coupling constant” between the CSS-potential and the conserved3-form current J associated with the closed membrane that constitutes the physical boundary of the bag: S θ = 13! 3 θ π Z d x J λµν∂V Tr (cid:0) A [ λ D µ A ν ] (cid:1) , (4) J νρλ∂V ( x ) = Z ∂V δ [ x − y ] dy ν ∧ dy ρ ∧ dy λ , (5) ∂ µ J µνρ∂V = 0 . (6)As anticipated in the Introduction, this deceptively simple procedure clearly identifies the variables that governthe formation and dynamics of glue bags. To the extent that such variables arise through a reformulation of the”topological charge density”, they are an integral part of the dynamics of YM-theory.
The form of the full Lagrangianfor gluodynamics includes a kinetic term for the field strength F introduced above S A ≡ − × Z d x (cid:0) ∂ [ µ A νρσ ] (cid:1) . (7)The remarkable properties of the above Lagrangian will be highlighted in the following subsection. B. The meaning of the CSS-Lagrangian
The computation of the quantum vacuum pressure that determines the structure of the ground state of stronginteractions must take into account the contribution of zero-point oscillations of a rank-three gauge field, A µνρ [31].As we discuss below, this field is known to have no radiative degrees of freedom in four dimensions, the only dynamicaleffect being a static long-range force within finite vacuum domains, or glue bags. The overall effect of this force isa constant but otherwise arbitrary pressure within the bags. This remarkable feature was exploited to associate the A µνρ -field with the “bag constant” of the hadronic vacuum as a kind of Casimir effect for strong interactions [31, 32].In other words, the gauge field A µνρ does not correspond to a massless particle but gives an energy proportional tothe volume when external bags are minimally coupled to it .The gauge potential A µνρ gives rise to a ”Maxwell field of the fourth kind” in the sense that the corresponding fieldstrength F µνρσ ≡ ∂ [ µ A νρσ ] satisfies the generalized Maxwell equation ∂ µ F µνρσ = 0 , (8)and is invariant under the extended gauge transformation δA µνρ = ∂ [ µ λ νρ ] . (9) This form of the Lagrangian is reported in Ref.[17] as Eq.(5). The mathematical and physical properties of this Lagrangian system havebeen studied extensively [19, 20, 30], not only in connection with the dynamics of hadronic bags [18, 24, 25] but also in connection witha production mechanism of dark matter [29] and the cosmological inflationary scenario of the early universe [23, 27]. To be precise,the complete Lagrangian also includes a kinetic term for the closed membrane with or without the presence of gravity. The mostgeneral case was discussed in Ref.[18]. Clearly, in the case of
QCD the above Lagrangian is gravity independent and the coupling of theCSS-potential is to external membranes nucleated out of the vacuum energy background provided by the F -field, as discussed in thetext. With hindsight, this work is complementary to that of Ref.[12] which focuses, instead, on the volume dependence of the topologicalsusceptibility in YM-theory.
The “gluonic” gauge potential, or CSS-potential (1), is subject to the same extended gauge transformation (9) inthe sense that, when the YM-potential is gauge rotated U ≡ e i Λ ( x ) = e i Λ a ( x ) T a , Λ ( x ) ≡ Λ a ( x ) T a , (10) A µ ′ = U − A µ U − ig U − D µ U , (11) A ′ µνρ = T r (cid:0) A [ µ F νρ ] (cid:1) ′ , (12)it transforms as follows A ′ µνρ = A µνρ − ig T r (cid:2) (cid:0) D [ µ U (cid:1) U − F νρ ] (cid:3) , (13) D µ U = i D µ Λ U , (14) δA µνρ = 1 g T r (cid:2) (cid:0) D [ µ Λ (cid:1) F νρ ] (cid:3) ≡ g ∂ [ µ Λ νρ ] . (15)The gauge transformation (9) is instrumental in determining the dynamics of the CSS-potential since it requires the coupling of the potential to the rank-three current density J µνρ ( x ) that we have anticipated in the previoussubsection. Thus, it seems noteworthy that, unlike the phenomenological bag models of hadrons, the spatial extensionof glueballs is a dynamical consequence of the hidden gauge invariance of the Chern-Simons term in QCD .Summing up our discussion so far, the full dynamics of A µνρ ( x ), in the presence of external membranes, is governedby the Lagrangian density L = − · (cid:0) ∂ [ µ A νρσ ] (cid:1) − κ J µνρ A µνρ = − · F λµνρ F λµνρ + 14! F λµνρ ∂ [ λ A µνρ ] − κ J µνρ∂V A µνρ . (16)The remarkable property of the system (16,5) is that its dynamics is exactly solvable. In the absence of coupling the solution is immediate since Maxwell’s equation (8) is identically satisfied with F µνρσ = f ǫ µνρσ . The arbitrary integration constant f is physically associated with the vacuum energy density[18, 22] through the energy momentum tensor T µν = 13! F µαβγ F αβγν − · δ µν F αβγδ F αβγδ , (17)which, in view of the given expression of the field strength, reduces to the simple form T µν = f δ µν . (18)When the coupling to external bags is switched on, as in (16,5), the dynamics is still exactly solvable and one wayto obtain the general solution is through the following steps. First, the field equations δLδF λµνρ = 0 −→ F λµνρ = ∂ [ λ A µνρ ] , (19) δLδA µνρ = 0 −→ ∂ λ F λµνρ = κ J µνρ ( x ) , (20)imply that the current is conserved, a property that is consistent with the invariance of the system under the extendedgauge transformation (9): δA µνρ = ∂ [ µ λ νρ ] ←→ ∂ µ J µνρ ( x ) = 0 . (21) It has been shown that in higher dimensional models this constant is ” quantized ”. This result could provide a solution to the long-standing cosmological constant puzzle [36, 37].
In this instance, the conserved membrane current J can be written as the divergence of a rank four antisymmetric bulk current K J µνρ ( x ) ≡ ∂ λ K λµνρ , (22)where K λµνρ ( x ) ≡ Z V δ [ x − z ] dz λ ∧ dz µ ∧ dz ν ∧ dz ρ . (23)On the other hand, dz λ ∧ dz µ ∧ dz ν ∧ dz ρ = ǫ λµνρ d z , (24)so that one can write K λµνρ ( x ) as K λµνρ ( x ) = ǫ λµνρ Θ V ( x ) , (25)where Θ V ( x ) = Z V d z δ [ x − z ] , (26)stands for the (generalized) unit step-function of the V manifold, namely, Θ V ( P ∈ V ) = 1 , Θ V ( P / ∈ V ) = 0.By inverting Eq.(22) one can also express the bulk-current K in terms of the boundary current J∂ λ K λµνρV = J µνρ∂V ( x ) −→ K λµνρV = ∂ [ λ ∂ J µνρ ] ∂V , (27)so that the general solution of Maxwell’s field equation (20) takes the following form F λµνρ = f ǫ λµνρ + κ ∂ [ λ ∂ J µνρ ] ∂V = ǫ λµνρ ( f + κ Θ V ( x ) ) , (28)where f is, again, the constant solution of the homogeneous equation.The above solution indicates that the interior and exterior regions of a ”vacuum domain” are characterized by adifferent value of the vacuum energy density and pressure. This is the main feature of most phenomenological “bagmodels” of hadrons. The essential difference is that, in our case, this property is a dynamical consequence of thecoupling of the 3-index potential to a relativistic test bubble, as dictated by the requirement of gauge invarianceunder the transformation (9). In other words, the overall effect of the coupling is that the A -field is responsible fora long-range static interaction that results in the existence of a ground state with distinct phases. It seems to usthat this dynamical property of the CSS-Lagrangian is a reminder of the topological structure of the ground state ofgluodynamics. In other words, the many vacuums that arise in the topological sector of
QCD as a consequence ofinstanton effects reappear in the guise of ”hadronic vacuum-domains,” or glue bags, as a consequence of the vacuumpolarization effects due to the CSS-potential.
C. The CSS-potential satisfies the “Wilson-loop” criterion for confinement
We have mentioned in the Introduction that YM-instantons alone do not generate a confining force between colorcharges. In contrast, one of the advantages of the field theory approach is that the CSS-Lagrangian has a built-inmechanism for confinement, i.e., it gives rise to a confining potential between infinitesimal surface elements of agluebag. The crux of the argument is the calculation of the ”Wilson loop” for the three-index potential coupled tothe boundary current of a bag W [ J ] = D exp (cid:18) − κ Z d xA µνρ J µνρ (cid:19) E = Z [ J ] Z [ 0 ] ≡ exp ( − Γ [ J ] ) . (29)From the above expression it is possible to derive the expression of the static potential following the standardprescription V ( R ) ≡ − lim T →∞ T ln W [ ∂B ] . (30)In the case of a spherical bag, the calculations were carried out explicitly in Ref.[32]. For our present purposes, it issufficient to recall the following general result: in order to extract the static potential V ( R ) from Eq.(30) one mustcompute the contribution of the currents associated with a pair of antipodal points P and P on the surface of the bag Z B d x J µνρ ∂ J µνρ = Z ∂B Z ∂B dy µ ∧ dy ν ∧ dy ρ ∂ dy ′ µ ∧ dy ′ ν ∧ dy ′ ρ == 14 π Z T dτ Z T dτ ′ Z S (2) d σ Z S (2) d ξ × y µνρ ( τ , σ ) 1[ y ( τ , σ ) − y ( τ ′ , ξ ) ] y µνρ ( τ ′ , ξ ) δ [ ξ − σ ] . (31)In the above expression, (cid:0) σ , σ (cid:1) and (cid:0) ξ , ξ (cid:1) are two independent sets of world coordinates on the boundarymanifold. The world-history of these two points, in Euclidean time, constitutes the “Wilson loop”.Close inspection of the double integral (31) reveals that the interaction between two diametrically opposite surfaceelements is mediated by a Coulomb force that operates in the bulk of the bag. This corresponds to the explicit formof the Green function in Eq. (31). In the case of a spherical bubble, the radial dependence of the surface elementscan be calculated explicitly and leads to the following result V ( R ) ≡ − lim T →∞ T ln W [ ∂B ] = π κ R . (32)This expression represents an attractive potential that is proportional to the volume enclosed by the membrane.Again, it seems noteworthy that, unlike the phenomenological bag models where confinement is imposed at theoutset, the result (32)is the combined effect of bulk and boundary dynamics of the CSS-Lagrangian (16) through theintermediary 3-index gauge potential. The result (32) is also the basis of the noted correspondence with
QED . In order not to break the continuity of ourdiscussion, the correspondence with the two-dimensional case will be discussed in more quantitative terms in SectionIV. However, it may be helpful to anticipate here the basis of that correspondence: in Section IV, we argue that thezero-charge sector of
QED may also be interpreted as a bag model in (1 + 1) -dimensions. Once this unconventionalinterpretation is clarified, it is almost straightforward to show that the same dynamics is effectively realized in
QCD .Thus, we note that the “volume law” encoded in Eq.(32) represents a natural extension of the “area law” for theWilson loop of a quark-antiquark pair bounded by a string. In the string case, it is well known that the definition(30) leads to a linear potential between two test quarks V ( R ) ∝ R . (33)The behavior of the Coulomb potential in (1 + 1)-dimensions meets this requirement. The linearly rising potential(33) is consistent with the “string-picture ” of charge confinement, namely, that opposite charges are connected by a“flux-tube”, or string, of constant energy per unit length. However, in one spatial dimension, strings and membranesreduce to a pair of points.
This geometric degeneracy is removed in (3+1)-dimensions. Evidently, it is the “membrane-bag” sector of
QED that is effectively realized in QCD since the three-index potential is minimally coupled to closedmembranes which, in turn, constitute the boundary of a bag.We further observe that Eqs.(32) and (33) describe the same geometric effect. In both cases the static potentialis proportional to the “volume” of the manifold connecting the two test charges. In Eq.(33), R is essentially the”linear volume” enclosed by the two external sources. In our case, R is proportional to the volume enclosed by theexternal membrane. In either case, the result reflects the basic underlying idea that ”confinement”, at the classicallevel, requires an infinite amount of energy in order to separate the two sources. III. EXPLICIT COMPUTATION OF THE STATIC POTENTIAL
Inspired by the preceding observations, the purpose of this Section is to further elaborate on the physical contentof the Chern-Simons term as a gauge invariant interaction associated with a Maxwell field of the fourth kind. To dothis, we will work out the static potential for the 4-D generalization of the Schwinger model, as originally introducedin Ref. [21], via a path-integral approach. In effect, the initial point of our analysis is the bosonized form of theSchwinger model in D=4, that is, L = 12 ( ∂ µ φ ) + 12 m φ φ + g √ π ∂ µ φ ε µνρσ A νρσ − F µνρσ , (34)where g is a coupling constant and m φ refers to the mass of the scalar field φ .According to usual procedure, integrating out the φ field induces an effective theory for the A νρσ field. It is nowimportant to recall that the A νρσ field can also be written as A νρσ = ε νρσλ ∂ λ ξ [31, 32], where ξ refers to anotherscalar field. This then leads to the following effective theory for the model under consideration: L = 12 ξ ∆ g (cid:14) π (cid:16) ∆ − m φ (cid:17) ∆ ξ , (35)where ∆ = ∂ µ ∂ µ .We are now ready to compute the interaction energy between static pointlike sources. We start off our analysis bywriting down the functional generator of the Green’s functions, that is, Z [ J ] = exp (cid:18) − i Z d xd yJ ( x ) D ( x, y ) J ( y ) (cid:19) , (36)where, D ( x, y ) = R d k (2 π ) D ( k ) e − ikx , is the propagator. In this case, the corresponding propagator is given by D ( k ) = − m φ M ! k ( k + M ) + m φ M k , (37)where M = m φ − g / π .Enlisting the standard representation Z = e iW [ J ] and employing Eq. (36), W [ J ] takes the form W [ J ] = − Z d k (2 π ) J ( k ) ∗ ( − m φ M ! k ( k + M ) + m φ M k ) J ( k ) . (38)Next, for J ( x ) = (cid:2) Qδ (3) (cid:0) x − x (1) (cid:1) + Q ′ δ (3) (cid:0) x − x (2) (cid:1)(cid:3) , we obtain that the interaction energy of the system is givenby V = − QQ ′ Z d k (2 π ) g / πg / π − m φ k (cid:16) k + g / π − m φ (cid:17) − m φg / π − m φ k e i k · r , (39)where r = x (1) − x (2) .This, together with Q ′ = − Q , yields finally V = Q π g / π (cid:16) g / π − m φ (cid:17) L (cid:18) − e − q g /π − m φ L (cid:19) + m φ (cid:16) g / π − m φ (cid:17) L , (40)where L = | r | . One immediately sees that the above static potential profile is analogous to that encountered in thetwo-dimensional Schwinger model. As a matter of fact, in order to put our discussion into the proper context, itis useful to summarize the relevant aspects of the two-dimensional Schwinger model. In such a case, we begin byrecalling the bosonized form of the model under consideration [38]: L = − F µν + 12 ( ∂ µ φ ) − e √ π ε µν F µν φ + m X (cos (2 πφ + θ ) − , (41)where P = (cid:16) e π / (cid:17) e γ E with γ E the Euler-Mascheroni constant and θ refers to the θ -vacuum.Consequently, by using the gauge-invariant but path-dependent variables formalism which is known to provide aphysically-based alternative to the Wilson loop approach [39, 40], the static potential reduces to V = Q √ πe (cid:16) − e − e √ π L (cid:17) , (42)for the massless case. On the other hand, for the massive case ( θ = 0), the static potential then becomes V = Q λ (cid:18) πm P λ (cid:19) (cid:0) − e − λL (cid:1) + q (cid:18) − e / π λ (cid:19) L, (43)where λ = e π + 4 πm P . The above results clearly show that the 4-D generalization of the Schwinger model isstructurally identical to the 2-D Schwinger model.In this perspective it seems worth recalling that there is an alternative way of obtaining the Lagrangian density (35),which provides a complementary insight into the physics of confinement. In fact, we refer to a theory of antisymmetrictensor fields that results from the condensation of topological defects as a consequence of the Julia-Toulouse mecha-nism. More precisely, the Julia-Toulouse mechanism is a condensation process dual to the Higgs mechanism proposedin [41]. This mechanism describes phenomenologically the electromagnetic behavior of antisymmetric tensors in thepresence of magnetic-branes (topological defects) that eventually condensate due to thermal and quantum fluctua-tions. Using this phenomenology, we have discussed in [42, 43] the dynamics of the extended charges (p-branes) insidethe new vacuum provided by the condensate. Actually, in [42] we have considered the topological defects coupledboth longitudinally and transversally to two different tensor potentials, A p and B q , such that p + q + 2 = D , where D = d + 1 space-time dimensions. The technical details are given in Ref. [42]. The net result is that, after thecondensation, the Lagrangian density turns out to be L = ( − q q + 1)! [ H q +1 ( B q )] + eB q ε q,α,p +1 ∂ α Λ p +1 + ( − p +1 p + 2)! [ F p +2 (Λ p +1 )] + ( − p +1 ( p + 1)!2 m Λ p +1 , (44)showing a B ∧ F type of coupling between the B q potential with the tensor Λ p +1 carrying the degrees of freedom ofthe condensate. Following our earlier procedure [42], the effective theory that results from integrating out the fieldsrepresenting the vacuum condensate, is given by L = ( − q +1 q + 1)! H q +1 ( B q ) (cid:18) e ∆ − m (cid:19) H q +1 ( B q ) . (45)Hence we see that this expression with p = − q = 3 becomes L = 12 × F µνρλ ( A ) (cid:18) e ∆ − m (cid:19) F µνρλ ( A ) . (46)It is straightforward to verify that Eq. (46) reduces to Eq. (35).In this way we establish a new connection among different effective theories. From this discussion it should be clearthat the above connections are of interest from the point of view of providing unifications among diverse models aswell as exploiting the equivalence in explicit calculations. IV. FINAL REMARKS
The Schwinger model of two-dimensional ”electrodynamics”, because of its kinematical constraints, has a built-inmechanism of confinement that has inspired many subsequent models of ”hadronization” since the very inception of
QCD as the leading theory that describes the interaction among colored quarks and gluons.The possibility of generalizing
QED to four dimensions has always been met with skepticism, mostly becausethe ”obvious” generalization of the model has been known for a long time, namely, the theory of ordinary quantumelectrodynamics, or QED . As a matter of historical fact, the ”Schwinger model” was conceived by restricting thefamiliar Maxwell-Dirac Lagrangian to (1 + 1)-dimensions with an eye on the relationship between gauge invarianceand mass. This original procedure of ”descending” from four to two dimensions, while keeping the same form of theMaxwell-Dirac Lagrangian, has generated the widespread notion that QED is the unique extension of QED . It hasbeen recognized, however, that this is not a one-to-one correspondence so that, by the reverse procedure of ”ascending”from two to four dimensions, the “electrodynamic” interpretation of the Schwinger model seems purely formal in the0sense that the physical content of QED and QED is drastically different. Thus, in order to underscore this difference,in Section IV we have itemized the essential physical properties of the Schwinger model. Those properties, we haveargued, pertain to a theory of spatially extended objects, membranes and bags to be precise, and can be realized intwo as well as in four dimensions . It is a remarkable fact, and the main conclusion of this paper, that the verydynamics of Quantum Chromodynamics in the large distance limit possesses the same kinematical constraints, andtherefore the same confining and screening potential that are found in the two-dimensional case. This result hingeson a reinterpretation of the Chern-Simons term in QCD as a gauge invariant interaction associated with a Maxwellfield of the fourth kind. The gauge invariant coupling of the CSS-potential requires the existence of relativistic closedmembranes. We have treated such membranes as an approximation to the physical boundary of non perturbativegluebags. The dynamics of this system has been shown to be exactly solvable. The static interaction potential, unlikethe instanton configurations of the old ”topological” interpretation, satisfies the Wilson-loop criterion for confinementwhile the quantum axial anomaly in
QCD triggers the same screening mechanism of mass generation that operatesin the two-dimensional Schwinger model.
V. ACKNOWLEDGEMENTS
P. Gaete was partially supported by FONDECYT (Chile) grant 1130426, DGIP (UTFSM) internal project USM111458. One of us (PG) wants to thank the Abdus Salam ICTP for hospitality and support. This work was carriedout in part at the Physics Department, Harvey Mudd College, and in part at the Physics Department of the Universityof Trieste. One of us (AA) wishes to thank John Townsend (HMC) and Euro Spallucci (U of T) for the hospitalityextended to him while on sabbatical leave from California State Polytechnic University-Pomona.
VI. APPENDIXA. Quantum anomaly and mass generation: The massive case
The purpose of this Appendix is to highlight some other aspects of the strict correspondence between the two-dimensional case and the four-dimensional one. Two interesting questions arise naturally. First, what happens if theCSS-potential acquires a mass? And, concomitantly, what is the physical mechanism that may induce a mass termfor A µνρ ?Summarizing our previous and subsequent discussion, the following properties of A µνρ constitute the crux of the generalized Schwinger mechanism in QCD :a) When massless, A µνρ represents nothing more than a constant background field since, as we have shown,in (3+1)–dimensions A µνρ does not possess radiative degrees of freedom. Here, for later reference, we wish to addthat this property, even though peculiar, is shared by all d –potential forms in ( d + 1)–spacetime dimensions [18, 22].For instance, the best known case is in two dimensions: F µν = ∂ [ µ A ν ] = ǫ µν f while in four dimensions, F µνρσ = ǫ µνρσ f , and f represents a constant vacuum energy density in both cases by virtue of the field equations.b) If the field acquires a mass, then it describes massive pseudoscalar particles, in two a well as in four di-mensions. The two dimensional case is well known from the Schwinger model; in four dimensions, the free fieldequation for A µνρ in the massive case [30, 32] ∂ λ ∂ [ λ A µνρ ] + m A µνρ = 0 , = ⇒ ∂ µ A µνρ = 0 , (47)imposes the divergence free constraint on the four components of A µνρ leaving only one propagating degree offreedom. In other words, the introduction of a mass term “excites” a dynamical (pseudoscalar) particle of matter outof the constant energy background. An interesting cosmological application of this transmutation was discussed in Ref.([29]) in connection with the problem of “darkmatter/energy in the universe. There, we also discussed the subtle issue of the presence of a mass term in an otherwise gauge invarianttheory through the use of Stueckelberg’s formalism. quantum anomaly term in QCD in perfect analogy with the two dimensional case ofthe Schwinger model. In the two dimensional case the occurrence of mass appears in the form of a pole in thepropagator of the gauge field; in
QCD , it manifests itself as the “Kogut-Susskind” pole in the correlation function ofthe topological charge density [33].
B. From two to four dimensions
At the quantum level, the result of the previous section is altered by the phenomenon of “screening.” Quantummechanically, rather than supplying the large amount of energy that is required in order to separate two test charges,it is energetically more favorable to create quark-antiquark pairs within the volume separating the two test charges.This phenomenon was anticipated long ago by Kogut and Susskind [34], again in connection with the Schwinger modelof
QED . The net physical result of the quantum anomaly in QED , is the “hadronization process,” namely thebinding of quarks into physical hadrons through screening of the original test charges and the emergence of mass,corresponding to a pole in the propagator of the gauge field .A qualitative argument that anticipates the existence of a similar mechanism in the large distance limit of QCD is based on the existence of a pole, the so called Kogut-Susskind pole, in the correlation function of the topologicalcharge density Q : χ = i lim q → q µ q ν Z d x e iqx h | T [ K µ ( x ) K ν (0) ] | i . (48)In Gabadadze’s work, this correlation function is referred to as vacuum topological susceptibility . The expressionof χ quoted above is derived by recalling that the topological charge density Q is the derivative of the Chern-Simonscurrent Q = ∂ µ K µ and substituting this identity into the expression for the correlation function of the topologicalcharge density.We have noticed in subsection VI A that, in agreement with Luscher’s observation, the correlation function of twoboundary current is non zero and corresponds to the existence of a Coulomb-type potential within the bulk of thehadronic vacuum domain V . This implies that the above expression for χ is non zero and this can only happen if thecorrelation function of two Chern-Simons currents develops a pole in the limit of vanishing momentum. This is oneway in which the “Schwinger mechanism” of mass generation is implemented in an otherwise gauge invariant theory.In this Subsection, following the example of QED which we shall discuss in Section IV.D, we wish to show thatthe screening of the classical potential (32), extracted above from the Wilson loop, takes place when the effect of thequantum axial current anomaly in QCD is taken into account.For the sake of comparing our results, it seems useful to recall, at this point, the main properties that hold in 2 D as we discuss them in the “bag interpretation” of QED .In the presence of fermions, it is well known that the vector and axial-vector currents cannot be simultaneouslyconserved. If one insists on the conservation of the vector current, then the divergence of the axial-vector current isnot zero. Without digressing on these well known results, the properties of bubble-dynamics in 2 D may be distilledinto the following points:1. There is a vector current, j µ ( x ), coupled to a vector gauge potential A µ ( x ).2. The Maxwell field F µν = ∂ [ µ A ν ] , in this case a differential two-form in two spacetime dimensions, is dual to azero-form: F µν = ǫ µν f .In the absence of coupling, f amounts to an arbitrary integration constant that is physically related to abackground vacuum energy density. When coupled to a test bubble, or, in electrodynamics parlance, to a dipolecharge distribution, the gauge field A gives rise to a confining potential that is proportional to the linear volumeof the bag.3. The axial current is dual to the vector current: j µ ( x ) ≡ ǫ µν j ν ( x ).4. The axial current j µ ( x ) is “anomalous”, i.e., its divergence is proportional to the dual field strength of A µ ( x ),that is, ∂ µ j µ ( x ) ∝ ǫ νρ ∂ [ ν A ρ ] ∝ ǫ νρ F νρ .2The purpose of this brief synopsis of the salient features of QED is to motivate the calculations described belowfor the CSS-Lagrangian. Indeed, a strong argument for the viability of the Schwinger mechanism in QCD is thatthe three-index gauge potential A µνρ possesses the same physical properties as the gauge vector potential A µ intwo dimensions. Thus, in order to underscore the stringent similarity between the two cases, we list below thecorresponding items that we wish to implement in QCD :1. There is a three-index tensor current, J µνρ ( x ), coupled to a gauge three-form potential A µνρ ( x ).2. The Maxwell field F µνρσ = ∂ [ µ A νρσ ] in this case a differential four-form in four dimensions, is dual to a zero-form: F µνρσ = ∂ [ µ A νρσ ] = ǫ µνρσ f .In the absence of coupling, as we have seen, f amounts to an arbitrary integration constant that is physicallyrelated to a background energy density. When coupled to a test bubble, the gauge field A gives rise to a confiningpotential that is proportional to the volume of the bag.3. The role of “axial current” is played by the Hodge dual of the tensor current, namely, j µ ( x ) ≡ ǫ µνρσ j νρσ ( x ) . (49)4. The axial current j µ ( x ) is “anomalous”, i.e., its divergence is proportional to the dual field strength of A λµν ( x ),that is, ∂ µ j µ ( x ) ∝ ǫ µνρσ ∂ [ λ A ν ρ σ ] ∝ ǫ λµνρ F λµνρ . C. Screening and mass generation in
QCD
Our immediate objective is to translate the above correspondence into explicit computational steps and show thatthe same screening mechanism that generates mass in two dimensions is also active in the case of the CSS-Lagrangian.Thus, according to the above code of correspondence, one needs two currents that are dual to each other and anexpression of the quantum anomaly in terms of the field strength of the abelian CSS-potential. It is a remarkablefact that these necessary ingredients emerge naturally in the field theoretical interpretation of the topological chargedensity. Indeed, we recall from Section II that the θ -term in the YM-action can be rewritten as an interaction termbetween the CSS-potential and the boundary current (4).However, that expression is equivalent to S θ = κ Z d x J µνρ∂V ( x ) A µνρ , κ ≡ θ π Λ QCD , (50)and the membrane current, being the boundary current of a bag, is divergence-free ∂ µ J µνρ∂V = 0 . (51)On the other hand, from QCD we know that the color singlet axial current is anomalous ∂ µ J µ = g π ǫ λµνρ Tr ( F λµ F νρ ) , (52) F aλµ = ∂ [ λ A aµ ] + g f abc (cid:2) A bλ , A cµ (cid:3) . (53)Therefore, in terms of the CSS-potential, the anomaly equation takes the form ∂ µ J µ = g Λ QCD π ǫ λµνρ ∂ [ λ A µνρ ] . The net result of the above manipulations is that we reproduce the pattern of dual currents that exists in the2-dimensional case. Then, in exact analogy with the case of
QED , we proceed to encode the constraints (49), (53)in the generating functional as follows : Z D [ A ] exp ( − iS [ A ] − i S θ [ A , J ∂V ] ) ≡ Z [ J ] , (54) From a mathematical point of view, it is important to realize that, once the expression of the anomaly is given, one can by-pass theintegration over the fermionic degrees of freedom in the path-integral so long as one takes into account the constraints listed above.More specifically, the summation in the path integral must take into account the duality relation between J and J and the anomalousdivergence as constraints on the integration measure [ DJ ]. In other words, we take stock of the fact that, in the large distance limitof QCD , as well as in
QED , the only remnant of quark-dynamics is encoded in the given anomaly equation. Z = Z [ DA ][ DJ ] δ " J λ − Λ QCD ǫ λµνρ J µνρ∂V δ (cid:20) ∂ µ J µ − g π ǫ λµνρ T r ( F λµ F νρ ) (cid:21) × exp (cid:18) iS [ A ] + i θ π Z d xJ λµν∂V Tr (cid:0) A [ λ D µ A ν ] (cid:1) (cid:19) . (55)Integration over J µ implements the anomaly relation J µνρ∂V ≡ g π Λ QCD ǫ µνρσ ∂ σ ∂ Tr (cid:0) F αβ ∗ F αβ (cid:1) , (56)while an integration by parts leads to the following expression for the generating functional Z = Z [ DA ] exp iS [ A ] + i θg π Λ QCD Z d x (cid:20) ǫ λµνρ Tr ∂ [ λ (cid:0) A µ D ν A ρ ] (cid:1) − ∂ Tr (cid:0) F βγ ∗ F βγ (cid:1) (cid:21) ! . (57)At large distance, or in the infrared limit, the second term in the above expression dominates over the Yang-Millsterm when θ g π Λ QCD | k | > g , (58)Equation (58) defines the distance where the anomaly induced action dominates over the kinetic Yang-Mills term,i.e., it defines the infrared range of momenta for which the topological term is dominant Z ≈ Z [ DA ] exp i θg π Λ QCD Z d x (cid:20) ǫ λµνρ ∂ λ Tr (cid:0) A [ µ D ν A ρ ] (cid:1) − ∂ Tr (cid:0) F βγ ∗ F βγ (cid:1) (cid:21) ! . (59)Thus, in the infrared domain the Yang-Mills field enters the generating functional only through the abelian CSS-potential Z = Z [ DA µ ] [ DA ρστ ] δ " A µνρ − QCD (cid:18) A a [ µ ∂ ν A aρ ] + 2 g f abc A a [ µ A bν A cρ ] (cid:19) × exp i θg π Λ QCD Z d x (cid:20) ǫ λµνρ ∂ [ λ Tr (cid:0) A µ D ν A ρ ] (cid:1) − ∂ Tr (cid:0) F βγ ∗ F βγ (cid:1) (cid:21) ! = Z D [ A ρστ ] exp (cid:18) i × Z d x ∂ [ λ A µνρ ] g κ − ∂ ∂ [ λ A µνρ ] (cid:19) . (60)Note that the Hodge dual of J µ is a rank-three, totally anti-symmetric current J µνρ∂V so that in the non-perturbative(strong-coupling) regime of QCD we recover the same duality relationship that holds true in
QED J µ = 2 g κ ǫ µνρλ J νρλ∂V . (61)With the above results in hand, we conclude that the complete effective action in the large distance limit of QCD , including the effects of the quantum anomaly, is as follows Z = Z [ DA... ] exp (cid:18) i Z d x (cid:20) − × ∂ [ λ A µνρ ] (cid:18) g κ − ∂ (cid:19) ∂ [ λ A µνρ ] (cid:21) (cid:19) . (62)The above expression represents the effective gauge invariant action for a massive three-index potential anticipated insubsection II C. The physical spectrum consists of massive pseudo-scalar particles in exact analogy to the 2-dimensionalcase. This is the Schwinger mechanism that operates in the strong coupling limit of QCD .4 D. On the analogy between the “topological” terms in D and in D . The “topological term” in two dimensions S θ = θe π Z V d x ǫ µν F µν , (63)is also given by a total divergence S θ = θe π Z V d x ∂ µ A µ , (64)where A µ ≡ ǫ µν A ν . From Stokes theorem it follows that S θ = θe π I ∂V dy µ A µ ( y ) = θe π Z l ds dy µ ds A µ ( y ) , y µ ( l ) = y µ (0) . (65)The above equation can be re-written as S θ = θe π Z d x J µ∂V A µ , (66)where we have introduced the boundary current J µ∂V ( x ) = I ∂V dy µ δ (2) ( x − y ) . (67)In 2 D the topological term is equivalent to the coupling between A µ and the boundary current J µ∂V . At largedistance, or in the infrared limit, it follows from the effective action for QED that the scale at which the topologicalterms dominates over the Maxwell term is given by e | k | >> −→ | k | >> e . (68)In this large distance regime we have that J µ∂V = ǫ µν ∗ J ν = πθ J µ , (69)and ∗ J ν is directed along the normal to ∂V , ∂ µ ∗ J µ = ∂ µ h | J µ | i = e π ǫ αβ F αβ . (70)This allows us to express the v.e.v. of the vector current in terms of the field strength J µ∂V = e π ǫ µν ∂ ν ∂ ǫ αβ F αβ . (71)Therefore, the topological action becomes S θ = − e π Z d xF µν − ∂ F µν . (72)Finally, adding a gauge invariant kinetic term for the field A µ gives us the total action S tot ≡ S + S θ = − Z d xF µν − ∂ + e /π − ∂ F µν . (73) [1] P. Hasenfratz and J. Kuti, Phys. Rept. , 75 (1978). [2] G. Gabadadze, Phys. Rev. D , 094015 (1998).[3] M. Luscher, Phys. Lett. B , 465 (1978).[4] R. Jackiw and C. Rebbi, Phys. Rev. Lett. , 172 (1976).[5] C. G. . Callan, R. F. Dashen and D. J. Gross, Phys. Lett. B , 334 (1976).[6] A. A. Belavin, A. M. Polyakov, A. S. Schwartz and Yu. S. Tyupkin, Phys. Lett. B , 85 (1975).[7] K. G. Wilson, Phys. Rev. D , 2445 (1974).[8] A. Aurilia, Phys. Lett. B , 203 (1979).[9] J. M. Maldacena, Adv. Theor. Math. Phys. , 231 (1998).[10] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Phys. Rept. , 183 (2000).[11] E. Witten, Adv. Theor. Math. Phys. , 505 (1998).[12] G. Gabadadze, Phys. Rev. D , 055003 (1998).[13] J. S. Schwinger, Phys. Rev. , 397 (1962).[14] J. S. Schwinger, Phys. Rev. , 2425 (1962).[15] A. Casher, J. B. Kogut and L. Susskind, Phys. Rev. Lett. , 792 (1973).[16] S. R. Coleman, R. Jackiw and L. Susskind, Annals Phys. , 267 (1975).[17] A. Aurilia and F. Legovini, Phys. Lett. B , 299 (1977).[18] A. Aurilia, D. Christodoulou and F. Legovini, Phys. Lett. B , 429 (1978).[19] A. Aurilia and D. Christodoulou, J. Math. Phys. , 1446 (1979).[20] A. Aurilia and D. Christodoulou, Phys. Lett. B , 589 (1978).[21] A. Aurilia, Y. Takahashi and P. K. Townsend, Phys. Lett. B , 265 (1980).[22] A. Aurilia, H. Nicolai and P. K. Townsend, Nucl. Phys. B , 509 (1980).[23] A. Aurilia, G. Denardo, F. Legovini and E. Spallucci, Phys. Lett. B , 258 (1984).[24] A. Aurilia, G. Denardo, F. Legovini and E. Spallucci, Nucl. Phys. B , 523 (1985).[25] A. Aurilia, R. S. Kissack, R. B. Mann and E. Spallucci, Phys. Rev. D , 2961 (1987).[26] A. Aurilia, M. Palmer and E. Spallucci, Phys. Rev. D , 2511 (1989).[27] S. Ansoldi, A. Aurilia, R. Balbinot and E. Spallucci, Class. Quant. Grav. , 2727 (1997).[28] S. Ansoldi, C. Castro and E. Spallucci, Phys. Lett. B , 174 (2001).[29] S. Ansoldi, A. Aurilia and E. Spallucci, Phys. Rev. D , 025008 (2001).[30] A. Aurilia and Y. Takahashi, Prog. Theor. Phys. , 693 (1981).[31] A. Aurilia and E. Spallucci, Phys. Rev. D , 105004 (2004).[32] A. Aurilia and E. Spallucci, Phys. Rev. D , 105005 (2004).[33] J. B. Kogut and L. Susskind, Phys. Rev. D , 697 (1974).[34] J. B. Kogut and L. Susskind, Phys. Rev. D , 3501 (1974).[35] K. Fujikawa, Phys. Rev. D , 285 (1984).[36] R. Bousso and J. Polchinski, JHEP , 006 (2000).[37] G. Dvali, Phys. Rev. D , 025018 (2006).[38] D. J. Gross, I. R. Klebanov, A. V. Matytsin and A. V. Smilga, Nucl. Phys. B , 109 (1996).[39] P. Gaete and I. Schmidt, Phys. Rev. D , 125002 (2000).[40] P. Gaete and I. Schmidt, Phys. Rev. D , 027702 (2001).[41] F. Quevedo and C. A. Trugenberger, Nucl.Phys. B , 143 (1997).[42] P. Gaete and C. Wotzasek, Phys.Lett. B , 108, (2004).[43] P. Gaete and C. Wotzasek, Phys.Lett. B634