The bosonized version of the Schwinger model in four dimensions: a blueprint for confinement?
Antonio Aurilia, Patricio Gaete, José A. Helayël-Neto, Euro Spallucci
aa r X i v : . [ h e p - t h ] F e b The bosonized version of the Schwinger model in four dimensions: a blueprint forconfinement?
Antonio Aurilia, ∗ Patricio Gaete, † Jos´e A. Helay¨el-Neto, ‡ and Euro Spallucci § Department of Physics, California State Polytechnic University-Pomona, Pomona, California 91768, USA Departmento de F´ısica and Centro Cient´ıfico-Tecnol´ogico de Valpara´ıso-CCTVal,Universidad T´ecnica Federico Santa Mar´ıa, Valpara´ıso, Chile Centro Brasileiro de Pesquisas F´ısicas (CBPF), Rio de Janeiro, RJ, Brasil Dipartimento di Fisica Teorica, Universit`a di Trieste and INFN, Sezione di Trieste, Italy (Dated: October 21, 2018)For a (3 + 1)-dimensional generalization of the Schwinger model, we compute the interactionenergy between two test charges. The result shows that the static potential profile contains a linearterm leading to the confinement of probe charges, exactly as in the original model in two dimensions.We further show that the same 4-dimensional model also appears as one version of the B ∧ F modelsin (3 + 1) dimensions under dualization of Stueckelberg-like massive gauge theories. Interestingly,this particular model is characterized by the mixing between a U (1) potential and an Abelian 3-formfield of the type that appears in the topological sector of QCD. PACS numbers: 14.70.-e, 12.60.Cn, 13.40.Gp
I. INTRODUCTION
It is generally agreed that two-dimensional field-theorymodels may provide an excellent and rich framework totest ideas in gauge theories. In fact, the interest in study-ing these models is basically connected to the possibilityof obtaining exact solutions, which are believed to beshared by their more realistic counterparts in four di-mensions. Of these, the Schwinger model, also known asQuantum Electrodynamics in (1 + 1)-space-time dimen-sions, or
QED [1, 2] has probably enjoyed the greatestpopularity due to some special features that it possesses.For example, the energy spectrum contains a massivemode in spite of the gauge invariance of the originalLagrangian, the charge is screened and confinement isenforced by the explicit occurrence of a rising Coulombpotential. To our mind, these special features representthe essential ingredients of a mechanism by which onehopes to understand the phenomenon of quark-bindinginto physical hadrons. These issues were first analyzedin QED in Refs. [1–4].Unfortunately, against this suggestive two-dimensionalperspective, it seems to us that a convincing analyticalproof of color confinement in quantum chromodynamics(QCD) still eludes us. The root of the problem is wellknown: while asymptotic freedom is a well establishedproperty of the perturbative dynamics of QCD, the tran-sition to infrared slavery is problematic because of non-perturbative effects that dominate in the large distancelimit of the theory. Once this ?large distance limit? is de-fined in terms of some phenomenological scale of distance, ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] the immediate problem is that of identifying the dynami-cal variables that operate in that limit. A hint about thenature of those hidden dynamical variables comes fromthe phenomenological bag models of hadrons: the partialsuccess of those models indicate that, in the large dis-tance limit of QCD, the spatial extension of hadrons andthe bag degrees of freedom must somehow be includedamong those new dynamical variables. It is clear that, inorder to speak meaningfully of a ?QCD-solution? of theconfinement problem, one would expect that such vari-ables should arise from the very dynamics of QCD andcontrol the mechanisms of color confinement [5]. Thisis where the extrapolation of results from two to fourspacetime dimensions may play a significant role in theunderstanding of the confinement mechanism in QCD.For instance, the correspondence between the colorlesstopological sector of QCD and the zero-charge sector of
QED was noted long ago in Ref.[6] but never fully ex-ploited; The extrapolation from two to four dimensions,at least for the bosonized version of the Schwinger model,was considered in [7] while a general ”gauge mixing mech-anism for the generation of mass” was proposed in [8].Motivated by these observations, the general purposeof the present discussion is to communicate a deeperunderstanding of the physical content of the (3 + 1)-dimensional generalization of the Schwinger model. Themany avenues of research that are open to us were out-lined in a research proposal by the authors [9]. However,it seems clear that the first line of inquiry is to explorein more detail the role of the Abelian 3-form field amongthe physical observables of the model. It has long beenknown that this 3-form field does not support any prop-agating degree of freedom, its sole physical effect consist-ing of a static interaction between two probe charges.Thisremarkable property is entirely analogous to the two di-mensional case where in QED there are no ”photons”associated with the electromagnetic field [6]. Then, if theSchwinger model has any relevance in the issue of con-finement in four dimensions, then the static potential in-duced by the Abelian 3-form field must also exhibit thesame behavior found in the two-dimensional case. Wefind that this reasonable expectation is fully supportedby the explicit calculation of the interaction energy be-tween two external test charges.A second objective of this work is to elucidate theremarkable interplay between guge invariance and theappearance of mass in the physical spectrum of theSchwinger model. With hindsight, the emergence of thismassive mode can be traced back directly to the dimen-sionality of the coupling constant in QED which sets amass scale in the model. Evidently this is not the case in QED but a similar phenomenon takes place, at least inthe bosonized version of the Schwinger model in (3 + 1)dimensions. We illustrate how this same generalization ofthe S-model basically amounts to a Stueckelberg-like for-mulation of a massive gauge theory characterized by themixing between a U (1) potential and an Abelian 3-formfield.Our work is organized according to the following out-line: in Section II, we recall the salient features of dual-ization in terms of two simple Lagrangian systems andshow their equivalence to different representations of amassive Proca field. In Section III, using a path-integralapproach, we compute the interaction energy, and hencethe analytic form of the static potential in the bosonizedversion of the Scwinger model in four spacetime dimen-sions. Finally, some Concluding Remarks are cast in Sec.IV.Throughout the following discussion, the signature ofthe metric is (+1 , − , − , − II. DUALIZATION, GAUGE INVARIANCE ANDMASS GENERATION
Let us start our considerations by recalling that thestudy of duality symmetry in gauge theories has been ofconsiderable importance in order to provide an equivalentdescription of physical phenomena by distinct theories.As well-known, duality refers to a physical equivalencebetween two field theories which formulated in terms ofdifferent dynamical variables [10].In order to put our discussion into context, we alsorecall that the dualization of Stueckelberg-like massivegauge theories and B ∧ F models follows from a general p dualization of interacting theories in d spacetime di-mensions [11–15]. In particular, in the case of (3 + 1)dimensions, the following B ∧ F models are found: L (1) = − F µν ( A ) + 112 H µνρ ( B ) + m ε µνρσ B µν ∂ [ ρ A σ ] , (1) L (2) = − H µν ( B ) + 112 F µνρ ( A ) + m ε µνρσ B µ ∂ [ ν A ρσ ] , (2) L (3) = 12 ( ∂ µ ϕ ) − F µνρσ ( A )+ m ε µνρσ ϕ∂ [ µ A νρσ ] . (3)At this point, it is instructive to make a brief re-examination of equations (1) and (2). For this pur-pose, we observe that the Lagrangian density (1) maybe rewritten as L (1) = − F µν −
12 ˜ H σ ˜ H σ − m H σ A σ , (4)where we have made use of ˜ H µ = / ε µνλρ ∂ ν B λρ .Next, in order to eliminate the dual-field H σ care mustbe taken, for it satisfies the constraint ∂ µ ˜ H µ = 0 (Bianchiidentity). Thus, to take into account the constraint, weshall introduce a Lagrange multiplier χ . In such a case,the corresponding effective Lagrangian density (4) reads L (1) = − F µν −
12 ˜ H σ ˜ H σ − m H σ A σ + χ∂ σ ˜ H σ . (5)By defining Z σ ≡ A σ + m ∂ σ χ , with Z µν = F µν , we readilyverify that L (1) = − Z µν −
12 ˜ H σ ˜ H σ − m H σ Z σ . (6)By a further definition of the fields, W σ ≡ ˜ H σ + m Z σ ,we find that the Lagrangian density (1) can be broughtto the form L (1) = − Z µν + 12 µ Z µ , (7)with µ ≡ m / . We immediately see that the La-grangian density (7) exhibits a Proca-type mass term.We now turn our attention to the Lagrangian density(2). It is convenient to rewrite this equation in the alter-native form L (2) = 14 ˜ H µν + 112 F µνρ + m
24 ˜ H ρσ A ρσ , (8)where ˜ H µν = / ε µνλρ H λρ .It is worthy to notice that the B µ - field appears onlythrough ˜ H µν . Again, in order to eliminate the dual-field˜ H µν care must be taken, for it satisfies the constraint ∂ µ ˜ H µν = 0. As before, we shall introduce a Lagrangemultiplier χ ν . It gives rise to the following Lagrangiandensity, L (2) = 14 ˜ H µν + 112 F µνρ + m
24 ˜ H µν A µν −
12 ˜ H µν χ µν , (9)where χ µν = ∂ µ χ ν − ∂ ν χ µ . Now, letting Z µν = A µν − m χ µν , we obtain L (2) = 14 ˜ H µν + 112 F µνρ + m
24 ˜ H µν Z µν . (10)It should be further noted that, by defining W µν = ˜ H µν + m Z µν , equation (10) reduces to L (2) = 112 F µνρ − µ Z µν , (11)where we have written, µ = m , and F µνρ = Z µνρ . Thus L (2) describes a massive field of spin 1, exactly a Procaequation, although Z µν ∈ [(1 , ⊕ (0 , III. INTERACTION ENERGY
Inspired by the preceding observation, we shall nowconsider the (3 + 1)-dimensional generalization of theSchwinger model, as originally introduced in Ref.[6]. Aswe have already noticed, we will work out the static po-tential for this (3 + 1) generalization, via a path-integralapproach. To this end, we consider the bosonized formof the Schwinger model in D=(3 + 1), that is, L = 12 ( ∂ µ φ ) + 12 m φ φ + g √ π ∂ µ φ ε µνρσ A νρσ − F µνρσ , (12)where g is a coupling constant and m φ refers to the massof the scalar field φ .We readily verify that when, m φ →
0, equation (12)reduces to equation (3).According to usual procedure, integrating out the φ field induces an effective theory for the A νρσ field. It isnow important to recall that the A νρσ field can also bewritten as A νρσ = ε νρσλ ∂ λ ξ [16, 17], where ξ refers toan another scalar field. This then leads to the followingeffective theory for the model under consideration: L = 12 ξ ∆ g (cid:14) π (cid:16) ∆ − m φ (cid:17) ∆ ξ , (13)where ∆ = ∂ µ ∂ µ .We are now ready to compute the interaction energybetween static pointlike sources. We start off our analysisby writing down the functional generator of the Green’sfunctions, that is, Z [ J ] = exp (cid:18) − i Z d xd yJ ( x ) D ( x, y ) J ( y ) (cid:19) , (14)where, D ( x, y ) = R d k (2 π ) D ( k ) e − ikx , is the propagator. Inthis case, the corresponding propagator is given by D ( k ) = − m φ M ! k ( k + M ) + m φ M k , (15)where M = m φ − g / π . By means of expression Z = e iW [ J ] and employing Eq.(14), W [ J ] takes the form W [ J ] = − Z d k (2 π ) J ∗ ( k ) (cid:16) − m φ M (cid:17) k ( k + M ) J ( k ) − Z d k (2 π ) J ∗ ( k ) m φ M k J ( k ) . (16)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 isgiven by V = − QQ ∗ Z d k (2 π ) (cid:18) g /πg /π − m φ (cid:19)(cid:16) k + g / π − m φ (cid:17) e i k · r + QQ ∗ Z d k (2 π ) m φg / π − m φ ! k e i k · r , (17)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) + Q π m φ (cid:16) g / π − m φ (cid:17) L, (18)where L = | r | . One immediately sees that the abovestatic potential profile is analogous to that encounteredin the two-dimensional Schwinger model. Incidentally, inorder to put our discussion into context it is useful tosummarize the relevant aspects of the two-dimensionalSchwinger model. In such a case, we begin by recallingthe bosonized form of the model under consideration [18]: L = − F µν + 12 ( ∂ µ φ ) − e √ π ε µν F µν φ + m X (cos (2 πφ + θ ) − , (19)where P = (cid:16) e π / (cid:17) e γ E with γ E the Euler-Mascheroniconstant and θ refers to the θ -vacuum.Consequently, by using the gauge-invariant butpath-dependent variables formalism which provides aphysically-based alternative to the Wilson loop approach[19, 20], the static potential reduces to V = Q √ πe (cid:16) − e − e √ π L (cid:17) , (20)for the massless case. On the other hand, for the massivecase ( 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, (21)where λ = e π + 4 πm P . The above results clearly showthat the (3 + 1)-D generalization of the Schwinger modelis structurally identical to the (1+1)-D Schwinger model.In this perspective it is worth recalling that there is analternative way of obtaining the Lagrangian density (13),which provides a complementary view into the physicsof confinement. In fact, we refer to a theory of anti-symmetric tensor fields that results from the conden-sation of topological defects as a consequence of theJulia-Toulouse mechanism. More precisely, the Julia-Toulouse mechanism is a condensation process dual tothe Higgs mechanism proposed in [21]. This mechanismdescribes phenomenologically the electromagnetic behav-ior of antisymmetric tensors in the presence of magnetic-branes (topological defects) that eventually condensatedue to thermal and quantum fluctuations. Using thisphenomenology we have discussed in [22, 23] the dynam-ics of the extended charges (p-branes) inside the newvacuum provided by the condensate. Actually, in [22]we have considered the topological defects coupled bothlongitudinally and transversally to two different tensorpotentials, A p and B q , such that p + q + 2 = D , where D = d + 1 space-time dimensions.We skip all the technical details and refer to [22] forthem. Thus, after the condensation, the Lagrangian den-sity 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 , (22)showing a B ∧ F type of coupling between the B q poten-tial with the tensor Λ p +1 carrying the degrees of freedomof the condensate. Following our earlier procedure [22],the effective theory that results from integrating out thefields representing 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 ) . Hence we see that this expression with p = − q = 3becomes L = 12 × F µνρλ ( A ) (cid:18) e ∆ − m (cid:19) F µνρλ ( A ) . (23)It is straightforward to verify that Eq. (23) reduces toEq. (13).In this way, we establish a new connection among dif-ferent effective theories. It must be clear from this dis-cussion that the above connections are of interest fromthe point of view of providing unifications among diversemodels as well as exploiting their equivalence in explicitcalculations. IV. CONCLUDING REMARKS
Finally, the point we wish to emphasize is that thereare two generic features that are common in the four-dimensional case and their upper/lower extensions, aswe shall show below. First, the existence of a linear po-tential, leading to the confinement of static charges. Thesecond point is related to the correspondence among di-verse effective theories. To see this, it should be notedthat by using the methodology illustrated in [12], we havethat one of the B ∧ F models in (4+1) dimensions is givenby the mixing between a U (1) potential and an Abelian3-form field by means of a topological mass term, that is, L (4+1) = − F µν ( A ) F µν ( A ) + αH µνκλ ( C ) H µνκλ ( C )+ βε µνκλρ A µ ∂ ν C κλρ , (24)with α = − and β = σ , where the parameter β hasmass dimension. This model was considered in [24], andthe main motivation to consider this model is based onthe possible connection with dark energy.However, we shall start from the five-dimensionalspacetime model L (4+1) = − F ˆ µ ˆ ν F ˆ µ ˆ ν + αH ˆ µ ˆ ν ˆ κ ˆ λ H ˆ µ ˆ ν ˆ κ ˆ λ + βε ˆ µ ˆ ν ˆ κ ˆ λ ˆ ρ A µ ∂ ν C ˆ κ ˆ λ ˆ ρ + 112 m C C ˆ µ ˆ ν ˆ ρ C ˆ µ ˆ ν ˆ ρ , (25)with the additional presence of a mass term m C for theAbelian 3-form field; this explicit mass term makes adifference: if it were not introduced, the model could bereduced to nothing but a Proca-type model in (4 + 1)dimensions. Next, we perform its dimensional reductionalong the lines of [24, 25]: A ˆ µ → (cid:0) A ¯ µ , A (cid:1) , A = φ , ∂ ( everything ) = 0, C ˆ µ ˆ ν ˆ κ = (cid:0) C ¯ µ ¯ ν ¯ κ , C ¯ µ ¯ ν (cid:1) and C ¯ µ ¯ ν = B ¯ µ ¯ ν . Carrying out this prescription in equation (25), wethen obtain L (3+1) = − F ¯ µ ¯ ν F µν + 12 ( ∂ ¯ µ φ ) + αH ¯ µ ¯ ν ¯ κ ¯ λ H ¯ µ ¯ ν ¯ κ ¯ λ − αG ¯ µ ¯ ν ¯ κ G ¯ µ ¯ ν ¯ κ − βε µ ¯ ν ¯ κ ¯ λ A ¯ µ ∂ ¯ ν B ¯ κ ¯ λ − βε ν ¯ κ ¯ λ ¯ ρ φ∂ ¯ ν C ¯ κ ¯ λ ¯ ρ + m C C ¯ µ ¯ ν ¯ ρ C ¯ µ ¯ ν ¯ ρ − m C B ¯ µ ¯ ν B ¯ µ ¯ ν , (26)where ¯ µ, ¯ ν, ¯ κ, ¯ λ, ¯ ρ = 0 , , ,
3. Making use of an addi-tional dimensional reduction, that is, A ¯ µ → (cid:0) A µ , A (cid:1) , ∂ ( everything ) = 0, B ¯ µ ¯ ν = ( B µν , C µ ) L (2+1) = − F µν F µν + 12 αG µν G µν − βε µνκ A µ ∂ ν C κ + m C C µ C µ , (27)where G µν = ∂ µ C ν − ∂ ν C µ . Next, after performing the in-tegration over C µ , the induced effective Lagrangian den-sity is given by L (2+1) = − F µν (cid:18) σ (∆ + m C ) (cid:19) F µν . (28)Again, by applying the gauge-invariant formalism, thecorresponding static potential for two opposite chargeslocated at y and y ′ turns out to be V = − q π K ( M L ) + q m C M L, (29)where L = | y − y ′ | and M = σ + m C . In summary,then: this potential displays the conventional screeningpart, encoded in the Bessel function, and the linear con-fining potential. As expected, confinement disappearswhenever m C → m C is non-trivial,but much smaller than the topological mass parameter, σ . A final consideration we would like to raise concernsthe presence of some sort of fundamental mechanism thatendows one of the gauge potentials, the p - or the ( p + 1)-form, with a Proca-type mass term: if only the usualfield-strength squared and the topological mass terms are present, a field reshuffling is always possible to be doneand one of the gauge potentials can be integrated overyielding, at the end, a Proca-like p -form or ( p + 1)-formmassive model; exactly like we have worked out for theLagrangians (1) and (2). However, if a more fundamentalmechanism is at work (like the Higgs mechanism, for ex-ample) that gives an explicit (non-topological) mass termto one of the gauge fields, then the simple equivalence toa p -form Proca field is no longer true and a confiningcontribution to the static interparticle potential shows.We would like to conclude our work by pointing out therelationship between the generation of a non-topologicalmass and the confinig profile of the interparticle poten-tial. V. ACKNOWLEDGMENTS
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