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USM-TH-165 On external backgrounds and linear potential in three dimensions Patricio Gaete∗ Departamento de F´ısica, Universidad Técnica F. Santa Mar´ıa, Valpara´ıso, Chile (Dated: February 1, 2008) For a three-dimensional theory with a coupling φεµνλvµFνλ, where vµ is an external constant background, we compute the interaction potential within the structure of the gauge-invariant but path-dependentvariablesformalism. Whileinthecaseofapurelytimelikevectorthestaticpotential remains Coulombic, in the case of a purely spacelike vector the potential energy is the sum of a Bessel and a linear potentials, leading to the confinement of static charges. This result may be considered as another realization of theknown Polyakov’s result. PACSnumbers: 11.10.Ef,11.15.-q 5 0 0 I. INTRODUCTION 2 n It is presently widely accepted that quantitative understanding of confinement remains as the major challenge in a J QCD. In this respect, the distinction between the apparently related phenomena of screening and confinement have 8 been of considerable importance in order to gain further insight and theoretical guidance into this problem. It is 1 worth recalling at this stage that field theories which yield a linear potential are relevant to particle physics, since those theories may be used to understand the confinement of quarks and be considered as effective theories of QCD. 1 According to this viewpoint, a simple effective theory in which confining potentials are obtained when a scalar field v φ is coupled to gauge fields via the 3+1 dimensional interaction term 6 3 g = φεµναβF F , (1) 1 I µν αβ L 8 1 0 has been investigated in Ref.[1]. The phenomenology of this model has shown that in the case of a constant electric 5 field strengthexpectationvalue the static potentialremains Coulombic,while in the case ofa constantmagnetic field 0 strength the potential energy is the sum of a Yukawa and a linear potential, leading to the confinement of static / h charges. More interestingly, similar results have been obtained in the context of the dual Ginzburg-Landau theory t [2], as well as for a theory of antisymmetric tensor fields that results from the condensation of topological defects as - p a consequence of the Julia-Toulouse mechanism [3]. Thus, from a phenomenological point of view, there is a class of e models which are good candidates for effective theories of QCD. h In order to give continuity to this program it would be interesting to verify the above scenario when a scalar field : v φ is coupled to gauge fields via the 2+1 dimensional interaction term i X g = φεµνλv F , (2) r LI 2 µ νλ a where v is an external constant background. Our motivation is mainly to explore the effects of the external back- µ ground constant on the confining and screening nature of the potential that could be useful for later developments in more realistic theories, such as describing low-energy properties in condensed matter physics [4]. Here we should mention that the interaction term (2) arises from the dimensional reduction of Maxwell electrodynamics with the (Lorentz-violating) Carroll-Field-Jackiw term [5, 6]. The Lorentz violating theories have been extensively dis- cussed in the last few years. For example, in connection with ultra-high energy cosmic rays [7], with space-time varying coupling constants [8], and in supersymmetric Lorentz-violating extensions [9]. In this Letter, however, we examine the effects of the Lorentz violating background on the interaction energy along the lines of Refs.[1, 3]. It must now be observedthat this approachprovides a physically-basedalternative to the usual Wilson loop approach, whereinthelattertheusualqualitativepictureofconfinementintermsofanelectricfluxtubelinkingquarksemerges naturally. As weshallsee,the static potentialremainsCoulombicforapurelytimelike vectorvµ. Onthe otherhand, adopting a purely spacelike vector vµ, the potential energy is the sum of a Bessel and a linear potential, that is, the confinementbetweenstaticchargesisobtained. Inthislastrespectwerecallthat,almosttwentyyearsago,Polyakov [10] showedthat compact Maxwelltheory in 2+1 dimensions confines permanently electric test charges. In this way ∗Electronicaddress: [email protected] 2 our calculation provides a complementary phenomenologicalpicture of the known Polyakov’sresult, in the hope that this will be helpful to understand better effective gauge theories in 2+1 dimensions. II. INTERACTION ENERGY As mentioned above,our principal purpose is to calculate explicitly the interactionenergy between static pointlike sources for a model one which contains the term (2). This model is similar to the studied in Ref.[1]. To this end we willcomputetheexpectationvalueoftheenergyoperatorH inthephysicalstate Φ describingthesources,whichwe | i will denote by H . The Abelian gauge theory we are considering is defined by the following generating functional h iΦ in three-dimensional spacetime: = φ Aexp i d3x , (3) Z D D L Z (cid:26) Z (cid:27) where the Lagrangiandensity is given by 1 g 1 m2 = F2 + φεµνλv F + ∂ φ∂µφ φ2, (4) L −4 µν 2 µ νλ 2 µ − 2 and m is the mass for the scalar field φ. As in [1] we restrict ourselves to static scalar fields, a consequence of this is that one may replace ∆φ = 2φ, with ∆ ∂ ∂µ. It also implies that, after performing the integration over φ in µ −∇ ≡ , the effective Lagrangiandensity is given by Z 1 g2 1 = F2 εµνλv F εσγβv F . (5) L −4 µν − 8 µ νλ 2 m2 σ γβ ∇ − By introducing Vνλ εµνλv , expression (5) then becomes µ ≡ 1 g2 1 = F2 VνλF VγβF . (6) L −4 µν − 8 νλ 2 m2 γβ ∇ − Atthisstagewenotethat(6)hasthesameformasthecorrespondingeffectiveLagrangiandensityinfour-dimensional spacetime. ThiscommonfeatureprovidesthestartingpointfortheexaminationoftheeffectsoftheLorentz violating background on the interaction energy. A. Spacelike background case We now proceed to obtain the interaction energy in the V0i = 0 and Vij = 0 (v = 0) case (referred to as the 0 6 spacelike background in what follows), by computing the expectation value of the Hamiltonian in the physical state Φ . Lagrangiandensity (6) then becomes | i 1 g2 1 = F2 V0iF V0kF A J0, (7) L −4 µν − 2 0i 2 m2 0k− 0 ∇ − where J0 is the external current, (µ,ν =0,1,2) and (i,k =1,2). Once this is done, the canonical quantization in the manner of Dirac yields the following results. The canonical momenta are Π0 =0, (8) and Π =D E , (9) i ij j where E F andD δ g2V 1 V . Since D is a nonsingularmatrix(detD =1 g2 V2 =0)with i ≡ i0 ij ≡ ij − i0∇2−m2 j0 − ∇2−m2 6 V2 Vi0Vi0, there exists t(cid:16)he inverse of D and fr(cid:17)om Eq.(9) we obtain ≡ 1 1 E = δ detD+g2V V Π . (10) i detD ij i0 2 m2 j0 j (cid:26) ∇ − (cid:27) 3 The corresponding canonical Hamiltonian is thus 1 g2 (V Π)2 1 H = d2x A ∂ Πi J0 + Π2+ · + B2 , (11) C (− 0 i − 2 2 ( 2 M2) 2 ) Z ∇ − (cid:0) (cid:1) where M2 m2+g2V2 and B is the magnetic field. Requiring the primary constraint (8) to be stationary, leads to the seconda≡ryconstraintΓ (x) ∂ Πi J0 =0. It is easily verifiedthat the preservationofΓ forall times does not 1 i 1 ≡ − giveriseto anyfurther constraints. The theoryis thus seentopossessonly twoconstraints,whicharefirstclass. The extended Hamiltonian that generates translations in time then reads H = H + d2x(c (x)Π (x)+c (x)Γ (x)), C 0 0 1 1 wherec (x)andc (x)arethe Lagrangemultiplierfields. Since Π =0foralltime andA˙ (x)=[A (x),H]=c (x), 0 1 0 R 0 0 0 whichiscompletelyarbitrary,wediscardA (x)andΠ (x)becausetheyaddnothingtothedescriptionofthesystem. 0 0 Then, the Hamiltonian takes the form 1 g2 (V Π)2 1 H = d2x Π2+ · + B2+c(x) ∂ Πi J0 , (12) (2 2 ( 2 M2) 2 i − ) Z ∇ − (cid:0) (cid:1) where c(x)=c (x) A (x). 1 0 − The quantization of the theory requires the removal of non-physical variables, which is done by imposing a gauge condition such that the full set of constraints becomes second class. A convenient choice is found to be [11] 1 Γ (x) dzνA (z) dλxiA (λx)=0, (13) 2 ν i ≡ ≡ Z Z Cξx 0 where λ (0 λ 1) is the parameter describing the spacelike straight path xi = ξi +λ(x ξ)i, and ξ is a fixed point (refere≤nce p≤oint). There is no essential loss of generality if we restrict our consideration−s to ξi =0. The Dirac brackets can now be obtained and the nontrivial Dirac bracket involving the field variables takes the form 1 A (x),Πj(y) ∗ =δjδ(2)(x y) ∂x dλxjδ(2)(λx y). (14) i i − − i − Z (cid:8) (cid:9) 0 We are now in a position to evaluate the interaction energy between pointlike sources in the model under consideration, where a fermion is localized at y and an antifermion at y. From our above discussion, we see that H ′ h iΦ reads 1 g2 (V Π)2 1 H = Φ d2x Π2+ · + B2 Φ . (15) h iΦ h | (2 2 ( 2 M2) 2 )| i Z ∇ − Next, as was first established by Dirac[12], the physical state can be written as y Φ Ψ(y)Ψ(y) =ψ(y)exp ie dziA (z) ψ(y) 0 , (16) i | i≡ ′   ′ | i (cid:12) (cid:11) yZ′ (cid:12)   where 0 is the physical vacuum state and the line integral appearing in the above expression is along a spacelike | i path starting at y and ending at y, on a fixed time slice. From this we see that the fermion fields are now dressed ′ by a cloud of gauge fields. From the foregoing Hamiltonian discussion, we first note that y′ Π (x) Ψ(y)Ψ(y′) =Ψ(y)Ψ(y′)Π (x) 0 +e dz δ(3)(z x) Φ . (17) i i i | i − | i Zy (cid:12) (cid:11) Combining Eqs.(15) and (17)(cid:12), we have H = H +V(1)+V(2), (18) h iΦ h i0 where H = 0 H 0 , and the V(1) and V(2) terms are given by: h i0 h | | i e2 y′ 1 y′ V(1) = d2x dz′δ(2)(x z′) 2 dziδ(2)(x z), (19) − 2 i − 2 M2∇x − Z Zy ∇x− Zy 4 and e2m2 y′ 1 y′ V(2) = d2x dz′δ(2)(x z′) dziδ(2)(x z), (20) 2 i − 2 M2 − Z Zy ∇x− Zy where the integrals over zi and z′ are zero except on the contour of integration. i The V(1) term may look peculiar, but it is just the familiar Bessel interaction plus self-energy terms. In effect, expression (19) can also be written as V(1) = e2 y′dz′∂z′ y′dzi∂iG(z′,z), (21) 2 i i z Zy Zy where G is the Green function 1 G(z′,z)= K (M z′ z). (22) 2π 0 | − | Employing Eq. (22) and remembering that the integrals over zi and z′ are zero except on the contour of integration, i expression (21) reduces to the familiar Bessel interaction after subtracting the self-energy terms, that is, e2 V(1) = K (M y y′ ). (23) −2π 0 | − | The task is now to evaluate the V(2) term, which is given by e2m2 y′ y′ V(2) = dz′i dziG(z′,z). (24) 2 Zy Zy By using the integral representation of the Green function (22) ∞ ∞ cos(xt) K (x)= cos(xsinht)dt= dt, (25) 0 √t2+1 Z Z 0 0 where x>0, expression (24) can also be written as ∞ e2m2 1 1 V(2) = dt (1 cos(MLt)), (26) 2πM2 t2√t2+1 − Z 0 where L y y′ . ≡| − | Now let us calculate integral (26). For this purpose we introduce a new auxiliary parameter ε by making in the denominator of integral (26) the substitution t2 t2+ε2. Thus it follows that → ∞ e2m2 dt 1 V(2) limV(2) = lim (1 cos(MLt)). (27) ≡ε→0 ε→02πM2 t2+ε2√t2+1 − Z 0 e A direct computation on the t-complex plane yields e2m2 1 e−MLε 1 V(2) = − . (28) 4M2 ε √1 ε2 (cid:18) (cid:19) − Taking the limit ε 0, expression (28)ethen becomes → e2m2 V(2) = y y′ . (29) 4M | − | From Eqs.(23) and (29), the corresponding static potential for two opposite charges located at y and y′ may be written as e2 e2m2 V(L)= K (ML)+ L, (30) −2π 0 4M 5 where L y y′ . ≡| − | It must now be observed that the rotational symmetry is restored in the resulting form of the potential, although theexternalbackgroundbreakstheisotropyoftheprobleminamanifestway. Itshouldberemarkedthatthisfeature is also shared by the corresponding four-dimensional spacetime interaction energy. We furthernotethattheresult(30)agreeswiththatofPolyakovbasedonthe monopoleplasmamechanism,inthe short distance regime. In this way the above analysis reveals that, although both models are different, the physical content is identical in the short distance regime. This behavior is also obtained in the context of the condensation of topological defects [13]. B. Timelike background case Now we focus on the case V0i = 0 and Vij = 0 (v = 0) (referred to as the timelike background in what follows). 0 6 6 The corresponding Lagrangiandensity reads 1 g2 1 = F Fµν VijF VklF A J0, (31) L −4 µν − 8 ij 2 m2 kl− 0 ∇ − (µ,ν =0,1,2) and (i,j,k,l=1,2). Here again, the quantization is carried out using Dirac’s procedure. We can thus write the canonical momenta Πµ = Fµ0, which results in the usual primary constraint Π0 = 0 and Πi = Fi0. Defining the electric and magnetic fields, as usual, by Ei =Fi0 and B2 = 1F Fij, the canonical Hamiltonian is thus 2 ij 1 1 g2 1 H = d2x E2+ B2 ε ε VijBm VklBn A ∂ Πi J0 . (32) C 2 2 − 16 ijm kln 2 m2 − 0 i − Z (cid:26) ∇ − (cid:27) (cid:0) (cid:1) Time conservation of the primary constraint Π0 = 0 leads to the secondary constraint Γ (x) ∂ Πi J0 = 0, and 1 i ≡ − the preservationof Γ (x) for all times does not give rise to any further constraints. Moreover,it is straightforwardto 1 see that the constrained structure for the gauge field is identical to the usual Maxwell theory. However, in order to put the discussion into the context of this paper, it is convenient to summarize the relevant aspects of the analysis described previously[14]. Therefore, we pass now to the calculation of the interaction energy. As in the previous subsection, our objective will be to calculate the expectation value of the Hamiltonian in the physical state Φ (Eq. (16)). In other words, | i 1 H = Φ d2x E2 Φ . (33) h iΦ h | 2 | i Z (cid:26) (cid:27) Then using the above Hamiltonian structure, we obtain the following form: y′ 2 e2 H = H + d2x dz δ(2)(x z) , (34) h iΦ h i0 2  i −  Z Zy     where H = 0 H 0 and, as before, the integrals over z are zero except on the contour of integrations. We 0 i h i h | | i also draw attention to the fact that the second term on the right-hand side of Eq.(34) is clearly dependent on the distance between the external static fields. In fact, this term can be manipulated in a similar manner to that in the three-dimensional case [15]. Accordingly, the potential for two opposite charges located at y and y reads ′ e2 V = ln(µL), (35) 2π where µ is a massive cutoff introduced to regularize the potential, and L y y′ . ≡| − | III. FINAL REMARKS In summary, we have considered the confinement versus screening issue for a three-dimensional theory with a coupling φεµνλv F , when the external constant vector v is pure spacelike or timelike, respectively. µ νλ µ 6 It was shown that in the case when the vector v is purely timelike no unexpected features are found. Indeed, µ the resulting static potential remains Coulombic. More interestingly, it was shown that when the vector v is purely µ spacelike the static potential displays a Bessel piece plus a linear confining piece. Effectively, therefore, the model studied leads to a confining potential between static charges for spacelike v . An analogous situation in the four- µ dimensional spacetime case may be recalled [1], where a constant expectation value for the gauge field strength F characterizes the external background constant v . Also, a common feature of these models (three and four- µν µ h i dimensional) is that the rotational symmetry is restored in the resulting interaction energy. We conclude by noting that our result can be considered as another physical realization of the Polyakov’s model. However, although both models lead to confinement, the above analysis reveals that the mechanism of obtaining a linear potential is quite different. IV. ACKNOWLEDGMENTS I would like to thank G. Cvetic for helpful comments on the manuscript. 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