We present an alternative derivation of the parity anomaly for a massless Dirac field in 2+1 dimensions coupled to a gauge field. The anomaly functional, a Chern-Simons action for the gauge field, is obtained from the non-trivial Jacobian corresponding to a non local symmetry of the Pauli-Villars regularized action. That Jacobian is well-defined, finite, and yields the standard Chern-Simons term when the cutoff tends to infinity.Facultad de Ciencias Exacta

Ciccolini, M. L.

Fosco, Cesar Daniel

Schaposnik, Fidel Arturo

Physics Letters B

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26 October 2000Physics Letters B 492 (2000) 214–218www.elsevier.nl/locate/npeNonlocal parity transformations and anomaliesM.L. Ciccolini a, C.D. Fosco a,∗, F.A. Schaposnik ba Centro Atómico Bariloche, Instituto Balseiro, Comisión Nacional de Energía Atómica, 8400 Bariloche, Argentinab Departamento de Física, Universidad Nacional de La Plata, C.C. 67, 1900, La Plata, ArgentinaReceived 2 July 2000; accepted 7 September 2000Editor: L. Alvarez-GauméAbstractWe present an alternative derivation of the parity anomaly for a massless Dirac field in 2 + 1 dimensions coupled to agauge field. The anomaly functional, a Chern–Simons action for the gauge field, is obtained from the nontrivial Jacobiancorresponding to a nonlocal symmetry of the Pauli–Villars regularized action. That Jacobian is well-defined, finite, and yieldsthe standard Chern–Simons term when the cutoff tends to infinity.  2000 Elsevier Science B.V. All rights reserved.1. IntroductionBecause of the presence of ultraviolet divergences,most quantum field theory models require the use of aregularization scheme at some intermediate step in therenormalization program. The regularization proce-dure, to be effective, will modify the large momentumbehaviour of the theory. In some cases, there are sym-metries which are particularly sensitive to that modifi-cation, leading to the existence of anomalies, namely,symmetries which remain broken even after the reg-ularization is removed. In other words, performing asymmetry transformation and renormalizing are non-commuting operations.In the context of the functional integral quantiza-tion, anomalies are traditionally attributed to the non-invariance of the integration measure under a classicalsymmetry transformation. The non-invariance of the* Corresponding author.E-mail addresses: ciccolini@ib.cnea.gov.ar (M.L. Ciccolini),fosco@cab.cnea.gov.ar (C.D. Fosco),fidel@athos.fisica.unlp.edu.ar (F.A. Schaposnik).integration measure shows up in that the Jacobian cor-responding to the classical symmetry transformation isnontrivial and, in general, ill-defined. When a properregularization is introduced, this Jacobian gives rise toan anomalous term in the effective action [1].In Ref. [2], an alternative procedure to study anom-alies within the functional integral approach was pre-sented. It amounts to considering the functional inte-gral for a regularized action, and to restrict the study tosymmetry transformations that leave that action invari-ant. For the cases of the conformal and chiral anom-alies, we have shown that those symmetry transfor-mations do exist, and that they tend to the usual oneswhen the cutoff is removed. The transformations arenecessarily nonlocal, but on a length scale of the orderof the inverse of the cutoff. The outcome of performingthose symmetry transformations is that, while the reg-ularized classical action is indeed invariant (even for afinite cutoff), the integration measure is not. Moreover,this noninvariance is explicit, since the correspond-ing Jacobian determinant is well defined and differentfrom 1, for any value of the cutoff. Contact with the0370-2693/00/$ – see front matter  2000 Elsevier Science B.V. All rights reserved.PII: S0370-2693(00) 01 05 9- 5M.L. Ciccolini et al. / Physics Letters B 492 (2000) 214–218 215usual anomalies is made by taking the infinite cutofflimit, what reproduces the known results.In this letter, we will point out that this procedurecan also be applied to the case of the parity anomalyin 2 + 1 dimensions, since the regularized actiondoes have a nonlocal symmetry, which coincideswith the classical parity symmetry in the infinitecutoff limit. Moreover, the associated Jacobian for thisdiscrete transformation is finite, and reproduces theparity anomaly, a Chern–Simons term in the effectiveaction [3,4]. This shows that the procedure we havedeveloped in Ref. [2] can also be applied to the case ofa discrete symmetry.The organization of this letter is as follows: In Sec-tion 2, we discuss the classical parity transformationsand its generalization to the (quantum) regularizedcase. The generalized transformations are then appliedto the calculation of the parity anomaly in Section 3.Section 4 contains our conclusions.2. Generalized parity transformationsLet us consider massive Dirac fermions coupledto an external Abelian gauge field Aµ, in 2 + 1dimensions. The (Minkowski space–time) action forthe system is(1)Sf [A] =∫d3xLf ,where the Lagrangian density Lf is given by(2)Lf = ψ̄(x)(i /∂ − e /A(x)−m)ψ(x).In our conventions, gµν = diag(1,−1,−1), and theDirac matrices verify {γµ, γν} = 2gµν , and γ †µ = γ µ.We recall that, in 2+ 1 dimensions, parity transfor-mations are, in fact, tantamount to spatial reflections:(3)x = (x0, x1, x2) P−→ xP = (x0,−x1, x2)since a spatial inversion is a rotation in π when thenumber of space coordinates is even.The gauge and spinor fields transform according tothe rules:A(x)= (A0(x),A1(x),A2(x))→AP (xP )= (A0(x),−A1(x),A2(x)),ψ(x)→ψP (xP )= γ1ψ(x),(4)ψ̄(x)→ SψP (xP )= ψ̄(x)γ1.Hence, under P , the classical action transforms asfollows:S[A]→ SPf [A]=∫d3xP ψ̄P (xP )(i /∂P − e /AP (xP )−m)ψP (xP )(5)=∫d3x ψ̄(x)(i /∂ − e /A(x)+m)ψ(x).Writing the mass dependence of the action explicitly,we have the relation:(6)SPf (m)= Sf (−m).This indicates, of course, that the massless classicaltheory is parity invariant.Regarding the regularized action, although not everyregularization method can be implemented in termsof an action, there are many important cases wherethis can be done. Examples of those are the Euclideancutoff, lattice, and Pauli–Villars (PV) regularizationmethods. We shall use the PV method, since it main-tains most of the symmetries, except parity (see [5]for a thorough discussion in odd-dimensional spaces).This greatly simplifies the discussion. In our (2+ 1)-dimensional example, the functional integral is ren-dered convergent by the addition of just one bosonicregulator spinor field φ, whose mass Λ plays the roleof a cutoff. The regularized action Srf is:Srf = i∫d3x[ψ̄(x)( /∂ + ie /A)ψ(x)(7)+ φ̄(x)( /∂ + ie /A+ iΛ)φ(x)].It is possible to write the regularized action (7) interms of just one fermionic field Ψ , although at theexpense of equipping that field with a nonlocal action.To that end, we consider the regularized functionalintegral Z r[A] corresponding to (7):Z r[A] =∫DψDψ̄DφDφ̄× exp{−∫d3x[ψ̄(x) /Dψ(x)(8)+ φ̄(x)( /D + iΛ)φ(x)]},where /D ≡ ( /∂ + ie /A).216 M.L. Ciccolini et al. / Physics Letters B 492 (2000) 214–218We first integrate out the regulator in (8):Z r[A] =∫DψDψ̄ det−1[1− i /DΛ](9)× exp(−∫d3xψ̄ /Dψ)(we have neglected an irrelevant constant factor). Wethen make the change of variables:(10)ψ(x)=[1− i /DΛ]−1/2Ψ (x),(11)ψ̄(x)= SΨ (x)[1− i /DΛ]−1/2under which the measure transforms according to:(12)Dψ Dψ̄ =DΨ DSΨ det[1− i /DΛ].The result of the two previous steps is that we mayrewrite (9) in the equivalent form:(13)Z r[A] =∫DΨ DSΨ exp(−Snlf )where Snlf denotes a nonlocal form of the regularizedaction,(14)Snlf =∫d3x d3y Ψ̄ (x)D(x, y)Ψ (y),with(15)D(x, y)=[/D1− i /D/Λ](x, y).We have found it convenient to adopt a ‘bracket’ likenotation to write the action (14),(16)Snlf = 〈Ψ |D|Ψ 〉, D=/D1− i /D/Λ,since it avoids writing operators kernels and integra-tions explicitly. Note that the ‘bra’ includes the γ0 fac-tor for the adjoint field.To understand the behaviour of (14) under paritytransformations, we previously need to know the D op-erator transformation properties. Denoting by /D(x, y)the kernel of /D, we see that(17)γ1 /DP(xP , yP)γ1 = /D(x, y).We can also express this result as:(18)γ1 /DP γ1 = /D,what in turn implies for D:(19)γ1DP (Λ)γ1 =D(−Λ).The parity transformed nonlocal regularized actionbecomes(20)SnlPf =〈ΨP∣∣DP (Λ)∣∣ΨP 〉= 〈ΨP ∣∣γ1D(−Λ)γ1∣∣ΨP 〉,where we need to plug in the parity transformed of Ψand SΨ . The transformation rules for the new fields,can be obtained as follows: from the relation betweenΨ and ψ , we learn that(21)ψP(xP)= [1− i /DPΛ]−1/2ΨP(xP)and(22)ψP(xP)=−γ1[1+ i /DΛ]−1/2γ1ΨP(xP).On the other hand, we have the relation(23)ψP(xP)= γ1ψ(x)= γ1(1− i /DΛ)−1/2Ψ (x).Then,ΨP(xP)= γ1√1+ i /D/Λ√1− i /D/ΛΨ (x)(24)= γ1〈x∣∣∣∣ D(Λ)√D(Λ)D(−Λ)∣∣∣∣Ψ 〉.We obtain an analogous relation for Ψ̄ (x). Usingthe compact notation, we see that(25)∣∣ΨP 〉= γ1 D(Λ)√D(Λ)D(−Λ) |Ψ 〉,(26)〈ΨP∣∣= 〈Ψ | D(Λ)√D(Λ)D(−Λ)γ1,and these are the generalized parity transformationswe were looking for. We note that they tend to thestandard parity transformations when the cutoff isremoved (Λ→∞).It is immediate to verify that, with these transforma-tion rules, the nonlocal form of the regularized actionremains invariant:(27)SnlPf = Snlf .Being the Pauli–Villars fields integrated out exactly,the effective action (14) incorporates the completeM.L. Ciccolini et al. / Physics Letters B 492 (2000) 214–218 217effect of the regularization, in the sense that thecontribution to Snlf of the determinant associated withthe Pauli–Villars field is totally included. For example,were one to use this effective action to compute thevacuum polarization, the corresponding diagrams willautomatically be convergent to all orders.3. Parity anomalyAlthough the regularized action is invariant underthe generalized parity transformations, the fermionicmeasure acquires a nontrivial Jacobian:(28)DΨPDSΨP =DΨDSΨ J ,whereJ = det[−(√D(Λ)D(−Λ)D(Λ))2](29)= det(i /D−Λi /D+Λ).The parity anomaly is obtained from the infinite-Λlimit of this Jacobian. This result is known, andcoincides of course with (twice) the leading term in aderivative expansion of the effective action I efff [A,Λ]:We see that:exp(iIeff[A,Λ] − iIeff[A,−Λ])(30)= det(/D + iΛ/D − iΛ)= J ,(31)Ieff[A,Λ] = −i ln det( /D + iΛ).Hence the Jacobian is expressed as a function of theeffective action. In the Λ→∞ one obtains [6,7]:(32)Ieff[A,Λ]→ Λ|Λ|SC−S,where the C−S action SC−S is defined by:(33)SC−S = e24π∫d3x εµνρAµ(x)∂νAρ(x).Thus we can write for the Jacobian:(34)J = exp{±2iSC−S[A]},which is the parity anomaly result. It is not renormal-ized by higher order correction, as a consequence ofthe Coleman–Hill theorem [8].For the case of a continuous symmetry transforma-tion, the Fujikawa Jacobian resulting from the associ-ated infinitesimal change of variables can be used tocalculate the potentially anomalous divergence of thecorresponding Noether current. This procedure doesnot, of course, have a direct parallel in the present, dis-crete symmetry case. Nevertheless, from the knowl-edge of the Jacobian (34), one can make explicit theanomalous behavior of the fermion current under aparity transformation. Indeed, from Eq. (13), we canwriteZ r[AP]≡ exp (−I reff[AP ])(35)=∫DΨDSΨ exp(−Snlf [AP , SΨ ,Ψ ]).Now, changing the fermion variables as in (25)–(26),and using the invariance (27) of the nonlocal action,we see thatZ r[AP]=J [AP ]∫ DΨP DSΨ P× exp(−Snlf [AP , SΨP ,Ψ P ])(36)=J [AP ]Z r[A]or(37)exp(−I reff[AP ]+ I reff[A])= J [AP ].Differentiation of the regularized effective action withrespect to the gauge field A yields the ground statecurrent 〈jµ[A]〉 so that one has(38)−e〈jµ[AP ]〉+ e〈jµ[A]〉= δ logJ [AP ]δAµ.Using relations (4) for the left-hand side and (34) forthe right-hand side, we can then write the anomalous,parity-odd vacuum current:〈jµ[A]〉odd = 12eδ logJ [AP ]δAµ(39)=±i e4π∫d3x εµνρ∂νAρ(x),which is the well-known result first obtained in [4].It is immediate to verify that all the steps leadingthe parity anomaly for the Abelian case can begeneralized to the non-Abelian case as well, with theonly modification that the Jacobian is now relatedto the non-Abelian massive determinant. The resultis of course that this Jacobian is the exponential of(2i times) the non-Abelian Chern–Simons action.218 M.L. Ciccolini et al. / Physics Letters B 492 (2000) 214–2184. ConclusionsWe have shown that the parity anomaly in 2 + 1dimensions in a theory of massless fermions coupledto an external gauge field can be obtained from theJacobian for a generalized symmetry transformation ofthe regularized theory. This procedure has the virtueof disentangling the symmetries and infinities, bylooking for transformations which are symmetries ofthe regularized action. This avoids involving modeswith an arbitrarily high momentum into the symmetrytransformation, and the bonus is that the resultingJacobian automatically takes care of the anomaly. Itis worth noting that it is precisely the fact that theregularization breaks the symmetry what renders thesymmetry transformations of the regularized actionnonlocal. Non-anomalous local symmetries have, inthis setting, a distinctive property: they are alwayslocal, even when acting on the regularized action.Our calculations required the use of a nonlocal formof the regularized action and we used ‘regularized’fermion fields Ψ . The nonlocality of the action, how-ever, is hidden in the nonphysical region of momentaabove the cutoff. For example, when A= 0, the D op-erator has a kernel:(40)/D(x, y)= /∂(1+ i /∂Λ)∫d3k(2π)3e−ik(x−y)1− (k/Λ)2which is nonlocal on a scale 1/Λ.AcknowledgementsC.D.F. would like to acknowledge Prof. F.D. Maz-zitelli, for useful conversations and correspondence.M.L.C. is supported by a CNEA fellowship. C.D.F.is a member of CONICET, and is partially supportedby ANPCyT (PICT 97/1040) and Fundación Antor-chas grants. F.A.S. is partially supported by CICBAas Investigador, and through grants CONICET (PIP4330/96), and ANPCyT (PICT 97/2285).References[1] K. Fujikawa, Phys. Rev. Lett. 42 (1979) 1195;K. Fujikawa, Phys. Rev. D 21 (1980) 2848.[2] C.D. Fosco, F.D. Mazzitelli, Phys. Lett. B 481 (2000) 129.[3] S. Deser, R. Jackiw, S. Templeton, Phys. Rev. Lett. 48 (1982)975;S. Deser, R. Jackiw, S. Templeton, Ann. Phys. 140 (1982) 372.[4] A.N. Redlich, Phys. Rev. Lett. 52 (1984) 18;A.N. Redlich, Phys. Rev. D 29 (1984) 2366.[5] L. Alvarez-Gaumé, S. Della Pietra, G. Moore, Ann. Phys.(NY) 163 (1985) 288.[6] For a review, see: G.V. Dunne, Lectures at the 1998 Les Hou-ches Summer School: Topological Aspects of Low DimensionalSystems, hep-th/9902115.[7] R.E. Gamboa Saraví, G.L. Rossini, F.A. Schaposnik, Int.J. Mod. Phys. A 11 (1996) 2643.[8] S. Coleman, B. Hill, Phys. Lett. B 159 (1985) 184.

Nonlocal parity transformations and anomalies

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Nonlocal parity transformations and anomalies

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