On the ultraviolet finiteness of parity-preserving U(1)×U(1) massive QED 3
aa r X i v : . [ h e p - t h ] A ug The parity-preserving U (1) × U (1) massive QED : ultraviolet finiteness and no parityanomaly W.B. De Lima, ∗ O.M. Del Cima, † and E.S. Miranda ‡ Universidade Federal de Vi¸cosa (UFV),Departamento de F´ısica - Campus Universit´ario,Avenida Peter Henry Rolfs s/n - 36570-900 - Vi¸cosa - MG - Brazil.
The parity-preserving U A (1) × U a (1) massive QED is ultraviolet finiteness – exhibits vanishing β -functions, associated to the gauge coupling constants (electric and chiral charges) and the Chern-Simons mass parameter, and all the anomalous dimensions of the fields – as well as is parity andgauge anomaly free at all orders in perturbation theory. The proof is independent of any regulariza-tion scheme and it is based on the quantum action principle in combination with general theoremsof perturbative quantum field theory by adopting the Becchi-Rouet-Stora algebraic renormalizationmethod. I. INTRODUCTION
The quantum electrodynamics in three space-time dimensions (QED ) has attracted attention since the seminalworks by Deser, Jackiw, Templeton and Schonfeld [1] in view of a possible theoretical foundation for condensed mat-ter phenomena, such as high- T c superconductivity, quantum Hall effect and, more recently, graphene and topologicalinsulators. The massive and the massless QED can exhibit interesting and subtle properties, namely superrenormal-izability, parity violation, topological gauge fields, anyons and the presence of infrared divergences. Since then, theplanar quantum electrodynamics has been widely studied in many physical configurations, namely, small (perturba-tive) and large (non perturbative) gauge transformations, Abelian and non-Abelian gauge groups, fermions families,odd and even under parity, compact space-times, space-times with boundaries, curved space-times, discrete (lattice)space-times, external fields and finite temperatures.The massless U (1) QED is ultraviolet and infrared perturbatively finite, infrared and parity anomaly free at allorders [2], despite some statements found out in the literature that still support that parity could be broken evenperturbatively, called parity anomaly, which has already been discarded [2, 3]. The massive U (1) QED can beodd (odd fermion families number) or even (even fermion families number) under parity symmetry. The parity-evenmassive U (1) QED exhibits vanishing gauge coupling β -function, vanishing anomalous dimensions of all the fields,and besides that, is infrared and parity anomaly free at all orders in perturbation theory [4].The proposed issue is to show that the parity-even U A (1) × U a (1) massive QED [5] exhibits ultraviolet finiteness –vanishing β -functions of both gauge couplings and anomalous dimensions of all the fields – and is parity anomaly free atall orders in perturbation theory. The proof is done by using the BRS (Becchi-Rouet-Stora) algebraic renormalizationmethod in the framework of Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) subtraction scheme, which is basedon general theorems of perturbative quantum field theory [6–10], thus independent of any regularization scheme.Accordingly, the action of the model and its symmetries, the gauge-fixing and the antifields action are established inSection II. The issue of the extension of parity-even U A (1) × U a (1) massive QED in the tree-approximation to allorders in perturbation theory – its perturbative quantization – is organized according to two independent parts. First,in Section III, it is analysed the stability of the classical action – for the quantum theory, the stability correspondsto the fact that the radiative corrections can be reabsorbed by a redefinition of the initial parameters of the theory.Second, in Section IV, it is computed all possible anomalies through an analysis of the Wess-Zumino consistencycondition, furthermore, it is checked if the possible breakings induced by radiative corrections can be fine-tuned by asuitable choice of local non-invariant counterterms. Section V is left to the final comments and conclusions. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected]
II. THE MODEL AND ITS SYMMETRIES
The U A (1) × U a (1) gauge invariant action for the parity-preserving massive QED (the mass-gap graphene-likeplanar quantum electrodynamics [5]) is:Σ inv = Z d x (cid:26) − F µν F µν − f µν f µν + µǫ µρν A µ ∂ ρ a ν + iψ + /Dψ + + iψ − /Dψ − − m ( ψ + ψ + − ψ − ψ − ) (cid:27) , (1)where /Dψ ± ≡ ( /∂ + ie /A ± ig /a ) ψ ± , the coupling constants e (electric charge) and g (chiral charge) are dimensionful withmass dimension , and, m and µ are mass parameters with mass dimension 1. The field strengths, F µν = ∂ µ A ν − ∂ ν A µ and f µν = ∂ µ a ν − ∂ ν a µ , are the associated to the electromagnetic field ( A µ ) and the chiral gauge field ( a µ ), respectively.The Dirac spinors ψ + and ψ − are two kinds of fermions where the ± subscripts are related to their spin sign [11], andthe gamma matrices are γ µ = ( σ z , − iσ x , iσ y ).The action (1) was built up assuming invariance under parity, fixed posteriorly, and the gauge U A (1) × U a (1)transformations as follows: δψ + ( x ) = i [ θ ( x ) + ω ( x )] ψ + ( x ) , δψ − ( x ) = i [ θ ( x ) − ω ( x )] ψ − ( x ) ,δψ + ( x ) = − i [ θ ( x ) + ω ( x )] ψ + ( x ) , δψ − ( x ) = − i [ θ ( x ) − ω ( x )] ψ − ( x ) ,δA µ ( x ) = − e ∂ µ θ ( x ) , δa µ ( x ) = − g ∂ µ ω ( x ) , (2)from which the BRS field transformations shall be defined. In view of a forthcoming quantization of the model (1),a parity-even gauge-fixing action has been added. However, in order to follow the BRS procedure [7], two sorts ofghosts ( c and ξ ), antighosts ( c and ξ ) and Lautrup-Nakanishi fields ( b and π ) [12], the latter ( b and π ) playing the roleof the Lagrange multiplier fields for the gauge condition, have to be introduced. The gauge-fixing action, presentedbellow, belongs to the class of linear covariant gauges discussed by ’t Hooft [13]:Σ gf = Z d x (cid:26) b∂ µ A µ + α b + c (cid:3) c + π∂ µ a µ + β π + ξ (cid:3) ξ (cid:27) . (3)Hereafter, the BRS transformations of the quantum fields are now defined by: sψ + = i ( c + ξ ) ψ + , sψ + = − i ( c + ξ ) ψ + ; sψ − = i ( c − ξ ) ψ − , sψ − = − i ( c − ξ ) ψ − ; sA µ = − e ∂ µ c , sc = 0 ; sa µ = − g ∂ µ ξ , sξ = 0 ; sc = be , sb = 0 ; sξ = πg , sπ = 0 . (4)Together with the parity-even action term, Σ inv + Σ gf , another parity-even action, Σ ext , is introduced. There, thenonlinear BRS transformations are coupled to the antifields (BRS invariant external fields) in order to control at thequantum level the renormalization of those transformations:Σ ext = Z d x (cid:26) i Ω + c + ψ + − i Ω − c − ψ − + ic + ψ + Ω + − ic − ψ − Ω − (cid:27) , (5)where c + = c + ξ and c − = c − ξ . It should be pointed out that in spite of the Faddeev-Popov ghosts be massless,consequently serious infrared divergences could be stemmed from radiative corrections, they are free fields, theydecouple, thus no Lowenstein-Zimmermann mass term [10] has to be introduced for them. Now, the complete classicalaction to be perturbatively quantized reads Γ (0) = Σ inv + Σ gf + Σ ext . (6)The propagators are the key ingredient on spectral consistency and unitarity analyses at tree-level of the model ,as well as in the determination of ultraviolet and infrared dimensions of the fields. The tree-level propagators can be The issues about unitarity, spectral consistency and two-particle scattering potentials have been discussed in [14]. derived for all the quantum fields just by switching off the coupling constants ( e and g ) and taking the free part ofthe action Σ inv + Σ gf ((1) and (3)). In that case all the propagators in momenta space are given by:∆ ++ ψψ ( k ) = i /k − mk − m , ∆ −− ψψ ( k ) = i /k + mk − m , (7)∆ µνAA ( k ) = − i (cid:26) k − µ (cid:18) η µν − k µ k ν k (cid:19) + αk k µ k ν k (cid:27) , (8)∆ µνaa ( k ) = − i (cid:26) k − µ (cid:18) η µν − k µ k ν k (cid:19) + βk k µ k ν k (cid:27) , (9)∆ µνAa ( k ) = µk ( k − µ ) ǫ µλν k λ , (10)∆ µAb ( k ) = ∆ µaπ ( k ) = k µ k , (11)∆ cc ( k ) = ∆ ξξ ( k ) = − ik , (12)∆ bb ( k ) = ∆ ππ ( k ) = 0 . (13)Additionally, in order to establish the ultraviolet (UV) and infrared (IR) dimensions of any fields, X and Y , the UVand IR asymptotical behaviour of their propagator ∆ XY ( k ), d XY and r XY , respectively, are written as: d XY = deg k ∆ XY ( k ) and r XY = deg k ∆ XY ( k ) , (14)where the upper degree deg k gives the asymptotic power for k → ∞ whereas the lower degree deg k gives the asymptoticpower for k →
0. The UV ( d ) and IR ( r ) dimensions of the fields, X and Y , are chosen to fulfill the followinginequalities: d X + d Y ≥ d XY and r X + r Y ≤ r XY . (15)At this moment, it shall be called into attention that the non decoupled propagators (7)–(9) carrying physical degreesof freedom are all massive, so there were no infrared divergences that would arisen during the ultraviolet subtractionsin BPHZ method.In order to fix the UV dimensions of the spinor fields, ψ + and ψ − , and the vector fields, A µ and a µ , use has beenmade of the propagators (7)–(9) together with the conditions (15), that results: d ++ = − ⇒ d + ≥ ⇒ d + = 1 ; d −− = − ⇒ d − ≥ ⇒ d − = 1 ; (16) d AA = − ⇒ d A ≥ ⇒ d A = 12 ; d aa = − ⇒ d a ≥ ⇒ d a = 12 . (17)From the propagators (11) and the conditions (15) and (17), the UV dimensions of the Lautrup-Nakanishi fields, b and π , can be fixed as: d Ab = − ⇒ d A + d b ≥ ⇒ d b = 32 ; d aπ = − ⇒ d a + d π ≥ ⇒ d π = 32 . (18)By considering the propagators (12), the UV dimensions of the Faddeev-Popov ghosts, c and ξ , and antighosts, c and ξ , are constrained by: d ¯ cc = − ⇒ d c + d ¯ c ≥ d ¯ ξξ = − ⇒ d ξ + d ¯ ξ ≥ . (19)Furthermore, assuming that the BRS operator ( s ) is dimensionless and knowing that the coupling constants e and g have mass dimension , from the conditions (19), the UV dimensions of the Faddeev-Popov ghosts and antighostsresult: d c = 0 and d ¯ c = 1 ; d ξ = 0 and d ¯ ξ = 1 . (20)Finally, from the action of the antifields (Σ ext ) together with the UV dimensions of all the quantum fields previouslyfixed, it follows that: d Ω + = 2 and d Ω − = 2 . (21) A µ a µ ψ + ψ − c c b ξ ¯ ξ π Ω + Ω − d / / − − − − GP d ), ghost number (ΦΠ) and Grassmann parity ( GP ). Briefly, the UV dimension ( d ), the ghost number (ΦΠ) and the Grassmann parity ( GP ) of all the fields are gatheredin Table I. The statistics is defined in such a way that, the half integer spin fields with even ghost number, as well as,the integer spin fields with odd ghost number anticommute among themselves. Meanwhile, the other fields commutewith the formers and also among themselves.The BRS invariance of the action Γ (0) (6) is expressed in a functional way by the Slavnov-Taylor identity S (Γ (0) ) = 0 , (22)where the Slavnov-Taylor operator S is defined, acting on an arbitrary functional F , by S ( F ) = Z d x (cid:26) − e ∂ µ c δ F δA µ + be δ F δc − g ∂ µ ξ δ F δa µ + πg δ F δξ + δ F δ Ω + δ F δψ + − δ F δ Ω + δ F δψ + − δ F δ Ω − δ F δψ − + δ F δ Ω − δ F δψ − (cid:27) . (23)and the corresponding linearized Slavnov-Taylor operator is written as S F = Z d x (cid:26) − e ∂ µ c δδA µ + be δδc + − g ∂ µ ξ δδa µ + πg δδξ ++ δ F δ Ω + δδψ + + δ F δψ + δδ Ω + − δ F δ Ω + δδψ + − δ F δψ + δδ Ω + − δ F δ Ω − δδψ − − δ F δψ − δδ Ω − + δ F δ Ω − δδψ − + δ F δψ − δδ Ω − (cid:27) . (24)Thenceforward, the following nilpotency identities hold: S F S ( F ) = 0 , ∀F , (25) S F S F = 0 if S ( F ) = 0 . (26)In particular, the linearized Slavnov-Taylor operator S Γ (0) is nilpotent, namely S (0) = 0, due to the fact that theaction Γ (0) (6) obeys the Slavnov-Taylor identity (22). Moreover, the operation of S Γ (0) upon the fields and theexternal sources (antifields) is given by S Γ (0) φ = sφ , φ = { ψ + , ψ − , ψ + , ψ − , A µ , a µ , b, c, c, π, ξ, ξ } , S Γ (0) Ω + = − δ Γ (0) δψ + , S Γ (0) Ω + = δ Γ (0) δψ + , S Γ (0) Ω − = δ Γ (0) δψ − , S Γ (0) Ω − = − δ Γ (0) δψ − . (27)In addition to the Slavnov-Taylor identity (22), the classical action Γ (0) (6) satisfies the following gauge conditions,antighost equations and ghost equations: δ Γ (0) δb = ∂ µ A µ + αb , − i δ Γ (0) δc = i (cid:3) c + Ω + ψ + − Ω − ψ − + ψ + Ω + − ψ − Ω − , δ Γ (0) δc = (cid:3) c ; (28) δ Γ (0) δπ = ∂ µ a µ + βπ , − i δ Γ (0) δξ = i (cid:3) ξ + Ω + ψ + − Ω − ψ − + ψ + Ω + − ψ − Ω − , δ Γ (0) δξ = (cid:3) ξ . (29)Furthermore, the action Γ (0) (6) is invariant under the two rigid symmetries associated to U A (1) × U a (1): W e rig Γ (0) = 0 and W g rig Γ (0) = 0 , (30)where the Ward operators, W e rig and W g rig , read W e rig = Z d x (cid:26) ψ + δδψ + − ψ + δδψ + + Ω + δδ Ω + − Ω + δδ Ω + + ψ − δδψ − − ψ − δδψ − + Ω − δδ Ω − − Ω − δδ Ω − (cid:27) , (31) W g rig = Z d x (cid:26) ψ + δδψ + − ψ + δδψ + + Ω + δδ Ω + − Ω + δδ Ω + − ψ − δδψ − + ψ − δδψ − − Ω − δδ Ω − + Ω − δδ Ω − (cid:27) . (32)The U A (1) × U a (1) gauge invariant action Γ (0) (6) being even under the parity transformation ( P ) fixes its actionupon the fields and external sources: ψ + P −→ ψ P + = − iγ ψ − , ψ − P −→ ψ P − = − iγ ψ + , ψ + P −→ ψ P + = iψ − γ , ψ − P −→ ψ P − = iψ + γ ;Ω + P −→ Ω P + = − iγ ψ − , Ω − P −→ Ω P − = − iγ ψ + , Ω + P −→ Ω P + = i Ω − γ , Ω − P −→ Ω P − = i Ω + γ ; A µ P −→ A Pµ = ( A , − A , A ) ; φ P −→ φ P = φ , φ = { b, c, c } ; a µ P −→ a Pµ = ( − a , a , − a ) ; χ P −→ χ P = − χ , χ = { π, ξ, ξ } . (33) III. THE STABILITY CONDITION: IN SEARCH FOR COUNTERTERMS
The stability condition, i.e. the multiplicative renormalizability, is ensured if perturbative quantum corrections donot produce local counterterms corresponding to renormalization of parameters which are not already present in theclassical theory, therefore thus those radiative corrections can be reabsorbed order by order through redefinitions ofthe initial physical quantities – fields, coupling constants and masses – of the theory. Consequently, in order to verifyif the classical action Γ (0) (6) is stable under radiative corrections, it is perturbed by an arbitrary integrated localfunctional (counterterm) Σ c , namely Γ ε = Γ (0) + ε Σ c , such that ε is an infinitesimal parameter and the countertermaction Σ c has the same quantum numbers as the tree-level action Γ (0) (6). Provided that the deformed action Γ ε satisfies all the conditions fulfilled by the classical action Γ (0) (6), this leads to the counterterm Σ c be subjected tothe following set of constraints: S Γ (0) Σ c = 0 ; (34) W e rig Σ c = 0 , W g rig Σ c = 0 ; (35) δ Σ c δb = δ Σ c δc = δ Σ c δc = 0 , δ Σ c δπ = δ Σ c δξ = δ Σ c δξ = 0 ; (36) δ Σ c δ Ω + = δ Σ c δ Ω + = 0 , δ Σ c δ Ω − = δ Σ c δ Ω − = 0 . (37)At this point, it should be emphasized that, as far as rigid gauge invariance is concerned, since the symmetry group U A (1) × U a (1) is a non-semisimple Lie group, in principle rigid invariance could be broken at the quantum level,in other words, rigid symmetry might be anomalous. Nevertheless, none of both abelian factors are spontaneouslybroken as well as the conditions displayed in (30), W e rig Γ (0) = 0 and W g rig Γ (0) = 0, express indeed the conservation ofthe electric ( e ) and the chiral ( g ) charges, therefore the conditions exhibited in (35), W e rig Σ c = 0 and W g rig Σ c = 0, arestill valid [15, 16].The most general Lorentz invariant and vanishing ghost number field polynomial (Σ c ) fulfilling the conditions(34)–(37) with ultraviolet dimension bounded by d ≤
3, reads:Σ c = Z d x n α iψ + /Dψ + + α iψ − /Dψ − + α ψ + ψ + + α ψ − ψ − + α F µν F µν + α f µν f µν + α ǫ µρν A µ ∂ ρ a ν o , (38)where α i ( i = 1 , . . . ,
7) are at first arbitrary parameters. However, there are other restrictions owing to superrenormal-izability and parity invariance. Since all quantum fields are massive no infrared divergences arise from the ultravioletsubtractions in the Bogoliubov-Parasiuk-Hepp-Zimmermann (BPHZ) renormalization procedure, thereby there is noneed to use the Lowenstein-Zimmermann subtraction scheme [10] – which explicitly breaks parity in three space-timedimensions [2, 4] while the infrared divergences are subtracted – in order to subtract those infrared divergences. Inthis way, as a by-product the BPHZ method is an available parity-preserving subtraction procedure guaranteeingthat at each loop order the counterterm (Σ c ) shall be parity-even. Concerning the model superrenormalizability, thecoupling-constant-dependent power-counting formula [17]: δ ( γ ) = 3 − X Φ d Φ N Φ − N e − N g . (39)is defined for the UV degree of divergence ( δ ( γ )) of a 1-particle irreducible Feynman graph ( γ ) where N Φ is the numberof external lines of γ corresponding to the field Φ, d Φ is the UV dimension of Φ (see Table I), and N e and N g arethe powers of the coupling constants e and g , respectively, in the integral corresponding to the diagram γ . Thanks tofact that counterterms are generated by loop graphs, they are at least of order two in the coupling constants, namely, e , g or eg . Accordingly, the effective UV dimension of the counterterm (Σ c ) is bounded by d ≤
2, implying that, α = α = α = α = 0. Furthermore, since a parity-even subtraction scheme is available thus the counterterm hasto be parity invariant, resulting that α = − α = α , and the counterterm expressed byΣ c = Z d x n α ( ψ + ψ + − ψ − ψ − ) + α ǫ µρν A µ ∂ ρ a ν o = z m m ∂∂m Γ (0) + z µ µ ∂∂µ Γ (0) , (40)where z m = − αm and z µ = α µ are arbitrary parameters to be fixed, order by order in perturbation theory, by thenormalization conditions:Γ ψ + ψ + ( /p ) (cid:12)(cid:12)(cid:12) / p =+ m = 0 or Γ ψ − ψ − ( /p ) (cid:12)(cid:12)(cid:12) / p = − m = 0 and i ǫ µρν ∂∂p ρ Γ µνAa ( p ) (cid:12)(cid:12)(cid:12) p = κ = µ , (41)with κ being an energy scale. The counterterm (40) shows that, a priori , only the mass parameters m and µ can getradiative corrections. This means that, at all orders in ~ , the β -functions associated to the gauge coupling constants, e and g , are vanishing, β e = 0 and β g = 0, respectively, as well as the anomalous dimensions ( γ ) of the fields.In time, a subtle property of the Chern-Simons piece of action in Γ (0) (6):Σ CS = µ Z d x ǫ µρν A µ ∂ ρ a ν , (42)shall be put in evidence. The Chern-Simons action Σ CS (42) is not BRS local invariant, thus its invariance underBRS transformations is up to a surface term, i.e. total derivative, s Σ CS = s n µ Z d x ǫ µρν A µ ∂ ρ a ν o = − µe Z d x ǫ µρν ∂ µ ( c∂ ρ a ν ) , (43)indicating that at quantum level the β -function associated to the Chern-Simons mass parameter ( µ ) vanishes, β µ = 0[17, 18]. In summary, the counterterm finally readsΣ c = Z d x n α ( ψ + ψ + − ψ − ψ − ) o = z m m ∂∂m Γ (0) , (44)yielding that all β -functions associated to the gauge coupling constants ( e and g ) and the Chern-Simons mass pa-rameter ( µ ), and all anomalous dimensions ( γ ) of the fields, are vanishing, excepting the β -function correspondingto the fermions mass parameter ( m ). As an ultimate result, it can be concluded that the parity-even U A (1) × U a (1)massive QED [5] is ultraviolet finiteness at all orders in perturbation theory, provided no gauge anomalies emergejeopardizing its unitarity. The issue of gauge anomalies is hereafter analyzed by means of the Slavnov-Taylor identitybehavior at the quantum level. IV. THE UNITARITY CONDITION: IN SEARCH FOR ANOMALIES
In view of the fact that the stability condition does not assure the extension of the theory to the quantum level,it still remains to show the inexistence of gauge anomalies and also the parity anomaly, once the latter is sometimesclaimed in the literature as a typical anomaly of three dimensional space-times.The quantum vertex functional (Γ) coincides with the classical action (Γ (0) ) at zeroth-order in ~ ,Γ = Γ (0) + O ( ~ ) , (45)shall fulfill the same conditions (28)–(30) of the classical action.According to the Quantum Action Principle [6], the Slavnov-Taylor identity (22) acquires a quantum breaking: S (Γ) = ∆ · Γ = ∆ + O ( ~ ∆) , (46)where the Slavnov-Taylor breaking ∆ is an integrated local Lorentz invariant functional, with ghost number equal to1 and UV dimension bounded by d ≤ .Taking into consideration the Slavnov-Taylor quantum identity (46), the nilpotency identity (25) applied to thequantum vertex functional, i.e., S Γ S (Γ) = 0, and the equation S Γ = S Γ (0) + O ( ~ ) obtained from (24) and (45), thisall together implies the following consistency condition for the quantum breaking ∆: S Γ (0) ∆ = 0 , (47)Moreover, in addition to (47), calling into question the Slavnov-Taylor identity (22), the gauge, antighost and ghostequations (28)–(29), so as the rigid conditions (30), it is verified that the Slavnov-Taylor quantum breaking (∆) alsosatisfies the constraints: δ ∆ δb = Z d x δ ∆ δc = δ ∆ δc = W e rig ∆ = 0 and δ ∆ δπ = Z d x δ ∆ δξ = δ ∆ δξ = W g rig ∆ = 0 . (48)The Wess-Zumino consistency condition (47) is indeed a cohomology problem in the sector of ghost number one.Consequently, its solution can always be written as a sum of a trivial cocycle S Γ (0) b ∆ (0) , where b ∆ (0) has ghost numberzero, and of nontrivial elements b ∆ (1) lying in the cohomology of S Γ (0) (24) in the sector of ghost number one:∆ (1) = b ∆ (1) + S Γ (0) b ∆ (0) , (49)reminding that the Slavnov-Taylor quantum breaking ∆ (1) (49) has to satisfy the conditions imposed by (47) and (48).It should be highlighted that the trivial cocycle S Γ (0) b ∆ (0) can be absorbed order by order into the vertex functionalΓ, namely S Γ (0) (Γ − b ∆ (0) ) = b ∆ (1) + O ( ~ ∆ (1) ), as a noninvariant integrated local counterterm, − b ∆ (0) . The linearizedSlavnov-Taylor operator S Γ (0) (24) in combination with the Slavnov-Taylor quantum identity (46) results that thebreaking ∆ (1) exhibits UV dimension bounded by d ≤ . Nevertheless, being an effect of radiative corrections, theinsertion ∆ (1) possesses a factor, e , g or eg , at least, hence its effective UV dimension turns out to be bounded by d ≤ .Now, it has been verified that, as displayed in (48), from the antighost equations (28)–(29), the quantum breaking∆ (1) (49) fulfill the constraints: Z d x δ ∆ (1) δc = 0 and Z d x δ ∆ (1) δξ = 0 , (50)then it follows that ∆ (1) reads ∆ (1) = Z d x n K (0) µ ∂ µ c + X (0) µ ∂ µ ξ o , (51)where K (0) µ and X (0) µ are rank-1 tensors with zero ghost number and UV dimension bounded by d ≤ . Beyond that,the breaking ∆ (1) may be split into two pieces, one even and another odd under parity, thus K (0) µ and X (0) µ can bewritten as K (0) µ = X i =1 v k,i V iµ + X i =1 p k,i P iµ and X (0) µ = X i =1 v x,i Υ iµ + X i =1 p x,i Π iµ , (52)with v k,i , p k,i , v x,i and p x,i being fixed coefficients to be further determined. Moreover, V iµ and Υ iµ are defined asvectors, while P iµ and Π iµ as pseudo-vectors, in such a way that V iµ ∂ µ c and Π iµ ∂ µ ξ are parity-even, whereas P iµ ∂ µ c and Υ iµ ∂ µ ξ are parity-odd, since ∂ µ c is a vector and ∂ µ ξ a pseudo-vector. Taking into consideration that K (0) µ and X (0) µ have their UV dimensions given by d ≤ and ∆ (1) must fulfill the conditions (47) and (48), it is verified thateven though there are P iµ and Υ iµ surviving all of these constraints, namely, P µ = ǫ µρν ∂ ρ A ν and Υ µ = ǫ µρν ∂ ρ a ν ,their contributions in ∆ (1) (51) are all ruled out by partial integration, therefore effectively for the anomaly analysispurposes, {P iµ } = ∅ and { Υ iµ } = ∅ , thereby leaving only the parity invariant part (∆ (1)even ) of the Slavnov-Taylorquantum breaking ∆ (1) (51): ∆ (1)even = Z d x nX i =1 v k,i V iµ ∂ µ c + X i =1 p x,i Π iµ ∂ µ ξ o , (53)ending the proof that parity is not broken at the quantum level, there is no parity anomaly at all.It still remains to verify if the parity-even breaking ∆ (1)even (53) is a genuine gauge anomaly or just a trivial cocycle thatcould be reabsorbed into the quantum action as noninvariant counterterms. However, it lacks to find the candidatesfor V µ (vector) and Π µ (pseudo-vector) with UV dimensions given by d ≤ provided that ∆ (1)even (53) satisfies theconstraints (47) and (48). So, they read as follows: V µ = A µ A ν A ν , V µ = A µ a ν a ν , V µ = A ν a ν a µ , Π µ = a µ a ν a ν , Π µ = a µ A ν A ν , Π µ = a ν A ν A µ , (54)as a consequence the quantum breaking ∆ (1)even (53) becomes expressed by∆ (1)even = Z d x n v k, A µ A ν A ν ∂ µ c + v k, A µ a ν a ν ∂ µ c + v k, A ν a ν a µ ∂ µ c ++ p x, a µ a ν a ν ∂ µ ξ + p x, a µ A ν A ν ∂ µ ξ + p x, a ν A ν A µ ∂ µ ξ o . (55)At this moment, from the Wess-Zumino consistency condition (47), whether a gauge anomaly owing to radiativecorrections exists or not shall be proved by analyzing the breaking ∆ (1)even (55), consequently if it could be writtenunambiguously as a trivial cocycle S Γ (0) b ∆ (0) , with b ∆ (0) being a zero ghost number integrated local monomials, thenoninvariant counterterms, there is no gauge anomaly and the unitarity is guaranteed, otherwise, if there is at leastone nontrivial element b ∆ (1) belonging to the cohomology of S Γ (0) (24) in the sector of ghost number one, the gaugesymmetry is anomalous and unitarity is definitely jeopardized. In order to give the sequel on the issue of gaugeanomaly, it can be straightforwardly verified that: S Γ (0) b ∆ (0)1 = S Γ (0) Z d x A µ A µ A ν A ν = − e Z d x A µ A ν A ν ∂ µ c , (56) S Γ (0) b ∆ (0)2 = S Γ (0) Z d x a µ a µ a ν a ν = − g Z d x a µ a ν a ν ∂ µ ξ , (57) S Γ (0) b ∆ (0)3 = S Γ (0) Z d x A µ A µ a ν a ν = − e Z d x A µ a ν a ν ∂ µ c − g Z d x a µ A ν A ν ∂ µ ξ , (58) S Γ (0) b ∆ (0)4 = S Γ (0) Z d x A µ A ν a ν a µ = − e Z d x A ν a ν a µ ∂ µ c − g Z d x a ν A ν A µ ∂ µ ξ . (59)Besides that, by taking into consideration the four trivial cocycles above (56)–(59), it follows that the quantumbreaking (53) might be rewritten as:∆ (1)even = S Γ (0) n λ b ∆ (0)1 + λ b ∆ (0)2 + λ b ∆ (0)3 + λ b ∆ (0)4 o = S Γ (0) (cid:26) λ Z d x A µ A µ A ν A ν + λ Z d x a µ a µ a ν a ν + λ Z d x A µ A µ a ν a ν + λ Z d x A µ A ν a ν a µ (cid:27) = Z d x n − e λ A µ A ν A ν ∂ µ c − e λ A µ a ν a ν ∂ µ c − e λ A ν a ν a µ ∂ µ c + − g λ a µ a ν a ν ∂ µ ξ − g λ a µ A ν A ν ∂ µ ξ − g λ a ν A ν A µ ∂ µ ξ o , (60)where it can be checked that v k, = − e λ , v k, = − e λ , v k, = − e λ , p x, = − g λ , p x, = − g λ , p x, = − g λ , (61)thus finally demonstrating that the quantum breaking ∆ (1)even (53) is actually a trivial cocycle S Γ (0) b ∆ (0) :∆ (1)even = S Γ (0) b ∆ (0) = S Γ (0) n λ b ∆ (0)1 + λ b ∆ (0)2 + λ b ∆ (0)3 + λ b ∆ (0)4 o . (62)Therefore, as a final result, the integrated local monomials b ∆ (0)1 , b ∆ (0)2 , b ∆ (0)3 and b ∆ (0)4 can be absorbed as nonin-variant counterterms order by order into the quantum action, e.g. , at n -order in ~ : S Γ (0) (Γ − ~ n b ∆ (0) ) ≡ S Γ (0) (cid:16) Γ − ~ n λ b ∆ (0)1 − ~ n λ b ∆ (0)2 − ~ n λ b ∆ (0)3 − ~ n λ b ∆ (0)4 (cid:17) = 0 ~ n + O ( ~ n +1 ) , (63)which concludes the proof on the absence of gauge anomaly, meaning that the U A (1) × U a (1) local symmetry is notanomalous at the quantum level.In summary, the last result (63), about the Wess-Zumino condition, combined with the previous one (44), thestability condition analysis, completes the proof of vanishing β -functions associated to the gauge coupling constants( e and g ) and the Chern-Simons mass parameter ( µ ), and all anomalous dimensions ( γ ) of the fields, as well as theabsence of parity and gauge anomaly at all orders in perturbation theory. Finally, as a by-product, it can also bededuced that the parity-even U A (1) × U a (1) massive QED [5] is ultraviolet finiteness. V. CONCLUSION
In conclusion, the parity-even U A (1) × U a (1) massive QED [5] is parity and gauge anomaly free at all orders inperturbation theory. Beyond that, it exhibits vanishing β -functions associated to the gauge coupling constants ( e and g ) and the Chern-Simons mass parameter ( µ ), and all the anomalous dimensions ( γ ) of the fields as well. Theproof is independent of any regularization scheme or any particular diagrammatic calculation, it is based on thequantum action principle in combination with general theorems of perturbative quantum field theory by adopting theBRS (Becchi-Rouet-Stora) algebraic renormalization method in the framework of BPHZ (Bogoliubov-Parasiuk-Hepp-Zimmermann) subtraction scheme [6–10]. As a final comment, once the quantum perturbative physical consistencyof the mass-gap graphene-like planar quantum electrodynamics has been proven from the results demonstrated heretogether with those presented in [5], it should be newsworthy to deepen its analysis so as to apply in graphene-likeelectronic systems [19]. Acknowledgements
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