The Parity-Preserving Massive QED3: Vanishing Β-Function and No Parity Anomaly

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Physics Letters B 750 (2015) 1–5

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Physics Letters B

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The parity-preserving massive QED3: Vanishing β-function and no parity anomaly

O.M. Del Cima

Universidade Federal de Viçosa (UFV), Departamento de Física, Campus Universitário, Avenida Peter Henry Rolfs s/n, 36570-900 Viçosa, MG, Brazil

a r t i c l e i n f o a b s t r a c t

Article history: The parity-preserving massive QED3 exhibits vanishing gauge coupling β-function and is parity and Received 3 June 2015 infrared anomaly free at all orders in perturbation theory. Parity is not an anomalous symmetry, even for Received in revised form 10 August 2015 the parity-preserving massive QED3, in spite of some claims about the possibility of a perturbative parity Accepted 12 August 2015 breakdown, called parity anomaly. The proof is done by using the algebraic renormalization method, Available online 15 August 2015 which is independent of any regularization scheme, based on general theorems of perturbative quantum Editor: M. Cveticˇ ﬁeld theory. In honor of Prof. Raymond Stora © 2015 Published by Elsevier B.V. This is an open access article under the CC BY license (1930–2015) (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

The quantum electrodynamics in three space–time dimensions rections, are quite non-trivial. Therefore, in those cases previously (QED3) has raised a great deal of interest since the precursor mentioned, the massive QED3 shall exhibit distinct behaviours work by Deser, Jackiw and Templeton [1] in view of a possible and properties [7,8] as compared to the case presented in this theoretical foundation for condensed matter phenomena, such as work. high-Tc superconductivity, quantum Hall effect and, more recently, The proof presented in this letter on the absence of parity and graphene and topological insulators. The massive and the massless infrared anomaly, and the vanishing gauge coupling β-function, in QED3 can exhibit interesting and subtle properties, namely su- the parity-even massive QED3, is based on general theorems of perrenormalizability [2], parity violation, topological gauge fields, perturbative quantum field theory [9–12], where the Lowenstein– anyons and the presence of infrared divergences. The massless Zimmermann subtraction scheme in the framework of Bogoliubov– QED3 is ultraviolet and infrared perturbatively finite, infrared and Parasiuk–Hepp–Zimmermann–Lowenstein (BPHZL) renormalization parity anomaly free at all orders [3], despite some statements method [12] is adopted. The former has to be introduced, owing found out in the literature that still support that parity could be to the presence of massless gauge field, so as to subtract infrared broken even perturbatively, called parity anomaly, which has al- divergences that should arise from the ultraviolet subtractions. ready been discarded [3–6]. The massless QED3 is parity-even at The issue of the extension of parity-even massive QED3 in the the classical and quantum level (at least perturbatively), however, tree-approximation to all orders in perturbation theory is orga- at the classical level, the massive QED3 can be odd or even un- nized according to two independent parts. First, it is analyzed the der parity symmetry. For the parity-even massive QED3, if whether stability of the classical action – for the quantum theory, the sta- parity is a quantum symmetry or not, shall be definitely proved bility corresponds to the fact that the radiative corrections can be by using a renormalization method independent of any regular- reabsorbed by a redefinition of the initial parameters of the theory. ization scheme. The massive QED3 has also been studied in de- Second, it is computed all possible anomalies through an analy- tails in many other physical configurations, namely, large gauge sis of the Wess–Zumino consistency condition, furthermore, it is transformations, non-Abelian gauge groups, odd and even under checked if the possible breakings induced by radiative corrections parity, fermions families, compact space–times, space–times with can be fine-tuned by a suitable choice of local non-invariant coun- boundaries, external fields and finite temperatures – in all of these terterms. It shall be stressed that when massless fields are present, situations, the issues of parity breaking or parity preserving at infrared divergences may appear from non-invariant counterterms, the quantum level, renormalizability and finite temperature cor- called infrared anomalies. The gauge invariant action for the parity-preserving massive QED3, with the gauge invariant Lowenstein–Zimmermann (LZ) E-mail address: [email protected]. mass term added, is given by:

http://dx.doi.org/10.1016/j.physletb.2015.08.031 0370-2693/© 2015 Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 2 O.M. Del Cima / Physics Letters B 750 (2015) 1–5 (s−1) 3 1 μν By switching off the coupling constant (e) and taking the free = d x − F F + iψ+Dψ+ + iψ−Dψ− + inv μν / / (s−1) 4 part of the action, + ((1) and (2)), the tree-level propa- inv gf gators in momenta space, for all the fields, read: μ μρν −m(ψ+ψ+ − ψ−ψ−) + (s − 1) Aμ∂ρ Aν , (1) 2 k/ + m k/ − m ++(k) = i , −−(k) = i , (6) 2 − 2 2 − 2 LZ mass term k m k m μ ν ≡ + μν 1 k k where D/ψ± (/∂ ie/A)ψ±, e is a dimensionful coupling constant (k, s) =−i ημν − + with mass dimension 1 , and m is a mass parameter with mass di- AA k2 − μ2(s − 1)2 k2 2 mension 1. In action (1), Fμν is the field strength for Aμ, Fμν = − μ ν μ(s 1) μρν ξ k k ∂μ Aν − ∂ν Aμ, and, ψ+ and ψ− are two kinds of fermions where +i kρ + , (7) k2[k2 − 2(s − 1)2] k2 k2 the ± subscripts refer to their spin sign [13], also, the gamma μ μ μ matrices are γ = (σz, iσx, iσy). The Lowenstein–Zimmermann pa- μ k (k) = , bb(k) = 0 , (8) rameter s lies in the interval 0 ≤ s ≤ 1 and plays the role of an Ab k2 additional subtraction variable (as the external momentum) in the 1 cc(k) =−i . (9) BPHZL renormalization program, such that the parity-even massive k2 QED3 is recovered for s = 1. At this moment, in order to establish the ultraviolet (UV) and in- In the BPHZL scheme a subtracted (finite) integrand, R(p, k, s), frared (IR) dimensions of any fields, X and Y , we make use of the is written in terms of the unsubtracted (divergent) one, I(p, k, s), UV and IR asymptotical behaviour of their propagator, (k, s), as XY dXY and r XY , respectively: = − 0 − 1 R(p,k, s) (1 tp,s−1)(1 tp,s)I(p,k, s) = dXY deg(k,s) XY(k, s), (10) = (1 − t0 − t1 + t0 t1 )I(p,k, s), p,s−1 p,s p,s−1 p,s r = deg (k, s), (11) XY (k,s−1) XY where td is the Taylor series about x = y = 0to order d if d ≥ 0. x,y where the upper degree deg gives the asymptotic power for Thus, since the Lowenstein–Zimmermann mass term presented (k,s) (k, s) →∞ whereas the lower degree deg gives the asymp- in (1) breaks parity, by assuming s = 1, a subtracted integrand, (k,s−1) − → R(p, k, s), reads totic power for (k, s 1) 0. The UV (d) and IR (r) dimensions of the fields, X and Y , are chosen to fulfill the following inequalities: ∂ R p k 1 = I p k 1 − I 0 k 1 − pρ I 0 k 0 ( , , ) ( ,, ) ( ,, ) ρ ( , , ) . d + d ≥ 3 + d and r + r ≤ 3 + r . (12) ∂ p X Y XY X Y XY parity-even parity-even parity-odd terms In order to fix the UV and IR dimensions of the spinor fields ψ+ and ψ−, and the vector field Aμ, use has been made of the In order to quantize the model, represented by the action (1), propagators, (6) and (7) together with the conditions (12), then, aparity-even gauge-fixing action, gf, is added: the following relations stem: ξ 3 μ 2 d±± =−1 ⇒ 2d± ≥ 2 → d± = 1 , (13) gf = d x b∂ Aμ + b + cc , (2) 2 3 r±± = 0 ⇒ 2r± ≤ 3 → r± = ; (14) together with a parity-even action term, ext, coupling the non-lin- 2 ear Becchi–Rouet–Stora (BRS) transformations to external sources: 1 dAA =−2 ⇒ 2dA ≥ 1 → dA = , (15) 2 3 ext = d x +sψ+ − −sψ− + 1 r AA =−2 ⇒ 2r A ≤ 1 → r A = . (16) 2 − sψ+ + + sψ− − . (3) From the propagators (8) and the conditions, (12), (15) and (16), it The BRS transformations are given by: can fixed the UV and IR dimensions of the Lautrup–Nakanishi field b as follows: sψ+ = icψ+ , sψ+ =−icψ+ , 1 3 dAb =−1 ⇒ dA + db ≥ 2 , dA = → db = , (17) sψ− = icψ− , sψ− =−icψ− , 2 2 1 =− ⇒ + ≤ = 1 → = 3 sAμ =− ∂μc , sc = 0 , r Ab 1 r A rb 2 , r A rb . (18) e 2 2 1 The dimensions (UV and IR) of the Faddeev–Popov ghost (c) and sc = b , sb = 0 , (4) e antighost (c¯) are fixed, by considering the propagators (9), such that: where c is the ghost, c is the antighost and b is the Lautrup– Nakanishi field [14], playing the role of the Lagrange multiplier d¯ =−2 ⇒ d + d¯ ≥ 1 , (19) field. In spite of been massless, since the Faddeev–Popov ghosts are cc c c free fields, they decouple, therefore, no Lowenstein–Zimmermann rcc¯ =−2 ⇒ rc + rc¯ ≤ 1 . (20) mass term has to be introduced for them. − Also, assuming that the BRS operator s (4) is dimensionless and The complete action, (s 1), reads bearing in mind that the coupling constant e has dimension − 1 (s−1) (s 1) (mass) 2 , the UV and IR dimensions for the ghost and antighost = + gf + ext , (5) inv result: in such a way that the parity-preserving massive QED3 is recovered (s−1) taking s = 1, ≡ |s=1. dc = 0anddc¯ = 1 ; rc = 0andrc¯ = 1 . (21) O.M. Del Cima / Physics Letters B 750 (2015) 1–5 3

Table 1 In addition to the Slavnov–Taylor identity (23), the classical action − UV (d) and IR (r) dimensions, ghost number () and Grassmann parity (GP). (s 1) (5) is characterized by the gauge condition, the ghost equa- Aμ ψ+ ψ− c cb + − s − 1 s tion and the antighost equation:

d 1/211013/22211 (s−1) r 1/23/23/201 3/23/23/21 0 δ μ = ∂ Aμ + ξb , (29) 0001−10 −1 −10 0 δb GP 011110000 0 δ(s−1) = c , (30) δc Finally, from the action of the antifields (3), and the UV and IR − δ(s 1) dimensions of the fields fixed previously, it follows that: −i = ic + +ψ+ + ψ+ + + δc 3 − −ψ− − ψ− − . (31) d ± = 2andr ± = . (22) 2 − The action (s 1) (5) is invariant also with respect to the rigid In summary, the UV (d) and IR (r) dimensions – which symmetry are those involved in the Lowenstein–Zimmermann subtraction scheme [12] – as well as the ghost numbers () and the Grass- (s−1) W rigid = 0 , (32) mann parity (GP) of all fields are collected in Table 1. Notice that the statistics is defined as follows. The integer spin fields with odd where the Ward operator, W rigid, is defined by ghost number, as well as, the half integer spin fields with even ghost number anticommute among themselves. However, the other 3 δ δ δ δ W rigid = d x ψ+ − ψ+ + + − + + fields commute with the formers and also among themselves. δψ+ δψ+ δ + δ + The BRS invariance of the action is expressed in a functional δ δ δ δ way by the Slavnov–Taylor identity + ψ− − ψ− + − − − . (33) δψ− δψ− δ − δ − S (s−1) = ( ) 0 , (23) (s−1) The parity-preserving massive QED3 (s = 1) action, |s=1, where the Slavnov–Taylor operator S is defined, acting on an arbi- is invariant under parity (P ), its action upon the fields and external trary functional F, by sources is fixed as below: F F P P 1 δ 1 δ xμ −→ x = (x0, −x1, x2), S(F) = d3x − ∂μc + b + μ e δ Aμ e δc −→P P =− 1 −→P P = 1 F F F F ψ+ ψ+ iγ ψ− , ψ+ ψ+ iψ−γ , + δ δ − δ δ + P P 1 P P 1 δ + δψ+ δ + δψ+ ψ− −→ ψ =−iγ ψ+ , ψ− −→ ψ = iψ+γ , − − F F F F δ δ δ δ −→P P = − − + . (24) Aμ Aμ (A0, A1, A2), δ − δψ− δ − δψ− P φ −→ φ P = φ, φ={c, c¯, b} , The corresponding linearized Slavnov–Taylor operator reads P P 1 P P 1 + −→ + =−iγ − , + −→ + = i −γ , 1 δ 1 δ S = 3 − μ + + F d x ∂ c b P P 1 P P 1 e δ Aμ e δc − −→ − =−iγ + , − −→ − = i +γ . (34) δF δ δF δ δF δ δF δ + + − − + In order to verify if the action in the tree-approximation − δ + δψ+ δψ+ δ + δ + δψ+ δψ+ δ + ((s 1)) is stable under radiative corrections, we perturb it by an − δF δ δF δ arbitrary integrated local functional (counterterm) c(s 1), such − − + that δ − δψ− δψ− δ − − − − δF δ δF δ (s 1) = (s 1) + εc(s 1) , (35) + + . (25) δ − δψ− δψ− δ − where ε is an infinitesimal parameter. The functional c ≡ c(s−1) The following nilpotency identities hold: |s=1 has the same quantum numbers as the action in the tree-approximation at s = 1. (s−1) SF S(F) = 0 , ∀F , (26) The deformed action must still obey all the conditions − presented above, henceforth, c(s 1) is subjected to the following S S = S F = F F 0if ( ) 0 . (27) set of constraints: S 2 = (s−1) In particular, ( ) 0, since the action obeys the Slavnov– c(s−1) S = 0 , (36) Taylor identity (23). The operation of S upon the fields and the − − − external sources is given by δc(s 1) δc(s 1) δc(s 1) = = = 0 , (37) δb δc δc S = ={ } φ sφ, φ ψ±, ψ±, Aμ, c, c, b , c(s−1) W rigid = 0 . (38) (s−1) (s−1) δ δ − S + =− , S + = , The most general invariant counterterm c(s 1) – the most δψ+ δψ+ general field polynomial – with UV and IR dimensions bounded − − δ(s 1) δ(s 1) by d ≤ 3 and r ≥ 3, with ghost number zero and fulfilling the con- S − = , S − =− . (28) δψ− δψ− ditions displayed in Eqs. (36)–(38), reads: 4 O.M. Del Cima / Physics Letters B 750 (2015) 1–5

c(s−1) 3 implies the following consistency condition for the breaking : = d x α1iψ+D/ψ+ + α2iψ−D/ψ− + S = + α ψ+ψ+ + α ψ−ψ− + 0 , (45) 3 4 μν μρν + α5 F Fμν + α6 Aμ∂ρ Aν . (39) and beyond that, also satisfies the constraints: where αi (i = 1, ..., 6) are, in principle, arbitrary parameters. How- δ δ 3 δ = = d x = W rigid = 0 . (46) ever, there are other restrictions owing to the superrenormaliz- δb δc δc ability of the theory and its parity invariance – the parity-even The Wess–Zumino consistency condition (45) constitutes a co- massive QED3 recovered for s = 1. On account of the superrenor- homology problem in the sector of ghost number one. Its solution malizability, the coupling constant-dependent power-counting for- can always be written as a sum of a trivial cocycle S (0), where mula [8,15] is given by: (0) has ghost number 0, and of nontrivial elements belonging to S the cohomology of (25) in the sector of ghost number one: δ(γ ) d 1 (1) = (1) + S (0) = 3 − N − Ne , (40) . (47) ρ(γ ) r 2 It shall be stressed that there still remains a possible parity vi- for the UV (δ(γ )) and IR (ρ(γ )) degrees of divergence of a olation at the quantum level induced by parity-odd noninvariant 1-particle irreducible Feynman graph, γ . Here N is the num- counterterms. Due to the fact that the Lowenstein–Zimmermann ber of external lines of γ corresponding to the field , d and subtraction method breaks parity during the intermediary steps, (1) r are the UV and IR dimensions of , respectively, as given in the Slavnov–Taylor identity breaking, , is not necessarily parity (1) Table 1, and Ne is the power of the coupling constant e in the in- invariant. In any case, must satisfy the conditions imposed by (0) tegral corresponding to the diagram γ . Due to the fact that the (45) and (46). The trivial cocycle S can be absorbed into the − counterterms are generated by loop graphs, they are at least of or- vertex functional (s 1) as a noninvariant integrated local coun- (0) (1) der two in the coupling constant (e). Consequently, the effective terterm, − . On the other hand, a nonzero would repre- − UV and IR dimensions of the counterterm c(s 1) are bounded by (0) sent an anomaly. If by chance, there exist any parity-odd odd, d ≤ 2 and r ≥ 2, then, α1 = α2 = α5 = 0. Furthermore, the coun- a parity anomaly would be present induced by the noninvariant c ≡ c(s−1)| = (0) terterm s=1 is parity invariant, yielding that α6 0 counterterm, − . =− = odd and α3 α4 α. Finally, it can be concluded that the countert- Taking into account the Slavnov–Taylor operator S (25) and erm results as (1) the quantum breaking (43), it results that the breaking exhibits UV and IR dimensions bounded by d ≤ 7 and r ≥ 3. Never- c ≡ c(s−1)| = 3 { − } 2 s=1 d x α(ψ+ψ+ ψ−ψ−) , theless, being an effect of the radiative corrections, the insertion (1) possesses a factor e2 at least, then its effective UV and IR di- ∂ ≤ 5 ≥ = zmm , (41) mensions are bounded by d 2 and r 2, respectively. ∂m From the antighost equation: =−α where zm is an arbitrary parameter (as α is, zm m ), and (1) (s−1) 3 δ ≡ |s=1. The counterterm (41) shows that, a priori, only d x = 0 , (48) the mass parameter m can get radiative corrections. This means δc that the βe -function related to the gauge coupling constant (e) is it follows that (1) can be written as vanishing (βe = 0) to all orders of perturbation theory, so as the anomalous dimensions of the fields. (1) 3 μ = d x Kμ∂ c , (49) Owing to the fact that classical stability does not imply the possibility of extending the theory to the quantum level, it still where Kμ is a rank-1 tensor with ghost number 0, with UV and IR lacks to show the absence of gauge anomaly, infrared anomaly dimensions bounded by d ≤ 3 and r ≥ 1(the ghost c is dimension- and the claimed parity anomaly. This result, combined with the 2 (1) previous one (41), completes the proof of vanishing gauge cou- less), respectively. The breaking can be split into two pieces, K pling β-function and the absence of infrared and parity anomaly which are even and odd under parity, by writing μ as in parity-even massive QED3 at all orders in perturbation theory. − K = V + P At the quantum level the vertex functional, (s 1), which coin- μ rv μ rp μ , (50) − cides with the classical action, (s 1) (5), at 0th order in h¯ , in such a manner that Vμ is a vector and Pμ a pseudo-vector. K (s−1) = (s−1) + O ¯ Bearing in mind that μ has its UV and IR dimensions bounded (h), (42) ≤ 3 ≥ V by d 2 and r 1, it can be concluded that there are no μ sat- has to satisfy the same constraints as the classical action does, isfying these dimensional constraints and the conditions (45) and namely Eqs. (29)–(32). (46), therefore, {Vμ} =∅, which means the absence of a parity- In accordance with the Quantum Action Principle [9,11], the even Slavnov–Taylor breaking. However, still remains the odd sec- Slavnov–Taylor identity (23) gets a quantum breaking: tor represented by Pμ, and it follows that by a dimensional analy- sis a candidate for Pμ, which satisfies also the conditions (45) and (s−1) (s−1) S( )|s=1 = · |s=1 = + O(h¯ ) , (43) (46), shows up: where ≡ |s=1 is an integrated local functional, taken at s = 1, 1 P = = ρν with ghost number 1 and UV and IR dimensions bounded by d ≤ 7 μ Fμ μρν F . (51) 2 2 and r ≥ 3. (1) The nilpotency identity (26) together with It turns out that there is only one parity-odd candidate, odd, which could be a parity anomaly, surviving all the constraints S = S + O(h¯ ), (44) above: O.M. Del Cima / Physics Letters B 750 (2015) 1–5 5 (1) (1) rp 3 ρν μ = = d x μρν F ∂ c , (52) References odd 2 [1] S. Deser, R. Jackiw, S. Templeton, Ann. Phys. (N.Y.) 140 (1982) 372. however, integrating it by parts, leads that [2] R. Jackiw, S. Templeton, Phys. Rev. D 23 (1981) 2291. (1) = (1) ≡ [3] O.M. Del Cima, D.H.T. Franco, O. Piguet, M. Schweda, Phys. Lett. B 680 (2009) odd 0 . (53) 108; O.M. Del Cima, D.H.T. Franco, O. Piguet, Phys. Rev. D 89 (2014) 065001. Hence it follows that, there is no radiative corrections to the [4] S. Rao, R. Yahalom, Phys. Lett. B 172 (1986) 227. insertion describing the breaking of the Slavnov–Taylor identity, [5] R. Delbourgo, A.B. Waites, Phys. Lett. B 300 (1993) 241; { (1)} =∅, which means that there is no possible breaking to R. Delbourgo, A.B. 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Zimmermann, Commun. Math. Phys. 15 (1969) 208; J.H. Lowenstein, W. Zimmermann, Nucl. Phys. B 86 (1975) 77; ization scheme, nor any particular diagrammatic calculation, and is J.H. Lowenstein, Commun. Math. Phys. 47 (1976) 53; based on general theorems of perturbative quantum field theory. P. Breitenlohner, D. Maison, Commun. Math. Phys. 52 (1977) 55. [13] B. Binegar, J. Math. Phys. 23 (1982) 1511. Acknowledgements [14] N. Nakanishi, Prog. Theor. Phys. 35 (1966) 1111; N. Nakanishi, Prog. Theor. Phys. 37 (1967) 618; O.M.D.C. dedicates this work to his father (Oswaldo Del Cima, B. Lautrup, Mat. Fys. Medd. Dan. Vid. Selsk 35 (11) (1967). [15] O.M. Del Cima, D.H.T. Franco, J.A. Helayël-Neto, O. Piguet, Lett. Math. Phys. 47 in memoriam), mother (Victoria M. Del Cima, in memoriam), daugh- (1999) 265; ter (Vittoria) and son (Enzo). He also thanks the referee for useful O.M. Del Cima, D.H.T. Franco, J.A. Helayël-Neto, O. Piguet, J. High Energy Phys. comments and suggestions. 9804 (1998) 010.