View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Open Access Repository Physics Letters B 750 (2015) 1–5 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb 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 3 (1930–2015) (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 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 − μ2(s − 1)2] k2 k2 the ± subscripts refer to their spin sign [13], also, the gamma 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 dcc¯ =−2 ⇒ dc + dc¯ ≥ 1 , (19) field.