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Contents lists available atScienceDirect

Physics

Letters

B

www.elsevier.com/locate/physletb

The

parity-preserving

massive

QED

3

:

Vanishing

β

-function

and

no

parity

anomaly

O.M. Del Cima

UniversidadeFederaldeViçosa(UFV),DepartamentodeFísica,CampusUniversitário,AvenidaPeterHenryRolfss/n,36570-900Viçosa,MG,Brazil

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory: Received3June2015

Receivedinrevisedform10August2015 Accepted12August2015

Availableonline15August2015 Editor:M.Cvetiˇc

InhonorofProf.RaymondStora (1930–2015)

The parity-preserving massive QED3 exhibits vanishing gauge coupling β-function and is parity and infraredanomalyfreeatallordersinperturbationtheory.Parityisnotananomaloussymmetry,evenfor theparity-preservingmassiveQED3,inspiteofsomeclaimsaboutthepossibilityofaperturbativeparity breakdown, called parityanomaly. The proof is doneby usingthe algebraicrenormalization method, whichisindependentofanyregularizationscheme,basedongeneraltheoremsofperturbativequantum fieldtheory.

©2015TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense (http://creativecommons.org/licenses/by/4.0/).FundedbySCOAP3.

Thequantumelectrodynamicsinthreespace–timedimensions (QED3) has raised a great deal of interest since the precursor

work by Deser, Jackiw and Templeton [1] in view of a possible theoreticalfoundation forcondensed matterphenomena, such as high-Tc superconductivity,quantumHalleffectand,morerecently,

grapheneandtopologicalinsulators.Themassiveandthemassless QED3 can exhibit interesting and subtle properties, namely

su-perrenormalizability[2],parity violation, topological gauge fields, anyons and the presence of infrared divergences. The massless QED3 isultravioletandinfraredperturbativelyfinite,infraredand

parity anomaly free at all orders [3], despite some statements foundout in theliterature that still support thatparity could be brokeneven perturbatively, called parity anomaly, which has al-readybeen discarded [3–6]. The massless QED3 is parity-even at

theclassicalandquantum level(atleastperturbatively),however, atthe classical level, the massive QED3 can be odd or even

un-derparitysymmetry.Fortheparity-evenmassiveQED3,ifwhether

parityis a quantum symmetry ornot, shall be definitely proved by using a renormalization method independent of any regular-ization scheme.The massive QED3 has also been studied in

de-tails in many other physical configurations, namely, large gauge transformations, non-Abelian gauge groups, odd and even under parity,fermions families, compact space–times,space–times with boundaries,externalfieldsandfinitetemperatures–inallofthese situations, the issues of parity breaking or parity preserving at the quantum level, renormalizability and finite temperature

cor-E-mailaddress:oswaldo.delcima@ufv.br.

rections,arequitenon-trivial. Therefore,inthosecasespreviously mentioned, the massive QED3 shall exhibit distinct behaviours

and properties [7,8] as compared to the case presented in this work.

Theproofpresentedinthisletterontheabsenceofparityand infraredanomaly,andthevanishinggaugecoupling

β

-function,in the parity-even massive QED3, is based on general theorems of

perturbative quantumfieldtheory

[9–12]

,wheretheLowenstein– ZimmermannsubtractionschemeintheframeworkofBogoliubov– Parasiuk–Hepp–Zimmermann–Lowenstein(BPHZL)renormalization method [12] isadopted.The former hasto be introduced,owing tothepresenceofmassless gaugefield,so astosubtract infrared divergencesthatshouldarisefromtheultravioletsubtractions.

Theissueoftheextension ofparity-evenmassiveQED3 inthe

tree-approximation to all orders in perturbation theory is orga-nizedaccordingtotwoindependentparts.First,itisanalyzed the stability ofthe classicalaction–forthequantumtheory,the sta-bilitycorrespondstothefactthattheradiativecorrectionscanbe reabsorbedbyaredefinitionoftheinitialparametersofthetheory. Second, it is computedall possible anomalies through an analy-sis of the Wess–Zumino consistency condition, furthermore,it is checkedifthepossiblebreakings inducedby radiativecorrections canbefine-tunedbyasuitablechoiceoflocalnon-invariant coun-terterms.Itshallbestressedthatwhenmasslessfieldsarepresent, infrareddivergencesmayappearfromnon-invariantcounterterms, calledinfraredanomalies.

The gauge invariant action for the parity-preserving massive QED3, with the gauge invariant Lowenstein–Zimmermann (LZ)

masstermadded,isgivenby:

http://dx.doi.org/10.1016/j.physletb.2015.08.031

0370-2693/©2015TheAuthor.PublishedbyElsevierB.V.ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/).Fundedby SCOAP3.

(2)



(invs−1)

=



d3x



1 4F μνF μν

+

i

ψ

+D

/

ψ

+

+

i

ψ

D

/

ψ

+

m

+

ψ

+

− ψ

ψ

)

+

μ

2

(

s

1

)



μρνA μ

ρAν







LZ mass term



,

(1)

where

/

D

ψ

±

≡ (/∂ +

ieA

/

±,e is adimensionfulcouplingconstant withmassdimension 12,andm isamassparameterwithmass di-mension 1.Inaction (1), Fμν isthe fieldstrength for Aμ, Fμν

=

μ Aν

− ∂

ν Aμ,and,

ψ

+ and

ψ

aretwo kindsoffermionswhere the

±

subscripts refer to their spin sign [13], also, the gamma matricesare

γ

μ

= (

σ

z

,

iσx

,

iσy

)

.TheLowenstein–Zimmermann pa-rameter s lies inthe interval 0

s

1 and plays the role of an additionalsubtractionvariable(astheexternalmomentum)inthe BPHZLrenormalizationprogram,suchthattheparity-evenmassive QED3isrecoveredfors

=

1.

IntheBPHZLschemea subtracted(finite)integrand, R

(

p

,

k

,

s

)

, iswrittenin termsoftheunsubtracted(divergent) one, I

(

p

,

k

,

s

)

, as

R

(

p

,

k

,

s

)

= (

1

t0p,s−1

)(

1

t1p,s

)

I

(

p

,

k

,

s

)

= (

1

t0p,s1

t1p,s

+

tp0,s1t1p,s

)

I

(

p

,

k

,

s

) ,

wheretd

x,y istheTaylorseriesaboutx

=

y

=

0 toorderd ifd

0. Thus, since the Lowenstein–Zimmermann mass term presented in (1) breaks parity, by assuming s

=

1, a subtracted integrand,

R

(

p

,

k

,

s

)

,reads R

(

p

,

k

,

1

)

=

I

(

p

,

k

,

1

)

  

parity-even

  

I

(

0

,

k

,

1

)

parity-even

pρI

(

0

,

k

,

0

)







parity-odd terms

.

Inorderto quantizethemodel,representedby theaction (1), a parity-evengauge-fixingaction,



gf,isadded:



gf

=



d3x



b

μAμ

+

ξ

2b 2

+

c



c



,

(2)

togetherwithaparity-evenactionterm,



ext,couplingthe

non-lin-earBecchi–Rouet–Stora(BRS)transformationstoexternalsources:



ext

=



d3x

+s

ψ

+

s

ψ

+

s

ψ

+

+

+

s

ψ

.

(3)

TheBRStransformationsaregivenby:

s

ψ

+

=

ic

ψ

+

,

s

ψ

+

= −

ic

ψ

+

,

s

ψ

=

ic

ψ

,

s

ψ

= −

ic

ψ

,

s Aμ

= −

1 e

μc

,

sc

=

0

,

sc

=

1 eb

,

sb

=

0

,

(4)

where c is the ghost, c is the antighost and b is the Lautrup– Nakanishi field [14], playing the role of the Lagrange multiplier field.Inspiteofbeenmassless,sincetheFaddeev–Popovghostsare free fields,theydecouple, therefore,noLowenstein–Zimmermann masstermhastobeintroducedforthem.

Thecompleteaction,



(s−1),reads



(s−1)

= 

inv(s−1)

+ 

gf

+ 

ext

,

(5)

insuchawaythattheparity-preservingmassiveQED3isrecovered

takings

=

1,



≡ 

(s−1)

|

s=1.

Byswitching offthecouplingconstant (e)andtakingthe free partoftheaction,



inv(s−1)

+ 

gf ((1)and

(2)

),thetree-level

propa-gatorsinmomentaspace,forallthefields,read:

++

(

k

)

=

i

/

k

+

m k2

m2

,

−−

(

k

)

=

i

/

k

m k2

m2

,

(6)

μνA A

(

k

,

s

)

= −

i



1 k2

μ

2

(

s

1

)

2

η

μν

kμkν k2

+

+

i

μ

(

s

1

)

k2

[

k2

μ

2

(

s

1

)

2

]



μρνk ρ

+

ξ

k2 kμkν k2



,

(7)

μAb

(

k

)

=

k μ k2

,

bb

(

k

)

=

0

,

(8)

cc

(

k

)

= −

i 1 k2

.

(9)

At thismoment,inorderto establishtheultraviolet(UV)and in-frared(IR)dimensionsofanyfields, X andY ,wemakeuseofthe UV andIR asymptotical behaviour of their propagator,

X Y

(

k

,

s

)

, dX Y andrX Y,respectively:

dX Y

=

deg(k,s)

X Y

(

k

,

s

) ,

(10) rX Y

=

deg(k,s−1)

X Y

(

k

,

s

) ,

(11) where the upper degree deg(k,s) gives the asymptotic power for

(

k

,

s

)

→ ∞

whereasthelower degreedeg

(k,s−1) givesthe asymp-toticpowerfor

(

k

,

s

1

)

0.TheUV(d)andIR(r)dimensionsof thefields,X and Y ,arechosentofulfillthefollowinginequalities:

dX

+

dY

3

+

dX Y and rX

+

rY

3

+

rX Y

.

(12) In order to fixthe UV andIR dimensionsof thespinor fields

ψ

+ and

ψ

−,and thevector field Aμ, use hasbeen madeof the propagators, (6) and(7) together withthe conditions (12), then, thefollowingrelationsstem:

d±±

= −

1

2d±

2

d±

=

1

,

(13) r±±

=

0

2r±

3

r±

=

3 2

;

(14) dA A

= −

2

2dA

1

dA

=

1 2

,

(15) rA A

= −

2

2rA

1

rA

=

1 2

.

(16)

Fromthepropagators

(8)

andtheconditions,

(12)

,

(15)

and

(16)

,it canfixedtheUVandIRdimensionsoftheLautrup–Nakanishifield

b asfollows: dAb

= −

1

dA

+

db

2

,

dA

=

1 2

db

=

3 2

,

(17) rAb

= −

1

rA

+

rb

2

,

rA

=

1 2

rb

=

3 2

.

(18)

The dimensions(UV andIR)ofthe Faddeev–Popovghost (c)and antighost (c)

¯

are fixed, by considering the propagators (9), such that:

dcc¯

= −

2

dc

+

dc¯

1

,

(19) rcc¯

= −

2

rc

+

r¯c

1

.

(20) Also, assuming that the BRS operator s (4) is dimensionlessand bearing in mind that the coupling constant e has dimension

(

mass

)

12, the UV andIR dimensionsfor the ghost and antighost

result:

(3)

Table 1

UV(d)andIR(r)dimensions,ghostnumber()andGrassmannparity(G P ). ψ+ ψc c b + s−1 s

d 1/2 1 1 0 1 3/2 2 2 1 1

r 1/2 3/2 3/2 0 1 3/2 3/2 3/2 1 0

 0 0 0 1 −1 0 −1 −1 0 0

G P 0 1 1 1 1 0 0 0 0 0

Finally,from the action ofthe antifields (3), andthe UV and IR dimensionsofthefieldsfixedpreviously,itfollowsthat:

d ±

=

2 and r ±

=

3

2

.

(22)

In summary, the UV (d) and IR (r) dimensions – which are those involved in the Lowenstein–Zimmermann subtraction scheme

[12]

–aswell astheghostnumbers(



)andthe Grass-mannparity(G P )ofallfieldsarecollectedin

Table 1

.Noticethat thestatisticsisdefinedasfollows.Theintegerspinfieldswithodd ghost number, as well as, the half integer spin fields with even ghostnumberanticommuteamongthemselves.However,theother fieldscommutewiththeformersandalsoamongthemselves.

The BRS invariance of the action is expressed in a functional waybytheSlavnov–Tayloridentity

S

(

(s−1)

)

=

0

,

(23)

wheretheSlavnov–Tayloroperator

S

isdefined,actingonan arbi-traryfunctional

F

,by

S

(

F

)

=



d3x



1 e

μc

δ

F

δ

+

1 eb

δ

F

δ

c

+

+

δ

F

δ

+

δ

F

δψ

+

δ

F

δ

+

δ

F

δψ

+

+

δ

F

δ

δ

F

δψ

+

δ

F

δ

δ

F

δψ



.

(24)

ThecorrespondinglinearizedSlavnov–Tayloroperatorreads

S

F

=



d3x



1 e

μc

δ

δ

+

1 eb

δ

δ

c

+

+

δ

F

δ

+

δ

δψ

+

+

δ

F

δψ

+

δ

δ

+

δ

F

δ

+

δ

δψ

+

δ

F

δψ

+

δ

δ

+

+

δ

F

δ

δ

δψ

δ

F

δψ

δ

δ

+

+

δ

F

δ

δ

δψ

+

δ

F

δψ

δ

δ



.

(25)

Thefollowingnilpotencyidentitieshold:

S

F

S

(

F

)

=

0

,

F

,

(26)

S

F

S

F

=

0 if

S

(

F

)

=

0

.

(27) Inparticular,

(

S



)

2

=

0,sincetheaction



(s−1)obeystheSlavnov– Tayloridentity

(23)

.Theoperation of

S

 uponthefieldsandthe externalsourcesisgivenby

S



φ

=

s

φ ,

φ

= {ψ

±

, ψ

±

,

,

c

,

c

,

b

} ,

S



+

= −

δ

(s−1)

δψ

+

,

S



+

=

δ

(s−1)

δψ

+

,

S



=

δ

(s−1)

δψ

,

S



= −

δ

(s−1)

δψ

.

(28)

InadditiontotheSlavnov–Tayloridentity

(23)

,theclassicalaction



(s−1)(5)ischaracterizedbythegaugecondition,theghost equa-tionandtheantighostequation:

δ

(s−1)

δ

b

= ∂

μA μ

+ ξ

b

,

(29)

δ

(s−1)

δ

c

= 

c

,

(30)

i

δ

(s−1)

δ

c

=

i



c

+

+

ψ

+

+ ψ

+

+

+

ψ

− ψ

.

(31)

Theaction



(s−1) (5)isinvariantalsowithrespecttotherigid symmetry

Wrigid



(s−1)

=

0

,

(32)

wheretheWardoperator, Wrigid,isdefinedby

Wrigid

=



d3x



ψ

+

δ

δψ

+

− ψ

+

δ

δψ

+

+

+

δ

δ

+

+

δ

δ

+

+

+ ψ

δ

δψ

− ψ

δ

δψ

+

δ

δ

δ

δ



.

(33)

The parity-preservingmassiveQED3 (s

=

1)action,



(s−1)

|

s=1,

isinvariantunderparity( P ),itsactionuponthefieldsandexternal sourcesisfixedasbelow:

xμ

−→

P xμP

= (

x0

,

x1

,

x2

) ,

ψ

+

−→ ψ

P +P

= −

i

γ

1

ψ

, ψ

+

−→ ψ

P +P

=

i

ψ

γ

1

,

ψ

−→ ψ

P P

= −

i

γ

1

ψ

+

, ψ

−→ ψ

P P

=

i

ψ

+

γ

1

,

P

−→

AμP

= (

A0

,

A1

,

A2

) ,

φ

−→ φ

P P

= φ , φ = {

c

,

¯

c

,

b

} ,

+

−→

P +P

= −

i

γ

1

,

+

−→

P +P

=

i

γ

1

,

−→

P P

= −

i

γ

1

+

,

−→

P P

=

i

+

γ

1

.

(34)

In order to verify if the action in the tree-approximation (



(s−1)) isstableunderradiativecorrections, weperturbit byan arbitrary integrated local functional (counterterm)



c(s−1), such that



(s−1)

= 

(s−1)

+

ε



c(s−1)

,

(35)

where

ε

is an infinitesimal parameter. The functional



c



c(s−1)

|

s=1 has thesame quantum numbersas theaction inthe

tree-approximationats

=

1.

The deformedaction



(s−1) muststill obey all the conditions presentedabove,henceforth,



c(s−1) issubjectedtothefollowing setofconstraints:

S





c(s−1)

=

0

,

(36)

δ

c(s−1)

δ

b

=

δ

c(s−1)

δ

c

=

δ

c(s−1)

δ

c

=

0

,

(37) Wrigid



c(s−1)

=

0

.

(38)

The most general invariant counterterm



c(s−1) – the most general field polynomial – with UV and IR dimensions bounded byd

3 andr

3,withghostnumberzeroandfulfillingthe con-ditionsdisplayedinEqs.(36)–(38),reads:

(4)



c(s−1)

=



d3x

α

1i

ψ

+D

/

ψ

+

+

α

2i

ψ

D

/

ψ

+

+

α

3

ψ

+

ψ

+

+

α

4

ψ

ψ

+

+

α

5FμνFμν

+

α

6



μρνAμ

ρAν

.

(39)

where

α

i(i

=

1

,

. . . ,

6)are,inprinciple,arbitraryparameters. How-ever, there are other restrictions owing to the superrenormaliz-ability of the theory and its parity invariance – the parity-even massiveQED3 recovered fors

=

1.Onaccount ofthe

superrenor-malizability,thecouplingconstant-dependentpower-counting for-mula[8,15]isgivenby:



δ(

γ

)

ρ

(

γ

)



=

3







d r



N

1 2Ne

,

(40)

for the UV (

δ(

γ

)

) and IR (

ρ

(

γ

)

) degrees of divergence of a 1-particle irreducible Feynman graph,

γ

. Here N is the num-ber of external lines of

γ

corresponding to the field



, d and

r are the UV and IR dimensionsof



,respectively, asgiven in

Table 1,andNe isthepowerofthecouplingconstante inthe in-tegralcorresponding to the diagram

γ

.Due to the fact that the countertermsaregeneratedbyloopgraphs,theyareatleastof or-der two in the coupling constant (e). Consequently, the effective UVandIRdimensionsofthecounterterm



c(s−1) areboundedby

d

2 and r

2, then,

α

1

=

α

2

=

α

5

=

0. Furthermore,the

coun-terterm



c

≡ 

c(s−1)

|

s=1 is parityinvariant, yielding that

α

6

=

0

and

α

3

= −

α

4

=

α

.Finally,itcanbe concludedthat the

countert-ermresultsas



c

≡ 

c(s−1)

|

s=1

=



d3x

{

α

+

ψ

+

− ψ

ψ

)

} ,

=

zmm

m

 ,

(41)

where zm is an arbitrary parameter (as

α

is, zm

= −

mα), and



≡ 

(s−1)

|

s=1. The counterterm (41) shows that, a priori, only

the mass parameter m can get radiative corrections. This means thatthe

β

e-functionrelatedto thegaugecouplingconstant (e)is vanishing (

β

e

=

0) to all orders ofperturbation theory, soas the anomalousdimensionsofthefields.

Owing to the fact that classical stability does not imply the possibility of extending the theory to the quantum level, it still lacks to show the absence of gauge anomaly, infrared anomaly and the claimed parity anomaly. This result, combined with the previous one (41), completes the proof of vanishing gauge cou-pling

β

-function andthe absenceof infraredandparity anomaly inparity-evenmassiveQED3atallordersinperturbationtheory.

Atthequantumlevelthevertexfunctional,



(s−1),which coin-cideswiththeclassicalaction,



(s−1)(5),at0thorderinh,

¯



(s−1)

= 

(s−1)

+

O

(

h

¯

) ,

(42)

has to satisfy the same constraints as the classical action does, namelyEqs.(29)–(32).

In accordance with the Quantum Action Principle [9,11], the Slavnov–Tayloridentity

(23)

getsaquantumbreaking:

S

(

(s−1)

)

|

s=1

= · 

(s−1)

|

s=1

= +

O

(

h

¯

) ,

(43)

where

≡ |

s=1 isan integratedlocalfunctional,takenats

=

1,

withghostnumber1 andUVandIRdimensionsboundedbyd

72

andr

3.

Thenilpotencyidentity

(26)

togetherwith

S



=

S



+

O

(

h

¯

) ,

(44)

impliesthefollowingconsistencyconditionforthebreaking

:

S



=

0

,

(45)

andbeyondthat,

alsosatisfiestheconstraints:

δ

δ

b

=

δ

δ

c

=



d3x

δ

δ

c

=

Wrigid

=

0

.

(46)

The Wess–Zuminoconsistency condition

(45)

constitutesa co-homologyprobleminthesectorofghostnumberone.Itssolution canalwaysbewrittenasasumofatrivialcocycle

S





(0),where



(0)hasghostnumber 0,andofnontrivialelementsbelongingto thecohomologyof

S

(25)inthesectorofghostnumberone:

(1)

= 

(1)

+

S





(0)

.

(47)

Itshallbestressedthattherestillremainsapossibleparity vi-olation atthe quantum level induced by parity-oddnoninvariant counterterms. Due to the fact that the Lowenstein–Zimmermann subtraction method breaks parity during the intermediary steps, theSlavnov–Tayloridentitybreaking,

(1),isnotnecessarilyparity invariant.Inanycase,

(1)mustsatisfytheconditionsimposedby

(45)and

(46)

.Thetrivialcocycle

S





(0) canbeabsorbedintothe vertex functional



(s−1) asa noninvariantintegrated local coun-terterm,

−

(0).Onthe other hand,a nonzero



(1) would repre-sent an anomaly. If by chance, there exist any parity-odd



(odd0), a parityanomaly would be presentinduced by the noninvariant counterterm,

−

(odd0).

Taking into account the Slavnov–Taylor operator

S

 (25)and the quantum breaking (43), it results that the breaking

(1) ex-hibits UVandIRdimensionsboundedbyd

72 andr

3. Never-theless, being an effectof theradiative corrections, theinsertion

(1)possessesafactore2atleast,thenitseffectiveUVandIR

di-mensionsareboundedbyd

52 andr

2,respectively. Fromtheantighostequation:



d3x

δ

(1)

δ

c

=

0

,

(48)

itfollowsthat

(1)canbewrittenas

(1)

=



d3x

K

μ

μc

,

(49)

where

K

μ isarank-1 tensorwithghostnumber0,withUVandIR

dimensionsboundedbyd

32 andr

1 (theghostc is dimension-less),respectively.The breaking

(1) canbesplitintotwo pieces, whichareevenandoddunderparity,bywriting

K

μ as

K

μ

=

rv

V

μ

+

rp

P

μ

,

(50)

insuchamannerthat

V

μ isavectorand

P

μ apseudo-vector.

Bearinginmindthat

K

μ hasitsUVandIRdimensionsbounded

byd

32 andr

1,itcanbeconcludedthatthereareno

V

μ

sat-isfyingthesedimensionalconstraintsandtheconditions

(45)

and

(46), therefore,

{

V

μ

}

= ∅

, which means the absence of a

parity-evenSlavnov–Taylorbreaking.However,still remainstheodd sec-torrepresentedby

P

μ,anditfollowsthatbyadimensional

analy-sisacandidatefor

P

μ,whichsatisfiesalsotheconditions

(45)

and (46),showsup:

P

μ

=

=

1 2



μρνF

ρν

.

(51)

It turns out that there is only one parity-odd candidate,

(odd1), which could be a parity anomaly, surviving all the constraints above:

(5)

(1)

=

(odd1)

=

rp

2



d3x



μρνFρν

μc

,

(52)

however,integratingitbyparts,leadsthat

(1)

=

(odd1)

0

.

(53)

Hence it follows that, there is no radiative corrections to the insertion describing the breaking of the Slavnov–Taylor identity,

{

(1)

}

= ∅

, which means that there is no possible breaking to theSlavnov–Tayloridentity, andneither parityisviolated nor in-fraredanomalystemsbynoninvariantcountertermsthatcould be inducedduetotheLowenstein–Zimmermannsubtractionmethod, whichbreaksparity attheintermediary stagesoftheIR subtrac-tions.

Inconclusion,theparity-preservingmassiveQED3 exhibits

van-ishing gauge coupling

β

-function (

β

e

=

0), vanishing anomalous dimensionsofallthefields(

γ



=

0), andbesidesthat,isinfrared and parity anomaly free at all orders in perturbation theory. In fact, the latteris a by-product ofsuperrenormalizability and ab-sence of parity-odd noninvariant couterterms that could be in-duced by the IR divergences subtractions which break parity – there is no Chern–Simons term radiatively induced at anyorder assomeclaimsfoundoutintheliterature.Itshallbestressedthat thealgebraicrenormalizationmethoddoesnotinvolveany regular-izationscheme,noranyparticulardiagrammaticcalculation,andis basedongeneraltheoremsofperturbativequantumfieldtheory.

Acknowledgements

O.M.D.C.dedicatesthiswork to hisfather (OswaldoDel Cima,

inmemoriam),mother(VictoriaM.DelCima,inmemoriam), daugh-ter(Vittoria)andson (Enzo).Healsothankstherefereeforuseful commentsandsuggestions.

References

[1]S.Deser,R.Jackiw,S.Templeton,Ann.Phys.(N.Y.)140(1982)372. [2]R.Jackiw,S.Templeton,Phys.Rev.D23(1981)2291.

[3]O.M.DelCima,D.H.T.Franco,O.Piguet,M.Schweda,Phys.Lett.B680(2009) 108;

O.M.DelCima,D.H.T.Franco,O.Piguet,Phys.Rev.D89(2014)065001. [4]S.Rao,R.Yahalom,Phys.Lett.B172(1986)227.

[5]R.Delbourgo,A.B.Waites,Phys.Lett.B300(1993)241; R.Delbourgo,A.B.Waites,Aust.J.Phys.47(1994)465. [6]B.M.Pimentel,J.L.Tomazelli,Prog.Theor.Phys.95(1996)1217. [7]H.Leutwyler,Helv.Phys.Acta63(1990)660;

N.Brali ´c,C.D.Fosco,F.A.Schaposnik,Phys.Lett.B383(1996)199; G.Dunne,K.Lee,C.Lu,Phys.Rev.Lett.78(1997)3434; S.Deser,L.Griguolo,D.Seminara,Phys.Rev.D57(1998)7444.

[8]O.M.DelCima,D.H.T.Franco,J.A.Helayël-Neto,O.Piguet,J.HighEnergyPhys. 9802(1998)002.

[9]J.H.Lowenstein,Phys.Rev.D4(1971)2281; J.H.Lowenstein,Commun.Math.Phys.24(1971)1; Y.M.P.Lam,Phys.Rev.D6(1972)2145;

Y.M.P.Lam,Phys.Rev.D7(1973)2943;

T.E.Clark,J.H.Lowenstein,Nucl.Phys.B113(1976)109. [10]C.Becchi,A.Rouet,R.Stora,Commun.Math.Phys.42(1975)127;

C.Becchi,A.Rouet,R.Stora,Ann.Phys.(N.Y.)98(1976)287; O.Piguet,A.Rouet,Phys.Rep.76(1981)1.

[11]O.Piguet,S.P.Sorella,AlgebraicRenormalization,Lect.NotesPhys.,vol. m28, Springer-Verlag,Berlin–Heidelberg,1995.

[12]W.Zimmermann,Commun.Math.Phys.15(1969)208; J.H.Lowenstein,W.Zimmermann,Nucl.Phys.B86(1975)77; J.H.Lowenstein,Commun.Math.Phys.47(1976)53;

P.Breitenlohner,D.Maison,Commun.Math.Phys.52(1977)55. [13]B.Binegar,J.Math.Phys.23(1982)1511.

[14]N.Nakanishi,Prog.Theor.Phys.35(1966)1111; N.Nakanishi,Prog.Theor.Phys.37(1967)618;

B.Lautrup,Mat.Fys.Medd.Dan.Vid.Selsk35 (11)(1967).

[15]O.M.DelCima,D.H.T.Franco,J.A.Helayël-Neto,O.Piguet,Lett.Math.Phys.47 (1999)265;

O.M.DelCima,D.H.T.Franco,J.A.Helayël-Neto,O.Piguet,J.HighEnergyPhys. 9804(1998)010.

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