Chiralanomaly (OutlineConsiderthez(ATcaseconstPrwhere.

consider

J=to=fermion 4==-

simplicity·for& (iFrDaYnJ(iTQ4PR4

iTaAYr

Now let us computedeW(A o]eProjection

Prfor details

note

JiFRDA Pr dix

....

+R

toof computation ofthe Dirac=

VECO3-r

A is coupled,)@Fh84nBEOt

For the purpose-VconstJO584 e=.

=(ATROtRe==GAVR

see the additional:-iPAPr4)dix.
components

anomalywith values

only

we can,inwhere.

In deW(A) /A[SaiDpE JR).

V:

iTPr4=iF@41) diX+=O=o -by"Corentz" inv.

↑ideW(A]la=o->Jax OeJ"Jany=+fJA=JA.

j)

comeJJain (JAm EJ j").

,Odav ebik

by Lorentz
M-ePR4(a)51) [VePR4(3)()
trs(iv
Jaipfam(diyei(-1)*==

Ja & talks

19-~

i9 Fall 2

tr(e) Sirs (vPrPr

192!

quadratically divergent⑪

Pauli-Villars regularization

Introduce

3

regulators

I:I.

:iistatistics

Co=1)

originalThis replaces[ (9)fermibosefermi(=-1)

bose(G1)(G=-1)

regulators⑧ by:=its(UPIt turns

Thenoutthat

integral istheafter some computation,

[Y9)

,

convergent--hence/# OdiEalkevi9tryleces) 9,[*191gui.

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not

match

with

If fact

thiscan becancelled

toWCAT,

XWCA]

W(A)

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=

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for the claimed formula.

alocalcounter term

In fact

Jaxtr (A d""(C DOY An )+with

does59)]=This does*.

/(8592-99) (log - + log()=

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findwe+anditjob,

C+= logta D=t

We

alsomayadd

E W(A)
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dabasically(die-08)An]

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GW[A]

1Apal

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compute - diddeWA)/ A

let us,WIA]

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=

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dae ein-G GoW[A] 10,

andti(E((e el) (85g2-gng"),l (logg25ipisxlog)-2E)

JaxEPr4)diyiF. Pr4 /diz iPAPR4Sonn

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.

<+:(axfalk)try (eeg(Kexchange ;+g(k m).

=trs)9) Pr HaI-=+rs)rPm-I-

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VP PR-(p)+ m~Sig( (i):=k+mPr20 Pm

1VP PRMRV him).

-,mUI-54(p 9)+ mI-(9)I(p)+j( 9),;? ) try (Pr(/ Vit U9)#((k 9/2 1)(k 1= )((k p)2 12)+2997)

that vanish--++159p2 2pp9)

as+:

0,

9/1 :+--20.

+**9,Pr7

&=-(dix(a()[i(+9)

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-

=:

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E+- [tro leade e) exure 9 Pr
(5p2 pp) (logp-G +,the* + #, 2

Gd, de WIA] la=>[-setCancel-i(p + q(x-If(9): (+=192-gg),andwea

modulaClog1)]]

8"p = pops terms

findo2) tr /Ende e) Ex-9, Pr,iS tr (EdAndA)=log()

Jtr (Ed)AdA

O(A3) terms

.

+=A)

.

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By computing

Summary
weS

haveacA](A)

seen=(=Jeth)EdAdA)+

isconsistentalA]ala)
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d local (A].

moduloThis(dAdA)

with+ (FE))

JTn(ed(AdA [A))+"computer ~On,

tofix the higher

Butthatgeneral.

isnotorderterms

necessarystructureofif---O(A3).
weanomalies.

usetheThe general structure of anomalies①TheS

anomalydix

becauseitcomesfrome.

A,P),
which is,only for divergent diagrams- local

modify the action by

Thereanomaly AcCA PJ
,

is

a

freedomis,definedThus

only modulo.

JedezTaP-derde Dap=

Green Dad

(we've been considering right action)de Ae(A P1-de9e(A d),.

tolocal counter terms.

de local functional of [A PJ③,i.
,regularization procedure(Already discussed)

thelocalis.

polynomial of derivatives of (Enon-trivial&AECA PJ,=.

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(A P)
,

Let usconsider thea (A)

(i)

=

case

ofchiralanomaly with V 10).
=[A3)c(trun(Ed(AdA↑+ConstantSatisfies
(ii)

It

is

WE

thealsotheconsistency condition (Exercise.

unique solution tothe WE

condition(of course modulo de local (A7) with the"initial condition"

a (A)
Thus,

ourenough=x=

ismosStr(EdlAdA + A)

the relation

Ga.

proveaCA]

VIjO(A3)+computation for mor
toIn view of(Strm(EdAdA),acA)

this=between

alsoGCA] a acos(0 A) for·confirms(t (F).

Fujikawa’s method
alternative ,an. . .

directmethod

Axial

anomaly(Gunganged4

mussless

Dirac fermion:

unitaryS
Axial=@values

withina.

(Note right:=4 (eiers)"4action(Det energ OF 04= (Deteter( (==o2 Trliet)---A

regularization

ac(A)
Note:

Thus

DiDa-e2==divergent.

:Fr(t Ve )Dahas/.
GLie groupa& (Feiers) &(4):compute anomalies.

representation V of Compact

Jax =i) A4anomaly

tosincerealcan.

UNE-UM andD=-eigenvalues

andprovide

regularization

aDie (Vanitary).

hence.

theUseofplane

basis

wave

Thia()

fixes

eof V-valuedSpinors

the space( se : CVtoikX
=

basis

serSCS,evaluate the trace :GlA)

2 Tr( ==>

2/dix(@h(ui(x) EVYui,2(d+=jikx

eix+gix gitx··=>

re-E/)

=-j

*

x((x)

inxgihx(Sitr (vsVOS

yn(ikp==+bu Ap)+,yap(ihp (n Ap)(iku++Vip =* Er v3+,Lik++ Af)/ihr(ikn & Ap)(ihr++&Cr

0*, V-+Ar)=5Ar) - Ceu)+d+Gr + Ar)

+++1

*+=

*unFruFr=:X+)Af) (b AM)(fp+ike-C-CMagike=

As

Life~

41
1

=

Only

G(A)

the=+if0is

S&3

least two

attonecessaryPll-+m+2

with

51

=

1m =0

-

powers,survive(X/12)e (For/12)

terms-·2ortryNs-).
itsreordering2m

survives2(ax((x))
Use1m+ 2(12)e:.

zu F+.

kY12+h((X + Fr)/12)"

trgV Ms)-(X * V Fi)/12.

ofGui Fur/12·A-*-e=·+exp)-eih yein) -=

Ey-

=a.
the limit 1 + 00

tros(v)*/2)2)F+=>

Jaix((x)+re((Fr F(x) (VV--emvex

=
=

Note:

diE()t (F)tr(FanFa)

The axial

rotationit, VsgitsC.

theis

additiveala] +,itiI

integrates

finite axial rotations& (iv)=(A)
totheAbeianisgroup(E + (2) V5,einfinitesimal formula aandThus.

=(A)

/Et F

=ae(A].

anomaly formula for,4)

EF84

i=

DATP4

exp(i)+G (A]e(FE))

Chiralanomaly

VV

For the purpose ofaS

=thecomputing

Dirac fermion=V 90,values

with(dix(i) ↑DARY

anomalyWe.

may

considerVinDar;

O=+AirChira rotation :-I49(9Pr=843843

A

regularization:=>++gP)Pr) &F O4

Det (gPr + gP)8584

Tr(EPr-EP1)=5(Pr=+=Tr(EV)---divergent.

:-id(A)

4+==,Det((Pr g"Pc) 19Pr Pc)] 8484

=i@P1)4

Det (Pr + 9Pc] : Det (9Pm=:+Tr(EVs eSimilarYar/)computation(tr(Ed(AdA [A))++deloc(A).

Axial anomaly in a general even dimension

Gr pry-25=,(V !.. 2)2=( )Va :=P mussless

Axial==represented

MU = -d2non...

r

Dirac fermion;

on

VaIP

=1,

with

unitaryVervaluesrep==

D2-U"VainV ofa.
compact G(ax (i)πP

anomalyActivate)a (A)S"daSd2=2/x((x))tos(vasame"=j

Tr(eVane-E/]=:

id (A]

computation(Exercise (2)- + (i)IRaeiY)