I

Anomalies
symmetry ofAwhenitE.
.
gwhenisnotitclassical

theis

actionsymmetry of

a

theis

The

theorydoes

Absence oft HooftA

anomaly

you doit.

Nothing

notimportant
is

sense

is

importantmay
isbecomewrong

really gauge it.
andusefulunderinvariantanomaly of=>It

just thatissymmetry.

unacceptable !

ismakean.
procedureasaquantumtheory.

criterionof consistency.

anomaly

global symmetry

gauge

It

not.

theory itself.

Anomaly of gauge symmetry

anomalous

path-integral measure

the classicalquantum system losesbetoharmless.

iswith thewrong

theby regularization

brokenAnomaly of global symmetry

Nothingsaid

isanomalyofwithanomalous
theRather,

information ofthe RG

whenyoutheory itselfthe,

long asanomaly is anthetheory.

flow :

high energy (or elementary) theory
lowas

try to·energy (or effective) theory

2

Suppose we haveglobal symmetries

group G ofThis may be

madeby coupling

itIs

it=

If DasP"
Then

wecanJE=.
geG:

aS1A9 PST S(A P).
=,,

functional of A :e-

AHAS ?

S(as P)=then,A

makemeasureyesz(AT,dynamicalvariable-/Pape-,withgauge theory,,

underDAP

S(P]

S(A P]-(Obe==position dependent g(u)'s

field AgaugeaJoab=invariantz(AS)toS(p]:

underinvarianta,(or correlation function) is

The partitionzlaT(P S(P)) withclassical systemaSyn[A]eand

S[A9 P9],"S(A b),=z(A].

consider the(A D) :,2[A]A/gThisisthe

ganging.

Itispossible if ZCAS]=ZSA].

3

However

this,DPS
This=GeEP
SometimesInsuchtheisgis,,=deZCA]

This,P)is.

field independent-,=Ag(A)

.

case,

ZlAs]
~Ag[A P]

Dat idela PJ=AgCA,a

Dap e:

so

The infinitesimal formanomaly.

theis

may not bez(A) e
=:

9g(1)

z(A] i9e(A].

usual form of t'Hooft anomaly.

Remarks·&ELA P]

Ifis,

AcCa Pl,thenweCounter

can=variationde)

L (Aof some local expression,da,

p 00 --)dix--,,.

,

modify the system by adding the localterm

iOLIA of ... 0 20,,-,-)

tothe

Lagrangian:4S'(A q)=,so,de SCA P]

thatandS(A P] +,themodifiedSAb))(=

S[A P],-S'(AS paDas ps e,,=+=

After this modification,of local=0.

,

anomaly,

isand,. Sa pl-iaP)

Dape
DaPe
the,,

↑

beganged

only up to variation

expression,,

SA P]

symmetry candefinedac (A P) ~ac(A q] +,anomalyi@g(A P]

-Thereforeno:

S[A3 ps],,,hasJe(Dabep -)dix,ide(A P]=system-Finite versioni) of(A-,Ge)02(a--,,p --)dix.

.

5·The-G(DAPeforanIn-Ge (Dade,thethe,,Gabe=

Ja GA J.

.
A-A + SA1

De-S(Abecan,P]deA DE=

JaxDrE JM-

expressedinterms

JanEofcurrent

thei9e(p)anomaly=

JanDre J"

:=

the

Jax-

original global symmetryide (A#=.

anomaly=·S(A P]-=anomaly

anomaly ofsimplyP))

defined byfor the gange transformation,,The,S(A p)

ide(A P]
·S(Aisarbitrary variationparticular

Thus

J

current

symmetry

at=0,.

DrJY

isthe trivial garge field,b]

OpE Jncurrent.

-=-Jaxx=non-conservation

.

G J,

*

6

Two basic examples

:

axial

anomaly (global (chiralanomaly (gaugeor'tHooft)

Axial anomaly

Theory:gauge theory
fermion ↑S(A 4]in:

WCI-=axial

indeed25=Dirac+Jax(-iπDA4)

phase rotations of fermion

4- gitsp

recallmassless,

representation V of G:

aSin(A]=,Symmetry

gange group Jwithrivvirs,- jeiers,

Gr = -185

FAY -FeierDa4=,B=F Dat-Da=- E(X) :iF A4:Fr"OliEU)4The Noether current isjj=-↑v +, 4=Fr"4=-GEFrU4

7

anomaly:=&ji2.

a(A].

a:=

A

used

FarFa

0,A,4]
ENex +(()

162

differential forms

And , Fa==(Fr Fyx)-

JE tr(FanFa)=we= (A5Jax

=Here

F

=dA+:

&(A A),=&Fyndx ndx*Far Fex daedsedan da=-(n - xd4xRemarkOriginalWemay,A-unganged:=1internal--(As1234G.

ungauge

external:=external,4):~internal

aisfield-dependent

S field-independentgChiral anomaly

Theoryfree massless fermion:Symmetry=Jaxx)↑ R + 94R

:4.

+,

current

,

:

=.

Fra

=

inof G

TnQ4m :4)
-Fr942,SCA Ym 4n]

anomaly-inandUR and UL

representations

S↑m--Fig
Eg

Jdix(-iFRatr-iFcDate)iFUetRiFVety+:DrJ = E(trun(edm (And Ax &Arte Ax))+-tr (e On (And An*+&ArteAx)]]

field independent.
ie.

Az(A]
==(dxfall) iDiJa

Ja [tra(ed(AdA [A2)] Tr) - d(AdA =AY))++-SRemarkAxialexternal)

regarded asisspecialGot=

Vot=chiralcase ofWilly

anomaly

..

Was-a

=

regardedasa:G
representation ofWills charge

VotL

becanG*V(l)G-gauge potential A(in whichanomalyVH)---

Was+1G
representation ofaWills charge Thenthe axial,

regarded

as

NB

is

chird anomaly

forthethe variation
theanomaly (where AErt

background Ato

We needforthe=+=(e,((0 A),modification.
match0external)

by local counter termcanbe10

Computation of anomalies
We

thecomputeaxialchiral anomalies

averifyand

the above sintements.

These anomalies

arepotential is regardeddeZ(A]
Ifwe=

field-independent

external for axial anomaly)

Z(A)

=

Il,PlP)-=>=,Gz(A).

Thus

z(A] ideCA).

Jade-SCA
JobS(a Jawrite(if G-gauge

W(A]-,Sa j.

z(A]( GW(AJ)-:-GW(A](Jax=JA.

JAIn particularide(A)==GeZ(A)/ZIA)-deWCA)=(Jax

DrE-JY)A

thenIl

GGz(A)=(00e

S(t,P)(at JJaxA j--Ilin(z(A] G C-W(Al)=.

:-GdW(A]=z(A]( - G d W(A] + GW(A) d W(AS),,[Ja d. A J JadA 5)
.

.

,

A

Ja GA-J][JaGA-j]a-.

JJain=>·-

di-dWCA]id---indeCA]=>/Jan--dA

J (dis JA 5 A.

.

Saridit JJA

==,

,

coun.

,

com

di - dndEW(A]-/didit JA

DE JN.

.

.

com[Jar (GiAn ) JJ AJA.

,,co12Axial

anomaly(A)=-GW(as A) lago(Jaej)a=,iglo)=<Jakejs )*ida(a] lao

by "Lorentz inv= o.

>JaindEjs Jaiy JA J (3)=.

Conn&fj5[=Jai*JA J.

&EYWsV 4
*=

=

iF@4

=/diy GAvalE(x)iT Ve4JAva-MP( ↑(3) irrea 4(3)()](VU"() Tos⑭A=i didDa laylao=Jared d A J(x))

Do this for A=Jaf()(a(-):9,:::com=dieiA=deciJaitr (ii)-M=(x2)13(ax(((x)e-=i(P+ q)k-

E(X)i(p 9) , e

Ji Tos (VV"threei(p 9(x)+++ins~(9 9 01 (p b e)
,,,,

p+ 9F

frithmically divergent

Los/zPauli-Villars
Introducef(k m),regularization
a=regulator:(A)(a1mass,opposite statistics (bose)(9 9 015+ (P.

.

,b,9)

= 0

ifaie(x)e-=,twos(r) Verm)+i didaspinorai(p 9(2)+S-14Computef(k m),vs()=f(k m),=29=-* U5-us )( m)+++2m2truece") +rs)- Virt-+vshasthUm Oftrolece)[-NVs-mrmVtn)

zmVTh+=4) m)Vs((-+(VU"Comsre(l ms)+(R+ M2) (Cl-pl2 m2)+trs (rs( + m) +" (+my r))(h 1) mil (h2 my++ 2m

Trs (Vs(()++myr ( m)()((( 4) m))++(h+ /+ m (h mi ((h-p1 m2)+++(9 0)4 (1 9).

.

&exchange

15

Use

Vy-Uppips=*try (V> + -(4)=tr (VVVVert)·ofnumerator

4(=-afterthe*=3hx(k-Pym -=pu,Similarly

ofthe

3rdtrs (Vy(()+m)V" ((v [rV=4)

no*+uxPr

No contribution

0.

to theintegral+(Ur() +mirs(rU +yars)

4m exter 9 Po

kx & Px:P m)m)((((=+1)

term

mers+=contribution=,=try (V58 +(P)4 * &Pohaswernthe 2ndnumerator

=

(()"

+k-integration of the regularized system>·-

1st term-·9050933+25 v3)-+yu-ve +x)

16:S(f(m 0) f(u 1),-,Nulee=(21)

41 e

*

Exp+((k q)++12) (h[12) ((k p) 14-+(9 0) - (P 9),,

endoubles

h 14
==,

Omit

1+ 0-16 tr(eeJ&==th(ee) exee Pr=-,=

Join Exe=me-·:

did a (a) (a=ofor dA
Theseare&In---

tobe

comparedi(P+q)x,= inlee) PrGA=

des etip

withEntr
JaElkFminX

17X(az& X (a0=0=
=

a [0]

=

da=0.

(A)(a-=0

GGX /azo=

dif(aas>-((x -above :Try J Frip.

16T)2,

SaxEie-

GeFispe* 2--(i Mantiquad)
-

=

MMMM4

i(p+9)x-ee

19

Gippi+Din Plastine
Meet &nMa.

[MUM &n Pps.

=:diacallaGlA]=Produs) e Eit-=u

Jak Elk et F
at

O(A2=-*"9P,

18

Chiralanomaly

Considerz(ATthecaseconst.

J==

the Diracto==& (iFrDaYn

S(iTA4fermion 4=

iTA4rdeW(A o]eProjection

Up-

PR4=

simplicity·for

JiFRDA Pr dix-.
components

anomalywith values

only·,inV

where.

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

we canTe

iTPr4
....

iF@41) diX+iF@Pr4)dix+R

toof computation ofNow let us compute-VECO3-r

A is coupled,)@Fh84nBEOt

For the purposeconsider

VconstJO584 ePr

where=(ATROtRe

===VR

=O

=o -by"Corentz" inv.

↑19ideW(A]la=o->Jax OeJ"Jany=+=

JaidEa(x)]diy(-).

j)

comeJJain (JAm EJ j").

,-Mdave eien==

Jai Optacus=

Jax I faluse-M-PR4(3)FM)/diy:93i9klee)irs (vPrPr-i9 Fall 2

trtrues (V192** PR4(a)51)Vre3!

quadratically divergent⑪

Pauli-Villars regularization

inI

statistics

by Lorentz

O

tras (ivePr4()P(3)i@Pr4(3)5() (-JJA

~JA

fermi

fermi
Co=1)

original(=-1)(G=1)

regulatorsbose((y=-1).

28

Replace byIt turnsits:=[ (9)

Ie,1=

that

outO

is

sufficient toi= 1

makeitconvergent.

After some computation+GdeW(A)(a=i)

find [*(9)(89 _ 9) 2/10g92_ lo=-,we85(6EiMilogh ? 92)7-=Ealkeitrilee- Elig 9)* 0(2)

di-Thisdoes

In fact ,
term

not

thistolookcantomatch with the claimed

be cancelled.
W(A]by addingaanomay

local.

counter