ChiralanomalyConsiderthez(ATcaseconst.

Prconsider=

GA J=to=fermion 4=PR4

iTaAYr=S(iTA4ProjectionUp

Now let us computedeW(A o]e& (iFrDaYn-....

-.
components

anomaly,in

we canVwhere.

iTPr4.

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

iF@41) diX+

iTAPr4)dix

with values

only=+R

toof computation ofthe Diracsimplicity·for

JiFRDA Pr dix-r

A is coupled

VECO3,)@Fh84nBEOt

For the purpose-V

constJO584 e=.

=(ATROtRe==whereVR=O=o -by"Corentz" inv.

↑ideW(A]la=o->Jax OeJ"Jany=.

j)

comeJJain (JAm EJ j")+=JA.

,by Lorentz

OJaidEa(k)]diyis rep4insiFin) Pr4(3)=JaidEa(x)]diy(-)=

dave eien=

Jai Optacus

Jax-I faluse-M** PR4(a)51)Vre3 PR4(3)FM)(Vtrues/diy:93i9klee)irs (vPrPr-~-i n ePR4() P(y)i@APR4(3) ↑() (Stras-JJA=M

i9 Fall 2

tr192

!

⑪

quadratically divergent.

Pauli-Villars regularization3Introduce

I 11 I

namesoCo=1)(=fermi-1)(G=1)

bose((y=-1)

regulators

originalThis replaces:ii

bosefermi

statistics⑧ by:=[ (9)regulators

its(UPm~II

trs (V* PR (*+ 1i)(-PR((9) + Ni))(kSince:=&"Themaps

Sp to Scnumerator

and1) ((u p)-

Sc to SRPre)U"(9) U Pr*=,+PRUMPR

1)
.
= 0

trs (V* PR (*+ 1i)(-PR((9) + Ni))trs (V*PmVPR)-v=>+trs(PRAU" (9) (M)aPE=PrtrsParcrer"F==[hTh-91*

tr, Virer"=

28y9 ygney+Vivreut

28mj4+-Eig-2eMx

eb (h-9)8 + 24* (h)-K (h-9/x.

term will not contribute as kg & 9g
Rylx & #Ex + 99x
/and·

*

[(2)

=

199

=

integration (if convergent

after the(3mdyn

=0.

Just
(k 2) ((h 9))+(k 1i 2) ((h 9)-1,dx

So Ih(k93 152)]2=++-I

1+1i1, )+(k-xq)2

1 :2) + x((k+~x(l-2)'+=:Di*(e=*x)92 1-e=+-==(l+x9-9)"(l xq)+*21 -+0: )2"+(1 x9) (1 + xi+x(tx) (d*-9) j -( x9) (l 19 9)+-q2-299)+·+

l-linear-Exercise Show that

Ei

=eiM

·

the

1-integral

isconvergent

if(This holds by the set-up

0=oi= 1

From now

le·,

AAThe

we,

l-linear term

·

Thus

on

can=this.

docontributet ** 2=

be-not

*=lim

tothe

integralafter the integration.

.

replaced by+y-l+x((x)(j g 299)
-integral may be performed[ (9)Then,assume.

as2)dx (e*=0: )2e 12=

lim20*-e 12The

bounded -integral

canbedonefor

eachi:

Ste0:)212(l12+==2log Di +The underlined terms

F Ei

the assumption
we,

[M ()

cantakethe=0SIt,= dn= 8+

I0:0()nunder0

Oilog Dix((x)(vig2 2997) log0:i= 1 2 3,.

logi=Bilogdi

log1=+0()(a(6++=

which leaves us withlimit 100+For+aftervanish~ThenO

I ZilogWi+log-1i)lyhi+x(c9+90()]

=(ii) dx(x+(l-x)(28 922959 log(x(1 + 9)*((( )(2092-29 9) log1 :*+]

G5(1y1 x( 92)++Omit this below

Sodax((x) t=Jodi

(l+ )

logo

(1 x)
-=-E

= /210959597(loga E)-+ (* )
We found[Y9)=/(8592-99) (log - + log()+-59)]]

Thus-,

wefindGdeW(A)(a=i)=di Eack)

evi9tryleces) 9 [*,191
Il# O& Eililli-t9).

JAEIA]

This doesnotmatchwith

If fact

thiscan becancelled

Let'sconsider

to,WCA].

OW(A)de OW(A)=0by adding.

alocalcounter term+

Saix 2 tru (DES "(C D ) A )*=for the claimed formula

Jaxtr (A d""(C DOY An+JEXW(AS la+2

==S
==GA

2(di&.

Santru (dEd (C DO) JAx)*+=dae gi&Ens +Xtrylee) (C + D(-9Y)~

faltig" isGGz(W(AT OW(A])+Milli (9)

=>
ieC.

.

WemayNow==Consider

the abovelet us--2(C + D + 2Y)
DandE(aix r (An(5*+/azode DWCA)

hasWe shall

log

=

=

WIA]==02-08)An]

0.

W(A] + XW(A] +*'W[A]C and D and for some Ecompute-.

diddeWA) /oJan Datak) End ·Was
Apal

=.

Jax (GAn E]aI) & GWSs,forda)-add

E W(A)

for-=alsowhicho==

daei9aA=X:

dae eipI①wasessentially computed& EW(A])

==S,

in

JaDEW oJa OpFalwel(dixirfalie-Recall

D·iDes&LHS--ei9 tr (e) [=are:9==

straightforward totry (ee3(- [**↓=C D and E,=*tilee)[ 19)(9)obtain.

(9) + 28+The above

19*(C + D( 94)-2E(bE/314*(94 + gi97)]:9 tr(ee) (8ig2-gig2)x~ (logg25logh - E)

For,

&=

DW + &'Wofaddition

hasnoeffect.

/JaxiFEPr4 /diy iF: Pr4 )diziFAPR4)
:.

.

con&A
=

=

define,CA

dasse'eip==> /dia(dyei9] /diz(Ei) twos---(v ePr4) FBS Ve Pr413)T(z) Ve Pi 4. FM)*)+-=i) did Falsei(p + q)x,-#os(vereJi

I+--Ea(k)i(P 9) , ei(p+ q(x)exchange1i

Ir-23Pm-+1:

Vie'PR++1)

exchange+=-<+:(axfalk)try (eeg(K+exchange

whereg(k m).

=trs(PRmUP-mUPR-+m)

(9) Pr=((=((p)g(k m),=-(p)Pr-+m)Prrs)Pr Ha--Pr" Prm(P--=+rs)r-P---k+m-Pr + P)

VP PR1-(p)+ mI20 Pm

Ha + mIk+ m

VP PR

k+mk+-"Pm-I. mU Pr

I20 PmIPr)(p)PM+-+ m--Pu( ((9) + m) + m(I++-P> (9)=-B+mupPr-c m)
+I

- +m1

VP PR
-(p)+ mimm2) m)
9) + m)r Pm( + m)U)Pm(((p)+((k 9/2 m2) (k m2) ((k p)2++&8=~-++r (kinside trace+m)u((P-+y + mPx)mPr)) Pr(-#4) mPcmPr-==+y + mPz)((k m)uPr(() m)( - Pr(↑=+mPr((p)

mPr(9)vXUS-(a))=- mPr):g(him)=tis) roph--Ik+mI-"Pm--vPPR-

1VP PR

k+m+ m-(p)+ mv(V~((k 9/2 m2) (k m2) ((k p)2+ m2)+·Sig(: )++-(9) -I()=

where54(p 9),=:=? ) try (Pr(/ Vit U9)#
9/2 1)(k 1)((k 132 12): (k+ii) dydz Zei[fi-i= 1=>ii)

dydz

...

1+1+↓ 1) try (Pi(y(1y)9-1trg(Pr( y-(39+z4)29)z(1 z)p2 + 23294 + 1-((y z)E2/3==++-/V

where3+,+terms that vanish
20 mitV-(39*+ zbelow))v9)as

M :, %+:0Jdydz==in.

ytrg(Pr(/Vsin)(+=·-

59(0492-Sig(=-++-9) "(4 9)-) d3dz=

( ↑ +*-+(pz=9)(p 95-+2997): )-+)v9)-y ry(p 9)x(p--8- (p 9) )p 9)

=+.

-9)rY

159p2 2pp9)-2**9,Pr7(9) -I()2(89 99) (log - +log()-

g7 1+92)2(0 p2 prpt) (logp- +log()

1 + P2)-6++.

1079 2993)--

159p2 24p9)-2x9r]

= [(892-59) (log - log()(5p2 pps) (logp2-Slog()-req+ ex·=

- (dXEak),Pr]:( 9) tr(edee above++(dxfa(k)e-=: (P +(4b 9) 4+ (9 < p),,,q(x44)2[t(e(ee])[(892 593) (log - E log()(8p2 pops) (logp2-Eilogi)]
to leade e) exure Pr]--+9,,On the other hand-Jac (GAm E]a) & GW o,-[PegiP E(k)]a,*==19 tr (eeb) (dirgh-gig)~ (logg25xlogh - E)=-[deita]e)(dig2-gig2)x2Y
tri(f(x(e e'l#,

If we getEtermsofIn total,

=1

this,

D

we=>

TheseIn

todog-gig

the.
termfind(dis earewith

Similarly for the (1512).

Gd, de WIA] la-cancels

logh - E)(logg25: (+=o2) tr /Ende e) Ex-9, Pr,

comparedbe---.

withth(Ed)AdA+[A)) (a-=X

X(0)=0& X(1)(a=0= =GGX(A)la = Str (EGAddAz+dGAddA,)

=> true de disei)diedeim)=>==Thus

ISe
--if

ac(A)

i(( + 9)xtru(E(e e)**deWIA]

modula=o=:J+exchange

9xPr dix,=exchange )(e) EddPodd

d de Walla-+.

ideCAl,+ ((d)AdA

O(A3) terms

then+we= A)

found