Renormalization and linear symmetry
Considerclassical systemaJS(0)

Supposewe=have0regularizationa,weS(in))thisInholds,

wecanThat

is ,

we

the=0=

that respects

this:

0.

,

each

at

order

divergent terms

particularSo ,symmetry

Slarnov-Taylor identityhave

JP(bin)andlinearad = Mijdj;

5 (0p Thenwith

choosecan.
symmetry

in

also

are

counter terms

renormalize

the

loop

thatsystem

expansion .

symmetric .
are

also

while

symmetric

respecting

I e.
.

a linear symmetry is protected from renormalization.
What

about

gauge symmetry ?

.

Renormalization of gauge theories
Considerfermion ↑GE

4dain

(e]

The question:

respects

Remark·0,

fromThe,1So.

Dirac

Lagrangianthetheory ispower

possible

it is

a

Tm4+

viewpoint ofthedo

Violatingthe=,

tocounting.
renormalize.
preserving gauge symmetry

weYes

(m]

whetherwhile

whetherSymmetry=isthe system

Answerand,·:

G

gange group

Fr Fro -iFXAP

renormalizable.

withrepresentation Va=Dimensions.eI

theorygauge

longgangeasintroduce

togauge

Fregularization thatsymmetry.

canbe done

charged scalar fields

chiral fermionsneedcounter terms.

as

same

notfortheories

withGauge fixed Lagrangian:= LetiB . A&B

It has BRST
symmetry

JBAnSdoc

Drc==-+(c C],

AssumptionFa,

=-dbT

c4iB

=the:

JBB

o=cut-off.

The IPI effective action

functional,

that respects the BRST.
symmetry

regularization

a1

Claim

GBP,

Dac.

+e,

finiteis

1*G

as,

as

properly defined fields/couplingsof.

↑ (Ao 40 Co To Bo ; to Mo So ; 1)
,E=Ao=,,,,EaA,,

40 4 Co C

er = Coll, min),,,mo-mole mi 1),↑(A 4 , 5 B ; em 3 ; 1),,if

wechoose

To=EstBo,=EBB

Jo 50(e m 3i1)=,finite.

as.

1yc

Z := Zile min),Colem ; 1)appropriately.

:,,,Mole m ; 1)
,Bole m 3 ; 1),,,

Remark

ZA

24 Ec Eg,,different=G xGzX

V=V , Va

AoMo,,Somay

different forbe

components :GThenCo,,..

-,xGmVv...

(ExiAs FAzAz=,similarly

for

Co,Co,...,,decomposition

toirreducible

decomposition

simpleorabelianfactorsFAnAu)

Bo

Toler 4i, Erti- , Yu
,

Co...

=

mj =..

3For=.

have

V may not

Ep=-,

G

toconsider

beirreducible

simplicity

just-

(24).

is,

tobesingle

EA

assumedasimple,so,

Ec , E,that

20 , 50:

Mo =(Moilis.
understoodyou1PT effective
ExtendedX=k=(An 4.

,5B),(k" kt k k5 ,S(x k),-c,action

K dbX
5E(X] + ↑-=w(5 k),e.

↑ (X K],,,=(axe:=,theSign is opposite toLecture 4S(X k) + 5 X.

,

W(Jk[X] K] + Ju(X] X.

,whereJ=Ji(X)

is(5 K) = -X.

ofthe soln

,Then&(X K) =,(X k),

0 ==+1(X)E-=Xibosonic

Xi formonic(dBX :jx7 K(8(8xS(x,, +5I.

x)

+Zinn-Justin equation

o·==SetSoxS(X(oxeJ-=·Also.

F↑ (x k),=.

.

,+id Aik-+JB)

Then

idAp-ik
5B-=8

idAn + ik[d((ax( = B2+IC( 3B-=S(X K] > K? iB+,=,k

S(X k) + j x

Jx(X).

3B +,

Ba K

"A quantum correction

=,

N

-

[iBtok,iB ."An + 1

independentBa K

iB=terms.

iB)·"dependence.

(A)

o·

=

(ox(e-S(X=(oxe(A)

and=These holdaslinearinan

Kothey(** )=Xo.

fixing function<X(P)5x)

identities,

among
also

renormalized "notgauge

above

relation,

is

that

soEnfields

Eki Kifermionic

is0theastheI

interpreted as

fields,(Jd BANcondition

longappropriate=begauge fixing

To be precise55)-(** ) may

hold for

--

Jx(X]

14

(k)( d D,-L-,=(,(JBA ) + J:

5,S(x k) + 5 X.

-Ko.

Ex(EJHowever:&,,is

X(0)

=,k.

(***)

withZki's

hold for renormalized

whereX,K:

·Zi

Ifchosenare

Zxi Zk:that

soi-independentis--#[·K5 B dependent

terms,

unchangedremains

Bo -i BodAoptkiBo↳-ThenJEp

za==3,zi"Zgza=

1=zki=Zk:=Zx.. ZaZa

30

=za3

Zyr=zcso&

B-iB-An 1 iB+25 ZB·,zc#-=

1:

that

=#=Za Z.

Ge

[E==

F Fr-iFA4.

LESuppose&B+

CoMo

Zi

=

↑&,chosen

1 I effective

action4te,+-2yz)
4 -i&N -+-Y...

+min+z&PN-1

thethatsoismadeloop &N-1

finite+ "Mr+...

-termsofits:+tin'**+Tw+...

--

divergence comes

LetMwo beTN+++. . .

DpCCfiniteThe+...

m" + hm" +Potti=N 1

iB An + c d

1 + tzi = Aphave been*m + G=

Fm4.

-e + he" +=+=

TNa,

nfrom

m

possibly divergent.

theE·loop integrals

counter terms

divergent part of IN+

finite.

We shall determine
showIn·e mthatMr,cTwoin +=(izA 4 c)andbecan,,counter term

newindependent of Bo KI

canbe chosen=,ais

the integral ofto(dix Li
polynomial of fields

choseninvariantcoefficients of4
flavor-g4,

symmetry4344,:

linearunder

rigid gange symmetryO-g8g".

"local expression"

ao,

withTw

finite

is

is

Two·

possible form of Twinthatso·afor5-hivge O

O=-+Th,

Xmass.

k:constant

A , c 5,,10

dimsymmetries:

5g

their derivatives&*-K K,

kis

commuting

- W,

with,

TB-g
C , mE-hi

·fields

Canonical dimension ofGN,=[x] + 1ApYC[KM

k4k1

3/2(I23/2Tw

en

[GBX]

,=o

at

iso

quadratic

mostDijkK"BiK"+in2

K"sK-indep+m↳·Ghost number

KYAKF#"KYKYC

[⑧O

I

-1

forbids

CNThis,n=+is,

K** K*

symmetry

ApThis·may

include& K:

at:,,c17 An,-2k5.

linearmostTrn(A 4--in

Kis(i = Ap 4.

,4 c) :,]+

(K*. (4+<c)+

K : GotheseGu , Ar

GOP,putThen

haveones).
GrnC,dricB

0 ,:-finite

==0.

,Trin
Also

droT,

Gro C

We alsowe(modulo

local expressions

define,we=Pr(A 4,definedroF

Zinn-Justin
at:=equ,c,] +"variation

Kd.o"

:

draXi

by

[inXi .
forMor=

PothP,*+..

+in*+Tn'+....

O(t*) :--

finiteWrite--

finiteas...

=.
0Then

·

Tr = Twis andTo

Recall+KibXi,

Tra=Tr(t 4 ,] + Kidoxi,= X PoXY:

Je=;

dbXi+ MX

thatis,=GotoTr.

↑Yno+(1)JBThn(2)didisX+:

draJe+=dradBXi

00==nontrivial only for
Xi

By

d =

=

A , 4,

O(ti)

g-JJ

Guid Ar(3)

Y(Xi

GinAn
=0Xi

=A 4,,

Mila 4 , []

:

*.

,

An

T-independent-~

also C-independent byghost # symmetry=

Fo[A 4]+,By

Zinn-Justin (2) :(4)EdvcX;One

canshow[OB N ] X == 0and,+Un

GBAn
c4,+(3u-1N Oc=-E(c ,(]

=constantsMo

andsomeaMu (Am <]=.

,symmetries

80 04,0that.

for:,,d An = 3 &C
=OdinAn.

See the additional+=note.

-x)dC4 dist

Un BBC

SN.

dimensionalanalysis,

Zinn-Justin (1) :Go

0 =+To=Je=Fra 4) +2. din An,

SECA 4) + B -iBOA,+=A

Fi(a 4] +Ap-E & didNiwAr-,

Ju . Sela 4)- IdinAr- &divid At

dbF(A 4)+,.

+c 0 D.

NoJe*=,.

(3x-4n) &C= 0S.

On the other hand&C.Se = D= BJax[enF* (AnAn]-iTA4]

Exercise:Fla 4] =-,+(n-1)(dx[tF (Ap An] -iPA4)*(dix2gauge

invariant,dim=4.

26 z↓=X F Fr - YNiYD44↑↑

numbernumber

FMNY+·↑

commutingmassmatrix

withgange/flavor sym

Grin = LG 2.
.

-(3x 4x)
-+ J.

en,

&" (3rd , C

14 /300p)++·+1!

Mn(Ap ())+,O(t)

terms

that

when

we,addZa

weca( [ (( )"Twif,-This,2n)i9 * 4,k4( 4x4)

Za 24-Mn(Ap C)++ClaimIndeed ,Fr (Ap An] + (5n,,canbecancelledappears

inS/Xo Ko ; eo mo 30]

o,appropriate

anby,

thenew,O(t) terms

toto , Mo.

writefor the Olt) effectOn
Exold =ofnew,and t(N)EAnew

in (similarly for Others):Ea

Exod=Zy(N)-Nh M
Za new O(ty 1)++new

Zpold +=z,+&zcold + new=,= (anZymo=(Zymolo+,+0(tt)Molnew++,

**Ot)

,

then

S[Xo Kojer mo 50],+,--,,,," t*,) from old/din &NEMFA
new-iznenFD4-ine4
g*(znewdC++ K++See

O (1 t=*k*(znewdpC++F(zumoSP+1znewtEnew) [Ap (3).

1znewtEnew) [Ap (3).

(Zheutnew) ( (P)-+c c.

K/ZnewtEnew)( * (p (3) 3

the additional

note.

+Oct**)·If(ZarEner==-Xx

3x-,z-=

newUn,Yv,(ZyMoer21<new - Su=- Ma-

↳,

(N)

heat Ener=-2then# For n + the new O(t) counter

terms=0.

I