Regularization and Renormalization
Divergences&=<&(p)P++m=&-

1=-+2

MM---- (yp( )p(3)b(3)b(3)P(y)P(x2),--(eTi)*5*(4. -Pr)+ind = 4~-

iP2(y x2)
-

mi=>

m

quadratically divergent

-=
=-13

Pl (3) $13)(3)-$13) Plaimm2)Pemic

quadratically divergent*
FD+l-p(b(X)b(xyP(x)P(x ))2pt++10

I=/&"4↑+O-&214I=>

33I2D+...

PR -It i&⑭+(253) + (27+4) +logarithmically divergent

42h-

The

integraldivergent-----P.

-

overatP2

loop momentale's

the141+canbe0ultra-violet ( = short distance) divergence...

)superficial degree of divergencepower of momentale ofthe(power(power in denominator==?

E=gfrom.

numerator)

in&th[,↳ # loops

=-

=from

net #=21 + E-=of#vertices ,momentum

integrals

Lecture 4d([ V + 1)

D4=,[- V+ 1

If Lin +&**;=

propagator

Vinternal lines#Theory of Scalar & in d-dimensions

Dintegral↑Vertex...

,external lines

#D-=D

:

=dL-2 I-do↑V2Id( V + 1) + (d--2)E4-E"theoryEdEEE

B=0=2-=4E2)

>
/
L
/

DX

/E
=D!
/:

:

:

:

D =4

quartic divergence

D=2

quadratic divergence

D=0logarithmic divergence

DSO(superficially) convergentmiFor E=0, 2i.e,↑9)theory,4atin&> P

the,

all orders

otherDO

Onlyadivergence

dforfiniteD:

perturbative expansion

ind + (d=-4)V- E

large enoughnumberfor any number V of vertices,

occursofV.

Feynman diagramsare(superficially) divergent.
d) PFor each E,

AnyCorrelator isatD30for.
large enough V(Superficially) divergent

sufficiently highorders

inperturbative expansion

How do we deal with such divergences?

At least, we need aregularization:a systematic change of the theory
so that the loop integrals are all finite.

.

Regularizations

Dchangek(PY)↳of propagator24

p2+ m2

Ph m2
1((x)

I=1x1

k(x)

2(10> xI1The1 , but

compared to1
esissame.

Sme.

Guemodifiedat.
1P11change of Lagrangian-me

P7 m2:-=sharp cut-off

Em> d,low/p/

p30

=d em

[E 1atPs 12-=>

original

thesignificantlyEeasultra-violet cut-oﬀ (UV cut-oﬀ):-.

theremains

propagatorom2p( 0+ m2) em P -P--

K2-5/12 "more generally

D(Cd)

Pauli-Villars regularization#re#-OneIN

power
E

choosecanintroduce,,&%the system withLagrangian(Pinternal2

propagatorsM& (x) E(y)+=P(y)b(3)(regulators (...

"F P )

areJ::...

foronly=.
possibleas.. :) J free part,

Thelarge

aslowermake thefield variables&E reg = (+ ++,=. . .

to-,of denominator

newand consider

11 91 Maideor,E=interaction

P + E : ::M+[7d )Pi(m0 (3)::Sh i m )②

Lattice

DER" + PM

mynex + OnxN⑤↑ a2)

Se reg[a(P (=,Advantage:

Momentumintegral isovercompact space2=yaDr-ipnaep()=Se 1) jet )gip(na-na)+mi③ Dimensional regularization
dimension

deXShisaydemD

Mif(h)mMDm4a:parameter ofdimension 1massMat vola g ==l(ii)Inte-=IR~11x)/12=

Vollage r er-()" =:.

VollS")
2125418

=

rider
[T()

I-(PT)* P(d/2)

M-ah f(k)Ch1crus psase)
2

This

makes

*sensealso fordeDeg.

ViaEDI

Sim=uddICo=VoVo=-dimim (12 -mi(log()=U&=pdD=Va-+---+1-V0 57721=for d+=p,

but for d = P-E :0()]clog()-v-1-Jjamlog(1 x() O(*)+

Min Y (up --==Euler's constanta-.

+=4+ d 4 -+divergent1g)):r) m 0()]:=-lyn)P(1-2)/z+regimdivergentfor d=P,

but for d = P-E :/ log(n) U-Jarlog() x(t)) O())

⑪ Exercise+-++Renormalization
After regularizationso(1cut-off

thefields

areD,

in

theaincouplings&,

finiteas,(1 +we0;on,removeavo;dependto(E MpR) in &correlation function ofthethatletwe,properly normalized
theE + o)

cutoff.

regularizationSn=[Sa (200 m ++P==Ena(Jdx)

Choose Zo(M)+ zo(n) (05,Mo(1)(P(X) -d(X)--We do thisorder

areby,+MoMz0(1)p2 + zop4)]n

Xo(1)

so

all finiteorder-cutot

inthatas1

isremoved

perturbation theory.

z(n)

=

1 + Xa (1) + x a (1)zoU)Mo(1)"zo(1[X()

GCo=

22,G+....

m 2+ Xb, (1) + x b(1)=X=+++---44(1) + x G(l) +G + 2.

...

+...

,

[(p)

=-,+-+=Exa (1)(64)+=

tranl(095+.

;&table+xbm)p+ d

called theDoperturbation theory

GreGo-2,5+=1-loop22 + 2-loop

iwithElops

tree

counter terms:

LinFindAn (1)
So

that

are

finite,G +L+ba(M),Cu(M)+-

recursively

the correlation functions of P's
ateachorder .

& I-loop-+--G-fin)(12 m2 log(=-(a+bemadeXa . Mp2-++log)x3Can these-+finize-↑C (1),finite ?

Yes ,G . (1)=b, (1)

=(1)

will

dofinite- (1mil)=thelog)job !

+finite+ finite

Xb (1),

ClaimFor

An(1)toSuchaeachn21bu(M,,Cn(1)theoryissaid#/ m/XA:

theorybeto

arefinite.

.
renormalizable

field/muss/coupling

renormalizable
is

20

when

theonly for

Superficialafinite&"theoryd4(loop contributions

LaM

of correlation functions.

numberg.

findfield/mass/coupling

renormalized

istofunctions of t

degree of divergence DE.

thatSobare:possibleisthe correlation

allPo/mols/Xo(1)Claimit,

Yes:

&4:

No=>

divergence at high enough loops=>d > 4:

No

renormalizable-supervenormalizable
notrenormalizable.

&mass dimension of couplings

Criterion :S=Sandmass-dimension

(S)

=(0)

=

The+

Soof·that

1:

(2]

=1

=a(m2j=2(xy==-(p)

d-P(t).

==d.

4- disrenormalizable)(coupling] 10

supervenormalizable#)(coupling >0renormalizable#(coupling)

0

not4)

+S makes senseso(in)

0

theoryJax ( + (b) -=<.

.

Recalle.

any diagram:

So ,

toto

makeasaas

thecarry1P]offunction

cut-off

effective

finite

is

Now,

anin.
g.

=

point :

thispossible

theis

of

choiceG , (1) =b (1)
.

=T(p,

finite. .

+finite-P(1)=. . .

,

offields/musses/couplings-

finite
mu

m ,

xbm,

there

renormalized

mu.

enoughisfiniteactionfunction

aimportant

Even whenvertices.

.
removedis

asit,renormalized,,1PIWithdiagramrenormalizationout

thetreeaTo 160 Mo(1) Xo(1) ; 1)g.

eis,is=1)
Xanas.
1-0

ambiguity-fields/masses/couplings.

To fix

ambiguity

theimpose renormalization condition,:

For example

P(p)=T(P m x = 1),,=PCPS

T( P,4) (pE=↑ (i...

--,in)(P ).

--TCP)

o-m+ ( P +)(pE-:

1=,-"On shell renormalization"mpp)/pip

Sim=Or&P( p)(p0,+ (p)(p=m2"intermediate renormalization"=0

P(p-P4)(pp=1=,= 0X

OrI(MT (p=.

some1)/ per(scalemassMm="another R C.

at( 1)/pn.

↑(P-P4)(pipWhen
the[Mi-the renormalization

ambiguityTit 1)
.

=

T

=pm

conditionis

imposed,completely fixed.

isLet usConfirm this.

1=="at-- (XHoop(f++30)-Q+++& (-For DT 4)

=,

2p+m

=
I+ m2- (lg)
M+Tilp P4)

Xa , (1)p2+2 (1)x==,-V-1-Sjaxlog(1-Piz

renormalization condition

0b (1)= (,-S

P , + P2

etc-(jaxlog)1 -(1)1-1)) 1x+=-v

R C.

+0()

2On0()+Sjar log (1 x(1-x)[M)++25 Arg( i)-intermediate:+-= (log() VO=x)-(4(1)

the,+x((244)

toSolutiona, (1)(2773)

0()

1 -r) +O , )(+-+Xb (1)+- (log()=-,mecut-off

momentum-.

"another" R C.
.

shell R C.

For dimensional regularization↑ (p p)pm2=.

- Am+↑ (PPP)=1a (t)=+O()]+(774)XG(t)renormalization condition0b ()= /E log(M)==,+/ log))+same0()log() -V-Sjaxlog(1 x))the,u +,+--Xb (t)

to,

P(t)

+=(253)+

Solution+Xa(t)p-log() 1asin

D-V-k++:

1-v+0())

O(t)]