Renormalization group
Choices of renormalization"On shell"
Al these>

originate fromweneeda.

g.

dictionary

condition #LagrangianPe

12 . . .

,PE Xe,...

.

I relation ?

4dp" theory.

↑( P--:-inclassical

same-"another R C

Sthe"Another (M)" , .

renormalization condition I
renormalizatione,:physics.

same-

But

"intermediate",conditions,".

1)(p m)=parametrized by
=+m2

=1

M

P(-p p)( m,=↑ (PrPucts,amassscaleformalizationMe44))pip [Mit=atMi P ,To describe the same physics,how are they related?

pointAnswer: so that the bare fields/couplings are the same.

z(m x M 1)p=

Molm X = m 1)

=MoXo(m X ; M 1)

=

40;

,.

,,,,P

z(mixin 1) pl=,=Molmix ; M , 1)=

Xo(mix ; M, 1)

To (9 Mo Xo ; 1),,/

↑ (PThem,,x;

IAM r)↑ (d) mi X = m' 1),change (P,,,M .

X)

+-,

(d'm X) for M- M'

the renormalization group (RG) transformation
theTheequalityA

betweenrelation

haslimita↑ P m Xim),,T(P,m ,isasrenormalized

the1-c,=andm,xiM 1) satisfies as,P(p , m, x ; m').

/fields/couplings

and,called,called the RG equation↑P==limx ; M)

is,

The RG transformation

bm'X
Putz(m x=be written

may,=m,m)b=

R (m X = ' ),,R(m x ; m ),,*Ngz (m Xin)(m,.

=,↑'R (mix Mim) /p M;=,=:=:-Um/m X ; M) m,,B(m xip),.

the infinitesimal RG transformation ( RG flow )MattandV(m Xim),=:↑ R (m Xi Mim) (p m,,*=,Thenas=-V(m x ; M)b,

Mum

=

Mix

=

-is,,Um(m x , M)m
,B(m x + ),,the infinitesimal RG equation

is[-VP-Umm ++ ]Pp xim)
,m,

= 0.

Computation
Recall·Exa+m+ +x b + + xby

1 + +x9

=

Zom :

=2 xo

=

theory+,,x++...

x C + -G+...

0b- (12 mi) log)) 1 v) m0()]=.

Thus0Zo*9=?

04d4in,====>+-(log()=to1-loop

MMm

MNatMixMm

Zo,

==

==

V

=--Jjdxlog(111 + Oft)

MP==-+-+,+x(-x)+01)

and0(t)Mm (b Mb )++M+↓(MN cOcti)-x

Ot+0(Y+,xMa)++0(t)O(tyV(m x m)·=,O(ti)=Um(m xim)=,

B(m xim)=,0(t)
-X

M

in

=>

=++Letus

considerAsX

aof ofIndeedZ

only,

in

V= 0,

theareOk

MDm

O(t)Mkm.
m + 0limitdimensionless,

B&U:

B

=B(x)

thislimit,

B=,V

at=V(x)Hoop.

arefunctions

The RG flow

Max:

Mind
The RG

Instead of,

x=-P()+M

tuse.

the RGflow[ (t)(=-and=p(t)the RGE=takese.

Writ+=eMo.

a+afx

P(x)

=Vix(ty):P(PC) J(t) eMo)
,0.

form

theSoMorsolution toa==,log(M/ o)=P(0)- )P(p xin)

V(x)p-P

V(x)9equi(1Thenp(x)=;t- independentisP↑) (pr--Pn,.

inXi m) P(p ).

--P(Pu)RGE n/taNWIt)
P(p=On the:=1,Pu,

hand ,

other(n)--↑(px-Pets :E Mo)

(P)

,

Col ;

the

=1,

(x]:

,

Peps etpe--,,Xi= 0,=eu)

(4)

4 -n

=

=are

0=>

e(-n) +

"dimensional
Combining

at-independentMo)

canonical dimensions(P(Ph >Pa X m)]--isP(px Pnxim).
=

analysis",

Pletpo--,,etpa,Two)iMo)

R-nas) Peppedim.

Gn-.

ePt-nat(1 Vict))+N(qx---Pa Tct) ; Mo),

Thismeans

DIfwePi
theuniformlythemomenta

asePi ,-

coupling in effectively changes↑ (0)&rescale(t)+isThedimension.

/"

* (t)

as

eﬀective coupling constant

the&-

1U(X)

>
-ofP

also

changedas

1 + VIT(t))
"is

has

theanomalous dimension"of P.

AtHloop=,144)2

=
↑ (t)

& (0)
=

I-10

TsOx- 0ast +-t4-M
The=

couplingoreTMoYo

isweaker

atlowerstronger

athigher energiesenergies.

·X)Mo)XIM)=1valid foris·3 log(N/M_M

ifevenMollog(M/Mo) ) may be large.

The Series
expansion↑ (p)hasax(MO)=( MOS log (N/)

Feynmann diagram

interpretation

:

Un...
&~

"RGsums

up

a+

xn

seriesof(log Mo/m)"Feynman diagrams"

variouspossibilities
*-O

B(x)

OT

Theinthe IR limit

infra-red free theorye.

.
g

P4 theory

4d& CDpX& FT I

just seen

We'vewithRED,

large # of flavors
nextTRuV>--&1+O

intheUVlimit

asymptotically free theory

B(x)

eg.

4d& (DpYang-Mills theory
withsmall # offlavors

nextI

X doesB(x)= 0notrun!

finite theoryes.

4d N P=Supersymmetric Yung-Mills
nextOther possibilities:UV fixed Point

0TER

J...

XUrO) etp ,a↑

---,

,

with+,

X++12e.

g.

X*V+---,

functions scales

V(X* ): =,Pn,

inXxiaMo)

simple way.

scale invariant theory+ (p p),=-iMo)g(4-n(1 V xt +(pCorrelation

E-X

IR fixed point

point
et PnIR>-non-trivial fixed point of RG flow

At such=IRE

Ev>-~UVXXconst.

(p2)1-**1&MP)=↑ ( p p),-[Aj, -P=

-

p2
=

X(P2)

--

=---+-- D-=+...

(l-(1-X)" ==(P(x)p(0)

+)As

a+2v

&<pyhu4)RG of 4d non-Abelian gauge theories~Considerandland/organge theory

Dirac fermion 4aAim

4daao

scalarComputeflow

RGIn

particularfindwe,

Drepresentation Vf

Drepresentation Vb).

B,V

that determines

andfields .

couplings

theB-function for the gauge,.
&chooseaEcCo ,renormalization condition

andoutZa Zp,

asa

gange group Gofcoupling constant
For thisinafunctions

thetheinwitha,function

renormalization,ofMorenormalized

point

M,

and

couplings

e, m,.
cut-off 1

What kind of renormalization condition

canweimpose?

We

imposemay,

Za Z4 Za,

to,

Mo,

loop

thein,,theon↑[A 4 C I B ; eaction,conditionam,,,

3 : 1]

fixedare142 effectiveatSo

eachthatorder

expansion .

Recall%S[Xo Kojer mo 50],,+O(1 t,--,,t t*) fromold/din &NEMFA
new-++=,O(tY )

iznenFD4-ine4
g*(znewdpC++F(zumoSP

1znewtEnew) (Ap (3) ]
.

+.

YA

renormalization condition· (dAu-drAph -iFQ4,

inT,

for example.

,

that

Fm4,constrains

joc,

the termsco"CAp <]
,

Let

T

write

us=

S[Apa(y)( 93( 4) Aub(P)+-+p,↑(p)[+ ( p 4)4(p)+,(p)Tgh(pp)((p)]

I -(199 PACS↑,+.....

M(p i) e End (VS),

Tghtp 1),N(-9 9 1),,=,

End (9)By Ward identities associated with symmetries (Lorentz,rigid G ghost #,these coefficient,

BRST...

,functions)areandadditional

constrained

as :arguments,

↑Rabp , p)([p(pTgh)↑[As-,1)(85 p2 pmpo)8=-

P2Tigu(42)=(p 9 9 4),A(pY B(pYOp 9),

anda((P 9)=.

satisfy*+Ngu(ph),,(p 9) ((p 9) + 9 D54 9)=-,,=A(MY)=Tgh(mCip 2).

,

commute

must,

goog"

=condition

we,

ez

1,

=-B(p)=

m,1,/p g= = (p + 9)

a==MionGpr 92,

P 9.

,03 Adg)s"↓-withD"CP 9) depends&renormalization

π((YT(P2A(pY4 + B(PY)=1 , 1)**-j 2

mayrakeRecall

Ste-P↑

mu& IPI vertex~↑Moab=,↑y(p p)=Tgh(p)=,--↑M(-9 9 4),t (d p2prpg*(1 9),

4 + mp---)(+ (Pf - x7 [P] + ... )-319~=At Pii~GetIPIt

Tree andOne-loop contributions

unlpwill=

1PIto+nummo+mS(---....

14 -[PI-- )..

--+:&CE-verticesmi -muto-mi......
Smi (~Y==E----------

E- --3...c-

14

7 scalar

3-2

=-++M

.......
S