IRenormalization 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

2

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,

3

The RG transformationbm'

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.

4Computation
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

M
Mm

MNatMixMm

Zo,====V=-1 + Oft)

Mm
M-Jjdxlog(11MP==-+-+(M +↓xOtx)+01)

and(MN cOcti)--0(t)++,+x(+0(Y+xb)

Ma)++0(t)O(ty5V(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 + 0limit

dimensionless,

B&U:

B

=B(x)

thislimit,

B

=,

V

at=V(x)Hoop.

arefunctions

S

The RG flow

Max:

Mind
The RG

Instead of,

x

=-P()

+Mtuse.

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)7

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) +"dimensionalCombining

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),

gThismeans

DIfwePi
theuniformly

themomenta

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

/"

* (t)

as

eﬀective coupling constant

the&-

1U(X)

>
-ofP

also

changedas

1 + VIT(t))"ishastheanomalous dimension"of P.

9At

Hloop=,144)2

=
↑ (t)

& (0)
=

I-10

TsO

x- 0

as

t +-t4-MThe=coupling
oreTMoYoisweaker

atlowerstronger

athigher energiesenergies.

10·X)Mo)

XIM)=1
valid 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...
&~

"RG

sums

up

a+

xn

seriesof(log Mo/m)"Feynman diagrams"

Itvariouspossibilities
*-O

B(x)

O

inthe IR limit

infra-red free theorye.

.
g

P4 theory

4d& CDp

X& FT I

just seen

We'vewithRED,

large # of flavors
next

1+O-intheUVlimitasymptotically free theoryeg.

4d& (DpYang-Mills theory
withsmall # offlavors

nextI

X doesB(x)= 0notrun!

finite theoryes.

4d N P=Supersymmetric Yung-Mills
next

12

Other possibilities:UV fixed Point0TERJ...

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 pointpointet PnIR>-

non-trivial fixed point of RG flow

At such=IREEv>-~UVXXconst.

(p2)1-**1&13MP)=↑ ( p p),-[Aj, -P=

-

p2
=

X(P2)

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

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

+)As

a+2v

&<pyhu4)

14

RG of 4d non-Abelian gauge theories~

Considerandland/organge theory

Dirac fermion 4aAim

4daao

scalarComputeflow

RGIn

particularfindwe,

Drepresentation Vf

Drepresentation Vb).

B,Vthat determinesandfields .

couplings

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

andoutZa Zp,

asa

gange group Gofcoupling constant
For thisinafunctions

thetheinwitha,

function

renormalization,ofMorenormalized

point

M,

andcouplings

e, m,.
cut-off 1

What kind of renormalization condition

canweimpose?

15We

imposemay,

Za Z4 Za,

to,

Mo,

loop

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

3 : 1]

fixedare142 effectiveatSoeachthatorderexpansion .

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) ].

+.

Y

A

renormalization condition· (dAu-drAph -iFQ4,inT,

for example.

,

that

Fm4,constrains

joc,

the termsco"CAp <]
,

16

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,global G ghost #,these coefficient,

BRST...

,

functions)areandadditional

constrained

as :arguments,

17↑

Rabp , p)([p(pTgh)-1),A(pY4 + B(PY)=

1 , 1)

P2Tigu(42)

=

A,

B,

C(P 9),a,Tigh(p 9/e ((p 9) + gie D(p 9)=-,+C,.

are

A(MY)

=Tgn(my((p 2).

functions ofcondition

we,

1ez

-1,

=-B(p)=m

1/p=g= (p + +==M)

,

-i

G

withcommutemust

renormalization+,D,

D(1 31aπ(m)T(P2-(p 9 9 4)↑As

(85 p2 pmpo)8=

**

ph 9,

p 9.
.

,

mayrake18

RecallSte-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

19

Tree andOne-loop contributions

unlpwill=1PI

to+num

mo+mS

(---

....

14 -

[PI-- )..

--+:&C

E-vertices

mi -muto-mi......
Smi (~Y==E----------

E- --3...c-

147 scalar

3-

2=-++

M

.......
S