The resultP

ub 7MaGut E-loopts*=-(0 p2-pmp)*A+X↑(2-2)

Try (2x-1).

((l -4)(2x 1) 2) Y-Emwhere(E-100p():=+-(x (l-x) p2 m2)/RMER+dC(f)=, daT(2-)E((d 2)xx-If

where:=--2dmy](x(l x)p2 (1 x)mj)/45-+-* 2

4 T(2-Exp-Pc.

--- 1-+

FTufre(l-a)&-gas

100pt.)=-2

arec-y + 1-d100py=

iade (2((p 9)+where:=

On:=

y(lEn

=Xv/kMan:+94SYOvRd)P(4-2+[13 7) IR2/, YI

R S"-,-y)92 + z(l z)p-in

andmostat

2yz94

2 pha 99,

linear,+,lineariny&z.

:The functions

I-loopat

are ,TT(p)=Ac=

B(Y=Ngu
Co(p)

that

2-=,+-G(V)a[(2-2) ** -2)-G(ViSa4(2-2) ** ammy +=renormalization condition

thein((jdx + (2 E)* (+-1appear

1+hat(2-e)-i-ih)dydz&+(-termscounter

TTMY

are&AIRY = -1,Co(pie)(p = g (p g=determines

them :+=m=BIRY,-iEzymo)"z"+ilzby theThe renormalizationterms.

=

contributions

the-Zie) ( (a)-The lastx+-+E)

one-loopcondition=my,

TIgn(nt=-1.

(E)(Emo)"==mG(Vfk4(2 f)5 +* (d 2)x(p m-=Z-do + (2- *) * /p=---=-GIViaT(2-4) amelperhidT(2-1) x(zm2

Zi + EL(3dzT(=>-2) - (+(4-d)(/p=Note:↑(2-2) ** (d + (d-419)===T(z)(E(3---(x+for d=a(p + 9)

PeEB)

V +O())() =log o(t))(aVlog5)d-2=+-+2B+0()+ +B)=M2

Let us firstConsider

O(mEy2 Mb/) terms,

renormalization
Thus ,

with

X

:=

#" =point.

Thenar+X ST+ 80(1)+Try (2x-s2Tv - hy+-Glute-G(Vf)Xam,*zhizt=ignorethe.
M- independent+...

-at-

M independent terms + O (C)-X(TV=logt's---=Eymo)"

andMy <Mall oflogm2=&=,,E-logy2,+zMy

the CaseI+-1x +...

I-Wi+...

hi(- 12x 15 2))-+=>

E=-X +...,

z

=- 2X

(i)

-+...

Hun-(e)" -X ( Try +Try fh 24)=+:---logm) (Ty -Tv - )
++...

+...

↑M-indep&(Eymo)"mo"
:

mo

=

=

zime

=

-

my

Using theseof

3 G(V +,the-+mo">)iXm+...

+3((Vt)(z logp)my-wecandetermine

renormalization

higher loop...

BaU functions

thegroup+,

atHoop:0=Me)=Me) +Ty ST+-Mate*-MMoo ==ZeNat0r=

6 GIV++-=+..

)

Umy=

Va=

Up

=

Uc=-z P.

MalogmtMlog(higher loop)

Umymf+z...

Upp

Me = (Ty + T - )

=h)

+...

-&--=

=

G((t)

2t+----(4π)2Mflug=Mlog=G(V)-th.

+..

--+...

For

general ofmy,(Tv F+ (mf m)

B+,

Fy (my m)=,Fy(my m),Un +=Upa80(1 4)jjax(2x 1)-x)p2

Class-20 M2x( + mq2(4-2x)

-

4/3

mfyO

mf M

4--x)(2 + my+-[+-x(lx(l4)-,

Garximx)(2 mix(l=Tro Fy(ms m)

myM

MbX14

[6G(mys-mf M

82xM2=GNA eptm- & GNMys

mf M

OVa,

U

remainn

Heavy fields (mrunning(Theyofmust

in>thethesame .

M)

do

gange

notcontributetothecoupling constant at scale M.

beintegrated out
the effective theory

andareatscaleabsentOnly the light fields (m ) contribute<>

M.

.

(Closer look atB-=-b,Mle) =0(ey+- = log(Mr)
e(m)

(Mo)
=

1 +b , log (MM) -b >0

Suppose.

M+n:

UVOr+...

e(p)smallergauge

0“asymptotic freedom”

free.....

equivalently
M+,

:

&(4)

coupling islargerstrongeratlower energies.

At

someM

=

1,

it

divergesblog()

1 +=:01 =expr)

characteristic energy of the system , the “dynamical scale”( NOT to be confused with the UV cut-oﬀ ! (...

&It determines

how theexp(-)
1,noteM1NOTIfab 30,(:...

appropriate parameter of the theory.

“dimensional transmutation”---Attheis,=gauge coupling runs,

SystemisWhenise(m)> 1,:"gauge theory" isgood description !

finding the low energy behaviour of thedifficult problem

ab,>o?

.

"Interesting theory"

More·

general systems

generalseab jo

GE

=TG

LE

·

=arbitrary basis , Fir=IeFapr,=+...

(T=-trees)

UNY

Gi

*

simple factors
with=Fb =VF·an:

FFbM(t)ForG

gauge group-Tri-Tust=V asa

representation ofGI=TTre=[RER:(a) Wila changes of V(·IfthereisjustasingleUll)b) =Ta=

(Dirac fermion 4->Majorana fermion (4OfimustbeatoTry4) :representation/R=anomaly must be absent
the fullin

Contribution=(4 410r4R) :chiral fermion
X·factor,issystem

halved.

Complex scalar P-

real scalar

Contribution

to(5Vomust

Tryis=behalved.

aP) :representation/R.

Examples·Charge Ri electronREDG=b,VII)=

Uf,-,Up==o0

Infrared free.

Pure Yang-Mills theoryG=b,=simpleVf,SUIN) RCD
D=hb

==V

==0.

- >0:

·=&:

·↳((bi)

SU(Nc)
No

Asymptotically freewithNf flavors

Vf /NayeNt=,Te,=+N 1 .
-:

-t:

=Tuf

Va,xNf=.

j((IN) 2Nf)-

Asymptotically free if NfCENC

·The standard

GmodelSU(3) x SU(2) x Ull)=

Leptons

Vs ,

=Vf R

=Va

=,.

((k 1,-,z)(B,,,5)]5) (BR=xTB

Rgen+-,5)73

Su 3 - 5.3
TerLI Bll-4=gen
=3,

r =

072 A

=gen

Th 1

Tr [(x (1 3)] x3
=

11+-4

:

1,

[(X(1 1)]x3.

50(2)Quarksii) Higgs

Try I (t 2)x3

:r,1) b (B 1-,

LTESU13)-I&((1 rgenerations-#3+3=,

Tus

=

x1

=

tv

bu-3-3Wil

2(1 R)

:

Tw

(2)

=

R-Tv

b,+2+(1 1)=1 +,[ (Hi(1 R)=(t)x2=-,

55=-(B R)2,gen(5) 3 2)xx.

(B 1)(B 1).

gen(5) x3 7 5) x3]x3,+=5t55=-

7=-6 833.

..

30IRfreeSOINc) &CD·G=Gb,SO(N) ,Vf

Nc-2

Ty=,+(N==-2)-=>(R-1=/EN-(Majorna)
Tv:

Ne

USp(Nc) &CD with Ne flavors
GUS (Na)=nVf===u+ 1 =(q )bN,

=Ub

S=-22-2Nf)

N-1<(N = 25 Nt,both=((Nc 2)++2,

Va=,

Ty =0==:even)(gU(d))g() (5))

Try =b0(xNex =E = IN

Asymptotically free if:·Nf flavors

with

Asymptotically freeif(((Nc+22-2Nt)

Ny < EN + 11=

·N

1=G=

simpleVy,Arana)=Tvy=b+ 5=hN /=TrybUp=0-340=

Asymptotically free.

SU(Nc) Supersymmetric &CD (S & CD) with=

G,t+:·(SYM)

Yang-Mills theory

Supersymmetric=SU(N1)=

Next,Vy=

Up=+xof (nona) (DNc)

NE((** (NX)ENf

N +,

Tr

=

(2 2)xNf+No - ) - Nf:

*No flavors=

Asymptotically free if3NN

Nf > 3 Nc··

N 1 SO(N) S & CD=

G SO(N1)V=,=YoRNeNt(alrana)

Vo ( *)=Try (Nc-2+=bN* Nf(xNf)x)

,

Tv

=

1x Ny

(Na 2) --tNf=.

-:·Ne flavors

with=G

1=

Asymptotically free if

USp(Nc) S & CD

USp(Na)

,

Vf

=

Vb

=with-3 (Nc 2)--

N+ <3 (N -2)=

Nf flavorso (Majorana)=(Nc No,Ne.

both even(D)(( )Ny+Tuf = Tv = Nb =.

:N-

Asymptotically free if

3(N2 -N=2Nf < 3(Nc 2) .
+

·N

=

SYM

2GSimpleV,Ya=Tufb,=N==SU(NcUt,VTryb==

Nc ++N+x-

Nf

===

Ye4"

240

Asymptotically freeSUINc) S & CD

2&Tub=-:·V,,==No flavors

with

Je EINeNeyTra=+((*(( )NNc + ( + 2)-(N ) -(N+:-++++Ny

Nf)

=

2Nc

-

Asymptotically free if Nf < INNe

·N 2 SO(N) S & CD

G SOIN1)=VTvy=Nc-2+=

Sco(( *(x N+ xtb , (N 2) - 2 (N=-=2(Nc 2)--N=G

2==2 +

NfTry,rana)=

N = 2 + 1xN ++) +(Ni 2 Nf)--+

Nf

Asymptotically free if:·(IRNyeN

Vf Ic

,

Nf flavors

with=USp(N ) S & CD.

USp(N ) Vf.

,

Vy

=with

Nf < 2(Nc-2)

Ny flavors·(Nc Ny both even.

/My
Yp((* )eN-=+

Try = Tr =b,=:

NNNNt2-Y
AsymptoticallyfreeifN+< 2 (Nc 2)+··

W= 4G=SimpleTvfb,SYM

==,4x

Vf

=

Vy

=2,+h -G&4(Majorana)yet (real)Tra=2-==GoG

4x3Sh

=0.

Noteforb, = 0:

W 1
=SACD,

SUNd

Ny,SO(Nc)

Nf,

USp(Nc) Nf==3 (N= 2)=3 (N+ 2)

=

2N--,

N

=

2 SOCD,SU(N)SO (Nc)USp (No)
N

=4,,,

N+Nf

Nf

3 Nc

2 (Nc 2)==-I

3B>oCIR free (Beexactly

oall loopa

2 (Nc + 2)

SYM

from higher loop

non-perturbatively

finite,scale

invariant.

XTheyarebelivedtobe

invariantalso under (super conformal transformations.

“(super) conformal field theories”