Quantum  Chromo  Dynamics  (QCD)  

Gauge group SU(3)

Gauge bosons Gluons Aµ

a=1..8

Fermions Quarks qa=1.,2,3 (3 "colours")

Sca4ering  in  QCD  

qg → qg

Colour  factors  

Ms

2→

Q,g  colour    averages  

Mt

2→

T (N )

C( A)

C(N )

u j T bT a( )ji

Xui u j T bT a( )ji

Xui( )†

⇒ Tr(T aT bT bT a )

fabcu jT c Xui fabd u jT d Xui( )†

⇒ fabc fabdTr(T cT d )

fabcu jT c Xui u j T bT a( )ji

Xui( )†

⇒ fabcTr(T cT aT b ) Mt Ms* ⇒ fabcTr[T aT bT c]

= 1

2fabcTr[T a[T b ,T c]]= 1

2fabc fbcdTr[T aT d ]

= i

4fabc fabc

= C( A)Tr[I]

= N .N

The  QCD  beta  func=on    

Rela=on  between  counter  terms:  

(Proof:  BRST  invariance  –  later)  

Beta  func=on  from  quark-­‐quark-­‐gluon  coupling  

Z2

Z1

Z3

Z2

= + finite

Feynman gauge, MS( )

Z1

=

from  Abelian  calc.  

Z1

f abcT cT b = 12 f

abc T c ,T b⎡⎣ ⎤⎦ =12 if

abc f cbdT d = − 12 iT A( )T a

Z1 ≠ Z2 ?

Z1

See  p442  Srednicki)  

I2 =

d 4l∫1l2

= 0I1 =

Z3

I2 =

,

(linear  terms  vanish)  

(d=4  for  divergent  terms)  

I2 =

=

I2 =

Z3

I3 =

I3 =

I4 =

⇒

Z3

Feynman gauge, MS( )

α ≡ g2

4π

ln Z3−1Z2

−2Z12( ) = Gn α( )

ε nn=1

∞

∑

dαd lnµ

= β α( ) =α 2G1 α( ) = − 113T A( )− 4

3nFT R( )⎡

⎣⎢⎤⎦⎥α 2

2π+O(α 3 )

!α = g !g

2π

dαd lnµ

= β α( ) =α 2G1 α( ) = − 113T A( )− 4

3nFT R( )⎡

⎣⎢⎤⎦⎥α 2

2π+O(α 3 )

T ( A) = N , T (R) ≡ T (N ) = 12

β(α ) = − 1

3 (11N − 2nf )α2

2π≡ b0α

2

SU (3) : For nf <17 β-function -ve (c.f. QED +ve)

α (µ1) =

α (µ0 )1− b0α (µ0 ) ln( µ1

µ0)

α (µ)→ 0 as µ→∞ Asympto=c  Freedom!