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!