Experimental studies of QCD
1. Elements of QCD
+
-
2. Tests of QCD in e e annihilation
3. Studies of QCD in DIS
4. QCD in pp ( p p ) collisions
1. Elements of QCD - SU(3) Theory
(i) Quarks in 3 color states: R, G, B
(ii) “colored” gluons as exchange vector boson
SU 3: 3×3 =8 1
B
R
B R
R
B
s
B
B
B
1 R R G G2B
B 6 B
Gluons of color octet:
R B , R G , G B , G R , B G , B R
1
R R G G
2
1
B B R R G G2
6
Ninth state = color singlet
does not take part in interaction
1
B B R RG G
3
To recapitulate the
previously discussed
material.
Quantum Electrodynamics
Lagrangian for free spin ½ particle:
L( x, t ) = iψ ( x, t )γ µ ∂ µψ ( x, t ) − mψ ( x, t )ψ ( x, t )
Applying the Euler-Lagrange formalism leads to the Dirac equation.
Invariance under Local Gauge Transformation
Demanding invariance under local phase transformation of the free
Langrangian (local gauge invariance):
ψ ( x ) → ψ ( x ) = e iα ( x )ψ ( x )
requires the substitution:
i∂ µ → i∂ µ + eAµ (x )
If one defines the tansformation of A under local gauge transformation as
Aµ ( x ) → Aµ ( x ) + ∂ µα ( x )
one finds
invariance of L:
iα ( x )
ψe
L( x ) ψ→
→ L( x )
To interpret the introduced field Aµ as photon field requires to
complete the Langrangian by the corresponding field energy:
L = ψ (iγ µ ∂ µ − m)ψ + eψ γ µψ Aµ −
1
Fµν F µν
4
Fµν = ∂ µ Aν − ∂ν Aµ
The requirement of local gauge invariance has automatically led to
the interaction of the free electron with a field.
Quantum Chromodynamics – SU(3) Theory
Lagrangian is constructed with quark wave functions
R
= G
B
Invariance of the Lagrangian under Local SU(3) Gauge Transformation
x ' x =U x x =e
i
k x k
2
x with any unitary (3 x 3) matrix U(x).
U(x) can be given by a linear combination of
8 Gell-Mann matrices 1 ... 8 [SU(3) group generators]
requires interaction fields – 8 gluons corresponding to these matrices
1.1 Color factors
e1
q Coupling strength:
1
c c 2 1 2 s
QCD
e1 e2 QED
q
Coupling strength
q
e2
q
CF
Examples:
B
1 R R G G2B
B 6 B
B
R
R
Color factor
B
R
1
R R G G2B
B 6
1
C F=
R
12
1 2 2 1
C F= =
2 6 6 3
R
R
R
1 R R G G 2
1
R
C =
F
4
Color factor for qq color singlet state (meson):
B
B
B
6 B
1
3×
B
R
B
R
C F=
C F =3
B
G
B
B G
G
B R
R R G G2B
B 1
C F=
3
1
B B R RG G
3
1
2
C F=
1
2
4
1 1 1 1 1
=
3 3 3 2 2 3
In the case of a color
singlet, each initial and
final state carries a
factor 1
Color singlet meson
is composed of 3
different possibilities
3
Triple and quadruple gluon Vertex
Gluons carry color charges:
important feature of SU(3)
R G
R B
G B
R
Color flow
G
B
Color factors
=
4
3
~ NC
~T F
=3
=
1
2
1.2 Evidence of colored spin ½ quarks
a) successful classification of mesons and baryons
b) clear two-jet event structure in
e e hadrons q q d
~ 1cos 2 d
c) R had =
ee hadrons indicates fractional charges and NC=3
ee d) Further indications for NC=3:
∆++ (Ωs) statistic problem:
Spin J(∆++)=3/2 (L=0), quark content |uuu>
→
++ =u u u forbidden by Fermi statistics
Solution is additional quantum number for quarks (color)
++ =
1
u u u i , j , k =color index
6 ijk i j k
• Triangle anomaly
γ
f
γ
f
γ
Divergent fermion loops
Divergence
[
2 1
~ Q f =1 11 N C 3
3 3
f
leptons
]
quarks
3 generations of u/d-type quark
cancels if NC = 3
1.3 Tests of QCD in different processes
Discussed in
Section 2 and 3
Tevatron / LHC
(optional)
2. Test of QCD in e+e- annihilation
2.1 Discovery of the gluon
Discovery of 3-jet events by the TASSO
collaboration (PETRA) in 1977:
3-jet events are interpreted as quark
pairs with an additional hard gluon.
# 3-jet events
0.15 ~ s
# 2-jet events
at √s=20 GeV
αs is large
2.2 Spin of the gluon
Ellis-Karlinger angle
Ordering of 3 jets: E1>E2>E3
Measure direction of jet-1 in the
rest frame of jet-2 and jet-3: θEK
Gluon spin J=1