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 Gluons of color octet: B R R B,R G,G B,G R,B G,B R R B 1 R RG G s 2 1 R R G G2B B B B 6 1 R RG G2B B Ninth state = color singlet B 6 B does not take part in interaction 1 R R G G B B 3
To recapitulate the Quantum Electrodynamics previously discussed material. Lagrangian for free spin ½ particle:
µ L(x,t) = iψ (x,t)γ ∂ 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) = eiα(x)ψ (x)
requires the substitution:
i∂µ → i∂µ + eA µ (x)
If one defines the tansformation of A under local gauge transformation as
Aµ (x) → Aµ (x) + ∂µα(x)
iα ( x ) one finds L(x) ψ→ψ e →L(x) invariance of L:
To interpret the introduced field A µ as photon field requires to complete the Langrangian by the corresponding field energy:
µ µ 1 µν L =ψ (iγ ∂µ − m)ψ + eψγ ψ Aµ − Fµν F 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
R Lagrangian is constructed with quark wave functions = G B
Invariance of the Lagrangian under Local SU(3) Gauge Transformation
k x i k x ' x =U x x =e 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 ... [SU(3) group generators] 1 8 requires interaction fields – 8 gluons corresponding to these matrices
1.1 Color factors
e1 q q Coupling strength Coupling strength: QED e e QCD 1 1 2 c c q q 2 1 2 s e2 C Examples: F Color factor B B 1 1 2 2 1 R R G G 2B B C F = = 6 2 3 B B 6 6
R R R R 1 1 R RG G R RG G2B B 2 6 1 1 C = R R C = R R F 12 F 4 1 Color factor for qq color singlet state (meson): R R G G B B 3
B B B R B G 1 3× R RG G2B B B R B G 6 B B B R B G 1 1 1 C = C = C F = F F 3 2 2 1 1 1 1 1 4 C =3 = F 3 3 3 2 2 3
In the case of a color Color singlet meson singlet, each initial and is composed of 3 final state carries a different possibilities factor 1 3
Triple and quadruple gluon Vertex Gluons carry color charges: important feature of SU(3) R G R B G B
R G Color flow B
Color factors
~ N C ~ T F 4 1 = =3 = 3 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 qq d ~1cos 2 d ee hadrons c) R = indicates fractional charges and N =3 had ee C
d) Further indications for N C=3:
∆++ Ω ( s) 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 Divergent fermion loops
f γ Divergence 2 1 ~ Q =111 N 3 f C [ 3 3 ] f leptons quarks 3 generations of u/d-type quark
cancels if N C = 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~ α is large # 2-jet events s s at √s=20 GeV 2.2 Spin of the gluon
Ellis-Karlinger angle
Ordering of 3 jets: E 1>E 2>E 3
Measure direction of jet-1 in the θ Gluon spin J=1 rest frame of jet-2 and jet-3: EK