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Quantum Chromodynamics Quantum Chromodynamics (QCD) is the theory of strong interactions i.e. the force field that binds quarks together in protons and neutrons. V. Hedberg Quantum Chromodynamics 1 Quantum Chromodynamics Interactions are carried out by massless spin-1 particles called gauge bosons. In quantum electrodynamics (QED), gauge bosons are photons and in QCD they are called gluons. Gauge bosons couple to conserved charges: QED: Photons couple to electric charges (Q) c c QCD: Gluons couple to colour charges (Y and I3). c Y is called colour hypercharge. c I3 is called colour isospin charge. The strong interaction acts the same on u,d,s,c,b and t quarks because the strong interaction is flavour-independent. V. Hedberg Quantum Chromodynamics 2 Quantum Chromodynamics c c The colour hypercharge (Y ) and colour isospin charge (I3) can be used to define three colour and three anti-colour states that the quarks can be in: c c c c Y I3 Y I3 r 1/3 1/2 r -1/3 -1/2 g 1/3 -1/2 g -1/3 1/2 b -2/3 0 b 2/3 0 All observed states (all mesons and baryons) have a total colour charge that is zero. This is called colour confinement. Zero colour charge means that the hadrons have the following colour wave-functions: 1 qq = (rr+gg+bb) 3 1 q q q = (r g b -g r b +b r g -b g r +g b r -r b g ) 1 2 3 6 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 V. Hedberg Quantum Chromodynamics 3 Quantum Chromodynamics c c The colour hypercharge (Y ) and colour isospin charge (I3) should not be confused with the flavour hypercharge (Y) and flavour isospin (I3) that were introduced in the quark model: Q Y I3 Q Y I3 d -1/3 1/3 -1/2 d 1/3 -1/3 1/2 u 2/3 1/3 1/2 u -2/3 -1/3 -1/2 s -1/3 -2/3 0 s 1/3 2/3 0 c 2/3 4/3 0 c -2/3 -4/3 0 b -1/3 -2/3 0 b 1/3 2/3 0 t 2/3 4/3 0 t -2/3 -4/3 0 After introducing colour, the total wavefunction of hadrons can now be written as: totalspacexspinxflavourxcolour V. Hedberg Quantum Chromodynamics 4 Quantum Chromodynamics Photons do not carry electric charge but gluons do carry colour charges themselves ! The gluons can in fact exist in 8 different colour states given by the following colour wave functions: C C C = r g I = 1 Y = 0 g1 3 C C = r g I = 1 Y C = 0 g2 3 C C C = r b I = 1/2 Y = 1 g3 3 C C = r b I = 1/2 Y C = 1 g4 3 C C C = g b I = 1/2 Y = 1 g5 3 C C = g b I =1/2 Y C = 1 g6 3 C C C 1/ 2 ( g g - r r ) I = 0 Y = 0 g7 = 3 C C C 1/ 6 ( g g - r r - 2 b b ) I = 0 Y = 0 g8 = 3 Gluons do not exist a free particles since they have colour charge. V. Hedberg Quantum Chromodynamics 5 Quantum Chromodynamics The colour hypercharge and colour isospin charge are additive quantum numbers like the electric charge. The gluon colour charge in the following process can therefore be easily calculated: C C u u C Example: I = 1/2 Y = 1/3 C _ 3 I3 = 0 Y = 2/3 gluon C C _ C I = 0 Y = 2/3 C 3 s s I3 = 1/2 Y = 1/3 1 Gluon: IC ==ICr – ICb --- 3 3 3 2 YC ==YCr – YCb 1 c g3= r b V. Hedberg Quantum Chromodynamics 6 Quantum Chromodynamics Gluons can couple to other gluons since they carry colour charge. This means that gluons can in principle bind together to form colourless states. These gluon states are called glueballs. V. Hedberg Quantum Chromodynamics 7 Quantum Chromodynamics The quark-antiquark potential The quark-antiquark potential can be described in the following simplified way: V(r) Breaking of the string Spring-like Vr= r (r 1fm r The strong couplings constant 4s Coulomb-like Vr= –--------- (r 0 1 fm 3 r V. Hedberg Quantum Chromodynamics 8 Quantum Chromodynamics The strong coupling constant The strong couplings constant s is the analogue in QCD of em in QED and it is a measure of the strength of the interaction. It is not a true constant but a “running constant” since it decreases with increasing Q2. What is Q2 ? Assume that the 4-vectors of the interacting quarks are P1 P2 given by u u P = (E,p) = (E,px,py,pz) g* q The 4-vector energy-momentum tranfer is then Q2 = q .q (i.e. the “mass” of the gluon) s s which can be calculated from the 4-vectors of the quarks P3 P4 , q = (Eq,q) = P 1 P2= (E1 - E2 P1 P2) V. Hedberg Quantum Chromodynamics 9 Quantum Chromodynamics The strong coupling constant 0.5 Theory Data α NLO NNLO In leading order of QCD, s(Q) Lattice Deep Inelastic Scattering is given by: e+e- Annihilation s 0.4 Hadron Collisions 12 Heavy Quarkonia = -------------------------------------------------------- Λ (5) α (Μ ) s 2 2 MS s Z 33– 2N lnQ 275 MeV 0.123 f QCD 0.3 0.119 α 4 ) { 220 MeV O( s 175 MeV 0.115 where 0.2 Nf: Number of allowed quark flavours : QCD scale parameter 0.1 that has to be determined experimentally 0.2 GeV) 1 10 100 Q [GeV] V. Hedberg Quantum Chromodynamics 10 Quantum Chromodynamics The principle of asymptotic freedom. At short distances the strong interaction is weaker and at large distance the interaction gets stronger. The combination of a Coulomb-like potential at small distances 2 and a small s at large Q (i.e. small distances) means that quarks and gluons act as essentially free particles and interactions can be described by the lowest order diagrams. At large distances the strong interaction can, however, only be described by higher order diagrams. Due to the complexity of the higher-order diagrams, the very process of confinement cannot be calculated analytically. Only numerical models can be used ! V. Hedberg Quantum Chromodynamics 11 Electron-positron annihilation The R-value At e+e- colliders one has traditionally studied the ratio of the number of events with hadrons to those with muons: e+e- hadrons R ------------------------------------------------ e+e- +- The cross section for hadron and muon production would be almost the same if it was not for quark flavours and colours i.e. 2 R = Nc eq where Nc is the number of colours (=3) and eq the charge of the quarks. V. Hedberg Quantum Chromodynamics 12 Electron-positron annihilation 2 22 2 2 2 R =Nc(eu+ed+es ) = 3 ((-1/3)+(-1/3)+(2/3)) = 2 if sm 22 2 2 R =Nc(eu+ed+es +ec)=10/3 if sm 22 2 2 2 R =Nc(eu+ed+es +ec+eb)=11/3 if sm s R (GeV) If the radiation of hard gluons is taken into account, an extra factor proportional Nc to s has to be added: Nc 2 sQ 2 ------------------- No colour R = 3eq1 + Nc 2 2 2 2 e2+e2+e2 2 2 q eu+ed+es+ec u d s+ec+eb 2 2 2 eu+ed+es V. Hedberg Quantum Chromodynamics 13 Electron-positron annihilation Jets of particles In the lowest order e+e- annihilation process, a photon or a Z0 is produced which then converts into a quark-antiquark pair. The quark and the antiquark fragment into observable hadrons. PETRA hadrons LEP hadrons (DESY) (CERN) * e e - e 0 e e q e+ e- q Z e+ 7-22 GeV q- 7-22 GeV 45-104 GeV q- 45-104 GeV hadrons hadrons Length: 2.3 km Length: 27 km (4184 magnets) Experiments: Tasso, Jade, Pluto, Mark J, Cello Experiments: DELPHI, OPAL, ALEPH, L3 V. Hedberg Quantum Chromodynamics 14 Electron-positron annihilation Jets of particles Since the quark and antiquark momenta are equal and counter- paralell, the hadrons are produced in two jets of equal energy going in the opposite direction. The direction of the jet reflects the direction of the corresponding quark. + e q jets of hadrons e- q e+ + e- hadrons Diagram for two-jet events. Two-jet event recorded by the Jade experiment at PETRA. V. Hedberg Quantum Chromodynamics 15 Electron-positron annihilation A study of the angular distribution of jets give information about the spin of the quarks. The angular distribution of e+ + e- + + - is d 2 2 where is the production angle with ---------------e+e- +- = ----------1 + cos dcos 2Q2 respect to the direction of the colliding electrons. The angular distribution of e+ + e- q + q is d + - 22 2 ---------------e e qq = N eq----------1 + cos if the quark spin = 1/2 dcos c 2Q2 d + - 22 2 ---------------e e qq = N eq----------1– cos if the quark spin = 0 dcos c 2Q2 where eq is the fractional quark charge and Nc is the number of colours (=3). V. Hedberg Quantum Chromodynamics 16 Electron-positron annihilation The experimentally measured angular distribution of jets is clearly following (1+cos2). The jets are therefore associated with spin 1/2 particles. Quarks have spin = 1/2 ! The angular distribution of the quark jets in e+e- annihilations, compared with models with spin=0 and 1/2.
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