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). Yc 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. V. Hedberg Quantum Chromodynamics 17 The discovery of the gluon
The accelerator: PETRA at the German laboratory DESY.
V. Hedberg Quantum Chromodynamics 18 The discovery of the gluon The experiment: TASSO
Muon detector Driftchamber
The central part of Magnet coil the TASSO experiment. Cherenkov detector Liquid Argon Calorimeter
V. Hedberg Quantum Chromodynamics 19 The discovery of the gluon
When the PETRA accelerator started up, one began to see three-jet events in the experiments. The interpretation was that the quark or antiquark emitted a high-momentum gluon that fragmented to a jet.
+ e q
g jets of hadrons e- q
A Tasso 3-jet event. A Jade 3-jet event. V. Hedberg Quantum Chromodynamics 20 The discovery of the gluon
The probability for a quark to emit a gluon is proportional to s and by comparing the rate of two-jet with three-jet events one can determine s.
At PETRA one measured: s=0.15 0.03 for s = 30-40 GeV
The process of turning quarks and gluons into hadrons is called hadronization:
V. Hedberg Quantum Chromodynamics 21 Electron-positron annihilation The spin of the gluon. It is possible to determine the spin of the gluon by measuring the angular distribution of jets in three-jet events.
This is done by measuring: gluon spin = 0 sin - sin3 cos = 2 sin1 gluon spin = 1
where the angles are defined in the following way.
3 Jet 2 E Jet 1 max 2 1 Jet 3 Emin Conclusion: Gluons have spin = 1 ! V. Hedberg Quantum Chromodynamics 22 Electron-proton scattering Electrons are good tools for investigating the properties of hadrons since electrons do not have a substructure. The wavelength of the exchanged photon determines how the proton is being probed. - >> rp Very low electron energies e Q2 The scattering is equivalent to that from a “point-like” * spin-less object. low
= rp Low electron energies e- The scattering is equivalent to that from an extended * charged object. - < rp High electron energies e The wavelength is short enough to make it possible to * interact with the valence quarks in the proton. - << r Very high electron energies e p * The electron can at these short wavelengths interact high with the sea of quarks and gluons.
V. Hedberg Quantum Chromodynamics 23 Electron-proton scattering Elastic scattering
Elastic scattering means that the same type of particles goes into and comes out of the collision.
lepton = e or P1 P2
* gauge boson = q = P1 P2 = ( , q)
P3 hadron = p P4 = P3 + q
Elastic electron-proton scattering can be used to measure the size of the proton.
V. Hedberg Quantum Chromodynamics 24 Electron-proton scattering Differential cross section
The angular distribution of the particles emerging from a scattering reaction is given by the differential cross section:
d ------where d= sin dd d dA d = ------L2
The total cross section of the reaction is obtained by integrating the differential cross section: 2 d d = ------d = ------sindd d d 0 0 V. Hedberg Quantum Chromodynamics 25 Electron-proton scattering Elastic scattering on a static point-like charge. electron
P electron e m Point-like charge M>>m
The Mott scattering formula describes the d 2 2 2 2 angular distribution of a relativistic electron ------= ------m + p cos --- of momentum p which is scattered by a d 2 M 4p4 sin4--- point-like electric charge e. 2
The Rutherford scattering formula describes 2 d m22 e the same for a non-relativistic electron ------= ------where = ------d 4 with a momentum p< V. Hedberg Quantum Chromodynamics 26 Electron-proton scattering Elastic scattering on an extended charged object. electron P2 er electron q = P1 r P1 P2 Static spherical charge distribution If the electric charge is not point-like, but spread out with a spherically symmetric density function (e e(r)) that is normalized 3 to one (O r d x = 1) then the Rutherford scattering formula has 2 2 to be modified by an electric form factor GE(q ): d d d m22 ------= ------G2 q2 where ------= ------ E d d d R R 4 4 4p sin --- 2 V. Hedberg Quantum Chromodynamics 27 Electron-proton scattering The electric form factor is the Fourier transform of the charge distribution with respect to the momentum transfer q: 3 2 = iq4 x GE q O r e d x The electric form factor has values between 0 and 1: Low momentum transfer: GE0 = 1 for q = 0 High momentum transfer: 2 q2 GEq 0 for Measurements of the cross-section can be used to determine the form-factor and hence the charge distribution. The mean quadratic charge radius is for example given by: 2 3 dG q r2 ==Or2r d x –6------E - E dq2 q2 = 0 V. Hedberg Quantum Chromodynamics 28 Electron-proton scattering Elastic electron-proton scattering n tro P3 lec P2 electron e P1 2 p q = P1 P2 Q = q . q The proton mass: M ro to n P4 Scattering of electrons on protons depends not only on the electric formfactor (GE) but also on a magnetic formfactor (GM) which is associated with the magnetic moment distribution: 2 d d 2 2 Q 2 2 ------= ------x G Q cos ---+ ------G Q sin --- 1 2 2 d d M 2 2M 2 2 2 Q G + ------G 2 2 2 E 4M M 2 2 G Q = ------G2Q = G 1 Q2 M 1 + ------4M 2 V. Hedberg Quantum Chromodynamics 29 Electron-proton scattering Measurement of the formfactors are conveniently divided up into three regions of Q2: i) low Q2 (Q< GE dominates the cross section and rE can be precisely measured: rE = 0,85 0,02 fm ii) Intermediate Q2 (0.02 < Q2 < 3 GeV): Both GE and GM give sizable contributions and the form- factors can be described by the parameterization: 2 2 GMQ 2 2 ------ GEQ p 2 + Q2 iii) High Q2 (Q2 > 3 GeV): GM dominates the cross section. V. Hedberg Quantum Chromodynamics 30 Electron-proton scattering The form factors are normalized so that Protons: GE0 = total charge = 1 GM0 = magnetic moment = p= +2.79 Neutrons: G 0 = magnetic moment = = -1.91 GE0 = total charge = 0 M n If the proton is a point particle then GE and GM do not depend 2 on Q and they should be constants with GE=1 and GM=2.79 . Point particle E M G Point particle G Measurements Conclusion: of GE and GM The proton of the proton has an gives: extended charge distribution ! V. Hedberg Quantum Chromodynamics 31 Electron-proton scattering Inelastic electron-proton scattering In inelastic electron-proton scattering, the proton is broken up into new hadrons: electron P1 P2 2 * q = P1 P2 = ( , q) Q = q . q P4 = P3 + q P3 M A new dimensionless variable called the Bjorken scaling variable (x) is introduced which can take values between 0 and 1: Q2 x = 2M where M is the mass of the proton. V. Hedberg Quantum Chromodynamics 32 Electron-proton scattering The differential cross section for inelastic electron-proton scattering can be written as: d 2 1 2 2 2 ------= ------. ---F xQ cos2--- + -----F xQ sin2--- 2 2 1 2 dE2d 2 4 --- M 4E1 sin 2 2 where two dimensionless structure functions F1(x,Q ) and 2 F2(x,Q ) parameterize the photon-proton interaction in the 2 2 same way a G1(Q ) and G2(Q ) do it in elastic scattering. An important concept is that of Bjorken scaling or scale invariance: 2 Q2 F12 xQ = F12 x when and x is fixed and finite. i.e. the structure functions are almost independent on Q2 when Q>>M. It is called scaling because structure functions at a given x remain unchanged if all particle masses, energies and momenta are multiplied by a scale factor. V. Hedberg Quantum Chromodynamics 33 Electron-proton scattering The discovery of quarks at the SLAC 2 mile LINAC 2 mile linac End station A V. Hedberg Quantum Chromodynamics 34 Electron-proton scattering The discovery of quarks The MIT-SLAC experiment The 20 GeV spectrometer The 8 GeV spectrometer Magnets Inside the shielding here were Cerenkov detectors, scintillators and detectors for e/separation. 8 GeV electrons were hitting a hydrogen target. The scattered electrons were selected by magnets at different angles and identified by detectors inside the brown shielding. V. Hedberg Quantum Chromodynamics 35 Electron-proton scattering The discovery of quarks: The measurements It is possible to calculate x and Q2 from the energies and scattering angle of the electron: 44 e - 2 2 e E2 Q = 4 E E sin --- 1 2 2 44 Hadrons e - 2 e Q x = ------E1 2 ME1 – E2 From a cross section measurement it is then possible to extract F2. The result that F2 does not depend on Q2 was later interpreted as the first evidence for the existance of quarks. V. Hedberg Quantum Chromodynamics 36 Electron-proton scattering Deep inelastic electron-proton scattering The scale invariance is explained in the parton model by the scattering on point-like constituents (partons) in the proton. These partons are identical to the quarks that were postulated by the quark model. 2 P1 electron P2 2 q . q 4 E E sin --- Q = = 1 2 2 0 q = 2 2 /Z P1 P2 Q Q x = = ------P3 u 2M 2 ME– E (1-x)P 1 2 proton d u 3 x P3 The parton model is valid if the proton momentum is sufficiently large so that the fraction of the proton momentum carried by the struck quark is given by the Bjorken x. V. Hedberg Quantum Chromodynamics 37 Electron-proton scattering Deep inelastic electron-proton scattering The structure function F1 depends on the spin of the partons (quarks) in the parton model: 2 F1xQ = 0 (spin-0) 2 2 The Callan-Gross relation: 2xF1xQ = F2xQ (spin-1/2) Measurements shows that the partons have spin 1/2: 2xF1 F2 Spin = 1/2 Spin = 0 Bjorken x V. Hedberg Quantum Chromodynamics 38 Electron-proton scattering 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 LINAC HERA e - 0 GeV e + 820-920 GeV (SLAC) e (DESY) e q q ee - g ee + g q q 1.6, 8, 20 GeV e g p 27.6 GeV e g p q q hadrons g hadrons g Length: 3 km Length: 6 km (1650 magnets) Experiments: SLAC-MIT Experiments: H1, ZEUS V. Hedberg Quantum Chromodynamics 39 Electron-proton scattering The HERA accelerator The HERA accelerator at the German DESY laboratory is the only large electron-proton collider ever built. It used PETRA as a pre-accelerator. H1 V. Hedberg Quantum Chromodynamics 40 Electron-proton scattering HERA, which was 6 km long, had a ring of superconducting magnets for the protons and a ring of warm magnets for the electrons. The center-of-mass energy of the collision of 28 GeV electrons on 920 GeV protons was 320 GeV. This is equivalent to a fix target accelerator with a 54 TeV electron beam. Proton accelerator Electron Cavities accelerator for the electrons V. Hedberg Quantum Chromodynamics 41 Electron-proton scattering The H1 Experiment The events at HERA were boosted in the proton direction due to the large difference in electron and proton beam energies. Tracker: Drift- and Muon detectors Proportional chambers Solenoid magnet Calorimeter: Liquid argon/lead - em Liquid argon/steel - had V. Hedberg Quantum Chromodynamics 42 Electron-proton scattering Measurement of structure functions i x = 0.000050, i = 21 + 2 2 H1 e p high Q ⋅ x = 0.000080, i = 20 2 6 10 x = 0.00013, i = 19 94-00 F x = 0.00020, i = 18 + x = 0.00032, i = 17 H1 e p low Q2 x = 0.00050, i = 16 10 5 96-97 x = 0.00080, i = 15 e x = 0.0013, i = 14 BCDMS p x = 0.0020, i = 13 4 10 NMC x = 0.0032, i = 12 x = 0.0050, i = 11 10 3 x = 0.0080, i = 10 x = 0.013, i = 9 x = 0.020, i = 8 2 10 x = 0.032, i = 7 x = 0.050, i = 6 x = 0.080, i = 5 A measurement of the cross section 10 x = 0.13, i = 4 + the energy and scattering angle of x = 0.18, i = 3 1 x = 0.25, i = 2 the electron made it possible to x = 0.40, i = 1 -1 10 measure F2. -2 x = 0.65, i = 0 10 H1 PDF 2000 No quark sub-structure was observed extrapolation -3 -18 10 H1 Collaboration 2 3 4 5 down to 10 m (1/1000th of a proton). 1 10 10 10 10 10 Q2 / GeV2 V. Hedberg Quantum Chromodynamics 43 SUMMARY: Electron-Positron interactions + e - l d 2 2 ------e+e- l+l- = ------1 + cos 2 dcos 2Q + e- l + e q d 22 2 jets ------e+e- qq = N e ------1 + cos dcos c q 2 e- q 2Q V. Hedberg Quantum Chromodynamics 44 SUMMARY: Elastic electron-proton scattering electron d 2 2 2 2 electron e ------= ------m + p cos --- P d 2 m M 4p4 sin 4--- Point-like charge 2 M>>m Static spherical charge distribution electron d d 2 2 electron ------= ------G q er P2 E P1 d d R r 2 d d 2 2 Q 2 2 P2 ------= ------x G Q cos ---+ ------G Q sin --- electron 1 2 2 P3 d d M 2 2M 2 electron 2 P1 2 Q G + ------G 2 M p 2 ro 2 E 4M M t 2 2 o G Q = ------G2Q = G n 1 Q2 M P 1 + ------4 4M 2 V. Hedberg Quantum Chromodynamics 45 SUMMARY: Inelastic electron-proton scattering E 2 P electron P ee 1 2 Hadrons 0 q = ee - /Z P1 P2 E 1 P3 u (1-x)P proton d u 3 x P3 Q2 = q . q Q2 x = Bjorken - x 2M 2 2 d 1 2 2 Q 2 ------= ------.--- .. F xQ cos --- + ------F xQ sin2--- 2 2 2 1 2 dE2d 2 4 --- xM 4E1 sin 2 V. Hedberg Quantum Chromodynamics 46