8. QCD

Quantum ChromodynamicsThe theory of strong interactionsVincent HedbergV. Hedberg Quantum Chromodynamics 1

Quantum ChromodynamicsInteractions by massless spin-1 particles: Gauge bosonsQuantum electrodynamics (QED): PhotonsQuantum chromodynamics (QCD): GluonsQED: Photons couple to electric charges (Q)QCD: Gluons couple to colour charges (Y c and I c ).3Y c : colour hypercharge.I c : colour isospin charge.3The strong interaction is flavour-independent:It acts the same on u,d,s,c,b and tV. Hedberg Quantum Chromodynamics 2

Quantum ChromodynamicsThe colour hypercharge (Y c ) + the colour isospin charge (I c )three colour and three anti-colour quark states3QCDY c Ic3r 1/3 1/2g 1/3 -1/2b -2/3 0Y c Ic3r -1/3 -1/2g -1/3 1/2b 2/3 0Colour confinement:Mesons and baryons total colour charge = 0.Colour wave-functions:QCD11qq = (rr+gg+bb)3q 1 q 2 q 3 = (r 1 g 2 b 3 -g 1 r 2 b 3 +b 1 r 2 g 3 -b 1 g 2 r 3 +g 1 b 2 r 3 -r 1 b 2 g 3 )6V. Hedberg Quantum Chromodynamics 3

Quantum ChromodynamicsQCD: Colour hypercharge (Y c ) and colour isospin charge (I c )3Quark model: Flavour hypercharge (Y) and flavour isospin (I 3 )Quark ModelQ Y I 3 Q Y I 3d -1/3 1/3 -1/2u 2/3 1/3 1/2s -1/3 -2/3 0d 1/3 -1/3 1/2u -2/3 -1/3 -1/2s 1/3 2/3 0c 2/3 4/3 0 c -2/3 -4/3 0b -1/3 -2/3 0 b 1/3 2/3 0t 2/3 4/3 0 t -2/3 -4/3 0The total wavefunction of hadrons:ψ total = ψ space x ψ spin x ψ flavour x ψ colourV. Hedberg Quantum Chromodynamics 4

Quantum ChromodynamicsPhotons electric charge = 0 !Gluons have colour charge !Gluons exist in 8 different colour states:χ C = r g g1 IC = 1 CY = 03QCDχ C = r g g2 IC = C1 Y = 03χ C = r b g3 IC = 1/2 CY = 13χ C = r b g4 IC = C1/2 Y = 13χCg5= g b I C = C1/2 Y = 13χ C = g b g6 IC =1/2 CY = 13χ C = 1/ 2 ( g g - r r ) I C = 0 CY = 0g7 3χ C = 1/ 6 ( g g - r r - 2 b b )g8I C = 0 CY = 03Color confinement Gluons do not exist a free particles.V. Hedberg Quantum Chromodynamics 5

Quantum ChromodynamicsColour hypercharge + colour isospin charge are additivequantum numbers (like the electric charge).Example:I C 3 = 1/2 CY = 1/3uuI C 3 = 0 C _Y = 2/3gluonI C 3 = 0 C _Y = 2/3ssI C 3 = 1/2 CY = 1/3Gluon:I 3CY CIC 3 ( r)I3 C 1= – ( b)= --2= Y C ( r) – Y C ( b)= 1χcg3 = r bV. Hedberg Quantum Chromodynamics 6

Quantum ChromodynamicsGluons can couple to other gluons(since they carry colour charge).QCD: Gluons can form colourless states.GlueballsExperiments: Difficult to prove existance of glueballs.Leptons have no colour charge.They do not interact strongly.V. Hedberg Quantum Chromodynamics 7

Quantum ChromodynamicsThe strong coupling constantQED: The electromagnetic coupling contant:α emQCD: The strong couplings constant: α sα s : “running constant” Decreases with increasing Q 2What is Q 2 ?The 4-vectors of the interacting quarks:P= (E,p) = (E,p x ,p y ,p z )uuThe 4-vectors of the quarks 4-vector of gluong*qq = (E q ,q) = P 1 P 2 = (E 1 - E 2, P 1 P 2 )P 1 P 2P 4P 3ssThe squared 4-vector energy-momentum tranfer:Q 2 = q . q (i.e. Q = the “mass” of the gluon)V. Hedberg Quantum Chromodynamics 8

α s=Quantum ChromodynamicsQCD prediction:-------------------------------------------------------12π( 33 – 2N f) ln( Q 2 ⁄ Λ 2 )N f : Number of allowed quarkflavoursThe strong coupling constant0.5α s (Q)0.40.30.2DataTheoryDeep Inelastic Scatteringe + e - AnnihilationHadron CollisionsHeavy QuarkoniaΛ(5)MSNLONNLOLatticeα s(Μ Z){275 MeV 0.123QCDO(α 4 220 MeV 0.119s)175 MeV 0.115Λ: QCD scale parameterDetermined experimentally(Λ≈0.2 GeV)0.11 10 100Q [GeV]V. Hedberg Quantum Chromodynamics 9

Quantum ChromodynamicsThe quark-antiquark potential (mesons)V(r)Breaking of the stringqrqSpring-likeVr ( ) = λr (r ≥ 1fm)rThe strong couplings constantrCoulomb-likeVr ( )=4-- α s– ----- (r < 0,1 fm)3 rV. Hedberg Quantum Chromodynamics 10r

Quantum ChromodynamicsThe principle of asymptotic freedom.QCD - Short distances - interaction is weakerQCD - Long distances - interaction is strongerSmall distance Coulomb-like potential quarks and gluonsSmall distance large Q 2 small α sfree particlesSmall distancesLarge distancesfirst order diagrams.higher order diagrams.Higher-order diagramsConfinement cannot be calculated analyticallyV. Hedberg Quantum Chromodynamics 11

Electron-positron collisions+ -eeV. Hedberg Quantum Chromodynamics 12

Electron-positron annihilationThe R-value from e + e - experimentsMeasurement:R σ ( e+ e - → hadrons)≡ -----------------------------------------------σ( e + e - → μ + μ - )rate of collisions that give hadronsrate of collisions that give muonsPrediction from theory:R = N c Σe2qN c is the number of colours (=3)e q the charge ofthe quarks.V. Hedberg Quantum Chromodynamics 13

Electron-positron annihilation2 2 22 2 2R =N c ( e u +e d +e s ) = 3 ((-1/3)+(-1/3)+(2/3)) = 22 2 2 2R =N c ( e u +e d +e s +e c )=10/32 2 2 2 2R =N c ( e u +e d +e s +e c +e b )=11/3ifififs

Electron-positron annihilationJets of particlese + e - annihilation process:A photon or a Z 0 is producedquark-antiquark pair.The quark and the antiquark fragment into observable hadrons.PETRA(DESY)hadronsLEP(CERN)hadronse - e +7-22 GeVqγ *Length: 2.3 kmExperiments: Tasso, Jade, Pluto, Mark J, Celloq-hadrons7-22 GeVe - e +45-104 GeVqZ 0hadrons45-104 GeVLength: 27 kmExperiments: DELPHI, OPAL, ALEPH, L3q-V. Hedberg Quantum Chromodynamics 15

Electron-positron annihilationTwo jets of particlesThe direction of the jet The direction of the quarkTwo-jet events in the Jade experimentTwo-jet events in theory:e +e - γ ∗ qqjets of hadronse + + e - → γ ∗ → hadronsEnergy and momentum conservationThe quark and anti-quark have equal energy and opposite directionJets have the same energy and opposite directionV. Hedberg Quantum Chromodynamics 16

Electron-positron annihilationThe angular distribution of 2 jetsand the spin of the quarks.e + + e - → γ ∗ → μ + + μ -dσd-------------- (cosθe + e- → μ + μ-)=---------πα 22Q 2 ( 1 + cos2 θ)eμθμee + + e - → γ ∗ → q + qdσd(cosθ→ qq ) = N ce2πα 2q --------- ( 1 + cos2 θ)2Q 2quark spin = 1/2dσd(cosθ→ qq ) = N ce2πα 2q --------- ( 1–cos2 θ)2Q 2quark spin = 0N c is the numbere q is the quark chargeof colours (=3)V. Hedberg Quantum Chromodynamics 17

Electron-positron annihilation1+cos 2 θ(spin 1/2)The experimental data1-cos 2 θ(spin 0)Conclusion: Quarks have spin = 1/2 !V. Hedberg Quantum Chromodynamics 18

The discovery of the gluonThe accelerator: PETRA at the German laboratory DESY.V. Hedberg Quantum Chromodynamics 19

The discovery of the gluonThe experiment: TASSOMuon detectorDriftchamberMagnet coilCherenkov detectorLiquid Argon CalorimeterV. Hedberg Quantum Chromodynamics 20

The discovery of the gluonJadeTassoThree-jet events in theory:e +e - γ ∗ qqgjets ofhadronsThree-jet events The third jet from a gluon !V. Hedberg Quantum Chromodynamics 21

The discovery of the gluonThe probability for gluon emission is proportional to α s .α s=Number of three-jet eventsNumber of two-jet events0.5α s (Q)0.4DataTheoryDeep Inelastic Scatteringe + e - AnnihilationHadron CollisionsHeavy QuarkoniaNLONNLOLatticePETRA:α s =0.15 ± 0.03fors = 30-40 GeV0.30.2Λ(5)MSα s(Μ Z){275 MeV 0.123QCDO(α 4 220 MeV 0.119s)175 MeV 0.1150.11 10 100Q [GeV]V. Hedberg Quantum Chromodynamics 22

Electron-positron annihilationAngular distribution of 3 jets and the spin of the gluon.E maxJet 1θ 3θ 2θ 1Jet 3Jet 2E mingluon spin = 0gluon spin = 1cosφ=-sinθ 2sinθ 3sinθ 1Angular distribution Gluons have spin = 1V. Hedberg Quantum Chromodynamics 23

Electron-positron annihilationHadronizationQCDfreeparticlesElectro weakV. Hedberg Quantum Chromodynamics 24

Electron-proton collisionseprotoneprotonV. Hedberg Quantum Chromodynamics 25

Electron-proton scatteringλ >> r p Very low electron energiesScattering from a “point-like” spin-less object.e -γ*Q 2lowλ = r p Low electron energiese -Scattering from an extended charged object.γ*λ < r pHigh electron energiesInteractions with the valence quarks in the proton.e -γ*λ

Electron-proton scatteringElastic scatteringElastic scattering: The same type of particles before and after.lepton = e or μP 1 P 2gauge boson = γ*q = P 1 P 2= ( ν, q)P 3hadron = pP 4 = P 3 + qElastic electron-proton scattering Size of the protonV. Hedberg Quantum Chromodynamics 27

Electron-proton scatteringDifferential cross sectionThe differential cross sectiondσ( θϕ , )---------------------dΩgives the angular distribution.sinθdθdϕdΩ=dA------L 2The total cross section by integration:σdσ---------------------( θϕ , )= dΩ =dΩπ 2πdσ---------------------( θϕ , )sinθdθdϕdΩ0 0V. Hedberg Quantum Chromodynamics 28

Electron-proton scatteringElastic scattering on a static point-like charge.electronPmelectrone θPoint-like chargeM>>mThe Rutherford scattering formulaNon-relativistic electronPoint-like electric charge e.-------dσdΩRm 2 α= --------------------------2 α4p 4 4 θsin --2=e 2-----4πThe Mott scattering formulaRelativistic electronPoint-like electric charge e.dσ------- dΩM=α 2-------------------------- m 2 p 2 2 θ+ cos --4p 4 4 θ 2sin (--)2V. Hedberg Quantum Chromodynamics 29

Electron-proton scatteringElastic scattering on an extended charged object.P 2eρ( r)electronP θ1relectronStatic sphericalcharge distributionMomentum transferq = P 1 P 2q 2 = q . qρ(r): a spherically symmetric density function withρ(r): describes how the charge e is spread out.3rdx = 1Not point-like interactionRutherford formula modified by electric form factor G E (q 2 )dσ-------dΩ=dσ-------dΩR G E 2 ( q 2 )q = P 1 P 2dσ-------dΩR--------------------------m 2 α 24p 4 4 θsin --V. Hedberg Quantum Chromodynamics 30=2

Electron-proton scatteringThe electric form factor is the Fourier transform of the chargedistribution with respect to the momentum transfer q:G Eq 2 = re iqxd3 xThe electric form factor has values between 0 and 1:Low momentum transfer:High momentum transfer:G E( 0) 1= forq = 0G E( q 2 ) → 0 for q 2 → ∞Measurements of the cross-section The form factorThe charge distribution.The mean quadratic charge radiusV. Hedberg Quantum Chromodynamics 31r E23= r 2 rdx =6 dG Eq2 – --------------------dq 2q 2 = 0

Electron-proton scattering-------dσdΩ=G 1Q 2 ( )P 3Pelectron2θP 1 q = P 1 P 2MQ 2 = q . qThe proton mass:Elastic electron-proton scattering-------dσdΩxMelectronprotonP 4Protons have charge + magnetic momentelectric formfactor (G E ) + magnetic formfactor (G M )G 1 Q 2 ( )Q 2cos2 Q 22θ-- +2----------G2M 2 2 ( Q 2 )θsin -- 2G 2 2+ ----------GE 4M 2 M= -------------------------------------Q12G 2 ( Q 2 2) = G M+ ----------4M 24-momentum transferV. Hedberg Quantum Chromodynamics 32

Electron-proton scatteringMeasurement of the formfactors:i) low Q 2 (Q

Electron-proton scatteringIf protons are point-particlesMeasurementsof G E and G Mof the protonG EPoint particleG MG E (Q 2 ) = 1G M (Q 2 ) = 2.79Point particleElectric chargeof a protonMagnetic momentof a protonConclusion:The protonhas anextendedchargedistribution !V. Hedberg Quantum Chromodynamics 34

Electron-proton scatteringInelastic electron-proton scatteringInelastic electron-proton scattering The proton is broken upP 1 P 2q = P 1 P 2P 3Mγ* = ( ν, q)P 4 = P 3 + qQ 2 =q . qelectron4-momentum transferBjorken scaling variablex =Q 22Mν0 < x < 1M is the mass of the proton.V. Hedberg Quantum Chromodynamics 35

Electron-proton scatteringThe differential cross section for inelastic ep scattering:----------------dσdE 2 dΩ--------------------------α 2.=2 θ4E -- 1 sin 4 21--Fν 2 xQ ,2θ( ) cos2 -- + ---- F2M 1( x,Q 2 ) 2θsin 2 --2Dimensionless structure functionsF 1,2 (x,Q 2 ) parameterize the photon-proton interaction in thesame way a G 1 (Q 2 ) and G 2 (Q 2 ) do it in elastic scattering.Bjorken scaling or scale invariance:F 1, 2( x,Q 2 ) = F 1, 2( x)when Q 2 → ∞ and x is fixed and finite.i.e. the structure functions are independent on Q 2 for Q>>M .Scaling: F 1,2 (x,Q 2 ) do not change if masses, energies andmomenta are multiplied by a scale factor.V. Hedberg Quantum Chromodynamics 36

Electron-proton scatteringThe discovery of quarks at the SLAC 2 mile LINAC2 mile linacEnd station AV. Hedberg Quantum Chromodynamics 37

Electron-proton scatteringThe discovery of quarksThe MIT-SLAC experimentThe 20 GeV spectrometerThe 8 GeV spectrometerMagnetsCerenkov detectorsScintillatorsDetectors for e/π separation8 GeV electrons hits a hydrogen targeteprotonV. Hedberg Quantum Chromodynamics 38

Electron-proton scatteringThe discovery of quarksCalculate x and Q 2 from the energy and angle of the electron.e - E 2 Q 2 θ= 4 ⋅ E 1⋅ E 2sin 2 --2e -HadronsθQ 2E γx = -----------------------------------------1 2 ⋅ M ⋅ ( E 1– E 2)Cross section measurement F 2F 2 does not depend on Q 2Protons have a sub-structure(partons)V. Hedberg Quantum Chromodynamics 39

Electron-proton scatteringDeep inelastic electron-proton scatteringThe parton model: Scale invarianceScattering on point-like constituents (partons) in the proton.The quark model: Partons = QuarksP 1 electron P 2γ/Z 0q = P 1 P 2P 3 uproton d uP 3(1-x)P 3Q 2 =x =q . qQ 22Mν==θ4 ⋅ E 1⋅ E -- 2sin 2 2Q 2-----------------------------------------2 ⋅ M ⋅ ( E 1– E 2)xParton/quark model Fraction of the proton momentum carriedby the struck quark is given by Bjorken x.V. Hedberg Quantum Chromodynamics 40

Electron-proton scatteringDeep inelastic electron-proton scatteringParton model F 1 depends on the spin of the partons (quarks)Prediction:The Callan-Gross relation:F 1 ( xQ , 2 ) = 0 quark spin =02xF 1 ( xQ ,2 ) = F 2 ( xQ ,2 ) quark spin=1/2Measurement:2xF 1F 2MeasurementTheory prediction:Spin = 1/2Spin = 0Bjorken xV. Hedberg Quantum Chromodynamics 41

Electron-proton scatteringPETRA(DESY)hadronsLEP(CERN)hadronse - e +7-22 GeVqγ *hadrons7-22 GeVLength: 2.3 kmExperiments: Tasso, Jade, Pluto, Mark J, Celloq-e - e +45-104 GeVqZ 0hadrons45-104 GeVLength: 27 km (4184 magnets)Experiments: DELPHI, OPAL, ALEPH, L3q-LINAC(SLAC)e -1.6, 8, 20 GeVhadronsγe -Length: 3 kmExperiments: SLAC-MITe0 GeVqgqgqgpHERA(DESY)Length: 6 km (1650 magnets)Experiments: H1, ZEUSV. Hedberg Quantum Chromodynamics 42e +27.6 GeVhadronsγe +e820-920 GeVqgqgqgp

Electron-proton scatteringThe HERA accelerator at DESYHERA accelerator: only large electron-proton collider ever built.Petra was pre-accelerator.6 km longH1V. Hedberg Quantum Chromodynamics 43

HERAElectron-proton scatteringProton accelerator: Super conducting magnets. Energy = 920 GeVElectron accelerator: Normal warm magnets. Energy = 28 GeVCollision energy = 320 GeV (54000 GeV fixed target)Proton acceleratorElectronacceleratorCavitiesfor the electronsV. Hedberg Quantum Chromodynamics 44

Electron-proton scatteringThe H1 ExperimentEvents at HERA were boosted in the proton directiondue to the large difference in electron and proton beam energies.Tracker:Drift- andProportionalchambersMuon detectorsSolenoidmagnetCalorimeter:Liquid argon/lead - emLiquid argon/steel - hadV. Hedberg Quantum Chromodynamics 45

Electron-proton scatteringMeasurement of structure functionsepF 2⋅ 2 i10 510 4x = 0.000050, i = 21x = 0.000080, i = 20x = 0.00013, i = 19x = 0.00020, i = 18x = 0.00032, i = 17x = 0.00050, i = 16x = 0.00080, i = 15x = 0.0013, i = 14x = 0.0020, i = 13x = 0.0032, i = 12H1 e + p high Q 294-00H1 e + p low Q 296-97BCDMSNMCx = 0.0050, i = 1110 3x = 0.0080, i = 10x = 0.013, i = 910 2x = 0.050, i = 610110 -110 6 1 10 10 2 10 3 10 4 10 5x = 0.020, i = 8x = 0.032, i = 7The cross section + the energy andscattering angle of the electronF 2x = 0.080, i = 5x = 0.13, i = 4x = 0.18, i = 3x = 0.25, i = 2x = 0.40, i = 1No quark sub-structure was observeddown to 10 -18 m (1/1000th of a proton)V. Hedberg Quantum Chromodynamics 4610 -210 -3H1 PDF 2000extrapolationx = 0.65, i = 0Q 2 / GeV 2H1 Collaboration

Summary ofscatteringformulasV. Hedberg Quantum Chromodynamics 47

SUMMARY: Electron-Positron interactionse +e - γ ∗ +ll -dσd-------------- cosθe + e- → l + l- πα= --------- 2 2Q 2 ( 1 + cos2 θ)e +e - γ ∗ qqjetsdσd-------------- (cosθe + e- → qq ) = N ce2πα 2q --------- ( 1 + cos2 θ)2Q 2V. Hedberg Quantum Chromodynamics 48

SUMMARY: Elastic electron-proton scatteringPmelectroneelectronθPoint-like chargeM>>m-------dσdΩM=α 2-------------------------- m 2 p 2 2 θ+ cos --4p 4 4 θ 2sin (--)2Static sphericalcharge distribution electron dσelectronPeρ( r)P1 θ 2r-------dΩdσ= -------dΩR G E 2 ( q 2 )Pelectron 2P 3electron θP 1MprotonP 4-------dσdΩ=G 1Q 2 ( )-------dσdΩV. Hedberg Quantum Chromodynamics 49xMG 1 Q 2 ( )Q 2cosθ-- ----------G2 2M 2 2 ( Q 2 )θ+sin -- 22 Q 22G 2 2+ ----------GE 4M 2 M= -------------------------------------Q12 G 2 ( Q 2 2) = G M+ ----------4M 2

SUMMARY: Inelastic electron-proton scatteringE 2ee -θE 1 γHadronsP 1 electron P 2γ/Z 0q = P 1 P 2P 3 uproton d uP 3(1-x)P 3xQ 2 =q . qx =Q 22MνBjorken - xdσ----------------dE 2 dΩ--------------------------α 2 1= .--.ν2 θ4E 1 sin 4 --2F 2 ( xQ ,2 )cos2θ-- Q 22+ ---------xM 2F 1( xQ ,2 )θsin 2 --2V. Hedberg Quantum Chromodynamics 50