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1. Test of QCD in E+E- Annihilation 2

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1. Test of QCD in E+E- Annihilation 2 V. Experimental studies of QCD 1. Test of QCD in e+e- annihilation 2. Structure of the proton 3. Structure functions and quark densities 4. Scaling violation in DIS 1. Test of QCD in e+e- annihilation 1.1 Two-jet events and production of colored q⎯q a) clear two-jet event structure in e+e− → hadrons (qq) σ (ee → hadrons ) b) R = indicates fractional charges and NC=3 had σ (ee → µµ) Additional indications for NC=3: • ∆++ statistic problem: Spin J(∆++)=3/2 (L=0), quark content |uuu> → ∆++ = u ↑ u ↑ u ↑ forbidden by Fermi statistic Solution is additional quantum number for quarks (color) 1 ∆++ = ε u ↑ u ↑ u ↑ i, j,k = color index 6 ijk i j k 1 Additional indications for NC=3: • Triangle anomalie γ ⎯f W Divergent fermion loops f W Divergences cancel if NC = 3: ⎡ 2 1 ⎤ 0 Q ( 1) ( 1) ( 1) N ( ) 3 = ∑ f = − + − + − + C ⋅ ⎢ − ⋅ ⎥ f ⎣ 3 3 ⎦ 3 generations of u/d-type quark 1.2 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 ~ αs α is large #2 − jet events s at √s=20 GeV 2 Spin of the gluon Ellis-Karlinger angle Ordering of 3 jets: E1>E2>E3 Measure direction of jet-1 in the Gluon spin J=1 rest frame of jet-2 and jet-3: θEK 1.3 Multi-jet events and gluon self coupling Non Abelian gauge theory (QCD) Casimir (color) factors: 4-jet events 4-jet cross section: FA,B,C,D,E are kinematic functions 3 Exploiting the angular distribution of 4-jets: • Bengston-Zerwas angle r r r r cos χBZ ∝ (p1 × p2 ) ⋅(p3 × p4 ) • Nachtmann-Reiter angle r r r r cosθNR ∝ (p1 − p2 ) ⋅(p3 − p4 ) Allows to measure the ratios TF/CF and NC/CF SU(3) predicts: TF/CF = 0.375 and NC/CF =2.25 NC/CF ≠ 0 → contribution from gluon self-coupling in the 4-jet events Confirms QCD prediction (SU(3)) and gluon self-coupling 1.4 Measurement of strong coupling αs 2 2 αs measurements are done at fixed scale Q : αs(Q ) a) αs from total hadronic cross section 2 QED ⎡ αs (s) αs (s) ⎤ σ had (s) = σ had (s)⎢1+ + 1.411⋅ 2 + ...⎥ ⎣ π π ⎦ σ (ee → hadrons ) ⎧ α α 2 ⎫ R 3 Q 2 1 s 1.411 s ... had = = ∑ q ⎨ + + 2 + ⎬ σ (ee → µµ) ⎩ π π ⎭ Not very precise. αs(s) b) αs from hadronic event shape variables σ3−jet 3-jet rate: R3 ≡ depends on αs σ had 3-jet rate is measured as function of a jet resolution parameter ycut 4 Jet Algorithm Hadronic particles are i and j grouped to a pseudo particle k as long as the invariant mass is smaller than the jet resolution parameter: 2 m ij < y s cut mij is the invariant mass of i and j. Remaining pseudo particles are jets. n-jet rate as function of ycut theo QCD calculations provide a theoretical prediction for R3 (αs , ycut) theo → fit R3 (αs , ycut) to the data to determine αs αs(s) Similarly other event shape variables (sphericity, thrust,…) can be used to obtain a prediction for αs c) αs from hadronic τ decays 2 αs(mτ ) τ Γ(τ →ντ + Hadrons) Rhad = ~ f (αs ) Γ(τ → ντ + eυe ) d) αs from DIS (deep inelastic scattering) 5 e) Running of αs Similar to QED there are propagator corrections to be taken into account screening anti-screening 2 Effective strong coupling αs(Q ) Conventionally α (µ 2 ) α (Q2 ) = s renormalization scale is s α (µ 2 ) Q2 s 2 2 1+ (33 − 2nf )log 2 12π µ µ = M Z n = active quark flavors 2 f For Q →∞ αs →0: µ2 = the renormalization scale At large Q2 quarks are asymptotically free Running of αs and asymptotic freedom α (µ 2 ) α (Q2 ) = s s α (µ 2 ) Q2 1+ s (33 − 2n )log 12π f µ 2 6 The Nobel Prize in Physics 2004 David J. Gross H. David Politzer Frank Wilczek "for the discovery of asymptotic freedom in the theory of the strong interaction" 1.5 Description of hadron production in e+ e- annihilation Described by Monte Carlo programs: •HERWIG: Parton shower (partons are associated to colorless clusters). Clusters with masses larger than MCl are split. • JETSET: String fragmentation. Color string between colored quarks. Gluons lead to “kinks”. Strings are fragmented 7 2. Structure of the proton θ − q 2 = 4EE′sin4 2.1 Elastic electron-proton scattering 2 dσ α 2 E′ General form of diff. = ⋅ {} e θ θ KK cross section dΩ 4EE′sin4 E 2 p M non pointlike r r Rutherford scattering partners k , E k ′, E ′ w/ spin e recoil q = k −k′ Pointlike target w/o spin elastic ⎛ 2 θ ⎞ {}KK Mott = ⎜cos ⎟ Mott scattering ⎝ 2 ⎠ p 2 elastic ⎛ θ q θ ⎞ Pointlike target w/ spin = ⎜cos2 − sin2 ⎟ {}KK eµ→eµ ⎜ 2 ⎟ ⎝ 2 2M 2 ⎠ 2 2 2 elastic ⎛G +τ G θ θ ⎞ q Extended proton w/ spin = ⎜ E M cos2 − 2τ G2 sin2 ⎟ mit τ = {}KK ep→ep ⎜ M ⎟ 2 ⎝ 1+τ 2 2 ⎠ 4M Size of the proton Rosenbluth Plot: dσ dσ θ = A(q 2 ) + B(q 2 )tan2 dΩ dΩ Mott 2 G2 −τ G2 q2 A(q2 ) = E M mit τ = 1−τ 4M2 2 2 B(q ) = - 2τ GM Allows the determination of electric and magnetic form factors (Sachs form factors) 8 p GM Electric and magnetic form factors can be described by dipole ansatz: 1 G(q2 ) = q 2 (1− )2 0.71 GeV2 p GE Dipole form factor is the Fourier transform of an exponential distribution → radius of the charge (magnetic moment) distribution: 2 dG r = − 6 2 dq q2 =0 n GM 1 1 2 2 2 2 −15 rel = rmag = 0.81⋅10 m Electric and magnetic size of proton and neutron are the same. 2.2 Inelastic ep scattering r r k , E k ′, E ′ e Event rate Event q p Invariant Proton breaks up mass W ← Q2 M Ehad E’ Observation: Large cross section at large Q2 Inelastic scattering: M≠W From elastic form factor one expects large 2 2 independent variables: suppression at large Q • ν = E - E′ = Ehad –M Elastic scattering: W=M • q 2 = W 2 − M 2 − 2Mν = Q 2 only one free variable Alternative: Bjorken-Variables Q2 W = M ⇒ = 1 Q2 ν 2Mν x = and y = Inelasti 2Mν E -city 9 a) Differential cross section and structure functions dσ α 2 ⎛ θ q2 θ ⎞ Elastic = ⋅ ⎜cos2 − sin2 ⎟ θ ⎜ 2 ⎟ dΩ 4E 2 sin4 ⎝ 2 2M 2 ⎠ eµ→eµ 2 2 dσ α ⎛ 2 2 θ 2 2 θ ⎞ = ⋅ ⎜W2(ν,q )cos + 2W1(ν,q )sin ⎟ ′ 2 4 θ dE dΩ 4E sin ⎝ 2 2 ⎠ Inelastic 2 ep→eX 2 dσ 4πα E′ ⎛ 2 2 θ 2 2 θ ⎞ = ⋅ ⎜W2(ν,q )cos + 2W1(ν,q )sin ⎟ dq2dν q 4 E ⎝ 2 2 ⎠ 2 2 With structure functions W1(ν,q ) and W2(ν,q ) . Scattering at point-like objects: − q2 q2 q2 2W point = δ(ν + ) W point = δ (ν + ) 1 2M 2 2M 2 2M W1 and W2 functions of only one variable b) First measurement of W2 SLAC & MIT Experiment Spectrometer at fixed θ ∆p ~ 0.1% ∆θ ~ 0.7mrad p e- beam, up to 20 GeV ∆p ∈[]0.1%; 0.25% p Dipole and quadrupole magnets Cherenkov counter for e/π separation 10 Structure function νW2 does not depend explicitly on Q2 but depends only on the dimensionless variable x: Q2 x = 2Mν → Scale invariance: “scaling” Indicates elastic scattering at point-like constituents of the proton: partons Real sensation in 1970 … 11 The Nobel Prize in Physics 1990 Jerome I. Friedman Henry W. Kendall Richard E. Taylor "for their pioneering investigations concerning deep inelastic scattering of electrons on protons and bound neutrons, which have been of essential importance for the development of the quark model in particle physics" c) Structure functions F1 and F2 Instead of W1 and W2 the structure functions F1 and F2 are used today: Q2 νW (ν,Q2 ) → F (x) = νW (x = ) 2 2 2 2Mν dimensionless Q2 MW (ν,Q2 ) → F (x) = MW (x = ) 1 1 1 2Mν 2 dσ 4πα E′ ⎛ 2 2 θ 2 2 θ ⎞ = ⋅ ⎜W2(ν,q )cos + 2W1(ν,q )sin ⎟ dq2dν q 4 E ⎝ 2 2 ⎠ dσ 4πα 2 E′ ⎛ q2 θ 2ν q2 θ ⎞ = ⋅ ⎜F ( )cos2 + F ( )sin2 ⎟ 2 4 ⎜ 2 1 ⎟ dq dν q Eν ⎝ 2Mν 2 M 2Mν 2 ⎠ dσ 4πα 2 E′ F (x) ⎛ θ 2xF (x) Q2 θ ⎞ = ⋅ 2 ⎜cos2 + 1 sin2 ⎟ 2 4 ⎜ 2 2 ⎟ dQ dx Q E x ⎝ 2 F2(x) 2M x 2 ⎠ 12 2.3 Parton Model for electron-nucleon scattering: ep → eX ee qi p σ = incoherent sum of all possible parton (quark) contributions 1. Nucleon consists of quasi-free point-like constituents (partons) 2. Lepton scatters elastically on free spin ½ partons 3. Scattered parton interacts strongly with the other constituents (spectators) to form observable hadrons Infinite momentum frame – Bjorken x Infinite momentum frame: • Proton carries infinite momentum pp q (transverse mom. + mass of quarks can be neglected) P • 4-mom fraction of parton: pp = xpP pp′ Infinite momentum frame: Proton rest frame: Consequence of elast. P = (M , 0) scattering: p + q = p′ p p q = (ν , qr) − q2 − q2 1= = xpP + q = pp′ 2xpMν 2mpν 2 2 2 Pq = Mν if formally m = x M pp + 2xpPq + q = pp′ p p 2 2 2 ⇒ q = −2xpPq − q Q Bjorken variable x = x p = = ≡ x momentum fraction of parton = −2xpMν 2Mν 2Mν 13.
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