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V. Experimental studies of QCD 1. Test of QCD in e+e- annihilation 2. Structure of the 3. Structure functions and 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 for (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

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 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 "

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. lead to “kinks”. Strings are fragmented

7 2. Structure of the proton θ − q 2 = 4EE′sin4 2.1 Elastic -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 (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 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 on and bound , which have been of essential importance for the development of the in "

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- 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 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