1. Lecture I: pQCD and spectra

2. 2 What is QCD? From: T. Schaefer, QM08 student talk

3. 3 QCD and hadrons Quarks and gluons are the fundamental particles of QCD (feature in the Lagrangian) However, in nature, we observe hadrons: Color-neutral combinations of quarks, anti-quarks Baryon multiplet Meson multiplet Baryons: 3 quarks I 3 (u,d content) S strangeness I 3 (u,d content) Mesons: quark-anti-quark

4. 4 Seeing quarks and gluons In high-energy collisions, observe traces of quarks, gluons (‘jets’)

5. 5 How does it fit together? S. Bethke, J Phys G 26, R27 Running coupling:  s decreases with Q 2 Pole at  =   QCD ~ 200 MeV ~ 1 fm -1 Hadronic scale

6. 6 Asymptotic freedom and pQCD At large Q 2, hard processes: calculate ‘free parton scattering’ At high energies, quarks and gluons are manifest But need to add hadronisation (+initial state PDFs) + more subprocesses

7. 7 Low Q 2 : confinement Lattice QCD potential  large, perturbative techniques not suitable Lattice QCD: solve equations of motion (of the fields) on a space-time lattice by MC String breaks, generate qq pair to reduce field energy Bali, hep-lat/9311009

8. 8 Singularities in pQCD Closely related to hadronisation effects (massless case) Soft divergence Collinear divergence

9. 9 Singularities in phase space

10. 10 Hard processes in QCD Hard process: scale Q >>  QCD Hard scattering High-p T parton(photon) Q~p T Heavy flavour production m >>  QCD Cross section calculation can be split into Hard part: perturbative matrix element Soft part: parton density (PDF), fragmentation (FF) Soft parts, PDF, FF are universal: independent of hard process QM interference between hard and soft suppressed (by Q 2 /  2 ‘Higher Twist’) Factorization parton densitymatrix elementFF

11. 11 Seeing quarks and gluons In high-energy collisions, observe traces of quarks, gluons (‘jets’)

12. 12 Fragmentation and parton showers large Q 2 Q ~ m H ~  QCD FF Analytical calculations: Fragmentation Function D(z,  ) z=p h /E jet Only longitudinal dynamics High-energy parton (from hard scattering) Hadrons MC event generators implement ‘parton showers’ Longitudinal and transverse dynamics

13. 13 NC: CC: DIS: Measured electron/jet momentum fixes kinematics Example DIS events

14. 14 Proton structure F 2 Q 2 : virtuality of the  x = Q 2 / 2 p q ‘momentum fraction of the struck quark’

15. 15 Factorisation in DIS Integral over x is DGLAP evolution with splitting kernel P qq

16. 16 Factorisation in pictures Or: LO matrix element, include gluon radiation in fragmentation: Evolve fragmentation Use NLO matrix element: Note: [DGLAP] evolution only keeps track of leading parton momentum

17. 17 Initial state and final state radiation Final state radiation Fragmentation function evolves Initial state radiation Parton density evolves Evolution can turn (leading) quarks into gluons and vice versa

18. 18 Parton density distribution Low Q 2 : valence structure Valence quarks (p = uud) x ~ 1/3 Soft gluons Q 2 evolution (gluons) Gluon content of proton rises quickly with Q 2

19. 19 pQCD illustrated CDF, PRD75, 092006 jet spectrum ~ parton spectrum fragmentation

20. 20 Note: difference p+p, e + +e - p+p: steeply falling jet spectrum Hadron spectrum convolution of jet spectrum with fragmentation e + + e - QCD events: jets have p=1/2 √s Directly measure frag function

21. 21 Fragmentation function uncertainties Hirai, Kumano, Nagai, Sudo, PRD75:094009 z=p T,h / 2√sz=p T,h / E jet Full uncertainty analysis being pursued Uncertainties increase at small and large z

22. 22 Global analysis of FF proton anti-proton pions De Florian, Sassot, Stratmann, PRD 76:074033, PRD75:114010... or do a global fit, including p+p data Universality still holds

23. 23 Heavy quark fragmentation Light quarks Heavy quarks Heavy quark fragmentation: leading heavy meson carries large momentum fraction Less gluon radiation than for light quarks, due to ‘dead cone’

24. 24 Dead cone effect Radiated wave front cannot out-run source quark Heavy quark:  < 1 Result: minimum angle for radiation  Mass regulates collinear divergence

25. 25 Heavy Quark Fragmentation II Significant non-perturbative effects seen even in heavy quark fragmentation

26. 26 Factorisation in perturbative QCD Parton density function Non-perturbative: distribution of partons in proton Extracted from fits to DIS (ep) data Matrix element Perturbative component Fragmentation function Non-perturbative Measured/extracted from e+e- Factorisation: non-perturbative parts (long-distance physics) can be factored out in universal distributions (PDF, FF)

27. 27 Hard processes in QCD Hard process: scale Q >>  QCD Hard scattering High-p T parton(photon) Q~p T Heavy flavour production m >>  QCD Cross section calculation can be split into Hard part: perturbative matrix element Soft part: parton density (PDF), fragmentation (FF) Soft parts, PDF, FF are universal: independent of hard process QM interference between hard and soft suppressed (by Q 2 /  2 ‘Higher Twist’) Factorization parton densitymatrix elementFF

28. 28 Two-body kinematics 2 -> 2 scattering Mandelstam variables: Invariants: independent of ref system NB: p are four-vectors Massless case: Good approximation if

29. 29 Centre-of-mass system

30. 30 p+p collision Incoming partons carry momentum fractions x 1, x 2 Lab CMS is not parton-parton CMS See Eskola paper for further practical details

31. 31 pQCD matrix elements Combridge, Kripfganz, Ranft, PLB70, 234 See also e.g. Owens, Rev Mod Phys 59,465

32. 32 Key topics today pQCD relies on factorization Separation of scales –Parton Density Functions (soft) –Hard Scattering –Fragmentation (soft) PDFs from DIS FF from e + e - Heavy flavour fragmentation is qualitatively different from light