Lecture I: pQCD and spectra

1. Lecture I: pQCD and spectra

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

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 S strangeness I3 (u,d content) I3 (u,d content) Mesons: quark-anti-quark Baryons: 3 quarks

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

5. How does it fit together? S. Bethke, J Phys G 26, R27 Running coupling: as decreases with Q2 Pole at m = L LQCD ~ 200 MeV ~ 1 fm-1 Hadronic scale

6. Asymptotic freedom and pQCD At high energies, quarks and gluons are manifest At large Q2, hard processes: calculate ‘free parton scattering’ + more subprocesses But need to add hadronisation (+initial state PDFs)

7. Low Q2: confinement a large, perturbative techniques not suitable Bali, hep-lat/9311009 Lattice QCD: solve equations of motion (of the fields) on a space-time lattice by MC Lattice QCD potential String breaks, generate qq pair to reduce field energy

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

9. Singularities in phase space

10. Hard processes in QCD • Hard process: scale Q >> LQCD • Hard scattering High-pT parton(photon) Q~pT • Heavy flavour production m >> LQCD Factorization • Cross section calculation can be split into • Hard part: perturbative matrix element • Soft part: parton density (PDF), fragmentation (FF) parton density matrix element FF QM interference between hard and soft suppressed (by Q2/L2 ‘Higher Twist’) Soft parts, PDF, FF are universal: independent of hard process

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

12. Fragmentation and parton showers mF MC event generators implement ‘parton showers’ Longitudinal and transverse dynamics High-energy parton (from hard scattering) Hadrons Q ~ mH ~ LQCD large Q2 Analytical calculations: Fragmentation Function D(z, m) z=ph/Ejet Only longitudinal dynamics

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

14. Proton structure F2 Q2: virtuality of the g x = Q2 / 2 p q ‘momentum fraction of the struck quark’

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

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

18. Parton density distribution Low Q2: valence structure Q2 evolution (gluons) Gluon content of proton risesquickly with Q2 Soft gluons Valence quarks (p = uud) x ~ 1/3

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

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

21. Fragmentation function uncertainties Hirai, Kumano, Nagai, Sudo, PRD75:094009 z=pT,h / 2√s z=pT,h / Ejet Full uncertainty analysis being pursuedUncertainties increase at small and large z

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

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

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

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. Hard processes in QCD • Hard process: scale Q >> LQCD • Hard scattering High-pT parton(photon) Q~pT • Heavy flavour production m >> LQCD Factorization • Cross section calculation can be split into • Hard part: perturbative matrix element • Soft part: parton density (PDF), fragmentation (FF) parton density matrix element FF QM interference between hard and soft suppressed (by Q2/L2 ‘Higher Twist’) Soft parts, PDF, FF are universal: independent of hard process

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

29. Centre-of-mass system

30. p+p collision Incoming partons carry momentum fractions x1, x2 Lab CMS is not parton-parton CMS See Eskola paper for further practical details

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

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