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Lecture I: pQCD and spectra

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

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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

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

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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

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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

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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

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

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9 Singularities in phase space

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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

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

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12 Fragmentation and parton showers large Q 2 Q ~ m H ~ QCD FF 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

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

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

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

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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

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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

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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

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

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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

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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

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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

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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’

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

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

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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)

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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

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

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29 Centre-of-mass system

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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

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

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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

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