Lecture I: pQCD and spectra

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What is QCD?

From: T. Schaefer, QM08 student talk

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QCD and hadronsQuarks 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

I3 (u,d content)

S stra

ngen

ess

I3 (u,d content)

Mesons: quark-anti-quark

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Seeing quarks and gluons

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

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How does it fit together?

S. Bethke, J Phys G 26, R27

Running coupling:s decreases with Q2

Pole at =

QCD ~ 200 MeV ~ 1 fm-1

Hadronic scale

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Asymptotic freedom and pQCD

At large Q2, hard processes: calculate ‘free parton scattering’

At high energies, quarks and gluons are manifest

gqqee

But need to add hadronisation (+initial state PDFs)

+ more subprocesses

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Low Q2: 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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Singularities in pQCD

Closely related to hadronisation effects

(massless case)

Soft divergence Collinear divergence

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Singularities in phase space

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Hard processes in QCD

• Hard process: scale Q >> QCD

• Hard scattering High-pT parton(photon) Q~pT

• 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 Q2/2 ‘Higher Twist’)

Factorization

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Seeing quarks and gluons

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

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Fragmentation and parton showers

large Q2 Q ~ mH ~ QCDF

Analytical calculations: Fragmentation Function D(z, ) z=ph/Ejet

Only longitudinal dynamics

High-energy

parton(from hard scattering)

Hadrons

MC event generatorsimplement ‘parton showers’

Longitudinal and transverse dynamics

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XpeXepe ee )( :CC , :NC

NC:

CC:

DIS: Measured electron/jet momentum fixes kinematics

Example DIS events

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Proton structure F2

Q2: virtuality of the x = Q2 / 2 p q‘momentum fraction of the struck quark’

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Factorisation in DIS

Integral over x is DGLAP evolution with splitting kernel Pqq

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Factorisation in pictures

Or: LO matrix element,include gluon radiation in fragmentation:Evolve fragmentation

qqgqqUse NLO matrix element:

Note: [DGLAP] evolution only keeps track of leading parton momentum

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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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Parton density distributionLow Q2: valence structure

Valence quarks (p = uud)x ~ 1/3

Soft gluons

Q2 evolution (gluons)

Gluon content of proton risesquickly with Q2

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pQCD illustrated

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CDF, PRD75, 092006

jet spectrum ~ parton spectrum

nTTT ppdp

dN

ˆ

1

ˆˆ

jet

hadronT

P

pz ,

fragmentation

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Note: difference p+p, e++e-

p+p: steeply falling jet spectrumHadron spectrum convolution of jet spectrum with fragmentation

e+ + e- QCD events: jetshave p=1/2 √sDirectly measure frag function

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Fragmentation function uncertaintiesHirai, 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

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Global analysis of FFproton anti-protonpions

De Florian, Sassot, Stratmann, PRD 76:074033, PRD75:114010

... or do a global fit, including p+p dataUniversality still holds

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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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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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Heavy Quark Fragmentation II

Significant non-perturbative effects seen even

in heavy quark fragmentation

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Factorisation in perturbative QCD

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Parton density functionNon-perturbative: distribution of partons in protonExtracted from fits to DIS (ep) data

Matrix elementPerturbative component

Fragmentation functionNon-perturbativeMeasured/extracted from e+e-

Factorisation: non-perturbative parts (long-distance physics) can be factored out in universal distributions (PDF, FF)

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Hard processes in QCD

• Hard process: scale Q >> QCD

• Hard scattering High-pT parton(photon) Q~pT

• 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 Q2/2 ‘Higher Twist’)

Factorization

c

chbbaa

abcdba

T

hpp

z

Dcdab

td

dQxfQxfdxdxK

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0

/222

)(ˆ

),(),( parton density matrix element FF

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Two-body kinematics

2 -> 2 scattering

Mandelstam variables:

Invariants: independent of ref systemNB: p are four-vectors

Massless case:

Good approximation if mpp 22

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Centre-of-mass system

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p+p collision

Incoming partons carry momentum fractions x1, x2

Lab CMS is not parton-parton CMS

sxxs 21ˆ

4321*ˆ yyy

See Eskola paper for further practical details

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pQCD matrix elementsCombridge, Kripfganz, Ranft, PLB70, 234

2

2

ˆˆ stdd s

See also e.g. Owens, Rev Mod Phys 59,465

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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