Quantum ChromoDynamics - CERNphysicschool.web.cern.ch/PhysicSchool/LatAmSchool/2009/... · Quantum ChromoDynamics Mike Seymour ... 15 March — 28 March 2009. QCD 2 Latin American

Quantum ChromoDynamics

Mike SeymourUniversity of Manchester/CERN TH

Latin American School of High Energy PhysicsRecinto Quirama, Colombia15 March — 28 March 2009

QCD 2 Latin American School of HEPRecinto Quirama, March 2009

Mike Seymour

Outline

1. Basics2. QCD Phenomenology at Tree Level3. Higher Order corrections4. Monte Carlo techniques


Mike Seymour

QCD Phenomenology – Tree Level

• The e+e– annihilation cross section• Jets in e+e– annihilation• Deep Inelastic Scattering

– the parton model

• Hadron collisions– the Drell—Yan process– jet production


Mike Seymour

e+e– Annihilation to ???• Most final states contain many hadrons• Feynman rules don't tell us anything about hadrons!• How to calculate:

• How to sum over all hadronic final states?• Symmetries can give us a clue…


Mike Seymour

• Matrix element to produce n hadrons:

• T is a guess (parametrization) of the unknown part.• Gives total cross section:

• Define:

1) Lorentz covariance:

2) Gauge invariance:


Mike Seymour

• Dimensions: B(s) dimensionless.• Gives fundamental prediction:

without knowing anything about interaction of hadrons!


Mike Seymour

• Space time picture of e+e– hadrons:

• Annihilation takes time » 1/ps• Confinement takes time » 1/mhadron

No time for confinement to affect cross section.

• Feynman rules do tell us how to calculate that.

Quark Parton Model


Mike Seymour

σ(e+e– hadrons)

• Start with photon exchange only:

where sum is over all quarks that can appear in the finalstate. 3 (Nc) colours per flavour

• From here on explicitly extract this factor:

• Kinematically allowed if ps > 2mq series of steps at2mc ~ 4 GeV, 2mb ~ 10 GeV, …


Mike Seymour


Mike Seymour

R(e+e-) at the Z0

• cf LEP average: 20.775±0.027• In general, sensitive to γ–Z interference:

• (actually 19.984 on Z peak)


Mike Seymour

τ decays

• Very closely-related process

• Rτ ´ B(τ! hadrons)/B(τ!µ)

τ ν


Mike Seymour

Jet production in e+e– hadrons

• Most e+e– events consist of two back-to-back jets• But some consist of three (or more) jets gluons


Mike Seymour

Three-jet cross section

• Two Feynman diagrams

• After some algebra…


Mike Seymour

Aside on Colour factor for e+e– qqbarg


Mike Seymour

Crash course in Three-Body Phase Space

• All massless:

• Momentum conservation:

• Define energy fractions xi´2pi/ps

• θ, φ, α Euler angles: generally uninteresting for QCD


Mike Seymour

1 x1

• phase space limits: 0·(pi+pj)2=2pi·pj=s(1-xk) ) xi·1x2

1

x3=2-x1-x2

1


Mike Seymour

Three-jet cross section

•• Note:

1. Total cross section infinite (see later…)2. Cross section depends on jet definition


Mike Seymour

• Simple concept

• Complicated in practice rarely used in e+e–

Jet Definitions: 1) cone (“Sterman—Weinberg”, “Snowmass”)

R E>E0


Mike Seymour

Jet definitions: 2) cluster (“Jade”, “Durham”, “kt”)

• Define– ‘closeness measure’ yij

– eg sij/s (JADE) or kt2ij/s (Durham)

– ‘combination scheme’ pij = pi pj

– eg pi+pj

• Iterate– find closest pair– combine them until all pairs are above ycut

• Many theoretical advantages


Mike Seymour

• Note: jet cross sectionis (strongly) a functionof jet definition

• Compare like withlike in theory andexperiment

• JADE n-jet rate as afunction of ycut


Mike Seymour

Deep Inelastic Lepton–Hadron Scattering

• Electron proton inelastic scattering = electron partonelastic scattering

Inelastic scattering tells us about internal structure• Q2 gives resolving power


Mike Seymour

Lorentz invariant variables

• Deep inelastic scattering neglect proton mass M

• Kinematic limits:– x>Q2/s– Q2<s

• HERA: s»105 GeV


Mike Seymour

General Form of Cross Section

• Phase space:• Matrix element: like e+e–, don't know about hadrons

)use Lorentz covariance, gauge invariance, etc:

• Hi: scalar fns of q·q=–Q2 and p·q=Q2/2x only

• Define: H1 = 4πF1 and H2 = 8πxF2 to give:


Mike Seymour

Other Forms

• Parameterizes cross section in terms of two unknownfunctions: structure functions

• Note: only s-dependence comes via y = Q2/sx• Often see other linear combinations:

where FT,L correspond to scattering of transverse andlongitudinally polarized photons

• Especially:


Mike Seymour

Parton Model• Suppose that proton is a bound state of partons• Look at proton in the Breit frame:

• Virtual photon sees tiny fraction of thin disk:partons have no time to interact with each otherphoton collides with single free parton

• Elastic electron—parton scattering

2R2R xM/Q


Mike Seymour

Parton Distribution Functions

• Suppose that parton type q carries a fraction η ofproton's momentum a fraction fq(η) of the time

• Provided partons are– pointlike (r2 ¿ 1/Q2)– dilute (f(η) ¿ Q2 R2

proton)

photons scatter incoherently off them:

• Outgoing parton on shell ) η=x• Therefore:


Mike Seymour

Callan Gross Relation

• Suppose partons = fermions:

partons = quarks!

• (partons = scalars ) FT = 0)


Mike Seymour

Bjorken Scaling

• Non-interacting pointlike partons structure functionsQ2-independent…


Mike Seymour

Neutrino DIS (i.e. Charged Current)

• Extra parity-violating terms…

•

• (Also get non-zero F3 for neutral current for Q2&Mz2)

ν e

W


Mike Seymour

Parton Distribution Functions

• Constrain pdfs using many different beams/targets:

etc…Global fits…


Mike Seymour


Mike Seymour

The Drell—Yan Process

• If parton model is correct, pdfs measured in DIS shouldbe applicable to hadron collisions,

• eg qqbar µ+µ–, “Drell—Yan pair production”


Mike Seymour


Mike Seymour

Jets in Hadron Collisions

• Sensitive to gluon pdf• Proportional to αs

2

• Fits data over 6 orders of magnitude in rate!


Mike Seymour

The ‘Underlying’ Event

• Incoming hadrons are colour singlets• Annihilation/scattering removes coloured parton• What happens to the hadron remnants?

• Non-perturbative ‘contamination’ of all jet cross sections


Mike Seymour

Summary

• Most QCD phenomenology can be understood from treelevel calculations, but…

• αs is not so small higher order corrections?• can we derive parton model from QCD?

Documents

Quantum ChromoDynamics - CERNphysicschool.web.cern.ch/PhysicSchool/LatAmSchool/2009/... · Quantum ChromoDynamics Mike Seymour ... 15 March — 28 March 2009. QCD 2 Latin American