Post on 28-Jun-2018
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
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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…
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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:
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
• Dimensions: B(s) dimensionless.• Gives fundamental prediction:
without knowing anything about interaction of hadrons!
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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, …
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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)
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
τ decays
• Very closely-related process
• Rτ ´ B(τ! hadrons)/B(τ!µ)
τ ν
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
Three-jet cross section
• Two Feynman diagrams
• After some algebra…
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
Aside on Colour factor for e+e– qqbarg
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
1 x1
• phase space limits: 0·(pi+pj)2=2pi·pj=s(1-xk) ) xi·1x2
1
x3=2-x1-x2
1
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
Three-jet cross section
•• Note:
1. Total cross section infinite (see later…)2. Cross section depends on jet definition
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
• Simple concept
• Complicated in practice rarely used in e+e–
Jet Definitions: 1) cone (“Sterman—Weinberg”, “Snowmass”)
R E>E0
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
Lorentz invariant variables
• Deep inelastic scattering neglect proton mass M
• Kinematic limits:– x>Q2/s– Q2<s
• HERA: s»105 GeV
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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:
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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:
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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:
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
Callan Gross Relation
• Suppose partons = fermions:
partons = quarks!
• (partons = scalars ) FT = 0)
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
Bjorken Scaling
• Non-interacting pointlike partons structure functionsQ2-independent…
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
Parton Distribution Functions
• Constrain pdfs using many different beams/targets:
etc…Global fits…
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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”
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
Mike Seymour
Jets in Hadron Collisions
• Sensitive to gluon pdf• Proportional to αs
2
• Fits data over 6 orders of magnitude in rate!
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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
QCD 2 Latin American School of HEPRecinto Quirama, March 2009
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?