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| Introduction to Quantum Chromodymanics |
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1.) Basics of QCD:
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(i) Introduction of color
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quark model: spin-statistics problem
magnetic moments of nucleons
introduction of color -> SU(3)
tests of color: pi^0 -> gamma gamma (explicit derivation)
e+ e- -> hadrons
(ii) Gluon Gauge Fields
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second color hypothesis: local symmetry -> gluons
construction of non-abelian Lagrangian
individual interactions: qqg, ggg, gggg
Feynman rules
(iii) Feynman-Path Integrals
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transistion amplitude in quantum mechanics
functionals
Green's functions: action functional
free action functional
path integral representation of field theory
fermions: Grassmann variables -> Grassman fields
fermion action functional
(iv) Gauge Fixing
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gauge fixing -> factorization of path integral
Hurwitz measure of SU(3)
Fadeev-Popov determinant
explicit examples: Lorentz gauge
axial gauge
gauge fixing term of Lagrangian
Fadeev-Popov ghosts
complete QCD Lagrangian (Lorentz gauge, axial gauge)
(v) Asymptotic Freedom
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Reminder: running QED coupling <- asymptotic diagrammatic discussion
screening
Transfer to QCD <- asymptotic diagrammatic discussion:
role of ghost loops
gauge dependence
resummation -> asymptotic freedom
introduction of QCD scale Lambda
renormalization group equation: beta_0, beta_1, beta_2
renormalization schemes: Dyson scheme
MS scheme
MSbar scheme
(discussed for fermion propagator)
quark masses: pole mass <-> MSbar mass
renormalization group equation and solution
examples: bottom, charm -> higher orders
(vi) Renormalization Group
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renormalization of amputated Green's functions <- only external
wave functions
explicit discussion of fermion propagator, qqg-vertex
derivation of general RGE for amputated Green's functions
explicit solution for large external momenta
explicit high-energy behaviour of amputated Green's functions:
leading log behaviour as a result of asymptotic freedom
(in contrast to power behaviour of fix-point theories)
2.) QCD at short Distances
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(i) Structure Functions of Nucleons
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derivation of elm. structure functions W_1, W_2 <-> F_1, F_2
as form factors of the hadron tensor decomposition
kinematical variables Q^2, \nu <-> x, y: boundaries
double differential cross section d sigma/dx dy
decomposition into virtual photon polarization states
transverse, longitudinal structure functions
experimental results: Bjorken scaling
Callan-Gross relation
historical plots
classical parton model: heuristic motivation from Q^2 behaviour
(ii) Parton Model of Deep Inelastic Lepton-Nucleon Scattering
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semiclassical discussion: incoherent superposition of individual
partonic processes
probabilistic picture:
- splitting process
- old-fashioned perturbation theory
- energy step: 0 < x < 1 dominant
basic assumptions of quark-parton model
hierarchy of interaction and hadronisation time scales
neutrino/electron-nucleon scattering: partonic cross sections
parton densities
x = momentum fraction
explicit results
spin 1/2 -> Callan-Gross
valence/sea densities
fractional quark charges
sum rules: baryon number
isospin
strangeness
-> Gross--Llewellyn-Smith
momentum sum rule -> gluons
(iii) Altarelli-Parisi Equations
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e+ e- -> mu+ mu- gamma: splitting probability in collinear regime
transfer to QCD: splitting kernels P_gq, P_qq, P_qg, P_gg
Altarelli-Parisi equations from splitting probabilities
solutions: singlet, non-singlet densities
Mellin moments
natural variable: s = log(log(Q^2)/log(Q_0^2))
moments of non-singlet densities -> log. scaling violation
interpretation
moments of singlet densities -> momentum sum rule
asymptotic behaviour
extraction of gluon density
historical plots
HERA plots
(iv) Factorization Theorems of QCD
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QCD corrections to deep inelastic lepton-nucleon scattering
introduction to dimensional regularization: momentum integrals
Clifford algebra
explicit NLO calculation of F_2: Bornterm
virtual corrections
renormalization
real corrections: n-dim. phase space
plus distributions
renormalization of parton densities
(MSbar, DIS schemes)
discussion of final result: scale/scheme choices
factorization theorem of QCD
(v) Drell-Yan Processes
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Bornterms for mu+ mu-, W, Z production
QCD corrections: results
(vi) e+ e- -> Hadrons: Total Cross Section
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Bornterm
QCD corrections: virtual, real, total result @ NLO
results @ NNNLO
(vii) Jets in QCD
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asymptotic freedom -> jet hypothesis
"SPEAR" jets: heuristic interpretation
3 jets: Daliz-plot -> discovery of gluons
measurements: gluon spin from Daliz-plot, Ellis-Karliner angle
gluon color from 4-jet events: Bengtsson-Zerwas angle
jet multiplicity: 2- and 3-jet fractions from Daliz-plot
shape variables: thrust, sphericity, masses,
C-parameter,...
jets in high-energy pp scattering at large pt: 2-jet rate @ LO
parton processes
Rutherford pole
quarkonium decays: leptonic decays -> wave function at origin
decays into ggg: Daliz-plot
jet energy distribution
3.) QCD at large Distances
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(i) Confinement Potential
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path integral: discretization
Monte Carlo integration, importance sampling
Wilson loop: connection to interquark potential
quarks on lattice points
gluons on links
discretized gluon action
results for potential
string tension: Richardson potential
meson string rotator
Chew-Frautschi plots
(ii) Chiral Invariance
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Noether theorem
Quantum Flavour Dynamics: SU(2), SU(3)
vector current
axial vector current
current algebra
current conservation: parity doublets
Nambu: quark condensates
Goldstone theorem -> pions, kaons
(iii) PCAC Hypothesis
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representation of pion field: divergence of axial vector current
pion pole dominance
nucleon form factors: Goldberger-Treiman relation
Gell-Mann/Oakes/Renner formula: pion mass from quark masses
generalization: soft pion theorems
(iv) Goldstone Theorem
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general proof for any global symmetry breaking
difference for local symmetry breaking