============================================== | | | Introduction to Quantum Chromodymanics | | | ============================================== 1.) Basics of QCD: ============== (i) Introduction of color --------------------- 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 ------------------ second color hypothesis: local symmetry -> gluons construction of non-abelian Lagrangian individual interactions: qqg, ggg, gggg Feynman rules (iii) Feynman-Path Integrals ---------------------- 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 ------------ 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 ------------------ 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 --------------------- 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 ====================== (i) Structure Functions of Nucleons ------------------------------- 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 -------------------------------------------------------- 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 -------------------------- 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 ----------------------------- 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 ------------------- Bornterms for mu+ mu-, W, Z production QCD corrections: results (vi) e+ e- -> Hadrons: Total Cross Section ------------------------------------- Bornterm QCD corrections: virtual, real, total result @ NLO results @ NNNLO (vii) Jets in QCD ----------- 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 ====================== (i) Confinement Potential --------------------- 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 ----------------- 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 --------------- 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 ----------------- general proof for any global symmetry breaking difference for local symmetry breaking