Modern Methods of Quantum Chromodynamics Christian Schwinn Albert-Ludwigs-Universit¨atFreiburg, Physikalisches Institut D-79104 Freiburg, Germany Winter-Semester 2014/15 Draft: March 30, 2015 http://www.tep.physik.uni-freiburg.de/lectures/QCD-WS-14 2 Contents 1 Introduction 9 Hadrons and quarks . .9 QFT and QED . .9 QCD: theory of quarks and gluons . .9 QCD and LHC physics . 10 Multi-parton scattering amplitudes . 10 NLO calculations . 11 Remarks on the lecture . 11 I Parton Model and QCD 13 2 Quarks and colour 15 2.1 Hadrons and quarks . 15 Hadrons and the strong interactions . 15 Quark Model . 15 2.2 Parton Model . 16 Deep inelastic scattering . 16 Parton distribution functions . 18 2.3 Colour degree of freedom . 19 Postulate of colour quantum number . 19 Colour-SU(3).............................. 20 Confinement . 20 Evidence of colour: e+e− ! hadrons . 21 2.4 Towards QCD . 22 3 Basics of QFT and QED 25 3.1 Quantum numbers of relativistic particles . 25 3.1.1 Poincar´egroup . 26 3.1.2 Relativistic one-particle states . 27 3.2 Quantum fields . 32 3.2.1 Scalar fields . 32 3.2.2 Spinor fields . 32 3 4 CONTENTS Dirac spinors . 33 Massless spin one-half particles . 34 Spinor products . 35 Quantization . 35 3.2.3 Massless vector bosons . 35 Polarization vectors and gauge invariance . 36 3.3 QED . 37 3.4 Feynman rules . 39 3.4.1 S-matrix and Cross section . 39 S-matrix . 39 Poincar´einvariance of the S-matrix . 40 T -matrix and scattering amplitude . 41 Unitarity of the S-matrix . 41 Cross section . 42 3.4.2 Perturbation theory and Feynman rules . 42 Feynman rules . 43 3.4.3 Feynman rules in QED . 44 3.4.4 Feynman rules for interactions with momenta . 45 3.5 Calculation of cross sections . 46 3.5.1 Example: e−e+ ! µ−µ+ ........................ 46 3.5.2 Remarks on the calculation of scattering amplitudes . 48 Crossing: e−µ ! e−µ .......................... 48 External photons and gauge invariance . 49 Remarks on calculations for many-particle final states . 50 4 Introduction to QCD 51 4.1 SU(3) ...................................... 52 4.1.1 Generators and Lie Algebra . 52 Lie Groups and generators . 52 Gell-Mann matrices . 53 Lie Algebras . 53 4.1.2 Representations . 54 Fundamental representations . 54 Combining representations . 55 Representations of Lie algebras . 57 Conjugate representation . 57 Adjoint representation . 57 Generators of product representations . 58 4.2 QCD as non-abelian gauge theory . 58 4.2.1 Non-abelian gauge invariance . 58 Covariant derivative . 58 Field-strength tensor . 59 Decomposition of gluon fields . 59 CONTENTS 5 Infinitesimal transformations . 60 4.2.2 QCD Lagrangian . 60 4.2.3 Gauge fixing . 61 4.3 Feynman rules . 62 4.4 Evaluation of colour factors . 64 Casimir operators . 64 Colour Fierz identity . 66 4.5 Examples . 66 4.5.1 Quark-antiquark potential . 66 4.5.2 qq¯ ! gg: Gauge invariance and ghosts . 68 Ward identity . 69 Relation to ghost diagrams . 70 5 Applications of the Parton model 73 5.1 e−e+ ! Hadrons . 74 5.1.1 Electromagnetic quark current . 74 5.1.2 Total cross section . 74 5.2 DIS . 77 DIS cross section in the naive parton model . 77 Field-theoretic treatment . 79 Derivation of the cross section . 80 5.3 The Drell-Yan process . 81 5.4 Dijet cross sections . 83 5.4.1 Four-quark processes . 84 II Multi-leg Born amplitudes 87 6 Spinor-helicity methods 89 6.1 Two-component spinors . 89 6.1.1 Weyl Spinors . 89 Lorentz Transformations . 89 6.1.2 Index notation . 90 Raising and lowering indices . 90 Spinor products . 91 Explicit expressions . 91 Conjugate spinors . 92 Dirac spinors . 92 Schouten Identity . 93 Braket notation . 93 6.2 Momenta and spinors . 93 6.2.1 Pauli matrices . 93 Raising and lowering indices . 94 6 CONTENTS Dirac algebra . 94 Fierz identities . 94 6.2.2 Four-momenta . 95 Factorization of momenta into spinors . 95 Constructing momenta from spinors . 95 Scaling . 96 6.2.3 Gluon polarization vectors . 96 6.2.4 Rules for calculations with Weyl spinors . 97 Properties of spinor products . 97 Momentum conservation . 97 External states . 98 6.3 Examples . 98 6.3.1 e−e+ ! qq¯ ................................ 99 6.3.2 e−e+ ! qqg¯ ............................... 100 7 Colour decomposition 103 7.1 Examples . 104 7.1.1 qq¯ ! gg ................................. 104 7.1.2 gg ! gg ................................. 105 7.2 Colour ordered Feynman rules . 106 7.2.1 Colour ordered vertices . 106 Three gluon vertex . 106 Four gluon vertex . 106 7.2.2 Colour ordered multi-gluon amplitudes . 106 8 Born amplitudes 109 8.1 General considerations . ..