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Introduction to QCD

Module PH2042

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Module version of SS 2022 (current)

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
SS 2022WS 2020/1WS 2019/20WS 2018/9WS 2010/1

Basic Information

PH2042 is a semester module in English language at Master’s level which is offered in summer semester.

This Module is included in the following catalogues within the study programs in physics.

  • Specific catalogue of special courses for nuclear, particle, and astrophysics
  • Complementary catalogue of special courses for condensed matter physics
  • Complementary catalogue of special courses for Biophysics
  • Complementary catalogue of special courses for Applied and Engineering Physics

If not stated otherwise for export to a non-physics program the student workload is given in the following table.

Total workloadContact hoursCredits (ECTS)
150 h 60 h 5 CP

Responsible coordinator of the module PH2042 is Antonio Vairo.

Content, Learning Outcome and Preconditions


The lecture will provide a basic introduction to the Quantumchromodynamics (QCD) and focus on its high energy (collider physics) and low energy (chiral perturbation theory) behaviour. The contents are as follows:

  • Strong interaction: historical overview and foundation of QCD
  • The QCD Lagrangian
    • SU(3) group and representations
    • Discrete symmetries: CPT
    • Continuous symmetries: gauge invariance
    • Gauge fixing and the Faddev-Popov ghost
    • QCD Lagrangian and Feynman rules
    • Exact symmetries and the θ term
    • Approximate symmetries
      • Isospin simmetry
      • Chiral simmetry
      • Heavy quark symmetry
  • Renormalization of QCD
    • Basics of dimensional regularization
      • The Adler-Bell-Jackiw anomaly in QCD
    • Renormalization and renormalization schemes
  • The running of α
    • The β function at one loop
    • Asymptotic freedom and dimensional transmutation
    • Confinement
      • The Wilson loop and the quark-antiquark static energy
    • Decoupling
  • QCD at high energies
    • e+e– → hadrons
      • Cross-section at LO
      • Threshold effects
      • Cross-section at NLO
      • Soft and collinear singularities
      • Bloch-Nordsieck and Kinoshita-Lee-Nauenberg theorems
      • Cross-section at higher orders and scale invariance
      • Jets, jet measures and event shape variables
    • Deep inelastic scattering
      • Kinematics
      • Structure functions
      • Bjorken limit and scaling
      • Parton distribution functions
      • Structure functions at NLO, splitting functions and scaling violation
      • DGLAP equations
  • QCD at low energies
    • Low-energy symmetries
      • Chiral symmetry and spontaneous symmetry breaking
      • Parity doubling, vector and axial vector spectral functions
      • Quark condensate
    • Low-energy degrees of freedom
      • Pions and kaons as Goldstone bosons
      • Non-linear realization of SU(Nf) X SU(Nf)
    • The chiral Lagrangian at order p²
      • Vector and axial vector currents
    • The chiral Lagrangian at order p² with a mass term
      • Gell-Mann Oakes Renner relations, Gell-Mann Okubo relation, light quark masses
    • The chiral Lagrangian at order p² with external fields
      • Coupling to electromagnetism and quark masses
    • π → μ ν and fπ
    • π π → π π

Learning Outcome

After successful completion of the module the students are able to:

  • Write the QCD Lagrangian and its Feynman rules
  • Describe its symmetries and degrees of freedom
  • Discuss the renormalization of QCD and compute the one loop running coupling constant
  • Compute the LO e+e- -> hadrons cross section
  • Explain what jets are and their basic properties
  • Describe the DIS, define PDF and derive the DGLAP equations
  • Introduce chiral perturbation theory
  • Derive basic equalities in chiral perturbation theory
  • Compute pion decays and scatterings


Quantum Mechanics 1 + 2 (PH0007 and PH1002) and some basic knowledge of Quantum Field Theory.

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

VO 4 Introduction to quantum chromodynamics Vairo, A. Tue, 14:00–16:00, PH 3343
Thu, 10:00–12:00, PH 3343

Learning and Teaching Methods

The blakboard lectures aim at introducing the students to the subject in a step by step approach where every result is carefully derived and consistently justified. Plenty of examples are presented. Figures and results are kept updated and reflect the status of the art. Further possible development beyond the Standard Model are suggested. Students are encouraged to participate actively in the lecture and encouraged to keep a critical open mind on the subject.


Blackboard, Powerpoint, webpage



F. Wilczek: QCD Made Simple, Phys. Today 53, August 22, (2000)

Running of α:

S. Bethke:  The 2009 World Average of α, Eur. Phys. J. C64 689, (2009)


N. Brambilla and A. Vairo: Quark Confinement and the Hadron Spectrum, 13th Annual Hampton University Graduate Studies at the Continuous Electron Beam Facility, JLab, e-Print: hep-ph/9904330

S. Scherer: Introduction to Chiral Perturbation Theory, Adv. Nucl. Phys. 7:277, (2003), e-Print: hep-ph/0210398


  • O. Nachtmann, A. Lahee & W. Wetzel: Elementary Particle Physics: Concepts and Phenomena, Springer Verlag, (1989)
  • F.J. Yndurain: The Theory of Quark and Gluon Interactions, Springer Verlag, (1983)
  • T. Muta: Foundations of Quantum Chromodynamics, World Scientific, (2009)
  • K. Huang: Quarks Leptons and Gauge Fields, World Scientific, (1982)
  • P. Pascual & R. Tarrach: QCD: Renormalization for the Practitioner, Springer Verlag, (1984)
  • R.K. Ellis, W.J. Stirling & B.R. Webber: QCD and Collider Physics, Cambridge University Press, (2010)
  • G. Sterman: Handbook of perturbative QCD, CTEQ Collaboration, (1993)
  • Yu.L. Dokshitzer, V.A. Khoze, A.H. Mueller & S.I. Troyan: Basics of perturbative QCD, Edition Frontières, (1991)
  • S. Pokorski: Gauge Field Theories (2nd Edition), Cambridge University Press, (2000)

Module Exam

Description of exams and course work

There will be an oral exam of 25 minutes duration. Therein the achievement of the competencies given in section learning outcome is tested exemplarily at least to the given cognition level using comprehension questions ans sample calculations.

For example an assignment in the exam might be:

  • Write the QCD Lagrangian and its Feynman rules.
  • What is meant with asymptotic freedom?
  • What are the experimental evidences for 3 colors?
  • Define jets.
  • Write the LO chiral lagrangian.

Exam Repetition

The exam may be repeated at the end of the semester.

Current exam dates

Currently TUMonline lists the following exam dates. In addition to the general information above please refer to the current information given during the course.

Exam to Introduction to QCD
Thu, 2024-03-21, 9:00 till 14:00 343
Dummy-Termin. Wenden Sie sich zur individuellen Terminvereinbarung an die/den Prüfer(in). Anmeldung für Prüfungstermin zwischen 25.03.2024 und 20.04.2024. // Dummy date. Contact examiner for individual appointment. Registration for exam date between 2024-Mar-25 and 2024-Apr-20. till 2024-03-13
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