Lattice Formulation of QCD

Part of the Springer Theses book series (Springer Theses)


In this chapter, we provide a detailed formulation of the lattice field theory in the context of QCD. Basic changes compared to the continuum formulation is introduced. Then, we formulate the Euclidean QCD action starting from a naive approach and improve it step-by-step until we have a suitable lattice action. We discuss the gauge and fermion sectors individually with their respective challenges and improvements. Steps of a typical application of the method are outlined in the closing of the chapter.


Lattice field theory Euclidean space-time Discrete action Fermion doubling Improvement program 


  1. 1.
    R.P. Feynman, Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20: 367–387 (1948).
  2. 2.
    K. Symanzik, Schrödinger representation and casimir effect in renormalizable quantum field theory. Nuclear Physics B 190(1): 1–44 (1981). ISSN 0550-3213. URL Volume B190 [FS3] No. 2 To Follow in Approximately Two Months
  3. 3.
    M. Lüscher, Schrödinger representation in quantum field theory. Nuclear Phys. B 254(0): 52–57 (1985). ISSN 0550-3213.
  4. 4.
    M. Luscher, R. Narayanan, P. Weisz, U. Wolff, The Schrodinger functional: a renormalizable probe for non-Abelian gauge theories. Nucl. Phys. B384:168–228 (1992).
  5. 5.
    S. Sint, On the Schrodinger functional in QCD. Nucl. Phys. B421: 135–158 (1994).
  6. 6.
    S. Sint, One loop renormalization of the QCD Schrodinger functional. Nucl. Phys. B451: 416–444 (1995).
  7. 7.
    S. Aoki, K.-I. Ishikawa, N. Ishizuka, T. Izubuchi, D. Kadoh, K. Kanaya, Y. Kuramashi, Y. Namekawa, M. Okawa, Y. Taniguchi, A. Ukawa, N. Ukita, T. Yoshie, 2+1 flavor lattice QCD toward the physical point. Phys. Rev. D79: 034503 (2009).
  8. 8.
    S. Aoki et al., Comparative study of full QCD hadron spectrum and static quark potential with improved actions. Phys. Rev. D60: 114508 (1999).
  9. 9.
    Y. Iwasaki, Renormalization group analysis of lattice theories and improved lattice action. II–four-dimensional non-abelian SU(N) gauge model (2011). arXiv:1111.7054
  10. 10.
    M. Luscher, P. Weisz, On-shell improved lattice gauge theories. Commun. Math. Phys. 97:59 (1985). [Erratum: Commun. Math. Phys. 98, 433 (1985)]
  11. 11.
    J.S. Bell, R. Jackiw, A PCAC puzzle: pi0 –> gamma gamma in the sigma model. Nuovo Cim. A60: 47–61 (1969).
  12. 12.
    S.L. Adler, Axial-vector vertex in spinor electrodynamics. Phys. Rev. 177: 2426–2438 (1969).
  13. 13.
    R. Gupta. Introduction to lattice QCD: Course, in Probing the standard model of particle interactions. Proceedings, summer school in theoretical physics, NATO Advanced Study Institute, 68th session, Les Houches, France, July 28–September 5, 1997. Pt. 1, 2, pp. 83–219 (1997).
  14. 14.
    K.G. Wilson, New phenomena in subnuclear physics: Part a, in New phenomena in subnuclear physics: part A, ed. by A. Zichichi (Springer, US, Boston, MA, 1977). ISBN 978-1-4613-4208-3.
  15. 15.
    L.H. Karsten, J. Smith, Lattice fermions: species doubling, chiral invariance and the triangle anomaly. Nuclear Phys. B 183(1): 103–140 (1981). ISSN 0550-3213.
  16. 16.
    M. Lüscher, S. Sint, R. Sommer, P. Weisz, Chiral symmetry and o(a) improvement in lattice QCD. Nuclear Phys. B 478(1): 365–397 (1996). ISSN 0550-3213.
  17. 17.
    J.B. Kogut, L. Susskind, Hamiltonian formulation of Wilson’s lattice gauge theories. Phys. Rev. D11: 395 (1975).
  18. 18.
    Herbert Neuberger, Exactly massless quarks on the lattice. Phys. Lett. B 417, 141–144 (1998)ADSMathSciNetCrossRefGoogle Scholar
  19. 19.
    D.B. Kaplan, A method for simulating chiral fermions on the lattice. Phys. Lett. B288: 342–347 (1992).
  20. 20.
    G. Curci, P. Menotti, G. Paffuti, Symanzik’s improved lagrangian for lattice gauge theory. Phys. Lett. B 130(3): 205–208 (1983). ISSN 0370-2693.
  21. 21.
    K. Symanzik. Some topics in quantum field theory, in Mathematical Problems in Theoretical Physics: Proceedings of the VIth International Conference on Mathematical Physics Berlin (West) ed. by R. Schrader, R. Seiler, D.A. Uhlenbrock, August 11–20, 1981 (Springer, Heidelberg, 1982), pp. 47–58. ISBN 978-3-540-38982-8.
  22. 22.
    K. Symanzik, Continuum limit and improved action in lattice theories. Nuclear Phys. B 226(1): 187–204 (1983a). ISSN 0550-3213.
  23. 23.
    K. Symanzik, Continuum limit and improved action in lattice theories. Nuclear Phys. B 226(1): 205–227 (1983b). ISSN 0550-3213.
  24. 24.
    P. Weisz, Continuum limit improved lattice action for pure yang-mills theory (i). Nuclear Phys. B 212 (1): 1–17 (1983). ISSN 0550-3213.
  25. 25.
    P. Weisz, R. Wohlert, Continuum limit improved lattice action for pure yang-mills theory (ii). Nuclear Phys. B 236(2):397–422 (1984). ISSN 0550-3213.
  26. 26.
    B. Sheikholeslami, R. Wohlert, Improved continuum limit lattice action for QCD with Wilson Fermions. Nucl. Phys. B259: 572 (1985).
  27. 27.
    M. Lüscher, P. Weisz, O(a) improvement of the axial current in lattice QCD to one-loop order of perturbation theory. Nuclear Phys. B 479(1): 429–458 (1996). ISSN 0550-3213.
  28. 28.
    R. Wohlert, Improved continuum limit lattice action for quarks. DESY-87-069 (1987)Google Scholar
  29. 29.
    S. Aoki et al. Nonperturbative O(a) improvement of the Wilson quark action with the RG-improved gauge action using the Schrodinger functional method. Phys. Rev. D73: 034501 (2006).
  30. 30.
    P.T. Matthews, A. Salam, The green’s functions of quantised fields. II Nuovo Cimento (1943–1954) 12(4): 563–565 (1954). ISSN 1827-6121.
  31. 31.
    P.T. Matthews, A. Salam, Propagators of quantized field. II Nuovo Cimento (1955–1965) 2(1): 120–134 (1955). ISSN 1827-6121.
  32. 32.
    C. Gattringer, C.B. Lang, Quantum chromodynamics on the lattice, Lecture Notes in Physics, vol. 788, 1 edn. (Springer, Heidelberg, 2010).
  33. 33.
    D. Chen, N.H. Christ, C. Cristian, Z. Dong, A. Gara, K. Garg, B. Joo, C. Kim, L. Levkova, X. Liao, R.D. Mawhinney, S. Ohta, T. Wettig, Qcdoc: A 10-teraflops scale computer for lattice QCD. Nuclear Phys. B - Proc. Suppl. 94(1): 825–832 (2001). ISSN 0920-5632.

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© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Strangeness Nuclear Physics Laboratory, Nishina CenterRIKENWakoJapan

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