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Nambu–Jona-Lasinio model

In quantum field theory, the Nambu–Jona-Lasinio model or more precisely: the Nambu and Jona-Lasinio model is a complicated effective theory of nucleons and mesons constructed from interacting Dirac fermions with chiral symmetry, paralleling the construction of Cooper pairs from electrons in the BCS theory of superconductivity The "complicatedness" of the theory has become more natural as it is now seen as a low-energy approximation of the still more basic theory of quantum chromodynamics, which does not work perturbatively at low energies


  • 1 Overview
  • 2 See also
  • 3 References
  • 4 External links


The model is much inspired by the different field of solid state theory, particularly from the BCS breakthrough of 1957 The first inventor of the Nambu–Jona-Lasinio model, Yoichiro Nambu, also contributed essentially to the theory of superconductivity, ie, by the "Nambu formalism" The second inventor was Giovanni Jona-Lasinio The common paper of the authors that introduced the model appeared in 1961 A subsequent paper included chiral symmetry breaking, isospin and strangeness At the same time, the same model was independently considered by Soviet physicists Valentin Vaks and Anatoly Larkin

The model is quite technical, although based essentially on symmetry principles It is an example of the importance of four-fermion interactions and is defined in a spacetime with an even number of dimensions It is still important and is used primarily as an effective although not rigorous low energy substitute for quantum chromodynamics

The dynamical creation of a condensate from fermion interactions inspired many theories of the breaking of electroweak symmetry, such as technicolor and the top-quark condensate

Starting with the one-flavor case first, the Lagrangian density is

L = i ψ ¯ ∂ / ψ + λ 4 [ ψ ¯ ψ ψ ¯ ψ − ψ ¯ γ 5 ψ ψ ¯ γ 5 ψ ] = i ψ ¯ L ∂ / ψ L + i ψ ¯ R ∂ / ψ R + λ ψ ¯ L ψ R ψ ¯ R ψ L }=\,i\,}\partial \!\!\!/\psi +}\,\left=\,i\,}_\partial \!\!\!/\psi _+\,i\,}_\partial \!\!\!/\psi _+\lambda \,\left}_\psi _\right\left}_\psi _\right}

The terms proportional to λ are the four-fermion interactions, which parallel the BCS theory The global symmetry of the model is U1Q×U1χ where Q is the ordinary charge of the Dirac fermion and χ is the chiral charge

There is no bare mass term because of the chiral symmetry However, there will be a chiral condensate but no confinement leading to an effective mass term and a spontaneous symmetry breaking of the chiral symmetry, but not the charge symmetry

With N flavors and the flavor indices represented by the Latin letters a, b, c, the Lagrangian density becomes

L = i ψ ¯ a ∂ / ψ a + λ 4 N [ ψ ¯ a ψ b ψ ¯ b ψ a − ψ ¯ a γ 5 ψ b ψ ¯ b γ 5 ψ a ] = i ψ ¯ L a ∂ / ψ L a + i ψ ¯ R a ∂ / ψ R a + λ N ψ ¯ L a ψ R b ψ ¯ R b ψ L a }=\,i\,}_\partial \!\!\!/\psi ^+}\,\left=\,i\,}_\partial \!\!\!/\psi _^+\,i\,}_\partial \!\!\!/\psi _^+}\,\left}_\psi _^\right\left}_\psi _^\right}

Chiral symmetry forbids a bare mass term, but there may be chiral condensates The global symmetry here is SUNL×SUNR× U1Q × U1χ where SUNL×SUNR acting upon the left-handed flavors and right-handed flavors respectively is the chiral symmetry in other words, there is no natural correspondence between the left-handed and the right-handed flavors, U1Q is the Dirac charge, which is sometimes called the baryon number and U1χ is the axial charge If a chiral condensate forms, then the chiral symmetry is spontaneously broken into a diagonal subgroup SUN since the condensate leads to a pairing of the left-handed and the right-handed flavors The axial charge is also spontaneously broken

The broken symmetries lead to massless pseudoscalar bosons which are sometimes called pions See Goldstone boson

As mentioned, this model is sometimes used as a phenomenological model of quantum chromodynamics in the chiral limit However, while it is able to model chiral symmetry breaking and chiral condensates, it does not model confinement Also, the axial symmetry is broken spontaneously in this model, leading to a massless Goldstone boson unlike QCD, where it is broken anomalously

Since the Nambu–Jona-Lasinio model is nonrenormalizable in four spacetime dimensions, this theory can only be an effective field theory which needs to be UV completed

See also

  • Gross–Neveu model


  1. ^ Nambu, Y; Jona-Lasinio, G April 1961 "Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity I" Physical Review 122: 345–358 Bibcode:1961PhRv122345N doi:101103/PhysRev122345 
  2. ^ Nambu, Y; Jona-Lasinio, G October 1961 "Dynamical Model of Elementary Particles Based on an Analogy with Superconductivity II" Physical Review 124: 246–254 Bibcode:1961PhRv124246N doi:101103/PhysRev124246 
  3. ^ Alexander Polyakov 1997 "13 A View from the Island" The Rise of the Standard Model: A History of Particle Physics from 1964 to 1979 Cambridge University Press p 244 ISBN 9780521578165 
  4. ^ Vaks, V G; Larkin, A I 1961 "On the application of the methods of superconductivity theory to the problem of the masses of elementary particles" PDF Sov Phys JETP 13: 192–193 

External links

  • Giovanni Jona-Lasinio and Yoichiro Nambu, Nambu-Jona-Lasinio model, Scholarpedia, 512:7487, 2010 doi:104249/scholarpedia7487

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