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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.
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, i.e., 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 terms proportional to λ are the four-fermion interactions, which parallel the BCS theory. The global symmetry of the model is U(1)Q×U(1)χ 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
Chiral symmetry forbids a bare mass term, but there may be chiral condensates. The global symmetry here is SU(N)L×SU(N)R× U(1)Q × U(1)χ where SU(N)L×SU(N)R 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), U(1)Q is the Dirac charge, which is sometimes called the baryon number and U(1)χ is the axial charge. If a chiral condensate forms, then the chiral symmetry is spontaneously broken into a diagonal subgroup SU(N) since the condensate leads to a pairing of the left-handed and the right-handed flavors. The axial charge is also spontaneously broken.
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.
- 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:1961PhRv..122..345N. doi:10.1103/PhysRev.122.345.
- 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:1961PhRv..124..246N. doi:10.1103/PhysRev.124.246.
- 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.
- 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.