In quantum field theory, the Nambu–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. It is named after Yoichiro Nambu and Giovanni Jona-Lasinio The model is much inspired by the different field of solid state theory, particularly from the BCS breakthrough of 1957. The model was introduced in a joint article of Yoichiro Nambu (who also contributed essentially to the theory of superconductivity, i.e., by the "Nambu formalism") and Giovanni Jona-Lasinio, published in 1961. A subsequent paper included chiral symmetry breaking, isospin and strangeness. Around that time, the same model was independently considered by Soviet physicists Valentin Vaks and Anatoly Larkin.
Overview 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 ) ] {\displaystyle {\mathcal {L}}=\ i\ {\bar {\psi }}\ \partial \!\!\!/\ \psi +{\frac {\ \lambda \ }{4}}\left[\ \left({\bar {\psi }}\ \psi \right)\left({\bar {\psi }}\psi \right)-\left({\bar {\psi }}\ \gamma ^{5}\ \psi \right)\left({\bar {\psi }}\ \gamma ^{5}\ \psi \right)\ \right]}
or, equivalently, decomposed into left and right chiral parts,
L = i ψ ¯ L ∂ / ψ L + i ψ ¯ R ∂ / ψ R + λ ( ψ ¯ L ψ R ) ( ψ ¯ R ψ L ) . {\displaystyle \ {\mathcal {L}}\ =\ i\ {\bar {\psi }}_{\mathsf {L}}\ \partial \!\!\!/\ \psi _{\mathsf {L}}\ +\ i\ {\bar {\psi }}_{\mathsf {R}}\ \partial \!\!\!/\ \psi _{\mathsf {R}}\ +\ \lambda \ \left({\bar {\psi }}_{\mathsf {L}}\ \psi _{\mathsf {R}}\right)\left({\bar {\psi }}_{\mathsf {R}}\ \psi _{\mathsf {L}}\right)~.}
… excerpt ends here. Continue reading the full article.
