View source for Nambu–Jona-Lasinio model

{{Short description|Effective field theory of nucleons}}
{{Refimprove|date=August 2009}}
 
In [[quantum field theory]], the '''Nambu–Jona-Lasinio model''' (or more precisely: ''the Nambu and Jona-Lasinio model'') is a complicated effective theory of [[nucleon]]s and [[meson]]s constructed from interacting [[Dirac fermion]]s with [[chiral symmetry]], paralleling the construction of [[Cooper pair]]s from [[electron]]s 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.

==Overview==
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.<ref name=Nambu1961I>
{{cite journal
 |first1=Y. | last1=Nambu
 |first2=G. | last2=Jona-Lasinio
 | date = April 1961
 | title = Dynamical model of elementary particles based on an analogy with superconductivity.&nbsp;I
 | journal = [[Physical Review]]
 | volume = 122  | issue=1 | pages = 345–358
 | doi = 10.1103/PhysRev.122.345 | doi-access = free
 | bibcode = 1961PhRv..122..345N
}}
</ref>
A subsequent paper included [[chiral symmetry breaking]], [[isospin]] and [[strangeness (particle physics)|strangeness]].<ref name=Nambu1961II>
{{Cite journal
 |first1=Y. | last1=Nambu
 |first2=G. | last2=Jona-Lasinio
 | date = October 1961
 | title = Dynamical model of elementary particles based on an analogy with superconductivity.&nbsp;II
 | journal = [[Physical Review]]
 | volume = 124 | issue=1 | pages = 246–254
 | doi = 10.1103/PhysRev.124.246 | doi-access = free
 | bibcode = 1961PhRv..124..246N
}}
</ref>
Around that time, the same model was independently considered by Soviet physicists [[Valentin Vaks]] and [[Anatoly Larkin]].<ref>
{{cite book
 |first=Alexander |last=Polyakov |author-link=Alexander Markovich Polyakov
 |year=1997
 |chapter=13. A view from the island
 |title=The Rise of the Standard Model: A history of particle physics from 1964 to 1979
 |publisher=Cambridge University Press
 |isbn=9780521578165 |page=244
 |chapter-url=https://books.google.com/books?id=klLUs2XUmOkC&pg=PA244 |via=Google
}}
</ref><ref>
{{cite journal
 |last1=Vaks   |first1=V.G.
 |last2=Larkin |first2=A.I.
 |year=1961
 |title=On the application of the methods of superconductivity theory to the problem of the masses of elementary particles
 |journal=[[Sov. Phys. JETP]]
 |volume=13 |pages=192–193
 |url=http://www.jetp.ras.ru/cgi-bin/dn/e_013_01_0192.pdf |via=jetp.ras.ru
}}
</ref>

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 [[fermionic condensate|condensate]] from fermion interactions inspired many theories of the breaking of [[electroweak symmetry]], such as [[Technicolor (physics)|technicolor]] and the [[top quark condensate|top-quark condensate]].

Starting with the one-[[flavor (physics)|flavor]] case first, the [[Lagrangian density]] is

:<math> \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] </math>

or, equivalently, decomposed into left and right chiral parts,

:<math>\ \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) ~.</math>

The terms proportional to <math>\ \lambda\ </math> are an attractive four-fermion interaction, which parallels the BCS theory [[phonon]] exchange interaction.
The [[global symmetry]] of the model is {{nobr|U(1)<sub>{{mvar|Q}}</sub> × U(1)<sub>{{mvar|χ}}</sub>}} where {{mvar|Q}} is the ordinary charge of the Dirac fermion and {{mvar|χ}} is the chiral charge. The parameter <math>\ \lambda\ </math> is equivalent to a reciprocal squared mass, <math>\ \lambda = \tfrac{ 1 }{\ M^2}\ ,</math> which represents short-distance physics or
the strong interaction scale, producing an attractive four-fermion interaction.

There is no bare fermion mass term because of the chiral symmetry. However, there will be a [[chiral condensate]] (but no [[color confinement|confinement]]) leading to an effective mass term and a [[spontaneous symmetry breaking]] of the chiral symmetry, but not the charge symmetry.

With {{mvar|N}} flavors and the flavor indices, represented by the Latin letters {{mvar|a}} and {{mvar|b}}, the Lagrangian density becomes

:<math> \mathcal{L}\ =\ i\ \bar{\psi}_a\partial\!\!\!/\ \psi^a + \frac{ \lambda }{\ 4\ N\ }\ \left[\ \left( \bar{\psi}_a\ \psi^b \right) \left( \bar{\psi}_b\ \psi^a \right) - \left( \bar{\psi}_a\ \gamma^5\ \psi^b \right) \left( \bar{\psi}_b\ \gamma^5\ \psi^a \right)\ \right]\ </math>
hence
:<math> \mathcal{L}\ =\ i\ \bar{\psi}_{\mathsf{L}\ a}\ \partial\!\!\!/\ \psi_\mathsf{L}^a\ +\ i\ \bar{\psi}_{\mathsf{R}\ a}\ \partial\!\!\!/\ \psi_\mathsf{R}^a + \frac{ \lambda }{\ N\ }\ \left( \bar{\psi}_{\mathsf{L}\ a}\ \psi_\mathsf{R}^b \right) \left(\bar{\psi}_{\mathsf{R}\ b}\ \psi_\mathsf{L}^a \right) ~.</math>

Chiral symmetry forbids a bare mass term, but there may be chiral condensates. The global symmetry here is {{nobr| SU({{mvar|N}})<sub>L</sub> × SU({{mvar|N}})<sub>{{sc|R}}</sub> × U(1)<sub>{{mvar|Q}}</sub> × U(1)<sub>{{mvar|χ}}</sub>}} where {{nobr|SU({{mvar|N}})<sub>{{sc|L}}</sub> × SU({{mvar|N}})<sub>{{sc|R}}</sub>}} 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)<sub>Q</sub> is the Dirac charge, which is sometimes called the baryon number and U(1)<sub>χ</sub> is the [[axial current|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.

The broken symmetries lead to (nearly) massless [[pseudoscalar (physics)|pseudoscalar]] bosons, e.g. [[pion]]s. See [[Goldstone boson]].

As mentioned, this model is sometimes used as a [[Phenomenology (particle physics)|phenomenological model]] of [[quantum chromodynamics]] in the [[chiral symmetry|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 complete]]d.

==See also==
*[[Gross–Neveu model]]

==References==
{{Reflist}}

== External links ==
* [[Giovanni Jona-Lasinio]] and [[Yoichiro Nambu]], [http://www.scholarpedia.org/article/Nambu-Jona-Lasinio_model Nambu-Jona-Lasinio model], Scholarpedia, 5(12):7487, (2010). 	doi:10.4249/scholarpedia.7487

{{Quantum field theories}}

{{DEFAULTSORT:Nambu-Jona-Lasinio Model}}
[[Category:Quantum chromodynamics]]
[[Category:Superconductivity]]