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quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines classical field theory, special relativity, and quantum mechanics. QFT is used in particle physics to construct physical models of subatomic particles a ...
, the Nambu–Jona-Lasinio model (or more precisely: ''the Nambu and Jona-Lasinio model'') is a complicated effective theory of
nucleon In physics and chemistry, a nucleon is either a proton or a neutron, considered in its role as a component of an atomic nucleus. The number of nucleons in a nucleus defines the atom's mass number (nucleon number). Until the 1960s, nucleons w ...
s and
meson In particle physics, a meson ( or ) is a type of hadronic subatomic particle composed of an equal number of quarks and antiquarks, usually one of each, bound together by the strong interaction. Because mesons are composed of quark subparticle ...
s constructed from interacting
Dirac fermion In physics, a Dirac fermion is a spin-½ particle (a fermion) which is different from its antiparticle. The vast majority of fermions – perhaps all – fall under this category. Description In particle physics, all fermions in the standard model ...
s with
chiral symmetry A chiral phenomenon is one that is not identical to its mirror image (see the article on mathematical chirality). The spin of a particle may be used to define a handedness, or helicity, for that particle, which, in the case of a massless particle ...
, paralleling the construction of
Cooper pair In condensed matter physics, a Cooper pair or BCS pair (Bardeen–Cooper–Schrieffer pair) is a pair of electrons (or other fermions) bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Coope ...
s from
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary partic ...
s in the
BCS theory BCS theory or Bardeen–Cooper–Schrieffer theory (named after John Bardeen, Leon Cooper, and John Robert Schrieffer) is the first microscopic theory of superconductivity since Heike Kamerlingh Onnes's 1911 discovery. The theory describes su ...
of
superconductivity Superconductivity is a set of physical properties observed in certain materials where electrical resistance vanishes and magnetic flux fields are expelled from the material. Any material exhibiting these properties is a superconductor. Unlik ...
. 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 In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a ty ...
, which does not work perturbatively at low energies.


Overview

The model is much inspired by the different field of
solid state theory Solid-state physics is the study of rigid matter, or solids, through methods such as quantum mechanics, crystallography, electromagnetism, and metallurgy. It is the largest branch of condensed matter physics. Solid-state physics studies how the l ...
, particularly from the BCS breakthrough of 1957. The first inventor of the Nambu–Jona-Lasinio model,
Yoichiro Nambu was a Japanese-American physicist and professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded half of the Nobel Prize in Physics in 2008 for the discovery in 1960 of the mechanism ...
, 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 In particle physics, chiral symmetry breaking is the spontaneous symmetry breaking of a chiral symmetry – usually by a gauge theory such as quantum chromodynamics, the quantum field theory of the strong interaction. Yoichiro Nambu was award ...
,
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. More specifically, isospin symmetry is a subset of the flavour symmetry seen more broadly in the interactions ...
and
strangeness In particle physics, strangeness ("''S''") is a property of particles, expressed as a quantum number, for describing decay of particles in strong and electromagnetic interactions which occur in a short period of time. The strangeness of a parti ...
. At the same time, the same model was independently considered by Soviet physicists
Valentin Vaks Valentin is a male given name meaning "strong, healthy, power, rule, terco". It comes from the Latin name ''Valentinus'', as in Saint Valentin. Commonly found in Spain, Romania, Bulgaria, France, Italy, Russia, Ukraine, Scandinavia, Latin America ...
and
Anatoly Larkin Anatoly Ivanovich Larkin (russian: Анатолий Иванович Ларкин; October 14, 1932 – August 4, 2005) was a Russian theoretical physicist, universally recognised as a leader in theory of condensed matter, and who was also a cele ...
. The model is quite technical, although based essentially on symmetry principles. It is an example of the importance of
four-fermion interactions In quantum field theory, fermions are described by anticommuting spinor fields. A four-fermion interaction describes a local interaction between four fermionic fields at a point. Local here means that it all happens at the same spacetime point. T ...
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 Condensate may refer to: * The liquid phase produced by the condensation of steam or any other gas * The product of a chemical condensation reaction, other than water * Natural-gas condensate, in the natural gas industry * ''Condensate'' (album ...
from fermion interactions inspired many theories of the breaking of
electroweak symmetry In particle physics, the electroweak interaction or electroweak force is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very differe ...
, such as
technicolor Technicolor is a series of color motion picture processes, the first version dating back to 1916, and followed by improved versions over several decades. Definitive Technicolor movies using three black and white films running through a special ...
and the top-quark condensate. Starting with the one- flavor case first, the
Lagrangian density Lagrangian may refer to: Mathematics * Lagrangian function, used to solve constrained minimization problems in optimization theory; see Lagrange multiplier ** Lagrangian relaxation, the method of approximating a difficult constrained problem with ...
is :\mathcal=\,i\,\bar\partial\!\!\!/\psi+\frac \,\left left(\bar\psi\right)\left(\bar\psi\right)-\left(\bar\gamma^5\psi\right)\left(\bar\gamma^5 \psi\right)\right\, i\,\bar_L\partial\!\!\!/\psi_L+\,i\,\bar_R\partial\!\!\!/\psi_R+\lambda \,\left(\bar_L \psi_R\right)\left(\bar_R\psi_L \right). 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 Spontaneous symmetry breaking is a spontaneous process of symmetry breaking, by which a physical system in a symmetric state spontaneously ends up in an asymmetric state. In particular, it can describe systems where the equations of motion or ...
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 :\mathcal=\,i \,\bar_a\partial\!\!\!/\psi^a+\frac \,\left left(\bar_a\psi^b\right)\left(\bar_b\psi^a\right)-\left(\bar_a\gamma^5\psi^b\right)\left(\bar_b\gamma^5 \psi^a\right)\right\,i\,\bar_\partial\!\!\!/\psi_L^a+\,i\,\bar_\partial\!\!\!/\psi_R^a+\frac \,\left(\bar_ \psi_R^b\right)\left(\bar_\psi_L^a \right). 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. The broken symmetries lead to massless
pseudoscalar In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. T ...
bosons which are sometimes called
pion In particle physics, a pion (or a pi meson, denoted with the Greek letter pi: ) is any of three subatomic particles: , , and . Each pion consists of a quark and an antiquark and is therefore a meson. Pions are the lightest mesons and, more gen ...
s. See
Goldstone boson In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries. They were discovered by Yoichiro Nambu in part ...
. As mentioned, this model is sometimes used as a
phenomenological model A phenomenological model is a scientific model that describes the empirical relationship of phenomena to each other, in a way which is consistent with fundamental theory, but is not directly derived from theory. In other words, a phenomenological ...
of
quantum chromodynamics In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a ty ...
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 Renormalization is a collection of techniques in quantum field theory, the statistical mechanics of fields, and the theory of self-similar geometric structures, that are used to treat infinities arising in calculated quantities by altering va ...
in four spacetime dimensions, this theory can only be an
effective field theory In physics, an effective field theory is a type of approximation, or effective theory, for an underlying physical theory, such as a quantum field theory or a statistical mechanics model. An effective field theory includes the appropriate degree ...
which needs to be
UV complete In theoretical physics, ultraviolet completion, or UV completion, of a quantum field theory is the passing from a lower energy quantum field theory to a more general quantum field theory above a threshold value known as the cutoff. In particula ...
d.


See also

*
Gross–Neveu model The Gross–Neveu (GN) model is a quantum field theory model of Dirac fermions interacting via four-fermion interactions in 1 spatial and 1 time dimension. It was introduced in 1974 by David Gross and André Neveu as a toy model for quantum ch ...


References


External links

* Giovanni Jona-Lasinio and
Yoichiro Nambu was a Japanese-American physicist and professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded half of the Nobel Prize in Physics in 2008 for the discovery in 1960 of the mechanism ...

Nambu-Jona-Lasinio model
Scholarpedia, 5(12):7487, (2010). doi:10.4249/scholarpedia.7487 {{DEFAULTSORT:Nambu-Jona-Lasinio Model Quantum chromodynamics Superconductivity