Nambu-Jona-Lasinio model
Encyclopedia
In quantum field theory
, the Nambu–Jona-Lasinio model (or more precisely: the Nambu and Jona-Lasinio model) is a theory of nucleon
s and meson
s constructed from interacting Dirac fermion
s with chiral symmetry
which parallels the construction of Cooper pair
s from electron
s in the BCS theory
of superconductivity
. In fact, the model was much inspired by recent progress in 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. They then included chiral symmetry breaking
, isospin
and strangeness.
The model is quite technical, although based essentially on symmetry principles. It is an example of the importance of four-fermion interactions. The model is defined in a spacetime with an even number of dimensions. It is still in use, particularly as a substitute for quantum chromodynamics
, which does not work perturbatively at low energies, and must be replaced there by models, which are effective, although not rigorous, at these conditions. The NJL model is of this kind.
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
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.
The broken symmetries lead to massless pseudoscalar bosons which are sometimes called pion
s. 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.
Quantum field theory
Quantum field theory provides a theoretical framework for constructing quantum mechanical models of systems classically parametrized by an infinite number of dynamical degrees of freedom, that is, fields and many-body systems. It is the natural and quantitative language of particle physics and...
, the Nambu–Jona-Lasinio model (or more precisely: the Nambu and Jona-Lasinio model) is a theory of nucleon
Nucleon
In physics, a nucleon is a collective name for two particles: the neutron and the proton. These are the two constituents of the atomic nucleus. Until the 1960s, the nucleons were thought to be elementary particles...
s and meson
Meson
In particle physics, mesons are subatomic particles composed of one quark and one antiquark, bound together by the strong interaction. Because mesons are composed of sub-particles, they have a physical size, with a radius roughly one femtometer: 10−15 m, which is about the size of a proton...
s constructed from interacting Dirac fermion
Dirac fermion
In particle physics, a Dirac fermion is a fermion which is not its own anti-particle. It is named for Paul Dirac. All fermions in the standard model, except possibly neutrinos, are Dirac fermions...
s with chiral symmetry
Chiral symmetry
In quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts of Dirac fields transform independently...
which parallels the construction of Cooper pair
Cooper pair
In condensed matter physics, a Cooper pair or BCS pair is two electrons that are bound together at low temperatures in a certain manner first described in 1956 by American physicist Leon Cooper...
s from electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s in the BCS theory
BCS theory
BCS theory — proposed by Bardeen, Cooper, and Schrieffer in 1957 — is the first microscopic theory of superconductivity since its discovery in 1911. The theory describes superconductivity as a microscopic effect caused by a "condensation" of pairs of electrons into a boson-like state...
of superconductivity
Superconductivity
Superconductivity is a phenomenon of exactly zero electrical resistance occurring in certain materials below a characteristic temperature. It was discovered by Heike Kamerlingh Onnes on April 8, 1911 in Leiden. Like ferromagnetism and atomic spectral lines, superconductivity is a quantum...
. In fact, the model was much inspired by recent progress in 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
Yoichiro Nambu
is a Japanese-born American physicist, currently a professor at the University of Chicago. Known for his contributions to the field of theoretical physics, he was awarded a one-half share of the Nobel Prize in Physics in 2008 for the discovery of the mechanism of spontaneous broken symmetry in...
, also contributed essentially to the theory of superconductivity, i.e., by the "Nambu formalism". The second inventor was Giovanni Jona-Lasinio
Giovanni Jona-Lasinio
Giovanni Jona-Lasinio , sometimes called Gianni Jona, is an influential Italian theoretical physicist, best known for his works on quantum field theory and statistical mechanics. He pioneered research concerning spontaneous symmetry breaking, and the Nambu–Jona-Lasinio model is named after him...
. The common paper of the authors that introduced the model appeared in 1961. They then included chiral symmetry breaking
Chiral symmetry breaking
In particle physics, chiral symmetry breaking is an example of spontaneous symmetry breaking affecting the chiral symmetry of gauge theory such as Quantum Chromodynamics. The origin may be described as a fermion condensate...
, isospin
Isospin
In physics, and specifically, particle physics, isospin is a quantum number related to the strong interaction. This term was derived from isotopic spin, but the term is confusing as two isotopes of a nucleus have different numbers of nucleons; in contrast, rotations of isospin maintain the number...
and strangeness.
The model is quite technical, although based essentially on symmetry principles. It is an example of the importance of four-fermion interactions. The model is defined in a spacetime with an even number of dimensions. It is still in use, particularly as a substitute for quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
, which does not work perturbatively at low energies, and must be replaced there by models, which are effective, although not rigorous, at these conditions. The NJL model is of this kind.
The dynamical creation of a condensate
Fermionic condensate
A fermionic condensate is a superfluid phase formed by fermionic particles at low temperatures. It is closely related to the Bose–Einstein condensate, a superfluid phase formed by bosonic atoms under similar conditions. Unlike the Bose–Einstein condensates, fermionic condensates are formed using...
from fermion interactions inspired many theories of the breaking of electroweak symmetry, such as technicolor
Technicolor (physics)
Technicolor theories are models of physics beyond the standard model that address electroweak symmetry breaking, the mechanism through which elementary particles acquire masses...
and the top-quark condensate
Top quark condensate
In particle physics, the top quark condensate theory is an alternative to the Standard Model in which a fundamental scalar Higgs field is replaced by a composite field composed of the top quark and its antiquark. These are bound by a four-fermion interaction, analogous to Cooper pairs in a BCS...
.
Starting with the one-flavor case first, the Lagrangian density is
The terms proportional to λ are the four-fermion interactions, which parallel the BCS theory.
The global symmetry
Global symmetry
A global symmetry is a symmetry that holds at all points in the spacetime under consideration, as opposed to a local symmetry which varies from point to point.Global symmetries require conservation laws, but not forces, in physics.-See also:...
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
Colour confinement
Color confinement, often simply called confinement, is the physics phenomenon that color charged particles cannot be isolated singularly, and therefore cannot be directly observed. Quarks, by default, clump together to form groups, or hadrons. The two types of hadrons are the mesons and the baryons...
) leading to an effective mass term and a spontaneous symmetry breaking
Spontaneous symmetry breaking
Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....
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.
The broken symmetries lead to massless pseudoscalar bosons which are sometimes called pion
Pion
In particle physics, a pion is any of three subatomic particles: , , and . Pions are the lightest mesons and they play an important role in explaining the low-energy properties of the strong nuclear force....
s. See Goldstone boson
Goldstone boson
In particle and condensed matter physics, Goldstone bosons or Nambu–Goldstone bosons are bosons that appear necessarily in models exhibiting spontaneous breakdown of continuous symmetries...
.
As mentioned, this model is sometimes used as a phenomenological model of quantum chromodynamics
Quantum chromodynamics
In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...
in the chiral limit
Chiral symmetry
In quantum field theory, chiral symmetry is a possible symmetry of the Lagrangian under which the left-handed and right-handed parts of Dirac fields transform independently...
. 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
Effective field theory
In physics, an effective field theory is, as any effective theory, an approximate theory, that includes appropriate degrees of freedom to describe physical phenomena occurring at a chosen length scale, while ignoring substructure and degrees of freedom at shorter distances .-The renormalization...
which needs to be UV completed.