HOME

TheInfoList



OR:

In
quantum field theory In theoretical physics, quantum field theory (QFT) is a theoretical framework that combines Field theory (physics), field theory and the principle of relativity with ideas behind quantum mechanics. QFT is used in particle physics to construct phy ...
, 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. Until the 1960s, nucleons were thought to be ele ...
s and
meson In particle physics, a meson () 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 subparticles, the ...
s constructed from interacting
Dirac fermion In physics, a Dirac fermion is a spin-½ particle (a fermion) which is different from its antiparticle. A vast majority of fermions fall under this category. Description In particle physics, all fermions in the standard model have distinct antipar ...
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 particl ...
, 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 Cooper. ...
s from
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary charge, elementary electric charge. It is a fundamental particle that comprises the ordinary matter that makes up the universe, along with up qua ...
s in the
BCS theory In physics, the Bardeen–Cooper–Schrieffer (BCS) 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 descr ...
of
superconductivity Superconductivity is a set of physical properties observed in superconductors: materials where Electrical resistance and conductance, electrical resistance vanishes and Magnetic field, magnetic fields are expelled from the material. Unlike an ord ...
. 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 study 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 type of ...
, 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 was a Japanese-American physicist and professor at the University of Chicago. Known for his groundbreaking contributions to theoretical physics, Nambu was the originator of the theory of spontaneous symmetry breaking, a concept that revoluti ...
(who also contributed essentially to the theory of superconductivity, i.e., by the "Nambu formalism") and
Giovanni Jona-Lasinio Giovanni Jona-Lasinio (born 1932), sometimes called Gianni Jona, is an Italian theoretical physicist, best known for his works on quantum field theory and statistical mechanics. He pioneered research concerning spontaneous symmetry breaking, and t ...
, published in 1961. A subsequent paper included
chiral symmetry breaking In particle physics, chiral symmetry breaking generally refers to the dynamical spontaneous breaking of a chiral symmetry associated with massless fermions. This is usually associated with a gauge theory such as quantum chromodynamics, the quant ...
,
isospin In nuclear physics and particle physics, isospin (''I'') is a quantum number related to the up- and down quark content of the particle. Isospin is also known as isobaric spin or isotopic spin. Isospin symmetry is a subset of the flavour symmetr ...
and
strangeness In particle physics, strangeness (symbol ''S'') is a property of particles, expressed as a quantum number, for describing decay of particles in strong and electromagnetic interactions that occur in a short period of time. The strangeness of a ...
. Around that time, the same model was independently considered by Soviet physicists Valentin Vaks and
Anatoly Larkin Anatoly Ivanovich Larkin (; 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 celebrated teacher of several generations of theorists. Bo ...
. 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 In particle physics, the electroweak interaction or electroweak force is the unified description of two of the fundamental interactions of nature: electromagnetism (electromagnetic interaction) and the weak interaction. Although these two force ...
, such as
technicolor Technicolor is a family of Color motion picture film, color motion picture processes. The first version, Process 1, was introduced in 1916, and improved versions followed over several decades. Definitive Technicolor movies using three black-and ...
and the top-quark condensate. Starting with the one- flavor case first, the
Lagrangian density Lagrangian field theory is a formalism in classical field theory. It is the field-theoretic analogue of Lagrangian mechanics. Lagrangian mechanics is used to analyze the motion of a system of discrete particles each with a finite number of degrees ...
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 or, equivalently, decomposed into left and right chiral parts, :\ \mathcal\ = \ i\ \bar_\mathsf\ \partial\!\!\!/\ \psi_\mathsf\ +\ i\ \bar_\mathsf\ \partial\!\!\!/\ \psi_\mathsf\ +\ \lambda\ \left( \bar_\mathsf\ \psi_\mathsf\right) \left(\bar_\mathsf\ \psi_\mathsf \right) ~. The terms proportional to \ \lambda\ are an attractive four-fermion interaction, which parallels the BCS theory
phonon A phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. In the context of optically trapped objects, the quantized vibration mode can be defined a ...
exchange interaction. The
global symmetry The symmetry of a physical system is a physical or mathematical feature of the system (observed or intrinsic) that is preserved or remains unchanged under some transformation. A family of particular transformations may be ''continuous'' (such ...
of the model is where is the ordinary charge of the Dirac fermion and is the chiral charge. The parameter \ \lambda\ is equivalent to a reciprocal squared mass, \ \lambda = \tfrac\ , 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 A fermionic condensate (or Fermi–Dirac 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 cond ...
(but no
confinement Confinement may refer to: * With respect to humans: ** An old-fashioned or archaic synonym for childbirth ** Postpartum confinement (or postnatal confinement), a system of recovery after childbirth, involving rest and special foods ** Civil confi ...
) 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 o ...
of the chiral symmetry, but not the charge symmetry. With flavors and the flavor indices, represented by the Latin letters and , 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 hence : \mathcal\ =\ i\ \bar_\ \partial\!\!\!/\ \psi_\mathsf^a\ +\ i\ \bar_\ \partial\!\!\!/\ \psi_\mathsf^a + \frac\ \left( \bar_\ \psi_\mathsf^b \right) \left(\bar_\ \psi_\mathsf^a \right) ~. Chiral symmetry forbids a bare mass term, but there may be chiral condensates. The global symmetry here is where 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 (nearly) 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. A pseudoscalar, when multiplied by an ordinary vector, becomes a '' pseudovector'' ...
bosons, e.g.
pion In particle physics, a pion (, ) or pi meson, denoted with the Greek alphabet, Greek letter pi (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 ...
s. See
Goldstone boson In 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 within the context of the BCS superco ...
. 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 study 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 type of ...
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, statistical field theory, and the theory of self-similar geometric structures, that is used to treat infinities arising in calculated quantities by altering values of these ...
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 degrees ...
which needs to be UV completed.


See also

*
Gross–Neveu model The Gross–Neveu model (GN) 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 qu ...


References


External links

*
Giovanni Jona-Lasinio Giovanni Jona-Lasinio (born 1932), sometimes called Gianni Jona, is an Italian theoretical physicist, best known for his works on quantum field theory and statistical mechanics. He pioneered research concerning spontaneous symmetry breaking, and t ...
and
Yoichiro Nambu was a Japanese-American physicist and professor at the University of Chicago. Known for his groundbreaking contributions to theoretical physics, Nambu was the originator of the theory of spontaneous symmetry breaking, a concept that revoluti ...

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