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The Gross–Neveu model (GN) is a
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 ...
model of
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 interacting via four-fermion interactions in 1 spatial and 1 time dimension. It was introduced in 1974 by
David Gross David Jonathan Gross (; born February 19, 1941) is an American theoretical physicist and string theorist. Along with Frank Wilczek and David Politzer, he was awarded the 2004 Nobel Prize in Physics for their discovery of asymptotic freedom. ...
and André Neveu as a
toy model A toy or plaything is an object that is used primarily to provide entertainment. Simple examples include toy blocks, board games, and dolls. Toys are often designed for use by children, although many are designed specifically for adults and ...
for quantum chromodynamics (QCD), the theory of strong interactions. It shares several features of the QCD: GN theory is asymptotically free thus at strong coupling the strength of the interaction gets weaker and the corresponding \beta function of the interaction coupling is negative, the theory has a dynamical mass generation mechanism with \mathbb_2 chiral symmetry breaking, and in the large number of flavor (N \to \infty) limit, GN theory behaves as 't Hooft's large N_c limit in QCD. It is made using a finite, but possibly large number, \ N\ , of
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 ...
wave function In quantum physics, a wave function (or wavefunction) is a mathematical description of the quantum state of an isolated quantum system. The most common symbols for a wave function are the Greek letters and (lower-case and capital psi (letter) ...
s \psi_1, \psi_2, \ldots, \psi_N, indexed below by Latin letter \ a ~. The model's
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=\bar \psi_a \left( i\ \partial\!\!\!/\ -\ m \right) \psi^a\ +\ \frac\left \bar\psi_a\ \psi^a \right2\ , where the formula uses
Einstein summation notation In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies s ...
. Each wave function \ \psi^a\ is a two component ( left / right)
spinor In geometry and physics, spinors (pronounced "spinner" IPA ) are elements of a complex numbers, complex vector space that can be associated with Euclidean space. A spinor transforms linearly when the Euclidean space is subjected to a slight (infi ...
and \ g\ is the interaction's
coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between tw ...
. If the mass \ m\ is zero, the model is ''chiral symmetric'' type, otherwise, for non-zero mass, it is ''classical mass'' type. This model has a U(N) global
internal 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 ...
. If one takes \ N = 1\ (which permits only one
quartic interaction In quantum field theory, a quartic interaction or ''φ''4 theory is a type of self-interaction in a scalar field. Other types of quartic interactions may be found under the topic of four-fermion interactions. A classical free scalar field \varph ...
) and makes no attempt to analytically continue the dimension, the model reduces to the massive Thirring model (which is completely integrable). It is a 2 dimensional version of the 4 dimensional Nambu–Jona-Lasinio model (NJL), which was introduced 14 years earlier as a model of dynamical chiral symmetry breaking (but no
quark confinement In quantum chromodynamics (QCD), color confinement, often simply called confinement, is the phenomenon that color-charged particles (such as quarks and gluons) cannot be isolated, and therefore cannot be directly observed in normal conditions b ...
) modeled upon 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. The 2 dimensional version has the advantage that the 4 fermi interaction is
renormalizable 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 the ...
, which it is not in any higher number of dimensions.


Features of the theory

Gross and Neveu studied this model in the large \ N\ limit, expanding the relevant parameters in a . After demonstrating that this and related models are asymptotically free, they found that, in the subleading order, for small fermion masses the bifermion condensate \ \overline_a\ \psi^a\ acquires a
vacuum expectation value In quantum field theory, the vacuum expectation value (VEV) of an operator is its average or expectation value in the vacuum. The vacuum expectation value of an operator O is usually denoted by \langle O\rangle. One of the most widely used exa ...
(VEV) and as a result the fundamental fermions become massive. They find that the mass is not analytic in the coupling constant \ g ~. The vacuum expectation value spontaneously breaks the chiral symmetry of the theory. More precisely, expanding about the vacuum with no vacuum expectation value for the bilinear condensate they found a tachyon. To do this they solve the renormalization group equations for the
propagator In quantum mechanics and quantum field theory, the propagator is a function that specifies the probability amplitude for a particle to travel from one place to another in a given period of time, or to travel with a certain energy and momentum. I ...
of the bifermion field, using the fact that the only renormalization of the coupling constant comes from the
wave function renormalization In quantum field theory, wave function renormalization is a rescaling (or renormalization) of quantum fields to take into account the effects of interactions. For a noninteracting or free field, the field operator creates or annihilates a single ...
of the composite field. They then calculated, at leading order in a but to all orders in the coupling constant, the dependence of the
potential energy In physics, potential energy is the energy of an object or system due to the body's position relative to other objects, or the configuration of its particles. The energy is equal to the work done against any restoring forces, such as gravity ...
on the condensate using the
effective action In quantum field theory, the quantum effective action is a modified expression for the classical action taking into account quantum corrections while ensuring that the principle of least action applies, meaning that extremizing the effective act ...
techniques introduced the previous year by
Sidney Coleman Sidney Richard Coleman (7 March 1937 – 18 November 2007) was an American theoretical physicist noted for his research in high-energy physics. Life and work Sidney Coleman grew up on the Far North Side of Chicago. In 1957, he received h ...
at the
Erice Erice (; ) is a (municipality) contiguous with the provincial capital Trapani, in western Sicily. Its historic core occupies the site of the ancient city of Eryx, one of the most significant archaeological and religious centres in pre-Roman w ...
International Summer School of Physics. They found that this potential is minimized at a nonzero value of the condensate, indicating that this is the true value of the condensate. Expanding the theory about the new vacuum, the tachyon was found to be no longer present and in fact, like the BCS theory of superconductivity, there is a
mass gap In quantum field theory, the mass gap is the difference in energy between the lowest energy state, the vacuum, and the next lowest energy state. The energy of the vacuum is zero by definition, and assuming that all energy states can be thought of ...
. They then made a number of general arguments about dynamical mass generation in quantum field theories. For example, they demonstrated that not all masses may be dynamically generated in theories which are infrared-stable, using this to argue that, at least to leading order in \tfrac\ the 4 dimensional \ \phi^4\ theory does not exist. They also argued that in asymptotically free theories the dynamically generated masses never depend analytically on the
coupling constant In physics, a coupling constant or gauge coupling parameter (or, more simply, a coupling), is a number that determines the strength of the force exerted in an interaction. Originally, the coupling constant related the force acting between tw ...
s.


Generalizations

Gross and Neveu considered several generalizations. First, they considered a Lagrangian with one extra quartic interaction :\mathcal = \bar\psi_a \left(i\ \partial\!\!\!/\ -\ m \right) \psi^a\ +\ \frac (\left \bar\psi_a\ \psi^a \right2\ -\ \left \bar\psi_a\ \gamma_5\ \psi^a\right2) chosen so that the discrete chiral symmetry \ \psi \mapsto \gamma_5 \psi\ of the original model is enhanced to a continuous U(1)-valued chiral symmetry \psi\rightarrow e^\psi ~.
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 ...
occurs as before, caused by the same VEV. However, as the spontaneously broken symmetry is now continuous, a massless
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 ...
appears in the spectrum. Although this leads to no problems at the leading order in the massless particles in 2 dimensional quantum field theories inevitably lead to infrared divergences, and so there appears to be no such theory. Two further modifications of the modified theory, which remedy this problem, were then considered. In one modification one increases the number of dimensions. As a result, the massless field does not lead to divergences. In the other modification, the chiral symmetry is gauged. Consequently, the Golstone boson is "eaten" by the
Higgs mechanism In the Standard Model of particle physics, the Higgs mechanism is essential to explain the Mass generation, generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles ...
as the
photon A photon () is an elementary particle that is a quantum of the electromagnetic field, including electromagnetic radiation such as light and radio waves, and the force carrier for the electromagnetic force. Photons are massless particles that can ...
becomes massive, and so does not lead to any divergences.


See also

*
Dirac equation In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin-1/2 massive particles, called "Dirac ...
* Nonlinear Dirac equation * Thirring model * Nambu–Jona-Lasinio model


References

{{DEFAULTSORT:Gross-Neveu model Quantum field theory