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
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
In the modeling of physics, a toy model is a deliberately simplistic model with many details removed so that it can be used to explain a mechanism concisely. It is also useful in a description of the fuller model.
* In "toy" mathematical models ...
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
function of the interaction coupling is negative, the theory has a dynamical mass generation mechanism with
chiral symmetry breaking, and in the large number of flavor (
) limit, GN theory behaves as t'Hooft's large
limit in QCD.
It consists of N Dirac fermions
. 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
:
.
Einstein summation notation
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of i ...
is used,
is a two component spinor object and
is the
coupling constant. If the mass
is nonzero, the model is massive classically, otherwise it enjoys a
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, ...
.
This model has a
U(N) global
internal symmetry
In physics, a 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 ''continuo ...
. If one takes N=1 (which permits only one quartic interaction) and makes no attempt to
analytically continue the dimension, the model reduces to the massive
Thirring model
The Thirring model is an exactly solvable quantum field theory which describes the self-interactions of a Dirac field in (1+1) dimensions.
Definition
The Thirring model is given by the Lagrangian density
:
\mathcal= \overline(i\partial\!\!\!/ ...
(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 ...
) modeled upon 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 sup ...
of superconductivity. The 2-dimensional version has the advantage that the 4-fermi interaction is renormalizable, which it is not in any higher number of dimensions.
Features of the theory
Gross and Neveu studied this model in the large
limit, expanding the relevant parameters in a
1/N expansion
In quantum field theory and statistical mechanics, the 1/''N'' expansion (also known as the "large ''N''" expansion) is a particular perturbative analysis of quantum field theories with an internal symmetry group such as SO(N) or SU(N). It cons ...
. After demonstrating that this and related models are asymptotically free, they found that, in the subleading order, for small fermion masses the bifermion condensate
acquires a
vacuum expectation value
In quantum field theory the vacuum expectation value (also called condensate or simply 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. ...
(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 of the bifermion field, using the fact that the only renormalization of the coupling constant comes from the
wave function renormalization of the composite field. They then calculated, at leading order in a 1/N expansion but to all orders in the coupling constant, the dependence of the
potential energy 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 ac ...
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 theoretical physics.
Life and work
Sidney Coleman grew up on the Far North Side of Chicago. In 1957, h ...
at the
Erice
Erice (; scn, Èrici) is a historic town and ''comune'' in the province of Trapani, Sicily, in southern Italy.
Geography
The main town of Erice is located on top of Mount Erice, at around above sea level, overlooking the city of Trapani, the ...
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 1/N, the 4-dimensional
theory does not exist. They also argued that in asymptotically free theories the dynamically generated masses never depend analytically on the
coupling constants.
Generalizations
Gross and Neveu considered several generalizations. First, they considered a Lagrangian with one extra quartic interaction
:
chosen so that the discrete chiral symmetry
of the original model is enhanced to a continuous U(1)-valued chiral symmetry
.
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 awar ...
occurs as before, caused by the same VEV. However, as the spontaneously broken symmetry is now continuous, a massless
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 par ...
appears in the spectrum. Although this leads to no problems at the leading order in the 1/N expansion, massless particles in 2-dimensional quantum field theories inevitably lead to
infrared divergence
In physics, an infrared divergence (also IR divergence or infrared catastrophe) is a situation in which an integral, for example a Feynman diagram, diverges because of contributions of objects with very small energy approaching zero, or equivalent ...
s and so the theory appears to not exist.
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. As a result, the Golstone boson is eaten by the
Higgs mechanism
In the Standard Model of particle physics, the Higgs mechanism is essential to explain the generation mechanism of the property "mass" for gauge bosons. Without the Higgs mechanism, all bosons (one of the two classes of particles, the other be ...
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, so they a ...
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- massive particles, called "Dirac par ...
*
Nonlinear Dirac equation
:''See Ricci calculus and Van der Waerden notation for the notation.''
In quantum field theory, the nonlinear Dirac equation is a model of self-interacting Dirac fermions. This model is widely considered in quantum physics as a toy model of self-i ...
*
Thirring model
The Thirring model is an exactly solvable quantum field theory which describes the self-interactions of a Dirac field in (1+1) dimensions.
Definition
The Thirring model is given by the Lagrangian density
:
\mathcal= \overline(i\partial\!\!\!/ ...
*
Nambu–Jona-Lasinio model
References
{{DEFAULTSORT:Gross-Neveu model
Quantum field theory