Coleman–Weinberg potential
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The Coleman–Weinberg model represents quantum electrodynamics of a scalar field in four-dimensions. The
Lagrangian 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 ...
for the model is :L = -\frac (F_)^2 + , D_ \phi, ^2 - m^2 , \phi, ^2 - \frac , \phi, ^4 where the scalar field is complex, F_=\partial_\mu A_\nu-\partial_\nu A_\mu is the electromagnetic field tensor, and D_=\partial_\mu-\mathrm i (e/\hbar c)A_\mu the covariant derivative containing the electric charge e of the electromagnetic field. Assume that \lambda is nonnegative. Then if the mass term is tachyonic, m^2<0 there is a spontaneous breaking of the
gauge symmetry In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
at low energies, a variant of 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 bein ...
. On the other hand, if the squared mass is positive, m^2>0 the vacuum expectation of the field \phi is zero. At the classical level the latter is true also if m^2=0. However, as was shown by
Sidney Coleman Sidney Richard Coleman (7 March 1937 – 18 November 2007) was an American theoretical physics, theoretical physicist noted for his research in high-energy theoretical physics. Life and work Sidney Coleman grew up on the Far North Side o ...
and
Erick Weinberg Erick J. Weinberg (born August 29, 1947) is a theoretical physicist and professor of physics at Columbia University. Weinberg received his undergraduate degree from Manhattan College in 1968. He obtained his Ph.D. from Harvard University in 1973 ...
, even if the renormalized mass is zero, spontaneous symmetry breaking still happens due to the radiative corrections (this introduces a mass scale into a classically conformal theory - model have a
conformal anomaly A conformal anomaly, scale anomaly, trace anomaly or Weyl anomaly is an anomaly, i.e. a quantum phenomenon that breaks the conformal symmetry of the classical theory. A classically conformal theory is a theory which, when placed on a surface wi ...
). The same can happen in other gauge theories. In the broken phase the fluctuations of the scalar field \phi will manifest themselves as a naturally light
Higgs boson The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the quantum excitation of the Higgs field, one of the fields in particle physics theory. In the Stand ...
, as a matter of fact even too light to explain the electroweak symmetry breaking in the minimal model - much lighter than
vector boson In particle physics, a vector boson is a boson whose spin equals one. The vector bosons that are regarded as elementary particles in the Standard Model are the gauge bosons, the force carriers of fundamental interactions: the photon of electromag ...
s. There are non-minimal models that give a more realistic scenarios. Also the variations of this mechanism were proposed for the hypothetical spontaneously broken symmetries including
supersymmetry In a supersymmetric theory the equations for force and the equations for matter are identical. In theoretical and mathematical physics, any theory with this property has the principle of supersymmetry (SUSY). Dozens of supersymmetric theories e ...
. Equivalently one may say that the model possesses a first-order
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
as a function of m^2. The model is the four-dimensional analog of the three-dimensional Ginzburg–Landau theory used to explain the properties of superconductors near the
phase transition In chemistry, thermodynamics, and other related fields, a phase transition (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of ...
. The three-dimensional version of the Coleman–Weinberg model governs the superconducting phase transition which can be both first- and second-order, depending on the ratio of the Ginzburg–Landau parameter \kappa\equiv\lambda/e^2, with a
tricritical point In condensed matter physics, dealing with the macroscopic physical properties of matter, a tricritical point is a point in the phase diagram of a system at which three-phase coexistence terminates. This definition is clearly parallel to the defin ...
near \kappa=1/\sqrt 2 which separates
type I Type 1 or Type I or ''variant'', may refer to: Health *Diabetes mellitus type 1 (also known as "Type 1 Diabetes"), insulin-dependent diabetes * Type I female genital mutilation * Type 1 personality *Type I hypersensitivity (or immediate hypersensit ...
from type II
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. Unlike ...
. Historically, the order of the superconducting phase transition was debated for a long time since the temperature interval where fluctuations are large ( Ginzburg interval) is extremely small. The question was finally settled in 1982. If the Ginzburg–Landau parameter \kappa that distinguishes type-I and type-II superconductors (see also
here Here is an adverb that means "in, on, or at this place". It may also refer to: Software * Here Technologies, a mapping company * Here WeGo (formerly Here Maps), a mobile app and map website by Here Television * Here TV (formerly "here!"), a TV ...
) is large enough, vortex fluctuations becomes important which drive the transition to second order. The tricritical point lies at roughly \kappa=0.76/\sqrt, i.e., slightly below the value \kappa=1/\sqrt where type-I goes over into type-II superconductor. The prediction was confirmed in 2002 by Monte Carlo computer simulations.


Literature

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See also

*
Quartic interaction In quantum field theory, a quartic interaction is a type of self-energy, 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 \varphi satis ...


References

Quantum field theory Quantum mechanical potentials {{quantum-stub