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 two static bodies to the "charges" of the bodies (i.e. the electric charge for electrostatic and the mass for Newtonian gravity) divided by the distance squared,
r2
G
F=Gm1
2 | |
m | |
2/r |
ke
F=keq1
2 | |
q | |
2/r |
l{L}
l{H}
l{L}
l{H}
T
V
l{L}=T-V
l{H}=T+V
V
T
T
V
T=\bar\psi(i\hbarc
\sigma\partial | |
\gamma | |
\sigma |
-mc2)\psi-{1\over4\mu0}F\muF\mu
V=-e\bar\psi(\hbarc\gamma\sigmaA\sigma)\psi
A coupling plays an important role in dynamics. For example, one often sets up hierarchies of approximation based on the importance of various coupling constants. In the motion of a large lump of magnetized iron, the magnetic forces may be more important than the gravitational forces because of the relative magnitudes of the coupling constants. However, in classical mechanics, one usually makes these decisions directly by comparing forces. Another important example of the central role played by coupling constants is that they are the expansion parameters for first-principle calculations based on perturbation theory, which is the main method of calculation in many branches of physics.
Couplings arise naturally in a quantum field theory. A special role is played in relativistic quantum theories by couplings that are dimensionless; i.e., are pure numbers. An example of a dimensionless such constant is the fine-structure constant,
\alpha=
e2 | |
4\pi\varepsilon0\hbarc |
,
In a non-abelian gauge theory, the gauge coupling parameter,
g
1{4g | |
2}{\rm |
Tr}G\mu\nuG\mu\nu,
g
e | |
\sqrt{\varepsilon0\hbarc |
In a quantum field theory with a coupling g, if g is much less than 1, the theory is said to be weakly coupled. In this case, it is well described by an expansion in powers of g, called perturbation theory. If the coupling constant is of order one or larger, the theory is said to be strongly coupled. An example of the latter is the hadronic theory of strong interactions (which is why it is called strong in the first place). In such a case, non-perturbative methods need to be used to investigate the theory.
In quantum field theory, the dimension of the coupling plays an important role in the renormalizability property of the theory,[1] and therefore on the applicability of perturbation theory. If the coupling is dimensionless in the natural units system (i.e.
c=1
\hbar=1
-2 | |
[G | |
N]=energy |
-2 | |
[G | |
F]=energy |
[F]=energy
One may probe a quantum field theory at short times or distances by changing the wavelength or momentum, k, of the probe used. With a high frequency (i.e., short time) probe, one sees virtual particles taking part in every process. This apparent violation of the conservation of energy may be understood heuristically by examining the uncertainty relation
\DeltaE\Deltat\ge
\hbar | |
2 |
,
In other formulations, the same event is described by "virtual" particles going off the mass shell. Such processes renormalize the coupling and make it dependent on the energy scale, μ, at which one probes the coupling. The dependence of a coupling g(μ) on the energy-scale is known as "running of the coupling". The theory of the running of couplings is given by the renormalization group, though it should be kept in mind that the renormalization group is a more general concept describing any sort of scale variation in a physical system (see the full article for details).
The renormalization group provides a formal way to derive the running of a coupling, yet the phenomenology underlying that running can be understood intuitively.[4] As explained in the introduction, the coupling constant sets the magnitude of a force which behaves with distance as
1/r2
1/r2
r
1/r2
1/r2
r
r
r
1/r2
1/r2
r
1/r
Since a running coupling effectively accounts for microscopic quantum effects, it is often called an effective coupling, in contrast to the bare coupling (constant) present in the Lagrangian or Hamiltonian.
See main article: Beta function (physics). In quantum field theory, a beta function, β(g), encodes the running of a coupling parameter, g. It is defined by the relation
\beta(g)=\mu
\partialg | |
\partial\mu |
=
\partialg | |
\partialln\mu |
,
The coupling parameters of a quantum field theory can flow even if the corresponding classical field theory is scale-invariant. In this case, the non-zero beta function tells us that the classical scale-invariance is anomalous.
If a beta function is positive, the corresponding coupling increases with increasing energy. An example is quantum electrodynamics (QED), where one finds by using perturbation theory that the beta function is positive. In particular, at low energies,, whereas at the scale of the Z boson, about 90 GeV, one measures .
Moreover, the perturbative beta function tells us that the coupling continues to increase, and QED becomes strongly coupled at high energy. In fact the coupling apparently becomes infinite at some finite energy. This phenomenon was first noted by Lev Landau, and is called the Landau pole. However, one cannot expect the perturbative beta function to give accurate results at strong coupling, and so it is likely that the Landau pole is an artifact of applying perturbation theory in a situation where it is no longer valid. The true scaling behaviour of
\alpha
In non-abelian gauge theories, the beta function can be negative, as first found by Frank Wilczek, David Politzer and David Gross. An example of this is the beta function for quantum chromodynamics (QCD), and as a result the QCD coupling decreases at high energies.
Furthermore, the coupling decreases logarithmically, a phenomenon known as asymptotic freedom (the discovery of which was awarded with the Nobel Prize in Physics in 2004). The coupling decreases approximately as
2) | |
\alpha | |
s(k |
\stackrel{def
k
Conversely, the coupling increases with decreasing energy. This means that the coupling becomes large at low energies, and one can no longer rely on perturbation theory. Hence, the actual value of the coupling constant is only defined at a given energy scale. In QCD, the Z boson mass scale is typically chosen, providing a value of the strong coupling constant of αs(MZ2) = 0.1179 ± 0.0010.[5] In 2023 Atlas measured the most precise so far.[6] [7] The most precise measurements stem from lattice QCD calculations, studies of tau-lepton decay, as well as by the reinterpretation of the transverse momentum spectrum of the Z boson.[8]
In quantum chromodynamics (QCD), the quantity Λ is called the QCD scale. The value is
Λ\rm=332\pm17MeV
Λ\rm=210\pm14
A remarkably different situation exists in string theory since it includes a dilaton. An analysis of the string spectrum shows that this field must be present, either in the bosonic string or the NS–NS sector of the superstring. Using vertex operators, it can be seen that exciting this field is equivalent to adding a term to the action where a scalar field couples to the Ricci scalar. This field is therefore an entire function worth of coupling constants. These coupling constants are not pre-determined, adjustable, or universal parameters; they depend on space and time in a way that is determined dynamically. Sources that describe the string coupling as if it were fixed are usually referring to the vacuum expectation value. This is free to have any value in the bosonic theory where there is no superpotential.