# Coupling constant

In physics, a coupling constant, usually denoted g, is a number that determines the strength of an interaction. Usually the Lagrangian or the Hamiltonian of a system can be separated into a kinetic part and an interaction part. The coupling constant determines the strength of the interaction part with respect to the kinetic part, or between two sectors of the interaction part. For example, the electric charge of a particle is a coupling constant.

A coupling constant 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 magentized iron, the graviational forces are more important than the magnetic forces because of the relative coupling constants. However, in classical mechanics one usually makes these decisions directly by comparing forces.

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## Fine structure constant

The coupling constant comes into its own in a quantum field theory. A special role is played in relativistic quantum theories by couplings constants which are dimensionless, ie, are pure numbers. For example, the fine-structure constant,

[itex]\alpha = \frac{e^2}{4\pi\epsilon_0\hbar c}[itex]

(where e is the charge of an electron and ε0 is the permittivity of free space) is such a dimensionless coupling constant that determines the strength of the electromagnetic force on an electron.

## Gauge coupling

In a non-Abelian gauge theory, the gauge coupling parameter, g, appears in the Lagrangian as

[itex]\frac1{4g^2}{\rm Tr}\,G_{\mu\nu}G^{\mu\nu}[itex]

(where G is the gauge field tensor). This should be understood to be similar to a dimensionless version of the electric charge defined as

[itex]\sqrt{4\pi\epsilon_0\alpha}.[itex]

The colour charge of quantum chromodynamics is precisely the gauge coupling.

## Weak and strong coupling

In a quantum field theory with a dimensionless coupling constant, g, if it is (much) smaller than one, then one says that the theory is 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. In such a case non-perturbative methods have to be used to investigate the theory.

## Running coupling

Missing image
Renormalized-vertex.png
Virtual particles renormalize the coupling

One can probe a quantum field theory at short times or distances by changing the wavelength or momentum, k of the probe one uses. With a high frequency, ie, short time probe, one sees virtual particles taking part in every process. The reason this can happen, seemingly violating the conservation of energy is the uncertainty relation

[itex]\Delta E\Delta t\ge\hbar[itex]

which allows such violations at short times. Such processes renormalize the coupling and make it dependent on the scale, k at which one observes the coupling. The phenomenon of scale dependence of the coupling, g(k) is called running coupling in a quantum field theory.

## Beta-function

The beta function of a quantum field theory measures the running of a coupling parameter. It is defined by the relation

[itex]\beta(g) = k\,\frac{\partial g}{\partial k} = \frac{\partial g}{\partial\log k}[itex]

For most theories the beta-function is positive, ie, the coupling increases as k increases (as the scale on which the theory is observed becomes shorter). This is also the case in quantum electrodynamics. At low energy, ie, long distances, one knows that α=1/137 (approximately). At the scale of the Z boson, ie, about 90 MeV, α is known to increase to about 1/127.

In a classical field theory in which a scale change is an invariance (symmetry) of the theory, the beta-function breaks this scale invariance. Since this is a quantum effect (arising directly from the uncertainty principle), a non-zero beta-function implies the existence of a scale anomaly in such a quantum field theory.

## Landau pole and asymptotic freedom

We noted that QED is weakly coupled at long distances, but the coupling increases at short distances. This increase was first noticed by Lev Landau who showed that QED becomes strongly coupled at high energy, and in fact the coupling becomes infinite at asympototically high energy. This phenomenon is called the Landau pole.

In non-Abelian gauge theories, the beta function is negative, as first found by Frank Wilczek, David Politzer and David Gross. As a result the coupling decreases at short distances. Furthermore, the coupling decreases logarithmically, a phenomenon known as asymptotic freedom. The coupling decreases approximately as

[itex] \alpha_s(k^2) \equiv \frac{g^2}{4\pi} = \frac1{\beta_0\log(k^2/\Lambda^2)}[itex]

where β0 is a constant computed by Wilczek, Gross and Politzer.

## QCD scale

The quantity Λ is called the QCD scale. The value is known pretty accurately to be

[itex]\Lambda_{MS} = 217^{+25}_{-23}{\rm\ MeV}[itex]

This value is to be used at a scale above the bottom quark mass of about 5 MeV. The meaning of ΛMS is given in the article on dimensional regularization.

## Charge, colour charge, etc

In quantum field theory, since the size of the interaction term is absorbed into the notion of the coupling constant (more correctly coupling parameter, since it runs), the word charge is freed up for another use. One says, for example, that the electrical charge of an electron is -1 and that of any observable particle is an integer multiple of this. The notion of charge is now exactly the same as the representation of the gauge group to which the particle belongs. Thus the colour charge of a quark is fixed at 4/3 since it belongs to the fundamental representation of SU(3), and the colour charge of a gluon is 8 since it belongs to the adjoint representation.

This difference in the notion of charge in classical and quantum field theory is alluded to in a shorthand phrase that is sometimes used: "charge in units of the positron charge".

## String theory

A remarkably different situation exists in string theory. Each perturbative description of string theory depends on a string coupling constant. However, in the case of string theory, these coupling constants are not pre-determined, adjustable, but universal parameters, but rather dynamical scalar fields that can depend on the position in space and time and whose values are determined dynamically.

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