# Coupling constant

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,

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 be used to investigate the theory.

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

which virtually allows such violations at short times. The foregoing remark only applies to some formulations of quantum field theory, in particular, canonical quantization in the interaction picture.

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).

Since a running coupling effectively accounts for microscopic quantum effects, it is often called an *effective coupling*, in contrast to the *bare coupling (constant)* presents in the Lagrangian or Hamiltonian.

In quantum field theory, a *beta function, β*(*g*), encodes the running of a coupling parameter, *g*. It is defined by the relation

where *μ* is the energy scale of the given physical process. If the beta functions of a quantum field theory vanish, then the theory is scale-invariant.

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, *α* ≈ 1/137, whereas at the scale of the Z boson, about 90 GeV, one measures *α* ≈ 1/127.

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.^{[4]}

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

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.

In quantum chromodynamics (QCD), the quantity Λ is called the **QCD scale**. The value is

The proton-to-electron mass ratio is primarily determined by the QCD scale.

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.