Beta function (physics)

Function that encodes the dependence of a coupling parameter on the energy scale

In theoretical physics, specifically quantum field theory, a beta function, β(g), encodes the dependence of a coupling parameter, g, on the energy scale, μ, of a given physical process described by quantum field theory. It is defined as

${\displaystyle \beta (g)=\mu {\frac {\partial g}{\partial \mu }}={\frac {\partial g}{\partial \ln(\mu )}}~,}$

and, because of the underlying renormalization group, it has no explicit dependence on μ, so it only depends on μ implicitly through g. This dependence on the energy scale thus specified is known as the running of the coupling parameter, a fundamental feature of scale-dependence in quantum field theory, and its explicit computation is achievable through a variety of mathematical techniques. The concept of Beta function was Introduced by Ernst Stueckelberg and André Petermann in 1953.^[1]

Scale invariance

If the beta functions of a quantum field theory vanish, usually at particular values of the coupling parameters, then the theory is said to be scale-invariant. Almost all scale-invariant QFTs are also conformally invariant. The study of such theories is conformal field theory.

The coupling parameters of a quantum field theory can run 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.

Examples

Beta functions are usually computed in some kind of approximation scheme. An example is perturbation theory, where one assumes that the coupling parameters are small. One can then make an expansion in powers of the coupling parameters and truncate the higher-order terms (also known as higher loop contributions, due to the number of loops in the corresponding Feynman graphs).

Here are some examples of beta functions computed in perturbation theory:

Quantum electrodynamics

The one-loop beta function in quantum electrodynamics (QED) is

• ${\displaystyle \beta (e)={\frac {e^{3}}{12\pi ^{2}}}~,}$

or, equivalently,

• ${\displaystyle \beta (\alpha )={\frac {2\alpha ^{2}}{3\pi }}~,}$

written in terms of the fine structure constant in natural units, α = e²/4π.^[2]

This beta function tells us that the coupling increases with increasing energy scale, and QED becomes strongly coupled at high energy. In fact, the coupling apparently becomes infinite at some finite energy, resulting in a 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.

Quantum chromodynamics

The one-loop beta function in quantum chromodynamics with ${\displaystyle n_{f}}$ flavours and ${\displaystyle n_{s}}$ scalar colored bosons is

${\displaystyle \beta (g)=-\left(11-{\frac {n_{s}}{6}}-{\frac {2n_{f}}{3}}\right){\frac {g^{3}}{16\pi ^{2}}}~,}$

or

${\displaystyle \beta (\alpha _{s})=-\left(11-{\frac {n_{s}}{6}}-{\frac {2n_{f}}{3}}\right){\frac {\alpha _{s}^{2}}{2\pi }}~,}$