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

$beta(g)\; =\; mu,frac\{partial\; g\}\{partial\; mu\}.$

Scale invariance

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

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

Quantum electrodynamics

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

• $beta(e)=frac\{e^3\}\{12pi^2\}$

or

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

written in terms of the fine structure constant, $alpha=frac\{e^2\}\{4pi\}$.

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and so it is likely that the Landau pole is an artifact of applying perturbation theory in a situation where it is no longer valid.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 $n\_f$ flavours is

• $beta(g)=-left(11-frac\{2n\_f\}\{3\}right)frac\{g^3\}\{16pi^2\}$

or

• $beta(alpha\_s)=-left(11-frac\{2n\_f\}\{3\}right)frac\{alpha\_s^2\}\{2pi\}$,

written in terms of $alpha\_s=frac\{g^2\}\{4pi\}$.

If $n\_fleq\; 16$, this beta-function tells us that the coupling decreases with increasing energy scale, a phenomenon known as asymptotic freedom. Conversely, the coupling increases with decreasing energy scale. This means that the coupling becomes large at low energies, and one can no longer rely on perturbation theory.

See also

References