Beta function (physics)

Quantum field theory
Feynman diagram
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In theoretical physics, specifically quantum field theory, a beta function or Gell-Mann–Low 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 by the Gell-Mann–Low equation^[1] or renormalization group equation, given by

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

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 first introduced by Ernst Stueckelberg and André Petermann in 1953,^[2] and independently postulated by Murray Gell-Mann and Francis E. Low in 1954.^[3]