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Concept
Beta function (physics)
Summary
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
:: \beta(g) = \frac{\partial g}{\partial \log(\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.
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 fi
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