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In [[theoretical physics]], specifically [[quantum field theory]], a '''beta function''', ''β(g)'', encodes the dependence of a [[Coupling constant|coupling parameter]], ''g'', on the [[energy scale]], ''μ'', of a given physical process described by [[quantum field theory]].
It is defined as
:: <math>\beta(g) = \mu \frac{\partial g}{\partial \mu} = \frac{\partial g}{\partial \ln(\mu)} ~,</math>
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 [[Coupling constant#Running coupling|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 E. C. G. Stueckelberg and A. Petermann at 1953.
==Scale invariance==
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