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:
Read more about this topic: Beta Function (physics)
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