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In these notes(at page 71) Bilal put some emphasis on that the running coupling on a certain scale ##\mu## is defined in such a way that the logarithms in the expansions of the vertex function small. Or equivalently it is defined such that running coupling on the scale in question is always a good approximation to the vertex function (and thus also to the amplitude)
$$\Gamma^{(4)}(p_i = \mu) \approx g_\mu.$$
He then defines the ##\beta##-function to be the function which express how the coupling must evolve in order for this to always be the case.
Later one introduces the Callan-Symanzik equation by observing that the bare vertex-function is independent of scale and there the beta function is just defined as the function which one obtains by differentiating the coupling with respect to scale and then multiplying by the scale
$$\beta (g) = \mu \frac{\partial g}{\partial \mu}.$$
It often seems like the Callan-Symanzik is used to find the running coupling, but how do we know that the Callan-Symanzik equations gives us the coupling 'appropriate to the scale ##\mu##'? Is the bare vertex function being independent of the scale ##\mu## somehow equivalent with the statement that that running coupling at the scale ##\mu## is a good approximation to the vertex function?
Why does one need the Callan-Symanzik equation anyway? The running coupling can always be found as long as one has found the vertex function by applying the method shown in Bilals notes.
$$\Gamma^{(4)}(p_i = \mu) \approx g_\mu.$$
He then defines the ##\beta##-function to be the function which express how the coupling must evolve in order for this to always be the case.
Later one introduces the Callan-Symanzik equation by observing that the bare vertex-function is independent of scale and there the beta function is just defined as the function which one obtains by differentiating the coupling with respect to scale and then multiplying by the scale
$$\beta (g) = \mu \frac{\partial g}{\partial \mu}.$$
It often seems like the Callan-Symanzik is used to find the running coupling, but how do we know that the Callan-Symanzik equations gives us the coupling 'appropriate to the scale ##\mu##'? Is the bare vertex function being independent of the scale ##\mu## somehow equivalent with the statement that that running coupling at the scale ##\mu## is a good approximation to the vertex function?
Why does one need the Callan-Symanzik equation anyway? The running coupling can always be found as long as one has found the vertex function by applying the method shown in Bilals notes.