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What is the motivation behind defining the beta function as the logarithmic derivative of the coupling constant with respect to scale and not just the regular derivative?

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Well, on the face of it, it hardly matters, since, for $g=g(\mu)$, $$ \beta(g)= \frac{\partial g}{\partial (\ln \mu)} \propto g^3 +... , $$ where the ellipsis corresponds to higher orders, could be written as $\beta =\mu \frac{\partial g}{\partial \mu }$, as well. So what you are asking is why is this particular function a function of just $g$ without explicit $\mu$ dependence.

There are deep and recondite and just-so-technical arguments for it, but, to my mind, the original realization of this fact, by Gell-Mann & Low (1954) is predicated on the structure of the (finite) renormalization group they discovered—actually they credit TD Lee with the specific formula below, in their appendix B. However, this is a just-so technical property that propagators and the rest of QFT obey. All I'll do below is display how the above infinitesimal (differential) form follows from the finite functional equation (Schroeder's equation 1870) below, without justifying that one.

$$g(\mu)=G^{-1}\left(\left(\frac{\mu}{M}\right)^d G(g(M))\right) $$ for some unspecified and undetermined function G (nowadays called Wegner's scaling function) and a constant d, in terms of the coupling g(M) at a reference scale M.

Gell-Mann and Low realized in these results that the effective scale can be arbitrarily taken as $\mu$, and can vary to define the theory at any other scale, $$g(\kappa)=G^{-1}\left(\left(\frac{\kappa}{\mu}\right)^d G(g(\mu))\right) = G^{-1}\left(\left(\frac{\kappa}{M}\right)^d G(g(M))\right).$$

This is the group property: as the scale $\mu$ varies, the theory presents a self-similar replica of itself, and any scale can be accessed similarly from any other scale, by group action, a formal "transitive conjugacy of couplings".

On the basis of this (finite) group equation and its scaling property, Gell-Mann and Low could then focus on infinitesimal transformations. They thus considered the above, $$ G(g(\mu))= e^{d~ (\ln \mu -\ln M)} G(g(M)), $$ and, I hope self-evidently, they differentiated w.r.t. $\ln \mu$ of your question, $$ \frac {\partial g(\mu)}{\partial \ln \mu} = d ~\frac{G(g(\mu))}{\partial G(g(\mu))/\partial g(\mu)}. $$ The r.h.s. only depends on $g(\mu)$, without any explicit $\mu$ dependence! They thus invented a computational method based on this mathematical flow function , in powers of $g$, so perturbation theory, $$ \displaystyle\frac{\partial g}{\partial \ln\mu} = \beta(g) .$$ Now this is a differential equation and can be solved routinely, $\ln(\mu/M)=\int\frac{dg}{\beta (g)} +c$, or even numerically.

As stated, the magic is in the sole dependence of $\beta$ on $g$ derived, but don't ask me to go beyond that. I just take the finite RG equation as a given.

(PS. Do you now see why Feynman hired Gell-Mann at Caltech in 1955, largely on the basis of this result?)

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    $\begingroup$ I want to add that a concise version of this argument is presented in Sec 12.4 of Weinberg vol. 1. $\endgroup$ – Hans Moleman Aug 11 at 19:11
  • $\begingroup$ Indeed, but W's (12.4.2), p 245, is vastly more general than the Schroeder equation here, which has a single argument, and trajectory. He goes beyond concision in justifying it, but, of course, it covers a broader range of theories. Concisely, all he is saying, in a hyper formal language, is that the topmost equation in the above answer is the only dimensionally consistent one for dimensionless g and dimensionfull μ, provided there is no other dimensionfull quantity in the problem: naive dimensional analysis dressed up in functional equations. $\endgroup$ – Cosmas Zachos Aug 12 at 21:09

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