Renormalization group resummation

Question

I'm having trouble in understeanding a mathematical feature of RG, namely how it provides a way to resum the perturbation series and how that's defined mathematically.

From a conceptual point of view one applies the RG flow to a 'theory' from a scale $\Lambda$ down to an energy scale $\mu$ and then reparametrizes it with a finite set of parameters more suitable for describing the physics at that energy scale (renormalization) $\{g^i_R(\mu)\}_{i=1........N}$, the physics must be indipendent of the parametrization we choose or equivalently it must be indipendent of the energy scale $\mu<<\Lambda$ which defines the parametrization, so given a set of observables $\{O_i\}_{i=1....M}$ it must hold (near a fixed point where the RG can be diagonalized): $$\frac{d\ O_i(g,\mu) }{d\log(\mu)}=(\frac{\partial}{\partial log(\mu)}-\beta(g)\frac{\partial}{\partial g}\ )\ O_i=0 \\ \beta(g)=-\ \frac{d \ g}{d \ log(\mu)}$$

($log\ \mu$ derivative so we don't introduce any new energy scale) Now if we calculate perturbatevly $\beta=\sum_n g^n\alpha_n\approx -\alpha g^2$ we get : $$log(\mu'/\mu)=\int \frac{dg}{ \alpha g^2}$$ we get the well known leading logarithm resummation: $$g(\mu')=\frac{g(\mu)}{1-\alpha \ g(\mu) \ log(\mu'/\mu)}$$ So if now we expand in a perturbative series $O_i$ at every order in $g(\mu')$ we have a resummation of the leading logs.

Moreover, consider $O_i(p/\mu,g(\mu))=\sum_n g^n(\mu)\Omega_n(p/\mu)$ in order to obtain a resummation the perturbative series must be redefined in terms of a new expansion parameter $g(\mu')=g_p=f(g(p),p/\mu)$ where $f$ is an exact soultion of the RG equation, such that (using RG flow invariance of the physical quantities):$$O_i(p/\mu,g(\mu))=O_i(1,g_p)=\sum_nf(g(p),p/\mu)^n\Omega_n(1)=\sum_n g_p^n \ \Omega_n(1)$$ Where the coefficents $\Omega_n(1)$ are free from large logarithms problems too. Now $g_p=g'=f(g,\mu'/\mu)$ comes from the implicit equation $$log(\mu'/\mu)=\int_g^{g'} \frac{d\tilde{g}}{\beta(\tilde{g})}$$ such that $f$ is invariant for reparametrization.

My problem now is (provided what i said before is correct and i apologize for the lenghty premise) how this redefinition of the perturbative series works? i.e. Why if i calculate $\beta(g)$ for a few terms i get a complete resummation of lelading,sub leading, sub sub leading terms and so on? Is there a way to define those concept of resummation and redefinition of the perturbative series in a clear and precise way which then can be applied to this particular case?

This post imported from StackExchange Physics at 2015-11-14 22:38 (UTC), posted by SE-user Fra