Renormalization differential equation ?

[tpm]

Renormalization differential equation ??

Let's suppose we have in perturbation theory the quantities

$$(m_0 , q_0 , G_0 (x,s))$$

With m,q, and G(x,s) the 'mass' 'charge' and 'Green function' (propagator)

and the sub-index '0' here stands for "free" theory (no interactions)

Then my question is if there is a PDE , ODE or similar that relates the 'renormalized' (finite values) of the interacting theory

$$(m_R , q_R , G_R (x,s))$$ (R=renormalization) and

$$(m_0 , q_0 , G_0 (x,s))$$

So we can use this PDE no matter if the theory is non-renormalizable or not to extract finite values, for several quantities as mass charge and so on.

 

[cygnus2]

you have to realize that these so called "beta functions" are usually calculated by perturbative methods.and you cannot trust your perturbation theory if your expansion parameter becomes large.they generally yield "asymptotic series".Now the point is that to what scales you can trust a solution generated from an eqn derived from pertubation theory. usually, as it happens, for non-renormalisable theories, the couplings et al blow up at large energy scales. but it is my guess that if you are working at low energy scales, as in an effective theory, you can still do that!

 

[tpm]

And how is this derived ??..i mean the renormalization group equation..and why can't be applied to NOn-renormalizable theories??, i have tried reading the Wikipedia article but it's rather 'fuzzy' without explaining they impose the condition:

$$\frac{d Z[\Lambda]}{d\Lambda}=0$$

after introducing a momentum cut-off so $$p \le \Lambda$$ and introducing a misterious function $$R_{\Lambda}$$