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 Λ down to an energy scale μ and then reparametrizes it with a finite set of parameters more suitable for describing the physics at that energy scale (renormalization) {giR(μ)}i=1........N, the physics must be indipendent of the parametrization we choose or equivalently it must be indipendent of the energy scale μ<<Λ which defines the parametrization, so given a set of observables {Oi}i=1....M it must hold (near a fixed point where the RG can be diagonalized): d Oi(g,μ)dlog(μ)=(∂∂log(μ)−β(g)∂∂g ) Oi=0β(g)=− d gd log(μ)
(log μ derivative so we don't introduce any new energy scale) Now if we calculate perturbatevly β=∑ngnαn≈−αg2 we get : log(μ′/μ)=∫dgαg2
we get the well known leading logarithm resummation:
g(μ′)=g(μ)1−α g(μ) log(μ′/μ)
So if now we expand in a perturbative series
Oi at every order in
g(μ′) we have a resummation of the leading logs.
Moreover, consider Oi(p/μ,g(μ))=∑ngn(μ)Ωn(p/μ) in order to obtain a resummation the perturbative series must be redefined in terms of a new expansion parameter g(μ′)=gp=f(g(p),p/μ) where f is an exact soultion of the RG equation, such that (using RG flow invariance of the physical quantities):Oi(p/μ,g(μ))=Oi(1,gp)=∑nf(g(p),p/μ)nΩn(1)=∑ngnp Ωn(1)
Where the coefficents
Ωn(1) are free from large logarithms problems too.
Now
gp=g′=f(g,μ′/μ) comes from the implicit equation
log(μ′/μ)=∫g′gd˜gβ(˜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 β(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