Concept behind mass renormalization

[clerk]

I am confused about the idea of mass renormalization in quantum field theory. Firstly, in case of charge renormalization there is a picture where you have a swarm of particle antiparticle pairs round the electron and hence depending on the energy of your probe , the charge gets renormalized .But somehow there is no such analogous picture in the mass renormalization. The way in which this is introduced in the texts that I have seen is that they calculate the corrections to the propagator coming from higher order feynman diagrams and include the cutoff dependent quantities that you get in the process inside the mass and field ..hence the renormalized mass becomes cutoff dependent . Does this imply that in a different scale, we will record a different mass ?? Another doubt is the distinction between bare and physical perturbation theory...in physical perturbation theory , we sort of perturb around the lagrangian with renormalized masses and coupling constants by including counterterms..but we don't know the renormalized masses accurately since we can only calculate the propagator corrections only to first few orders at best...also the counterterms sort of soak up the cutoff dependence of the mass, then how come it still remains a renormalized mass (dependent on cutoff)? I am really confused about this whole business.

 

[ofirg]

The physical mass of a particle that we measure (referred to as pole mass, since it is the pole of the exact propagator) doesn't depend on the energy scale we perform our experiment. It is a single number which is an intrinsic property of a particle.
In renormalized perturbation theory, we split our lagrangain into a lagrangian with the same form but renormalized parameters and counterterms. The values of these renormalized parameters are set by renormalization conditions which can be chosen.
The distinction is in perturbation theory. Instead of expanding the observables we wish to calculate into a power series in the bare parameters we do it in the renormalized parameters.
The renormalized parameters are usually more closely related to physical quantities and the renormalization condition (i.e., mostly energy scale) may be chosen so perturbation theory works better.

Hope that helps

 

[clerk]

Thanks a lot.. that was really very helpful. So even though I cannot get the exact propagator , I can get a rough estimate of the renormalized mass to any degree of accuracy by increasing the order of feynman diagrams at a given energy scale in a renormalizable theory.Next I perturb around these refined quantities .. Have I got it right?
I am still confused about the coefficients of the counterterms ..how are they fixed?

 

[ofirg]

The renormalized quantities is defined by some renormalization condition. The goal isn't to estimate them, they are parameters we use in perturbation theory. Renormalized perturbation theory is just a refined way of doing perturbation theory and calulating the same observable as before: cross section, energy shifts, etc...
The counterterms are fixed order by order, using the renormalization conditions.