[HELP]What is the difference between renormalized mass and the running mass

[piggy1983]

What is the difference between renormalized mass and the running mass? And physical mass,pole mass?

 

[blechman]

"renormalized mass" (also called "physical mass" sometimes) is the mass parameter AFTER you have subtracted your UV divergences.

"running mass" is a funny, and somewhat misleading, term. It is a mass that is a function of the energy. Roughly speaking, when you compute cross sections, you often find your answer involves logarithms that are large (for example, log(Q/m) where Q is the CM energy and m is the mass - if you're talking about scattering 1 GeV mass particles at the LHC, for example, this is a large ratio). These large logs can cause trouble, since if they get too large, they detroy perturbation theory! So using something called "Renormalization group" we can sum up these logs into functions we call "running couplings" (including mass).

As a simple example of how this works, think of calculating something in perturbation theory. You get an answer:

$$1+x+x^2+x^3$$

Now this is perturbation theory, so you know this should not make sense if x > 1. But you might also recognize this series as $(1-x)^{-1}$, which is valid for MUCH larger x (!) So you "resum" the series into this form and your calcuation will be valid for larger x. These resummed quantities are called "running couplings", but really they're just factors in cross sections and lifetimes.

Pole mass: this is a SPECIFIC renormalization scheme where you define your renormalized mass to be the place where the propogator $(p^2-m_{\rm pole}^2)^{-1}$ blows up.

Hope that helps!