# Does mass depend on energy scale?

1. May 31, 2013

### geoduck

In MS-bar scheme, the mass m is not the physical mass but runs with scale.

The physical mass should however be constant.

In that sense it seems you can't just regard mass as another coupling constant.

Physical coupling constants must be defined at some scale, while the physical mass is the physical mass at any scale.

What does it mean when m runs with scale? It doesn't mean the particle is getting heavier because the physical mass is always constant.

2. Jun 3, 2013

### Jolb

I'm not entirely sure of what the MS-bar scheme is, but it may help you to do some research on "mass renormalization."

Though you don't see it explained in many quantum field theory books, you will find in many condensed matter textbooks (such as Cardy's "Scaling and Renormalization in Statistical Physics" and in Chaikin and Lubensky's "Principles of Condensed Matter Physics") an explanation of how renormalization has everything to do with rescaling. Basically when you rewrite a Hamiltonian in some new system of units, you can make the change by keeping the Hamiltonian the same, but just changing the coupling constants--and if you rescale repeatedly by the right factors, you can massage your Hamiltonian into much prettier forms with "renormalized" constants. [Technically the process involves some more nontrivial steps than just changing units: for a lattice theory you might pick a block of pairwise interacting lattice sites and represent everything in each block by an equivalent single lattice site, etc.]

Whenever you renormalize, there's some rescaling going on behind the scenes, and in QFT, when you see any sort of renormalization, it can be interpreted in the context of rescaling, just as in condensed matter physics. (The details of this I'm not entirely sure of, but you could always look into the work done by Ken Wilson--he's the one who established all of this.) So basically when you see a "bare mass" (what you call the physical mass) and a "renormalized mass" (the coupling constant that's actually in your Hamiltonian), this is due to mass renormalization, and renormalization of any type has to do with scaling. (You can see the connection most easily in condensed matter/lattice theories but the same connection holds in field theories.)

But beware: this is very subtle stuff that took a while for people to figure out. In addition, there are some other perspectives you can take on this (for example, it's also legitimate to consider the coupling constant in the Hamiltonian to be the physical mass, and the experimentally measured mass to be the renormalized mass.) So be careful with this stuff: it can be confusing even for an expert.

One other way to interpret a "bare" vs. a "renormalized" quantity is in terms of screening, but this doesn't touch on the rescaling aspect of renormalization.

One kind of experiment that a lot of physicists studied to sort out the "bare" and "renormalized" masses is called "deep inelastic scattering", which probes the distinction between these two quantities.

Last edited: Jun 3, 2013