What is the significance of the renormalization scale μ?

[Jodahr]

Hello everybody,

I have a short question about the renormalization scale.

For dimensional regularization we introduce a scale μ with mass dimension to preserve the correct mass-dimension for the coupling and so on so that it is independent of the value of d = 4-2ε. But why can that μ have any arbitrary value. Why not just say it is 1 times mass dimension? In dimensional regularization they always do net tell someone much about the scale.

In the subtraction method it is somehow clearer since it appears after regularization only for the renormalization prescription. So for regularization that scale is only needed in the dim.reg. case, not for the subtraction method. Only then for the renormalization procedure it appears also in the approach.

It would be nice if someone can explain the case in dim. reg. and maybe can make a connection between the appearance in both cases (dim. reg. and subtraction ). And maybe someone knows good references to learn more about that arbitrary scales?

Thanks in advance!

Cheers,

Marcel

 

[king vitamin]

Choosing \mu to be 1 is fine, but what are your units? How are you measuring mass? Your equations will inevitably contain \mu in relation to other parameters - for example, at high energies you will usually have momentum dependence like \log\left( p^2/\mu^2 \right).

The main point is that by regulating your theory in the UV, you are essentially introducing a mass scale. You start with a "bare" coupling in the Lagrangian which does not know or care about renormalization procedures and is dimensionless and thus independent of energy scale. You then trade it for a "renormalized" coupling which depends on \mu. So you're trading a scale invariant parameter for two scale dependent parameters which depend on each other in a well-defined way.

This is all very clear for a finite cutoff method, but as you say it looks completely different in dim-reg. The point is that when you consider d<4, your coupling constant picks up a mass dimension, so the behavior of the term becomes dependent on scale. If your coupling has mass dimension 4 - \epsilon, the physics will depend on the ratio p^2/g^{2\epsilon-8}.

You asked for references on arbitrary scales. I would simply recommend studying the renormalization group. Any good modern QFT book should cover it extensively, and it's enormously useful across many fields of physics.