What is the significance of the renormalization scale μ?

In summary, the conversation discusses the use of a renormalization scale in dimensional regularization and the subtraction method. The scale, denoted as \mu, is introduced to preserve the correct mass-dimension for the coupling and is necessary for dim-reg but not for the subtraction method. The choice of \mu can be arbitrary, as long as it is consistent throughout the equations. The appearance of \mu in both methods is explained through the concept of renormalization and the dependence of physics on energy scale. Good references for further understanding are suggested to be found in the study of the renormalization group.
  • #1
Jodahr
10
0
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
 
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  • #2
Choosing [itex]\mu[/itex] to be 1 is fine, but what are your units? How are you measuring mass? Your equations will inevitably contain [itex]\mu[/itex] in relation to other parameters - for example, at high energies you will usually have momentum dependence like [itex]\log\left( p^2/\mu^2 \right)[/itex].

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 [itex]\mu[/itex]. 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 [itex]4 - \epsilon[/itex], the physics will depend on the ratio [itex]p^2/g^{2\epsilon-8}[/itex].

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.
 
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1. What is the renormalization scale μ and why is it important?

The renormalization scale μ is a parameter used in quantum field theory to regulate the divergent integrals that arise in calculations. It represents the energy scale at which the theory is being probed and has a direct effect on the predictions of the theory. Choosing an appropriate value for μ is crucial in order to obtain physically meaningful results.

2. How is the renormalization scale μ related to the running coupling constant?

The running coupling constant is a measure of the strength of the interaction between particles at a given energy scale. The value of the running coupling constant depends on the renormalization scale μ, with different values of μ resulting in different values for the coupling constant. As μ increases, the running coupling constant decreases, indicating that the interaction becomes weaker at higher energies.

3. What is the difference between the renormalization scale μ and the renormalization point Λ?

The renormalization scale μ is a continuous parameter that is chosen by the researcher, while the renormalization point Λ is a fixed value that is determined by the theory. The renormalization point represents the energy scale at which the theory is supposed to break down, while the renormalization scale is a way of rescaling the theory to make it applicable at all energy scales.

4. Can the choice of renormalization scale μ affect the physical predictions of a theory?

Yes, the choice of renormalization scale μ can have a significant impact on the physical predictions of a theory. Different values of μ can lead to different values for physical observables, such as particle masses and interaction strengths. In order to obtain accurate predictions, it is important to choose an appropriate value for μ that takes into account the energy scale of the system being studied.

5. How is the renormalization scale μ determined in practice?

In practice, the renormalization scale μ is often determined through a process called renormalization group evolution. This involves solving a set of equations that describe how the parameters of a theory change as the energy scale changes. By choosing an appropriate initial value for μ and using these equations, researchers can determine the optimal value of μ for a given calculation.

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