What is meant by a certain theory has a scale ?

[arroy_0205]

What is meant by "a certain theory has a scale"?

In the context of high energy physics (and in other branches too) I sometimes hear that a certain theory has "an intrinsic scale in it (the theory)". What is implied by that? How do I find out if a given theory has a scale in it? Can there be more than one scale in the same theory?
(From observation it seems to me that after solving a differential equation (describing some physical process) if the solution is exponential, then the constant part (not the numerical factors) in the exponential is the scale associated with it. Probably the same applies for other power laws also. Am I correct?)

 

[blechman]

Often in field theory, physical quantities depend on certain fixed energy (or length) scales. For example, if you are describing QED, your results will generally depend on the electron mass. Or when you are describing a ferromagnet, your answers depend on a correlation length which tells you how far away correlated spins are from each other.

So these theories have a "scale" associated with them.

The reason why this is important is that if there is a scale in your theory (such as the electron mass in QED), then by dimensional analysis, you know that your cross sections (which have units of 1/energy^2) must go like:

$$\sigma_{\rm QED}=\frac{1}{E^2}F(E/m_e)$$

where F is now a dimensionless function (this is called a "scaling argument"). Even if you know nothing else about the theory, this alone is a very powerful piece of information! For example, theories that have no scales (called "conformal field theories") must have F = constant.

As to whether or not a theory can have multiple scales: sure! The stardard model has many scales - the masses of all the quarks, leptons, gauge and Higgs bosons, as well as the QCD scale (defined as the place where QCD perturbation theory breaks down). As a rule, the more scales you have, the harder the problem is, since that "F" I mentioned earlier becomes a function of MANY ratios rather than just one.

Hope that helps!

ps: you might check out my webpage introduction to Effective Field Theory:

http://www.physics.wayne.edu/~ablechman/main/Research.html

for some more examples of how this stuff plays out in the real world. Sorry for the plug.