Running of the coupling

1. Jun 3, 2012

physicus

My course on QFT follows Srednicki's book. He derives the running of coupling constants in different theories in the following way: When he uses dimesional regularization (going to $x-\epsilon$ dimensions), he has to introduce a parameter $\mu$ with dimensions of mass in oder to keep the coupling, let's call it $\alpha$, dimensionless. Later, the running of the coupling is determined by demanding that no measurable quantity can depend on $\mu$. We get a function $\alpha(\mu)$.

Consider now a process. I want to know the coupling strength in that process. Apparently, $\mu$ is linked to the center of mass energy of the process considered. That's how I determine the coupling. My question is now the following: Why is $\mu$ linked to the energy scale of the process I am considering? In the derivation the magnitude of the parameter $\mu$ is completely arbitrary. I asked my professor the same question and he answered that $\mu$ must be linked to the center of mass energy, since this is the only parameter with dimesion of mass, that the considered process can depend on. Since ist depends on $\mu$ and the center of mass energy, these two quantities must be linked. I do not understand that argument: Didn't we demand earlier that physical processes must be independent of the unphysical parameter $\mu$? So the considered process cannot depend on $\mu$.

I would be very thankful if someone could clear that up for me.

Best regards, physicus

2. Jun 3, 2012

The_Duck

I think the idea is this: when you compute an amplitude for some process you will get an expression in terms of mu. You can evaluate this expression at any mu you like, as long as you choose the correct values of the coupling constants for the chosen value of mu.

However, some values of mu are better than others. In our amplitudes we generally find terms of the form ln(E/mu) where E is some typical energy scale of the process we are considering. If mu is very different from the scale E, this logarithm can be a large number. If it's large enough, it can actually start to hurt the convergence of the perturbative expansion. The presence of these large logarithms in the higher order terms can make them comparable in magnitude to the leading terms, whereas for good convergence we want the higher order terms to be much smaller than than the leading terms.

We can fix this by choosing mu to be comparable to the typical energy scales of the process we are considering. Then we have no large logarithms, and improved convergence of the perturbation series.