quantum field theory - Intuition for parameter $mu$ in dimensional regularization

In dimensional regularization, a dimensionless coupling $g$ is replaced by $\mu^{4-d}g$ so that it can remain dimensionless. $\mu$ is unphysical, though its choice affects the values of counterterms. By setting the derivatives of observables with respect to $\mu$ equal to zero, we get renormalization group equations.

I'm fine with this, but there's an additional claim slipped in: the renormalized couplings you get from using $\mu$ "describe physics at the energy scale $\mu$".

But when I first learned DR, $\mu$ was presented as a parameter that was just thrown in for convenience; it was totally meaningless, like the ghost particle mass in Pauli-Villars. Where does this interpretation of $\mu$ come from?

Answer

Dim. reg. is not very intuitive. You could say MS is not a very physical renormalization scheme. There are however several ways in which $\mu$ is connected to an actual physical energy scale in applications:

• 
  

  $\mu$ is arbitrary in general, however, in calculations you usually get logarithms of the form $log \left( \frac{\mu}{M}\right)$ where $M$ is some energy scale in your problem. Could be a momentum transfer for example. If you want your perturbative corrections to be small, you better choose $\mu \sim M$ otherwise the logs would be large and your perturbative correction would not be small. This is mainly the reason why $\mu$ is usually tought of as an energy scale in the problem, even though in principle it is arbitrary. If you follow this prescription for choosing $\mu$, you will find that it really is true that at high momentum transfer the QCD 2->2 scattering becomes weaker.

  
  


• 
  

  In problems with several interesting scales this leads to a problem, as you get several logarithms, say $log \left( \frac{\mu}{m} \right)$ and $log \left( \frac{\mu}{M} \right)$, with say $m \ll M$. In this case you can not choose a $\mu$ such that all logarithms are small. To solve problems of this kind with dim. reg. Effective Field Theory techniques are needed. That is you first construct an EFT valid for momenta smaller than $M$ and matching small momentum S matrix elements between the theories. For definiteness lets say $M$ is some heavy particle mass. In this case you would match the S matrix elements for the light particle between the full theory and the EFT witout heavy fields at some scale, say $\mu \sim M$, implementing the decoupling of the heavy particle BY HAND. The MS scheme does not satisfy the decoupling theorem, but you can put it in by hand. Similarly as with the former case, you match the theories at $\mu$ of order $M$ to avoid large logs in the matching.

  
  

In both cases, you put in the "interpretation" of $\mu$ by hand, to make your life easier, and make perturbative correction actually small. In this sense $\mu$ in applications is usually connected to some physical scale, even though in principle it could be arbitrary.