Questions about scale dependence and renormalization schemes

[Isaac0427]

TL;DR

How do the scale dependences of different renormalization schemes relate?

Hi,

I have several related questions about scale dependence in different renormalization schemes.

1. Is there scale dependence in the on-shell (OS) scheme? Peskin & Schroeder chapter 10 goes through on-shell renormalization without involving an auxiliary scale, but other sources (see https://arxiv.org/pdf/1901.06573 chapter 1) do include the scale from dimensional regularization in the on-shell scheme.

2. A similar question about the momentum-subtraction (MO), also called the "off-shell" scheme. This scheme is like the OS scheme, but instead of placing a condition on the propagator and vertex function at $p^2=m^2$ where m is the physical mass, the condition is placed at $p^2=-M^2$. In Peskin & Schroeder, M would be the scale. But, if you also included the scale from dimensional regularization, now we would have two scales. How does this work?

3. I have seen it said that you can mix different schemes (such as treating the propagator in the MS scheme but the vertex function in the MO scheme). If you do this, you would get multiple scales simultaneously. How does this work?

4. How do the scales in different schemes relate to each other, and how do they relate to the momenta used in a scattering experiment where you try to determine the value of the coupling constant at a particular scale?

Thank you in advance!

 

[Mordred]

OK I had hoped someone with more recent practice as well as having a copy of Peskin & Schroeder QFT book.

Forewarning its been years since I last went through this but I can offer some hints and suggestions to your questions above.
First and foremost the majority of your questions above is commonly answered by studying the the Pauli_Villars dimensional regularization scheme.

https://fma.if.usp.br/~burdman/QFT1/lecture_22.pdf

see 22.72 with regards to p^2=m^2

Now hopefully you have also studied the Langrangian counter term as the Feymann rules requires a propagator, a vertex, a propagator counter term and a counter term vertex. Though you also need to have the Bare Langrangian

In regards to MS shouldn't that be minimal subtraction and not momentum subtraction ?

https://web2.ph.utexas.edu/~vadim/Classes/2022f/ms.pdf

anyways hope that helps like I stated its been several years since I last looked at renormalization schemes

 

[Mordred]

Forgot to add the renormalization scheme used is largely a matter of convenience and choice. Though you will want to use one scheme per graph where you will typically have more than one graph involved.