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- 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!
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!