MS renormalization scheme /RG & Srednicki ch 27,28

[PJK]

Hi all,

I have two questions regarding chapter 27 and 28 in Srednicki's book. On page 163 he states:

"furthermore, the residue of the pole is no longer one. Let us call the residue R. The LSZ formula must now be corrected by multiplying its rightnad side by a factor of R^(-1/2) for each external particle...This is because it is the field $$R^{(-1/2)} \phi (x)$$ that now has unit amplitude to create a one-particle state."

But this would mean that
$$|<k|\phi |0>|^2 = R$$

I can not see why this is? I would expect that the result is R^2 because there is a factor of k^2 + m^2 in the LRZ formula...

My second question: On p.170 Srednicki states that bare parameters must be independent of $$\mu$$. Because if we "were smart enough, we would be able to compute the exact scattering amplitudes in terms of them". Why is this? After all bare parameters have no physical meaning at all (at least as far as I understand this), so why can't they be dependent on $$\mu$$? How would you calculate an exact amplitude just with arbitrary, unphysical bare parameters?

Hope anyone can help me and thanks for reading!

 

[eljose79]

Hi there,

Thank you for your questions about Srednicki's book. I am happy to provide some clarification on these topics.

Firstly, regarding the residue of the pole in the LSZ formula, the factor of R^(-1/2) comes from the fact that the field operator now has a different normalization due to the renormalization procedure. This means that the amplitude to create a one-particle state is now given by R^(-1/2) instead of just R. This is because the field itself has been rescaled by a factor of R^(-1/2). Therefore, the amplitude squared is just R, not R^2.

As for your second question, the reason why bare parameters must be independent of \mu is because the renormalization procedure is designed to remove the dependence on the arbitrary scale \mu. This allows us to calculate physical observables without any arbitrary scale dependence. If the bare parameters were dependent on \mu, then physical observables would also depend on \mu, which would make it difficult to compare results from different experiments. Additionally, as you mentioned, bare parameters have no physical meaning, so it makes sense for them to be independent of \mu.

I hope this helps clarify these concepts for you. Let me know if you have any further questions. Keep up the good work in your studies!

 

[Entropia]

The MS renormalization scheme and the renormalization group (RG) are important tools in theoretical physics, particularly in quantum field theory. Srednicki's book, specifically chapter 27 and 28, discusses these concepts in detail.

In chapter 27, Srednicki discusses the LSZ formula, which is used to calculate scattering amplitudes in quantum field theory. He points out that in the MS renormalization scheme, the residue of the pole is no longer equal to one, but is instead given by the parameter R. This means that the field used to create one-particle states must be corrected by a factor of R^(-1/2). This is because the amplitude for creating a one-particle state is no longer equal to one, but is instead given by R. This correction is necessary in order to obtain the correct scattering amplitudes.

In response to the first question, you are correct in saying that the result should be R^2, not R. The factor of k^2 + m^2 in the LSZ formula should be taken into account, resulting in an overall factor of R^2. This may be a typo in the book or a small mistake.

In chapter 28, Srednicki discusses the concept of renormalization group (RG) in more detail. The RG is a mathematical technique used to study how physical quantities change as we vary the energy scale at which we are studying the system. Srednicki points out that in the MS renormalization scheme, the bare parameters of the theory must be independent of the renormalization scale \mu. This is because if the bare parameters were dependent on \mu, then the RG equations would be more complicated and it would be more difficult to calculate the exact scattering amplitudes. By keeping the bare parameters independent of \mu, we can simplify the calculations and obtain exact results.

In response to the second question, it is true that bare parameters have no physical meaning. They are simply mathematical tools used in the renormalization process. However, they must be chosen in a specific way in order for the theory to be renormalizable and for the RG equations to be solvable. If the bare parameters were dependent on \mu, it would make the calculations more complicated and potentially lead to non-physical results.

In summary, the MS renormalization scheme and the RG are important tools in theoretical physics. They allow us to study how physical