Robert Klauber’s second order photon propagator

[Alhaurin]

Homework Statement

I have a question regarding Klauber's Student Friendly Quantum Mechanics, about renormalization in chapters 13 and 15. I have included all relevant equations in the attached document.

In equation 15-105 he obtains an expression for the PI_uv(k^2) term used when calculating the second order photon propagator correction, which has a divergent part, A(k,Lambda), and a convergent part, PI_c(k^2). However the "convergent" part actually diverges for the particular case k^2 = 0 (unless I am wrong in my calculations).

The problem with that is that in equation 13.52 it had been stated that the second order photon propagator D_uv(k) can be expressed as the first order propagator multiplied by 1 - eo^2 *A'(k, Lambda) - e0^2*PI_c(k^2). Then two pages later, in 13.62 the external photon polarization vector is conveniently renormalized by multiplying it by sqrt(1 - A'(A,k)), without the PI_c(k^2) term. He argues that it makes sense to remove the PI_c(k^2) term since any real photon (incoming or outgoing) will have k^2 = 0 and PI(k^2) can be expressed as a power expansion on k^2.

It is therefore implied that PI(k^2) evaluated at k^2 is 0, although the exact explanation is provided in the solutions book (problem 13.5) which of course I do not have. The thing is, PI_c(k^2) is clearly divergent at k^2 = 0 so I do not think that the divergence disappears by doing a power expansion. Maybe the answer is that we can simply make the assumption that the actual nth order PI(k^2) would have the divergence at k^2 = 0 removed, leading to a 0 value. Anyway, I would like somebody to help me with this.

Thanks in advance.

Homework Equations

See attached file.

The Attempt at a Solution

 

[eljose79]

Hello,

Thank you for your question. It is a common misconception that renormalization is a process of simply removing divergences in a theory. In reality, renormalization is a technique used to handle these divergences in a systematic way, and it involves making certain assumptions and approximations.

In the specific case you are referring to, the PI_c(k^2) term may indeed have a divergence at k^2 = 0. However, this divergence is removed through the process of renormalization, which involves rescaling certain parameters in the theory. This rescaling effectively removes the divergence at k^2 = 0, and the resulting value of PI_c(k^2) is finite.

In the context of Klauber's Student Friendly Quantum Mechanics, this is explained in more detail in problem 13.5 of the solutions book. Unfortunately, as you mentioned, you do not have access to this book. I would recommend consulting other sources on renormalization to get a better understanding of the process and how it applies to this specific case.

I hope this helps to clarify things for you. Please let me know if you have any further questions.