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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.

2. Relevant equations

See attached file.

3. The attempt at a solution

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# Robert Klauber’s second order photon propagator

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