$CP^N$ model in Peskin & Schroeder problem 13.3

In Peskin & Schroeder exercise 13.3 question d, it is asked to perform an expansion of the term $$iS =-N.tr\left[\log\left(-D^2-\lambda\right)\right]+\frac{i}{g^2}\int d^2x \lambda$$ where $$D_{\mu}=\left(\partial_{\mu}+iA_{\mu}\right)$$, and $$\lambda$$,$$N$$ and $$g$$ numbers. The expansion should be made around $$A_{\mu}=0$$, and we should use this result to prove the expansion is proportional to the vacuum polarization of massive scalar fields. In momentum space, the log can be written as

$$\int \frac{d^d x}{(2\pi)^d} \log\left(k^2+A^2-\lambda\right)$$ and my naive attempt to expand the log was

$$\log\left(k^2+A^2-\lambda\right)=\log\left[\left(k^2-\lambda\right)\left(1+\frac{A^2}{\left(k^2-\lambda\right)}\right)\right]=\log\left(k^2-\lambda\right)+\log\left(1+\frac{A^2}{\left(k^2-\lambda\right)}\right) \\ \approx\log\left(k^2-\lambda\right)+\frac{A^2}{\left(k^2-\lambda\right)}$$ but it did not help me so far since the second term vanishes. Can someone point me to the right direction?

• Have you checked out zzxianyu.files.wordpress.com/2017/01/peskin_problems.pdf ? Commented Aug 26, 2019 at 7:02
• I did, but I can't reproduce the result in this pdf. Also, most of the question has been left out. I found the following pdf, but what is done after equation (71) seems to be an overkill in my opinion, and I don't get all the steps. Commented Aug 26, 2019 at 13:05

I think you have an error, since actually $$D^2 = (\partial_{\mu} + iA_{\mu})^2 = \partial^2 - A^2 + i (\partial_{\mu}A^{\mu} + A^{\mu} \partial_{\mu}).$$ Then you can write $$\text{Tr}\log (-D^2 - \lambda) = \text{Tr}\log(-\partial^2-\lambda) + \text{Tr}\log \Big [ 1+ \frac{1}{-\partial^2 - \lambda} (A^2 - i(\partial_{\mu}A^{\mu} + A^{\mu} \partial_{\mu})) \Big ]$$ and expand the logarithm. The quadratic terms in $$A$$ will give you the kinetic term.
Alternatively you can also just reintroduce the scalars. The two one-loop diagrams with two external $$A$$ legs, i.e. vacuum polarization for $$A$$, will give you the kinetic term.
But be careful: You need to expand in $$p/\sqrt{\lambda}$$ when evaluating the loop integral in momentum space and drop terms $$O((p/\sqrt{\lambda})^4)$$.