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

Question

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?

Answer 1

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 $O((p/\sqrt{\lambda})^4)$.