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

[Manu_]

Homework Statement

Hello,

In Peskin & Schroeder exercise 13.3 question d, it is asked to perform an expansion of the term
$$iS=−N.tr[log(−D2−λ)]+ig2∫d2xλ$$

where $Dμ=(∂μ+iAμ)$, and $λ$,$N$ and $g$ numbers. The expansion should be made around $A_μ=0$, 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

$$∫ddx(2π)dlog(k2+A2−λ)$$

Relevant Equations

$$iS=−N.tr[log(−D2−λ)]+ig2∫d2xλ$$


$$∫ddx(2π)dlog(k2+A2−λ)$$

My naive attempt to expand the log was


$log(k2+A2−λ)=log[(k2−λ)(1+A2(k2−λ))]=log(k2−λ)+log(1+A2(k2−λ))≈log(k2−λ)+A2(k2−λ)$


but it did not help me so far since the second term vanishes. Can someone point me to the right direction?

 

[Nate810]

A better way to expand the log is to use the Taylor series expansion. The Taylor series of log(1+x) is$log(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \frac{x^5}{5} - \dots$Therefore, $log(k^2 - \lambda) + A^2(k^2 - \lambda) \approx log(k^2 - \lambda) + A^2(k^2 - \lambda) - \frac{A^4(k^2 - \lambda)^2}{2} + \frac{A^6(k^2 - \lambda)^3}{3} - \dots$