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- 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?

##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?