Logarithmic divergence of an integral

  • #1
1,344
33
I would like to prove that the following integral is logarithmically divergent.

$$\int d^{4}k \frac{k^{4}}{(k^{2}-a)((k-b)^{2}-x)((k-y)^{2}-a)((k-z)^{2}-a)}$$

This is 'obvious' because the power of ##k## in the numerator is ##4##, but the highest power of ##k## in the denominator is ##8##.

However, it is the highest power of ##k## in the denominator that is ##8##. There are other terms in ##k## in the denominator of the form ##k^7##, ##k^6##, etc.

I was wanting a more formal proof that the integral is logarithmically divergent.
 

Answers and Replies

  • #2
193
29
How does this sound?

Your denominator is a polynomial in k; so it can be written a product of terms like (k - zi) where the zi are zeros of the polynomial. If you break this product into partial fractions, you'll automatically get a log on integration.
 

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