Logarithmic divergence of an integral

[spaghetti3451]

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.

 

[John Park]

How does this sound?

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