Logarithmic divergence of an integral I would like to prove that the following integral is logarithmically divergent.
$$\int d^{4}k \frac{k^{4}}{(k^{2}-a)((k-x)^{2}-a)((k-y)^{2}-a)((k-z)^{2}-a)}$$
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.
 A: This integral is logarithmically divergent for large $k$. For small $|k|$ your integral should behave regularly. That is given the values of $a,x,y,z$ it should be not difficult to show that
$$\left|\int_{|k|\leq 1} d^{4}k \frac{k^{4}}{(k^{2}-a)((k-x)^{2}-a)((k-y)^{2}-a)((k-z)^{2}-a)}\right| \leq M$$
with $M$ some constant (not diverging).
For large $k$ you can approximate the integrand as $1/k^4$ with correction terms that are $O(k^{-3}).$ So introducing a ``cutoff'' $k^*$ for the integral such that we  only integrate over the ball $|k| \leq k^*$, your integral is given by
$$\left|\int_{1\leq |k|\leq k^*} d^{4}k \frac{k^{4}}{(k^{2}-a)((k-x)^{2}-a)((k-y)^{2}-a)((k-z)^{2}-a)} \right| \leq  \int_{1\leq |k|\leq k^*} \frac{d^4 k}{k^4} +  C \underbrace{\left|\int_{1\leq |k|\leq k^*}\frac{d^4 k}{k^3}\right|}_{\leq N}$$
with $N$ another constant (independent of $k^*$).
So in total, we have that  (for $k^*\to \infty$)
$$ \int_{|k|\leq k^*} d^{4}k \frac{k^{4}}{(k^{2}-a)((k-x)^{2}-a)((k-y)^{2}-a)((k-z)^{2}-a)} = \int_{1\leq |k|\leq k^*} \frac{d^4 k}{k^4}  + O(1)
= S_3 \log (k^*) +O(1)$$
with $S_3= 2\pi^2$ the surface of the 3-sphere.
A: It's been awhile since I've calculated these beastly integrals in QFT but a lot of them can be handled using the introduction of Feynman parameters. The basic version for two factors in the denominator is
$$ \dfrac{1}{AB} = \int_0^1 \int_0^1 dx dy \delta(x+y-1) \dfrac{1}{[xA+yB]^2}$$
For $n$ factors this generalizes to
$$ \dfrac{1}{A_1 A_2 \cdots A_n} = \int_0^1 \int_0^1 \cdots \int_0^1 dx_1 dx_2 \cdots dx_n \delta(\sum_i^{n}x_i-1) \dfrac{(n-1)!}{[x_1A_1+x_2A_2 + \cdots x_nA_n]^2}$$
