Superficial degree of divergence for Feynman diagrams

Question

The superficial degree of divergence for a diagram is defined as the power of $k$ in the numerator minus the power of $k$ in the denominator. It is written to be equal to $$4\times\text{number of loops} - \text{number of internal fermion lines} - 2 \times \text{number of internal boson lines}.$$

Why is it 4 and not 3? When you do a 4-dimensional integral in spherical coordinates, you get a factor of $k^3$ from your volume element. In particular, when you calculate the QED vertex, for instance, you get: $$\int_{\mathbb{R}^4} \frac{\mathrm{d}^4k}{(2\pi)^4} \frac{\gamma^\rho (\not\!{p_1}-\not\!{k}+m)\gamma^\mu(-\not\!{p_2}-\not\!{k}+m)\gamma_\rho}{k^2\left[\left(p_1-k\right)^2-m^2\right]\left[\left(p_2+k\right)^2-m^2\right]} \overset{k\to\infty}{\propto} \int_{\mathbb{R}^4} \frac{\mathrm{d}^4k\, k^2}{k^6} \overset{\text{spherical}}{\underset{\text{coordinates}}{\propto}} \int_0^\infty\frac{\mathrm{d}k}{k}.$$

Thus, I would say that the superficial degree of divergence of this diagram is $3\times1-2-2=-1$ and not $0$.

What am I missing?

Answer 1

No matter which coordinate system you choose, you must integrate over four coordinates, or $D$ when working in an arbitrary number of dimensions. This $d^Dk$ has units $[L]^{-D}$, where $[L]$ are units of length. The integral $$\int \frac{d^Dk}{k^n}$$ has units $[L]^{n-D}$ and is expected to diverge (as $k\rightarrow \infty$) whenever $D\geq n$. The limiting case $D=n$ might be a little confusing, but you can see that this diverges too. Take, for example, $D=n=3$: $$\int \frac{d^3k}{k^3}\propto \int \frac{dk}{k}=\log k\Biggr|_{0}^{\infty}=\text{divergent}$$ Here, it is said that the integral diverges logarithmically, for obvious reasons.