I'm studying QED from Mandl and Shaw's book. My question regards the regularization of the fermion self-energy correction $$ ie_0^2\Sigma(p) = \int_{\mathbb{R}^4} \frac{\text d^4 k}{(2\pi)^4} i\frac{(-g_{\mu\nu})}{k^2 + i\varepsilon} ie_0\gamma^\mu i\frac{p\!\!\!/ - k\!\!\!/ + m}{(p - k)^2 - m^2 + i\varepsilon} ie_0\gamma^\nu, $$ cf. eq. (9.20). This integral is clearly divergent for $k \to \infty$ as power counting gives $4-2+1-2=1$ a linear divergence. Applying the same logic to the limit $k\to0$, I get that this integral should not diverge in the infrared, both for $p^\mu$ on and off mass-shell, but the book states the opposite.
Question: why is this integral divergent is the infrared as well?
