two distinct variations. They differ in that the first kind of divergence appears in the very definition of the finite-$\mu$ integrals, while the latter is a byproduct of the cutting procedure and therefore more artificial.

The first type of singularity is related to the divergence of the fermionic propagator $1/((p_0+i\mu)^2+E_p^2)$ when $p_0=0$ and $E_p=\mu$, i.e. it appears along the original integration contour. It gets realized only in the limit where the $\theta(\mu-E_p)$ function in the integration measure gets saturated, but one might nevertheless worry that it makes the $p_0$ integrations ill-defined. The most natural resolution to this problem turns out to involve the use of an infinitesimal but nonzero temperature $T$ as a regulator. As we shall show in detail in appendix \[sec:matsu\], the effect of this prescription amounts to interpreting all $p_0$ integrations in the principal value sense, i.e. writing $$\begin{aligned}
\int_{-\infty}^\infty \frac{dp_0}{2\pi}\int_p &\to&\mathcal{P}\int_{-\infty}^\infty \frac{dp_0}{2\pi}\int_p\;=\; \lim_{\epsilon\to 0^+}\Bigg\{\int_{-\infty}^{-\epsilon} \frac{dp_0}{2\pi}+\int_{\epsilon}^{\infty} \frac{dp_0}{2\pi}\Bigg\}\int_p \nonumber \\
&=&\frac{1}{2}\,\Bigg\{\int_{-\infty+i0^+}^{\infty+i0^+} \frac{dp_0}{2\pi}+\int_{-\infty-i0^+}^{\infty-i0^+} \frac{dp_0}{2\pi}\Bigg\}\int_p\,, \label{principal}\end{aligned}$$ where we have assumed $p_0=0$ to be the only singular point on the real axis. While this does not affect the practical application of the cutting rules, it demonstrates that the integrand is well-defined on the entire integration contour, so that no imaginary parts 