can be generated in bubble graphs due to the divergence.

Another frequently occurring problem is the emergence of spurious poles in some of the cut graphs that would automatically cancel, should all of the $p_0$ integrations in the diagram be computed at the same time and the results added together. A simple example of this is seen in the integral $$\begin{aligned}
&&\int_{-\infty}^\infty\frac{dp_0}{2\pi}\int_{-\infty}^\infty\frac{dq_0}{2\pi}\frac{1}{p_0^2+E_1^2}\frac{1}{q_0^2+E_2^2}\frac{1}{(p_0-q_0)^2+E_3^2}\nonumber \\
&=& \frac{1}{2E_1}\frac{1}{2E_2}\frac{1}{(iE_1-iE_2)^2+E_3^2}+\frac{1}{2E_1}\frac{1}{2E_3}\frac{1}{(iE_1+iE_3)^2+E_2^2}+\frac{1}{2E_2}\frac{1}{2E_3}\frac{1}{(iE_2+iE_3)^2+E_1^2}\nonumber \\
&=&\frac{1}{4E_1 E_2 E_3(E_1+E_2+E_3)}\,,\end{aligned}$$ where the intermediate stage corresponds to the outcome of the cutting rules. Even though the initial integral as well as its final form are both perfectly well-defined for all real-valued $E_i$, we see that the intermediate result contains a sum of three terms that each diverge when the three energies satisfy the linear relation $E_1-E_2=\pm E_3$. This is clearly a deeply unphysical problem.

The simplest manifestation of the second type of divergence is seen in the two-loop sunset graph, introduced already in sec. \[sec:cut\], where we now set the mass of one of the two fermion lines to zero. Considering the two-cut contribution corresponding to eq. (\[sunset2\]), we are left to evaluate the integral $$\begin{aligned}
\!\!\!\!I_2^\text{2-cut}(\mu)= \int_p \frac{\theta(\mu-E_p)}{2E_p}\int_q \frac{\theta(\mu-q)}{2q} \Bigg[\frac{1}{(p_0-q_0)^2+(\mathbf{p}-\mathbf{q})^2}\Bigg]_{p_0\to iE_p,\,q_0\to iq}\,, \label{pvex}\end{aligned}$$ where the integrand $$\begin{aligned}
\Bigg[\frac{1}{(p_0-q_0)^2+(\mathbf{p}-\mathbf{q})^2}\Bigg]_{p_0\to iE_p,\,q_0\to iq}&=&\frac{1}{2E_p\, q-2\,\mathbf{p}\cdot\mathbf{q}-m^2}\end{aligned}$$ 