may be directly evaluated as a numerical integral.

Some insights into how the above integrals might become tractable can be gained by observing that the first term of eq. (\[I2int\]), containing no step functions, may be recast in the form of the original diagram evaluated at $\mu=0$, $$\begin{aligned}
&&\int_{p,q}\frac{1}{2|\mathbf{p}-\mathbf{q}|\,2E_p \,2E_q}\frac{2}{|\mathbf{p}-\mathbf{q}|+E_p+E_q} \nonumber \\
&=& \int_P \int_Q \frac{1}{\left(p_0^2+E_p^2\right)\left(q_0^2+E_k^2\right)\left((p_0-q_0)^2+(\mathbf{p}-\mathbf{q})^2\right)} = I_2(0).
\label{eq:mot1}\end{aligned}$$ Apart from being of limited physical interest, this term can be easily evaluated using the standard tools of perturbative zero-temperature field theory.

Perhaps even more interestingly, we note that the second term of eq. (\[I2int\]) is expressible in terms of a two-point function, $$\begin{aligned}
 \int_Q \frac{1}{q_0^2+E_q^2}\frac{1}{(p_0-q_0)^2+(\mathbf{p}-\mathbf{q})^2}\Big{|}_{p_0\to iE_p}=\int_q\frac{1}{2|\mathbf{p}-\mathbf{q}|\,2E_q}\frac{2 (|\mathbf{p}-\mathbf{q}|+E_q)}{(|\mathbf{p}-\mathbf{q}|+E_q)^2-E_p^2},\nonumber \\ \label{eq:I21}\end{aligned}$$ where the substitution $p_0\to iE_p$ is made only after performing the integral on the left-hand side (prior to this substitution, $p_0$ is considered real). Again, we note that the integral on the left-hand side is of a form often encountered in standard $T=0$ quantum field theory calculations. A similar result clearly exists for the third term of eq. (\[I2int\]) as well, via the substitution $\mathbf{p}\leftrightarrow\mathbf{q}$.

Example calculation: cutting rules
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The observations made in the previous subsection clearly suggest that the two-loop integral of eq. (\[eq:abc2\]) may be written in a 