\!\!\!\!\!=\int_P \int_Q \frac{1}{(p_0+i\mu)^2+E_p^2}\frac{1}{(q_0+i\mu)^2+E_q^2}\frac{1}{(p_0-q_0)^2+(\mathbf{p}-\mathbf{q})^2} \\
&=&\int_P \int_Q \int_K \frac{(2\pi)^4\delta^{(4)}(P-Q-K)}{\left((p_0+i\mu)^2+E_p^2\right)\left((q_0+i\mu)^2+E_k^2\right)\left(k_0^2+k^2\right)}\,,\quad E_k\equiv\sqrt{k^2+m^2}\,, \nonumber\end{aligned}$$ where the solid lines in the graph denote a massive fermion propagator and the dotted line a massless boson one. In order to reduce the integral to a more manageable form, we first perform the integrations over the 0-components of the different momenta, which is a rather straightforward task at such a low loop order.

The standard way of evaluating the $p_0$- and $q_0$-integrals proceeds by writing the $\delta$-function in eq. (\[eq:abc2\]) in the form [@Kapusta:2006pm] $$\label{eq:delta}
2\pi \delta(p_0-q_0-k_0)=\int_{-\infty}^\infty d\alpha\, e^{i\alpha(p_0-q_0-k_0)}\,,$$ which allows us to perform the $p_0$, $q_0$, and $k_0$ integrations independently using the Residue theorem. Choosing the integration contours to lie on the upper or lower half of the complex plane depending on the sign of the exponent in eq. (\[eq:delta\]), we obtain after quite some algebra $$\begin{aligned}
I_2(\mu)&=&\int_{p,q}\frac{1}{2|\mathbf{p}-\mathbf{q}|\,2E_p \,2E_q}\frac{2}{|\mathbf{p}-\mathbf{q}|+E_p+E_q}\nonumber\\
&&-\int_{p,q}\frac{\theta(\mu-E_p)}{2|\mathbf{p}-\mathbf{q}|\, 2E_p\, 2E_q}\frac{2 (|\mathbf{p}-\mathbf{q}|+E_q)}{(|\mathbf{p}-\mathbf{q}|+E_q)^2-E_p^2}\nonumber\\
&&-\int_{p,q}\frac{\theta(\mu-E_q)}{2|\mathbf{p}-\mathbf{q}|\, 2E_p\, 2E_q}\frac{2 (|\mathbf{p}-\mathbf{q}|+E_p)}{(|\mathbf{p}-\mathbf{q}|+E_p)^2-E_q^2}\nonumber\\
&&+\int_{p,q}\frac{\theta(\mu-E_p) \theta(\mu-E_q)}{2E_p\,2E_q}  \frac{1}{(\mathbf{p}-\mathbf{q})^2-(E_p-E_q)^2}\, . \label{I2int} \end{aligned}$$ While perfectly correct, this result is unfortunately rather unpractical, as the first three lines contain complicated UV divergent integrals of highly non-standard objects that we would need to determine in $3-2\epsilon$ dimensions. Only the last of the four terms is of a form that 