terms containing an even number of noises survive (in the Feynman representation, all products of diagrams containing an even number of dotted legs). Using this fact, and exploiting the Kronecher deltas of $T^{\left(0\right)}\left(\mathbf{p}_{f},s_{f};\mathbf{p}_{i},s_{i};t\right)$ (see Eq. (\[eq:cvhjcg\])), we can write: $$\label{eq:sdgrst}
P_{kj}\left(\mathbf{p}_{i},s_{i};t\right)
=
e^{\frac{i}{\hbar}(E_{i}^{\left(k\right)}-E_{i}^{\left(j\right)})t}\left[1+ I_{jk}^{\left(1\right)}\left(\mathbf{p}_{i},s_{i};t\right) + I_{j}^{\left(2\right)}\left(\mathbf{p}_{i},s_{i};t\right)+I_{k}^{\left(2\right)*}\left(\mathbf{p}_{i},s_{i};t\right)\right],$$ where we have defined: $$\begin{aligned}
 \label{eq:rtrtsfds}
I_{jk}^{\left(1\right)}\left(\mathbf{p}_{i},s_{i};t\right)
& \equiv &
\sum_{s_{f}}\sum_{\mathbf{p}_{f}=-\infty}^{+\infty}e^{\frac{i}{\hbar}(E_{f}^{\left(k\right)}-E_{i}^{\left(k\right)}-E_{f}^{\left(j\right)}+E_{i}^{\left(j\right)})t}\mathbb{E}\left[T_{k}^{\left(1\right)*}\left(\mathbf{p}_{f},s_{f};\mathbf{p}_{i},s_{i};t\right)T_{j}^{\left(1\right)}\left(\mathbf{p}_{f},s_{f};\mathbf{p}_{i},s_{i};t\right)\right], \nonumber \\
I_{j}^{\left(2\right)}\left(\mathbf{p}_{i},s_{i};t\right)
& \equiv & 
\mathbb{E}\left[T_{j}^{\left(2\right)}\left(\mathbf{p}_{i},s_{i};\mathbf{p}_{i},s_{i};t\right)\right].\end{aligned}$$ We focus our attention on $I_{jk}^{\left(1\right)}\left(\mathbf{p}_{i},s_{i};t\right)$. Using Eq. (\[eq:T1again\]), keeping in mind the spinor relation $\left(\overline{u}_{f}u_{i}\right)^{*}=\overline{u}_{i}u_{f}$, and performing the average over the noise, which brings in a Dirac delta in time which cancels one of the two time-integrals, one obtains: $$\begin{aligned}
 \label{eq:sdfgdr}
\lefteqn{\mathbb{E}\left[T_{k}^{\left(1\right)*}\left(\mathbf{p}_{f},s_{f};\mathbf{p}_{i},s_{i};t\right)T_{j}^{\left(1\right)}\left(\mathbf{p}_{f},s_{f};\mathbf{p}_{i},s_{i};t\right)\right]=} \qquad\qquad\nonumber \\
& = & 
\sqrt{\gamma_{m_{j}}\gamma_{m_{k}}}\frac{m_{j}m_{k}c^{4}}{\sqrt{E_{i}^{\left(j\right)}E_{f}^{\left(j\right)}E_{i}^{\left(k\right)}E_{f}^{\left(k\right)}}}\overline{u}(p_{f}^{(j)},s_{f})u(p_{i}^{(j)},s_{i})\overline{u}(p_{i}^{(k)},s_{i})u(p_{f}^{(k)},s_{f}) \nonumber \\
& & 
\cdot  \int_{0}^{t}dt_{1}e^{\frac{i}{\hbar}(E_{f}^{\left(j\right)}-E_{i}^{\left(j\right)}-E_{f}^{\left(k\right)}+E_{i}^{\left(k\right)})t_{1}}S\left(\mathbf{p}_{i},\mathbf{p}_{f}\right)\end{aligned}$$ where $p_{f}^{(j)} \equiv (E_{f}^{(j)}/c, {\bf p}_{f})$ and similarly for $p_{i}^{(j)}$. Moreover: $$\label{eq:zxgf}
S\left(\mathbf{p}_{i},\mathbf{p}_{f}\right)
\equiv
\frac{1}{L^{6}}\int_{-\frac{L}{2}}^{+\frac{L}{2}}d\mathbf{x}_{1}\int_{-\frac{L}{2}}^{+\frac{L}{2}}d\mathbf{x}_{2}\frac{e^{-\left(\mathbf{x}_{1}-\mathbf{x}_{2}\right)^{2}/4r_{C}^{2}}}{\left(\sqrt{4\pi}r_{C}\right)^{3}}e^{-\frac{i}{\hbar}\left[\left(\mathbf{p}_{f}-\mathbf{p}_{i}\right)\cdot\left(\mathbf{x}_{1}-\mathbf{x}_{2}\right)\right]}$$ (now we have explicitly indicated the integration volume). In order to compute $S\left(\mathbf{p}_{i},\mathbf{p}_{f}\right)$, we change integration variables as follows: $$\mathbf{y}=\left(\mathbf{x}_{1}+\mathbf{x}_{2}\right)\qquad\textrm{and}\qquad\mathbf{x}=\left(\mathbf{x}_{1}-\mathbf{x}_{2}\right),$$ and use the relation: $$\int_{-\frac{L}{2}}^{+\frac{L}{2}}dx_{1}\int_{-\frac{L}{2}}^{+\frac{L}{2}}dx_{2}f\left(x_{1},x_{2}\right)=\frac{1}{2}\int_{0}^{+L}dx\int_{-\left(L-x\right)}^{+\left(L-x\right)}dy\left[f\left(x,y\right)+f\left(-x,y\right)\right].$$ Accordingly, we have: $$S\left(\mathbf{p}_{i},\mathbf{p}_{f}\right)=\frac{1}{L^{3}}\int_{0}^{L}d\mathbf{x}\frac{e^{-\mathbf{x}^{2}/4r_{C}^{2}}}{\left(\sqrt{4\pi}r_{C}\right)^{3}}2\cos\left[\frac{1}{\hbar}\left(\mathbf{p}_{f}-\mathbf{p}_{i}\right)\cdot\mathbf{x}\right]\frac{1}{2^{3}}\prod_{i=1}^{3}2\left(1-\frac{x_{i}}{L}\right)$$ Let us now take the limit $L \rightarrow \infty$, which amounts to making the replacement: $$\sum_{\mathbf{p}_{f}=-\infty}^{+\infty}\longrightarrow\int d\mathbf{p}_{f}\qquad\textrm{and}\qquad\frac{1}{L^{3}}\longrightarrow\frac{1}{\left(2\pi\hbar\right)^{3}}.$$ In this limit, the term $x_i/L$ gives a vanishingly small contribution. Therefore we can write: $$S\left(\mathbf{p}_{i},\mathbf{p}_{f}\right)
=
\frac{1}{\left(2\pi\hbar\right)^{3}} \int_{-\infty}^{+\infty}d\mathbf{x}\frac{e^{-\mathbf{x}^{2}/4r_{C}^{2}}}{\left(\sqrt{4\pi}r_{C}\right)^{3}}e^{\frac{i}{\hbar}(\mathbf{p}_{f}-\mathbf{p}_{i})\cdot\mathbf{x}} 
= 
