Differential Cross-Section

[math]-i \mathfrak{M}_{e^-e^-}=-i \left ( \frac{e^2(\mathbf p_1^*+\mathbf p_{1}^{'*})_{\mu}(\mathbf p_2^*+\mathbf p_{2}^{'*})^{\mu}}{(\mathbf p_{2}^{'*}-\mathbf p_2^*)^2}- \frac{e^2(\mathbf p_1^*+\mathbf p_{2}^{'*})_{\mu}(\mathbf p_2^*+\mathbf p_{1}^{'*})^{\mu}}{(\mathbf p_{1}^{'*}-\mathbf p_2^*)^2} \right )[/math]

[math] \mathfrak{M}_{e^-e^-}= e^2\left ( \frac{\mathbf P_1^* \mathbf P_2^*+\mathbf P_{1}^{'*} \mathbf P_{2}^{'*}+\mathbf P_{1}^{'*} \mathbf P_2^*+\mathbf P_1^* \mathbf P_{2}^{'*}}{(\mathbf P_{2}^{'*}-\mathbf P_2^*)^2}- \frac{\mathbf P_1^* \mathbf P_2^*+\mathbf P_{2}^{'*} \mathbf P_{1}^{'*}+\mathbf P_{2}^{'*} \mathbf P_2^*+\mathbf P_1^* \mathbf P_{1}^*}{(\mathbf P_{1}^*-\mathbf P_2^*)^2} \right )[/math]

Using the fact that [math]\mathbf P_1^{'*} \mathbf P_2^{'*}=\mathbf P_1^* \mathbf P_2^* \quad \quad \mathbf P_1^{'*} \mathbf P_1^{*}=\mathbf P_2^{'*} \mathbf P_2^* \quad \quad \mathbf P_1^{*} \mathbf P_2^{'*}=\mathbf P_2^* \mathbf P_1^{'*}[/math]

[math] \mathfrak{M}_{e^-e^-}= e^2\left ( \frac{2\mathbf P_1^* \mathbf P_2^*+2\mathbf P_{1}^{'*} \mathbf P_2^*}{(\mathbf P_{2}^{'*}-\mathbf P_2^*)^2}- \frac{2\mathbf P_1^* \mathbf P_2^*+2\mathbf P_1^* \mathbf P_{1}^{'*}}{(\mathbf P_{1}^{'*}-\mathbf P_2^{'*})^2} \right )[/math]

[math] \mathfrak{M}_{e^-e^-}= e^2\left ( \frac{ (\mathbf P_1^{*2}-2 \mathbf P_1^* \mathbf P_2^{'*}+ \mathbf P_2^{'*2})-(\mathbf P_1^{*2}+2 \mathbf P_1^* \mathbf P_2^*+ \mathbf P_2^{*2})}{t}- \frac{(\mathbf P_1^{*2}-2 \mathbf P_1^* \mathbf P_1^{'*}+ \mathbf P_1^{'*2})-( \mathbf P_1^{*2}+2 \mathbf P_1^* \mathbf P_2^*+ \mathbf P_2^{*2})}{u} \right )[/math]

[math]\mathfrak{M}_{e^-e^-}=e^2 \left (\frac{u-s}{t}+\frac{t-s}{u} \right )[/math]