2. Draw all the diagrams that correspond to the elastic scattering amplitude electron-electron at order $e^4$, if the electrons have incoming momenta $p_1$ and $p_2$, and outgoing momenta $q_1$ and $q_2$.
3. Derive the vertices corresponding to the following interaction hamiltonians: $g\Phi(x)\bar\psi(x)\psi(x)$, $g(\partial_\mu A^\mu)^4$, $g \Phi^3$.
4. Calculate the scattering amplitude $\Phi(p_1)\Phi(p_2)\rightarrow\Phi(q_1)\Phi(q_2)$ in a $\lambda\Phi^4$ theory.
5. Show that $s+t+u=\sum m^2_{ext}$ where $m_{ext}$ are the masses of the in and out particles.
6. Calculate directly the cross section of the annihilation process $e^+ e^-\rightarrow \mu^+\mu^-$ for $s\rightarrow\infty$ for non polarised beams.
7. Caculate the cross section for $e^+ e^-\rightarrow \mu^+\mu^-$ though the exchange of a massive photon, which has a propagator equal to $D_{\mu\nu}(k)=-i\frac{g_{\mu\nu}-\frac{k_\mu k_\nu}{M^2}}{k^2-M^2+iM\Gamma}$, where $\Gamma$ is the decay width of the massive photon. What is the expression of thecross section at resonance ($s=M^2$)?