Problems QFT 6

1. Calculate directly the cross section of the annihilation process [math]\displaystyle{ e^+ e^-\rightarrow \mu^+\mu^- }[/math] for [math]\displaystyle{ s\rightarrow\infty }[/math] for non polarised beams.
2. Caculate the cross section for [math]\displaystyle{ e^+ e^-\rightarrow \mu^+\mu^- }[/math] though the exchange of a massive photon, which has a propagator equal to [math]\displaystyle{ D_{\mu\nu}(k)=-i\frac{g_{\mu\nu}-\frac{k_\mu k_\nu}{M^2}}{k^2-M^2+iM\Gamma} }[/math], where [math]\displaystyle{ \Gamma }[/math] is the decay width of the massive photon. What is the expression of thecross section at resonance ([math]\displaystyle{ s=M^2 }[/math])?
3. Calculate directly the cross section for [math]\displaystyle{ \gamma\gamma\rightarrow e^+ e^- }[/math] in the non polarised case in the ultrarelativistic limit [math]\displaystyle{ m_e \ll \sqrt{s} }[/math].
4. Gauge invariance tells us that [math]\displaystyle{ A_\mu }[/math] and [math]\displaystyle{ A_\mu+\partial_\mu\Lambda }[/math] give equal results. In [math]\displaystyle{ k }[/math] space, this means that the polarisation vector of a photon of momentum [math]\displaystyle{ k_\mu }[/math], [math]\displaystyle{ \epsilon_\mu }[/math], and [math]\displaystyle{ \epsilon_\mu+c k_\mu }[/math] give identical results. Show that this is the case for the Compton scattering amplitude.
5. Imagine that the photon is a massless scalar [math]\displaystyle{ \phi }[/math] that interacts with electrons via [math]\displaystyle{ {\mathcal L}_{int}=g\bar\psi\psi\phi }[/math]. How would that affect the Compton cross section?