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Problems QFT 5

From IFPA wiki
  1. Using Wick's theorem, write the diagrams that describe the scattering of 2 photons into 2 photons.
  2. Draw all the diagrams that correspond to the elastic scattering amplitude electron-electron at order [math]e^4[/math], if the electrons have incoming momenta [math] p_1[/math] and [math] p_2[/math], and outgoing momenta [math] q_1[/math] and [math] q_2[/math].
  3. Derive the vertices corresponding to the following interaction hamiltonians: [math]g\Phi(x)\bar\psi(x)\psi(x)[/math], [math]g(\partial_\mu A^\mu)^4[/math], [math]g \Phi^3[/math].
  4. Calculate the scattering amplitude [math]\Phi(p_1)\Phi(p_2)\rightarrow\Phi(q_1)\Phi(q_2)[/math] in a [math]\lambda\Phi^4[/math] theory.
  5. Show that [math] s+t+u=\sum m^2_{ext}[/math] where [math] m_{ext}[/math] are the masses of the in and out particles.
  6. Calculate directly the cross section of the annihilation process [math]e^+ e^-\rightarrow \mu^+\mu^-[/math] for [math]s\rightarrow\infty[/math] for non polarised beams.
  7. Caculate the cross section for [math]e^+ e^-\rightarrow \mu^+\mu^-[/math] though the exchange of a massive photon, which has a propagator equal to [math] 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]\Gamma[/math] is the decay width of the massive photon. What is the expression of thecross section at resonance ([math] s=M^2[/math])?

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