Problems QFT 5 (Redirected from Exercices 5)

1. Massless fermions can be described by Weyl spinors. Using the lagrangian we derived for left-handed spinors, derive the Feynman rules for
   1. propagators;
   2. external lines;
   3. vertices.
2. If one writes a Dirac spinor as [math]\displaystyle{ \psi_D=\psi_R+\psi_L }[/math], show that the above rules for Weyl spinors lead to the Feynman rules for Dirac spinors.
3. Using Wick's theorem, write the diagrams that describe the scattering of 2 photons into 2 photons.
4. Draw all the diagrams that correspond to the elastic scattering amplitude electron-electron at order [math]\displaystyle{ e^4 }[/math], if the electrons have incoming momenta [math]\displaystyle{ p_1 }[/math] and [math]\displaystyle{ p_2 }[/math], and outgoing momenta [math]\displaystyle{ q_1 }[/math] and [math]\displaystyle{ q_2 }[/math].
5. Derive the vertices corresponding to the following interaction hamiltonians: [math]\displaystyle{ g\Phi(x)\bar\psi(x)\psi(x) }[/math], [math]\displaystyle{ g(\partial_\mu A^\mu)^4 }[/math], [math]\displaystyle{ g \Phi^3 }[/math].
6. Calculate the scattering amplitude [math]\displaystyle{ \Phi(p_1)\Phi(p_2)\rightarrow\Phi(q_1)\Phi(q_2) }[/math] in a [math]\displaystyle{ \lambda\Phi^4 }[/math] theory.
7. Show that [math]\displaystyle{ s+t+u=\sum m^2_{ext} }[/math] where [math]\displaystyle{ m_{ext} }[/math] are the masses of the in and out particles.
8. Show that [math]\displaystyle{ p^\mu\sigma_\mu=\left(p^\mu\sigma_\mu+m\over \sqrt{2(E+m)}\right)^2 }[/math]