Non-relativistic M\oller scattering: e−e−→e−e−. If the electron and photon were spinless, we could write the Lagrangian as
L=−21ϕe(□+me2)ϕe−21A0□A0+emeA0ϕeϕe,(7.124)
where A0 is the scalar potential and the factor of me comes from the non-relativistic limit as in Section 5.2 (or by dimensional analysis!).
(a) Draw the three tree-level e−e−→e−e− diagrams following from this Lagrangian.
(b) Which one of the diagrams would be forbidden in real QED?
(c) Evaluate the other two diagrams, and express the answers in terms of s, t and u. Give the diagrams an extra relative minus sign, because electrons are fermions.
(d) Now let us put back the spin. In the non-relativistic limit, the electron spin is conserved. This should be true at each vertex, since the photon is too soft to carry off any spin angular momentum. Thus, a vertex can only allow for ∣↑⟩→∣↑;γ⟩ or ∣↓⟩→∣↓;γ⟩. This forbids, for example, ∣↑↓⟩→∣↑↑⟩ from occurring. For each of the 16 possible sets of spins for the four electrons (for example ∣↑↓⟩→∣↑↑⟩), which processes are forbidden, and which get contributions from the s-, t- or u-channels?
(e) It is difficult to measure electron spins. Thus, assume the beams are unpolarized, meaning that they have an equal fraction of spin-up and spin-down electrons, and that you do not measure the final electron spins, only the scattering angle θ. What is the total rate dcosθdσ you would measure? Express the answer in terms of ECM and θ. Sketch the angular distribution.