(a) Find the amplitude, M, for electron-muon scattering, in this theory.
(b) Calculate the spin-averaged quantity, ⟨∣M∣2⟩.
(c) Determine the differential cross section for electron-muon scattering in the CM frame. Assume the energy is high enough so that the electron and muon masses can be neglected: me,mμ→0. Express your answer in terms of the incident electron energy, E and the scattering angle, θ.
(d) From your result in (c), calculate the total cross section, assuming the ``photon'' is extremely heavy, mγc2≫E.
(e) Going back to (b), consider now the case of low-energy scattering from an extremely heavy ``muon'': ∣pe∣/c≪me≪mγ≪mμ. Find the differential cross section in the lab frame (muon at rest), assuming the muon does not recoil appreciably. Compare the Rutherford formula (Example 7.7), and calculate the total cross section.
[Note: This problem pertains to the massive scalar photon model described before Problem 7.42.]