Suppose that M>2m, so that the scalar can decay to an electron-positron pair.
a) Compute the decay rate, summed over final spins.
b) Compute ∣T∣2 for decay into an electron with spin s1 and a positron with spin s2. Take the fermion three-momenta to be along the z axis, and let the x-axis be the spin-quantization axis. You should find that ∣T∣2=0 if s1=−s2, or if M=2m (so that the outgoing three-momentum of each fermion is zero). Discuss this in light of conservation of angular momentum and of parity. (Prerequisite: 40.)
c) Compute ∣T∣2 for decay into an electron with helicity s1 and a positron with helicity s2. (See section 38 for the definition of helicity.) You should find that the decay rate is zero if s1=−s2. Discuss this in light of conservation of angular momentum and of parity.
d) Now consider changing the interaction to L1=igφΨγ5Ψ, and compute the spin-summed decay rate. Explain (in light of conservation of angular momentum and of parity) why the decay rate is larger than it was without the iγ5 in the interaction.
e) Repeat parts (b) and (c) for the new form of the interaction, and explain any differences in the results.