Field Theory I Final Assignment
Due December 13, 2011
Q1. (15 points) 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 |M |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 |M|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.
c) Compute the rate for decay into an electron with helicity s1 and a positron with helicity s2. You should
find that the decay rate is zero if s1 = −s2. Discuss this in light of conservation of angular momentum and
of parity.
̅ 𝛾 5 𝜓, and compute the spin-summed decay
d) Now consider changing the interaction to ℒ𝑖𝑛𝑡 = 𝑖𝑔𝜑𝜓
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.
Q2. (10 points) Problem 7.1 of Peskin and Schroeder.