Standard Model and Beyond · Problem 7.6
The Standard Electroweak Theory
Physics — Particle Physics · polarized-muon-decay-spin-projection
Problem
Use the spin projection in (2.178) on page 37 to verify the dependence of the expression (7.53) for polarized decay. Hint: show how the calculation leading to (7.49) is modified by the spin projection.Intuitive explanation
The key insight is that the muon's spin state constrains which electron momenta are kinematically allowed in the decay. By inserting the spin projection operator into the decay amplitude calculation, you weight different electron directions by their overlap with the muon's spin orientation—this is why the final rate depends on , the angle between the electron and muon spin directions. Rather than averaging over all possible electron directions, you're now selecting preferentially those aligned (or anti-aligned) with the muon's polarization, which modifies the angular distribution that emerged in the unpolarized case.
Intuitive explanation
The key insight is that the muon's spin state constrains which electron momenta are kinematically allowed in the decay. By inserting the spin projection operator into the decay amplitude calculation, you weight different electron directions by their overlap with the muon's spin orientation—this is why the final rate depends on , the angle between the electron and muon spin directions. Rather than averaging over all possible electron directions, you're now selecting preferentially those aligned (or anti-aligned) with the muon's polarization, which modifies the angular distribution that emerged in the unpolarized case.