Past exam of the theoretical physics major option course of the University of Oxford / 2015 / 6 / a / Solution

Evaluate the two-body Lorentz-invariant phase space in the decaying particle's rest frame, $p=(m,0)$. The spatial delta function sets ${\mathbf{q}}_2=-{\mathbf{q}}_1$. Both massless particles have energy $q=|{\mathbf{q}}_1|$, leavingHere the factor $1/2$ in the second equality is the Jacobian of the energy delta function. Integrating the solid angle gives $\int d{\Pi }_2=1/(8\pi )$. If the decay is isotropic, the squared amplitude is angle independent, soThis uses distinguishable final particles or an amplitude/phase-space convention with any required identical-particle symmetry factor already included. Two genuinely identical final bosons require an additional $1/2!$ if the full angular domain is integrated without such a factor. The complex particle and antiparticle in the following parts are distinct, so no such division is required there.