Past exam of the particle physics course of the University of Oxford / 2021 / 1 / a / Solution

In a relativistic two-body decay, four-momentum conservation in the parent's rest frame requires ${\mathbf{p}}_1^{\ast }=-{\mathbf{p}}_2^{\ast }$ andFor fixed daughter masses, this fixes a unique $p^{\ast }$, independent of direction. In the Lorentz-invariant phase space integral, the Dirac delta distribution enforcing conservation of energy therefore performs the radial momentum integration. Only the solid angle remains to be integrated; the resulting $p^{\ast }$ can be taken outside that integral, even if the matrix element depends strongly on direction.

For three or more final particles, four-momentum conservation generally leaves continuously variable daughter energies and momenta, so the decay rate includes integrations over them, or equivalently over independent invariant masses in a Dalitz plot. Integrating over a variable daughter invariant mass is also necessary when an unstable daughter is treated with a finite decay width rather than a fixed mass.