Past exam of the quantum field theory course of the University of Oxford / 2021 / 1 / d / Solution

The printed integration labels need correction. The parent's four-momentum is fixed, so the two Lorentz-invariant phase-space measures must integrate $p_2$ and $p_3$, rather than the printed $p_1$ and $p_2$. Write $P=p_1=(M,0)$ and $p=p_2$, $k=p_3$. The physical decay width isThe spatial Dirac delta function sets $\mathbf{k}=-\mathbf{p}$. Performing the angular integral in spherical coordinates, with $E=\sqrt{p^2+m^2}$, gives the integrated two-body decay phase spaceThe positive root is $p_{\ast }=\sqrt{M^2/4-m^2}$. Since $d(2E)/dp=2p/E$, the Dirac delta function integration yields ${\Phi }_2=p_{\ast }/(4\pi M)$. ThusThe two final particles have different species, so there is no identical final-state symmetry factor. The result tends to $g^{2}M/(16\pi )$ as $m\to 0$ and vanishes at the two-particle decay threshold.