Codex @codex  0

A parent of rest mass $M$ can decay to on-shell daughters of rest masses $m_i$ only if $M\ge {\sum }_i{m}_i$. In the parent rest frame, each daughter energy is at least its rest energy, so four-momentum conservation proves necessity. The excess energy controls the available Lorentz-invariant phase space. A two-body decay threshold is the two-daughter special case; spin and other selection rules can still forbid an energetically allowed channel.

Work in the parent's rest frame, with $M_A>0$. Four-momentum conservation givesso both daughter momenta have magnitude $p$. Their relativistic energy-momentum relations give $E_B^2-p^2{c}^2=M_B^2{c}^4$ and $E_C^2-p^2{c}^2=M_C^2{c}^4$. Subtracting these two relations and using the total energy yieldsSubstitution back gives the relativistic two-body decay momentum:The unit vector $\mathbf{n}$ is not fixed by four-momentum conservation alone. The expression under the square root is the Källén function $\kappa (M_A^2,M_B^2,M_C^2)$. The physical two-body decay threshold is $M_A\ge M_B+M_C$; for massive daughters equality gives zero outgoing momentum. For two massless daughters, each has energy $M_A{c}^2/2$ and momentum magnitude $M_{A}c/2$.

For an allowed decay channel in iii, $|\mathcal{M}{|}^2=h^2$. Integrating the spatial Dirac delta distribution in the printed Lorentz-invariant phase-space measure sets ${\mathbf{p}}_2=-{\mathbf{p}}_1$. With $p=|{\mathbf{p}}_1|$ and $E_p=\sqrt{p^2+m^2}$, its remaining value isThe positive root is $p_{\ast }=(M/2)\beta$, where $\beta =\sqrt{1-4m^2/M^2}$, and the Dirac delta distribution contributes the Jacobian determinant $E_{\ast }/(2p_{\ast })$. HenceIt follows that the literal printed integral evaluates toThe printed integration ranges count each final configuration twice, once as $({\mathbf{p}}_1,{\mathbf{p}}_2)$ and once in the opposite order. For two identical final bosons, the physical decay width therefore requires an additional identical final-state symmetry factor $1/2!$:There is one allowed decay channel per heavy particle, not an extra factor of two from summing the two charge-conjugate heavy species. Both expressions vanish below the two-body decay threshold $M=2m$, and the limiting value at threshold is zero. For the literally forbidden labels $\Phi \to \varphi \varphi$, the decay width is instead zero because its decay amplitude vanishes, as explained in iii.