Relativistic decays and phase space

Place a parent with mass $M$ at rest. For a two-body decay, momentum conservation makes the daughter three-momenta equal in magnitude and opposite in direction. Combining energy conservation with each on-shell condition fixes both daughter energies. There remains an orientation to integrate over, but not an independently adjustable daughter energy. If $M<m_1+m_2$, no real on-shell final momenta exist; a nonzero vertex cannot overcome this threshold.

The square root in the momentum formula is often written with the Källén polynomial. Its factorized form is useful because one factor vanishes at the physical threshold. For equal daughters, the momentum becomes $\sqrt{M^2/4-m^2}$. As the masses approach $M/2$ from below, the available momentum shrinks continuously. Additional angular-momentum selection rules can suppress the rate by further powers of this momentum, so threshold suppression is not always just one square root.

Choose single-particle normalization $\langle p'|p\rangle=2E(2\pi)^3\delta^3(\mathbf p'-\mathbf p)$. In the convention below the four-dimensional delta function carries its own factor $(2\pi)^4$. Some references place that factor outside their definition of phase space, so copying a formula without its definition can create an entire power of $2\pi$ error. Average over the parent spin when appropriate and sum over unobserved final spins.

For distinguishable daughters and an angle-independent spin-summed or spin-averaged amplitude, integration gives $\Gamma=p_*\overline{|\mathcal M|^2}/(8\pi M^2)$. An identical-particle final state needs its symmetry factor, and an angularly dependent amplitude must actually be integrated. A width has dimensions of energy; the amplitude's mass dimension depends on the number of external fields. This dimensional check is especially helpful when comparing decay and scattering formulas with different flux factors.

The total width sums exclusive partial widths, and a branching fraction divides one partial width by that total. In the exponential-decay approximation the mean proper lifetime is $1/\Gamma$. A boosted parent travels a mean laboratory distance $\beta\gamma/\Gamma$ in natural units. Restore a factor of $\hbar c$ when converting an inverse energy to meters. Proper lifetime and laboratory lifetime therefore answer different measurement questions.

A measured branching fraction also depends on selecting and correcting event samples. Daughter acceptance can change with the parent boost or decay angle; observing twice as many candidates does not automatically mean twice the underlying partial width. Near broad resonances or overlapping channels, a naive production-times-branching-factor description can fail. State the narrow-width and exponential assumptions before interpreting an apparent reconstructed peak as an independently produced unstable particle.