Count final states with an invariant measure

Use the module's $\hbar=c=1$ and $(+,-,-,-)$ conventions. The four-dimensional momentum volume is invariant under proper Lorentz transformations. Impose the mass shell with a delta function and retain only positive energy. Integrating over energy uses the derivative of $E^2-\mathbf p^2-m^2$, which gives the factor $2E$. This produces $d^3\mathbf p/(2E)$ rather than the ordinary three-dimensional volume. The positive-energy restriction is preserved by proper orthochronous transformations, the frame changes relevant here.

Relativistically normalized one-particle states introduce additional factors of two pi. Different references place the overall four-momentum delta-function factor inside or outside their definition of phase space. Both conventions can be consistent, but a rate formula and a phase-space formula from different conventions cannot be combined without translation. We include the full four-dimensional factor in our definition below. The positive-energy on-shell energy is always the energy of that same particle; substituting its mass for E in a moving frame destroys invariance.

For total timelike momentum P with invariant mass M, choose its rest frame. The spatial delta function sets the daughter momenta equal and opposite. The remaining energy delta function imposes $E_1(p)+E_2(p)=M$. Its derivative with respect to p is $pM/(E_1E_2)$ at the allowed root. Combining that Jacobian with the radial momentum volume leaves one factor of p and an angular measure. Invariance makes the final expression valid for computing the integral regardless of the original laboratory boost.

The allowed momentum is determined by the Källén function. Factoring it as $[M^2-(m_1+m_2)^2][M^2-(m_1-m_2)^2]$ makes the ordinary positive-energy threshold visible. At $M=m_1+m_2$ the two-body measure collapses as the momentum vanishes. For two massless daughters, p equals M/2 and the fully integrated measure is $1/(8\pi)$. That dimensionless result is a useful normalization check, but it is not itself a decay width or an angular probability.

For a decay, multiplying our measure by $|\mathcal M|^2/(2M)$ gives the rest-frame differential width, with appropriate spin treatment. The amplitude can weight different angles or invariant masses unequally; phase space alone does not determine that physics. For n identical final particles, integration over the complete labeled phase space counts each physical configuration n factorial times, so a corresponding symmetry factor is needed. Alternatively, an explicitly ordered region can count each configuration once. Using both remedies would undercount the rate.

A simulation also needs to distinguish its sampling coordinates from the desired measure. Drawing momentum components uniformly generally introduces a nonuniform Jacobian on the constrained surface. A generator can sample conveniently and attach weights, or design coordinates with a simpler measure, but the weighting must be documented. The same principle applies when comparing data with a phase-space shape: finite detector acceptance and momentum resolution distort the observed population. A discrepancy from an uncorrected uniform plot is not by itself a measurement of a new interaction amplitude.