Lorentz-invariant phase space ($d{\Phi }_2$)

For two outgoing particles the Lorentz-invariant phase space is $d{\Phi }_2=(2\pi {)}^4{\delta }^{(4)}(p-k-k^{\prime })d^{3}k/((2\pi {)}^{3}2E_k)d^3{k}^{\prime }/((2\pi {)}^{3}2E_{k^{\prime }})$. With massless daughters in their center-of-mass frame, $d{\Phi }_2=d\Omega /(32{\pi }^2)$ and $\int d{\Phi }_2=1/(8\pi )$. This fixes normalization factors in decay widths and scattering rates.

A display of three-body final states in two independent pairwise squared invariant masses. For a parent of mass $M$ decaying into daughters of masses $m_1,m_2,m_3$, four-momentum conservation gives $s_{12}+s_{13}+s_{23}=M^2+m_1^2+m_2^2+m_3^2$. A narrow intermediate two-body resonance fixes one pairwise invariant mass and appears as a band.

At fixed $s=s_{13}$, use $E_2^{\ast }=(M^2-m_2^2-s)/(2\sqrt{s})$ and $E_3^{\ast }=(s+m_3^2-m_1^2)/(2\sqrt{s})$ in the 13-pair rest frame. The endpoint values of its relative cosine give the displayed Dalitz plot boundary. The momenta are fixed by the Källén function; $(m_1+m_3{)}^2\le s\le (M-m_2{)}^2$. Integrating $s_{23}^{+}-s_{23}^{-}$ gives the invariant-coordinate area, whose phase-space normalization still contains $M^{-2}$.

After integrating the overall orientation of a scalar-parent three-body decay, the Lorentz-invariant phase space is constant in two pairwise squared invariant masses. The two-body factorization $d{\Phi }_3=dsd{\Phi }_2(M;\sqrt{s},m_2)d{\Phi }_2(\sqrt{s};m_1,m_3)/(2\pi )$ and the relative-angle Jacobian give the displayed density. The event density includes $|\mathcal{M}{|}^2/(2M)$ and need not be flat: a resonance produces bands. Identical-particle counting supplies its usual global symmetry factor.

For three massive daughters with a smooth nonzero decay amplitude at the decay threshold, the integrated Lorentz-invariant phase space vanishes as $Q^2$, where $Q$ is the parent's mass minus the sum of daughter masses. The spatial Dirac delta function leaves six independent momentum coordinates. Rescaling each by $Q^{1/2}$ contributes $Q^3$, while the conservation of energy Dirac delta function contributes $Q^{-1}$. The factors $1/(2E)$ are smooth nonzero constants at the decay threshold. Thus the decay width also scales as $Q^2$ when the decay amplitude has a nonzero constant term. A vanishing decay amplitude, a massless daughter, or a threshold singularity can change this law. In three spatial dimensions the same argument gives $Q^{(3N-5)/2}$ for $N$ massive daughters.

For one daughter of mass $m$ and two massless daughters, set $r=m^2/M^2$, where $M$ is the parent mass. In natural units, integrating over all directions and all kinematically allowed final momenta givesThis follows by splitting the final state into a massless spectator and a two-particle cluster of invariant squared mass $s$, and using $d{\Phi }_3=d{\Phi }_2(M;\sqrt{s},0)d{\Phi }_2(\sqrt{s};m,0)ds/(2\pi )$. The expression counts phase space alone; a physical decay rate must weight it with the squared decay amplitude. Near threshold the dimensionless factor vanishes as $(1-r{)}^3/3$.

For two distinct massless final particles in their centre-of-momentum frame, integrating the energy-momentum delta function gives $d{\Phi }_2=d\Omega /(32{\pi }^2)$. A massless two-particle initial state has flux $2s$, hence $d\sigma /d\cos \theta =\overline{|\mathcal{M}{|}^2}/(32\pi s)$. Average only over the initial spin states physically present.

For two distinct massless final particles, $d{\Phi }_2/d\Omega =1/(32{\pi }^2)$ in the centre-of-mass frame. The invariant initial flux is $4p_1\cdot p_2=2s$. Their ratio gives the displayed differential cross-section. Equivalently the denominator is $128{\pi }^2{p}_1\cdot p_2$. Initial spins are averaged and final spins and colors summed when defining an inclusive unpolarized cross-section; averaging final colors would change the observable.