Relativistic two-body decay

For a rest-frame decay $1\to 2+3$,The decay is allowed exactly when $m_1\ge m_2+m_3$.

In the centre-of-momentum frame with total energy $\mathcal{E}$, four-momentum conservation for a final massive particle of rest mass $m>0$ and one photon gives equal and opposite three-momenta. The energy–momentum relation and $|{\mathbf{p}}_{\gamma }|=E_{\gamma }/c$ give the displayed energies, andThe photon has positive energy when $\mathcal{E}>m{c}^2$. Kinematics fixes the speed and the opposition of the two final directions, while an angular distribution requires information about the interaction.

An isotropic two-photon decay of a massive parent with laboratory Lorentz factor $\gamma >1$ has the displayed normalized opening-angle probability density, with $\beta =\sqrt{1-{\gamma }^{-2}}$ and $2\arcsin (1/\gamma )<A<\pi$. To derive it in units $c=1$, the laboratory photon energies are $E_{1,2}=\gamma M(1\pm \beta z)/2$, where $z$ is uniform on $[-1,1]$. The diphoton invariant mass gives ${\sin }^2(A/2)=1/[{\gamma }^2(1-{\beta }^2{z}^2)]$. Apply the change of variables for probability densities to both signs of $z$. Its minimum-angle peak is an inverse-square-root singularity. At $\gamma =1$, the probability distribution is concentrated at $A=\pi$.

In the rest frame of the parent particle, four-momentum conservation requires $M=E_1+E_2\ge m_1+m_2$. For two equal masses the daughter momentum magnitude is $\frac{1}{2}\sqrt{M^2-4m^2}$; the two-body decay phase space closes at $M=2m$. For a finite constant decay amplitude, the decay width tends to zero at this threshold.

For $C\to B+t$ with massless $t$, use $p_C=p_B+p_t$ in the B rest frame. The mass-shell identity gives $E_t=(m_C^2-m_B^2)/(2m_B)$ and ${\mathbf{p}}_C={\mathbf{p}}_t$. The denominator differs from the familiar daughter-momentum formula in the parent's rest frame. Units here have $c=1$.

A stationary particle of rest mass $m$ absorbing a photon of energy $E$ acquires total invariant massThis follows from four-momentum conservation, since the incoming photon has momentum magnitude $E/c$. If the system splits into two equal particles, the largest kinematically permitted daughter rest mass is $M/2$, attained when both are at rest in the centre-of-momentum frame. Their common laboratory speed at threshold is $cE/(m{c}^2+E)$. Above this daughter-mass threshold there is insufficient centre-of-momentum energy; below it, relative kinetic energy is available.

The Källén function controls two-body relativistic kinematics. In the rest frame of total invariant mass $\sqrt{s}$, the magnitude of each final momentum is $\sqrt{\kappa (s,m_1^2,m_2^2)}/(2\sqrt{s})$. It also determines the invariant flux factor in a two-particle collision.