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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 $A=\pi$.

From c, $\sin (A/2)\ge 1/\gamma$, with equality when $z=0$: the photons are emitted perpendicular to the Lorentz boost axis in the center-of-momentum frame. ThereforeAt the minimum, the map $z\mapsto A$ is stationary: writing $A=A_{\min }+\delta$, its leading behavior is $\delta =\beta z^2/\gamma +O(z^4)$. A nonzero interval of uniformly distributed $z$ is therefore compressed near $A_{\min }$, producing an integrable inverse-square-root singularity in the opening-angle probability density. No differentiation of that probability density is needed. For finite $\gamma >1$, the probability density vanishes at $A=\pi$. For $\gamma =1$ the center-of-momentum frame is the laboratory and every event has $A=\pi$: the probability distribution becomes a Dirac delta distribution, rather than the singular $\beta =0$ substitution in c. At large $\gamma$, $A_{\min }\sim 2/\gamma$ in radians.