Lemma: Two-body phase space and angular moments

For timelike future $p$, $s=p^2>4m^2$, write

In the rest frame of $p$,

and the angular distribution of $\mathbf{k}=K\mathbf{n}$ is uniform. Its normalized moments are

Proof. The spatial delta sets $\mathbf{q}=-\mathbf{k}$. The energy delta $\delta (\sqrt{s}-2{\omega }_{\mathbf{k}})$ has radial Jacobian ${\omega }_{\mathbf{k}}/(2K)$. Combining the measure factors gives $v/(8\pi )$, with normalized angular measure $\mathrm{d}\Omega /(4\pi )$.

Rotation invariance makes the second moment a multiple of ${\delta }_{ij}$, determined by $|\mathbf{n}{|}^2=1$. The fourth moment is a multiple of the three displayed Kronecker products; double contraction determines $1/15$.