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Use the Lorentz-invariant phase-space measure from the original PDF, including the $1/(2E_{p^{\prime }})$ and $1/(2E_{q^{\prime }})$ factors missing from the local TeX. In the center of mass frame put $E=\sqrt{s}/2$ and $k=\sqrt{s-4m^2}/2$, assuming $s>4m^2$. The spatial Dirac delta distribution sets ${\mathbf{q}}^{\prime }=-{\mathbf{p}}^{\prime }$, leavingSince $d(2E_{k^{\prime }})/d{k}^{\prime }=2k^{\prime }/E_{k^{\prime }}$, the radial integral givesThe invariant flux factor is $2\sqrt{\kappa (s,m^2,m^2)}=2\sqrt{s(s-4m^2)}$, using the Källén function. Dividing the phase space by this flux yields $d\sigma /d\Omega =|\mathcal{M}{|}^2/(64{\pi }^{2}s)$, with the final-state labels retained as in the printed formula.

Let $x=\cos \theta$. Of the Mandelstam variables, $t$ isThe azimuthal integral contributes $2\pi$, and the angular endpoints are thereforeChanging variables gives the requested expression from the formula supplied in the paper:There is a normalization qualification: the displayed starting formula integrates over labeled final momenta and contains no $1/2!$. For the physical cross-section of two indistinguishable outgoing quanta of this real scalar field, the identical-particle factor in a final-state phase-space integral divides the full integral by two. With that convention $X=1/(32\pi )$; equivalently, integrate only one representative of each exchanged pair. The boxed value $1/(16\pi )$ follows the given formula exactly.

The interaction $\lambda \Phi F_{\mu \nu }{F}^{\mu \nu }/4$ gives the displayed on-shell scattering amplitude, up to an overall phase, for a scalar field decaying to two photons. Substituting either photon four-momentum for its polarization vector makes the expression vanish, verifying its Ward identity. In the parent rest frame, summing the two physical transverse photon polarizations gives $\sum |\mathcal{A}{|}^2=|\lambda {|}^2{M}^4/2$. Including the identical-particle factor in a final-state phase-space integral gives $\Gamma =|\lambda {|}^2{M}^3/(64\pi )$.

For uncorrelated incoming beams with equally populated spin states, the measured cross section averages over initial polarizations and sums over unobserved final polarizations. The displayed average applies to two massive incoming particles with spins $s_1,s_2$. It is distinct from a prepared-polarization channel and from the final-state identical-particle factor in a final-state phase-space integral. A polarization channel with an amplitude that grows faster than the others can reach a perturbative bound before the averaged quantity does.