Diphoton invariant mass ($M^2{c}^4=2E_1{E}_2(1-\cos \theta )$)

Adding two null photon four-momenta gives the displayed invariant mass, with opening angle $\theta$ measured in the same frame as the energies. A nonzero mass of the pair does not give either photon a rest mass. At fixed energies it vanishes for parallel photons and is greatest for opposite directions. A reconstructed mass peak must be assessed with background, resolution and identification controls before assigning a parent particle.

Logarithmic differentiation of the diphoton mass gives $dM/M=(d{E}_1/E_1+d{E}_2/E_2)/2$ at fixed angle. If $s_i$ are fractional energy standard deviations and $\rho$ their correlation coefficient, covariance propagation gives the displayed formula. A common multiplicative calibration error transfers directly to the reconstructed mass; independent errors combine differently. Angular errors, if present, add the corresponding derivative and covariance terms.