Introduction

The invariant mass resolution of mesons that undergo $2\gamma$ decay is determined by the single shower energy and position resolutions. In the case where spatial contribution can be neglected, one can extract the single shower energy resolution by measuring the invariant mass resolution in narrow energy bins. The spatial contribution is expected to be small when the two showers are well separated. In the case where two showers are closer in space, shower centroid resolution can have a larger impact on the mass resolution. Assuming that the spatial resolution is not important for $\eta (2\gamma )$ decays, we extracted the shower energy resolution function without assuming any functional form. This model-independent solution is then compared with the standard parametrization of the lead glass energy resolution. In order to describe the $\pi ^0$ mass with the same parameters, one has to take the spatial resolution into account. We introduced one additional parameter that scales the spatial contribution to the mass resolution. All parameters have been determined by simultaneous analysis of the $\eta$ and $\pi ^0$ data.