where and are corresponding photon energies, while is their angular separation. The expression for the variance of the squared invariant mass in terms of the variances of energy and shower position in the LGD is given by

where index counts photons. Taking all derivatives of Eq. (1) one can re-write Eq. (2) in the form

where spatial derivatives and variances are grouped into the term . The usual expression for the energy resolution function is given by [1]

where parameters and have to be determined. Spatial derivatives can be expressed in terms of measured photon momenta utilizing the expression

For example

The other spatial derivatives have a similar form and they can be obtained by the proper variable substitution. The position of the shower maximum is not directly measured. However, it is correlated with the energy and this has been taken into account within the shower-depth and nonlinearity corrections [2]. For the purpose of this analysis a common -plane is fixed at . The fluctuations in the shower position that determine , and are also affected by the energy. The energy dependence is proportional to with a proportionality constant that depends on the size of the LGD block [3]. This term alone would make spatial errors cylindrical and uniform across the face of the LGD. However, one can expect that the uncertainty in the shower position increases with polar angle , as , where is the radiation length of the lead glass. This makes shower centroid errors elliptical. The orientation of the ellipse depends on the shower azimuthal angle , and is connected with the correlation term . Combining all together,

with mmGeV , and mm[4]. In this study, we analyze events with two reconstructed showers. The sample, consisting of events, is selected by limiting invariant mass to . The corresponding invariant mass distribution is shown on the left side of Fig. 1. The sample is obtained by selecting pairs with shower separation . This distance is approximately twice the average shower size in the LGD plane. The corresponding mass distribution, shown on the right side of Fig. 1, contains events. Small peak around 0.2 represents the remnants that have large shower separation. The and peaks are fitted with a Gaussian over a polynomial background. In order to check the influence of spatial resolution on the overall mass resolution, the distribution of is plotted in Fig. 2 (solid line), calculated with the help of Eqs. (6) and (7). The distribution is obtained from the inclusive sample without restrictions on or . The dotted and dashed lines represent pairs with invariant mass within a window around the and peaks, respectively. Table 1 shows the Gaussian mean and

By measuring the mass or squared mass resolution in narrow energy bins one can extract the energy resolution function Eq. (4) and the corresponding parameters. The energy bins should be narrower than the detector energy resolution.