  The MUonE experiment proposes a novel approach to determine the leading
hadronic contribution to the muon g-2, from a precise measurement of the
differential cross section of the $\mu-e$ elastic scattering, achievable by
using the CERN SPS muon beam onto atomic electrons of a light target. The
detector layout is modular, consisting of an array of identical tracking
stations, each one made of a light target and silicon strip planes, followed by
an electromagnetic calorimeter made of $PbWO_4$ crystals with APD readout,
placed after the last station, and a muon filter. The scattering particles are
tracked without any magnetic field, and the event kinematics can be defined in
a large phase space region from the expected correlation of the outgoing
particle angles. The ambiguity affecting a specific region, with electron and
muon outgoing with similar deflection angles, can be solved by identifying the
electron track as the one with extrapolation matching the calorimeter cluster
or the muon track by associating it to hits in the muon filter. The role of the
calorimeter will be important for background estimate and reduction, and to
assess systematic errors, providing some useful redundancy and allowing for
alternative selections. Beam tests are carried out at CERN with a prototype
calorimeter to determine its calibration with both high energy (20-150 GeV) and
low energy electrons (1-10 GeV). In late summer a pilot run is scheduled with
up to three tracking stations and the calorimeter integrated within a common
triggerless readout system. The main motivations for the MUonE calorimeter are
discussed, and the status and first performance results will be presented.
