Researchers studied ways to correct a known weakness of crystal electromagnetic calorimeters: they measure electromagnetic energy very well but respond differently to hadronic energy, which hurts the measurement of particles like pions. Using detailed simulations of a prototype called CRILIN — a compact, longitudinally segmented Cherenkov crystal calorimeter made of PbF2 crystals read out by silicon photomultipliers (SiPMs) — the team shows that offline software corrections can recover much of the lost information and improve hadron energy reconstruction. A simple correction based on the shower size helps, and a machine‑learning method that looks at the full three‑dimensional shower shape does even better in these studies.

CRILIN is a semi-homogeneous calorimeter concept with five longitudinal layers. Each layer is a 7×7 matrix of small PbF2 crystals, each 1.3×1.3×4.0 cm3, for a total depth of about 20 cm. That compact design gives precise timing and fine spatial readout. The device was developed for a muon collider environment and is also being evaluated for future electron–positron Higgs factories where excellent jet energy resolution is required.

The authors simulated single charged pions with energies between 15 and 100 GeV using the Geant4 toolkit (QGSP_BERT physics list). They estimated the Cherenkov light yield from the Frank–Tamm formula in the 350–550 nm range, converted emitted photons to an energy scale using an electron calibration (about 24.3 photons per MeV), and applied a conservative photoelectron yield of 0.2 photoelectrons per MeV with Poisson fluctuations. To judge how much energy escapes the crystal block, they placed a virtual detector downstream and computed the true energy inside CRILIN as the beam energy minus that escaping energy.

Two kinds of software compensation were tested. First, simple event‑by‑event corrections used global shower shape variables such as the transverse root-mean-square width and the longitudinal center of gravity. Those already gave a substantial improvement over a plain sum of energies. Second, a ParticleNet graph neural network (GNN) that used the full 3D pattern of calorimeter hits produced a significantly better result. With realistic assumptions about a downstream hadronic calorimeter (HCAL), the GNN reconstruction reduced CRILIN’s contribution to the combined ECAL+HCAL resolution to about (1 GeV/E ⊕ 12%/√E ⊕ 2.5%), preserving excellent combined performance. The authors also checked that the result does not strongly depend on the assumed HCAL resolution over the range they considered.