Hierarchy of angular effects in a universal-mesh representation of a sampling calorimeter

Abstract

Fast simulation of electromagnetic calorimeter showers via deep generative models requires learning a distribution conditioned on the incident energy and angles (θi,ϕi) of the primary particle. We show that for the LEMURS universal-mesh representation of the FCCeeCLD silicon–tungsten sampling calorimeter (a co-rotating cylindrical grid aligned with the photon direction), this three-parameter conditioning problem reduces to one parameter plus a deterministic post-processing step. Analysing the complete FCCeeCLD subset of 939 010 Geant4 events, we identify a hierarchy of four residual angular effects, set against the dominant energy dependence of the normalised shower shape, whose Kullback–Leibler divergence DKL between extreme energy bands is of order 10−1. Against this scale, the normalised longitudinal profile depends weakly on the polar angle, at the DKL≲7×10−3 level. This residual factorises multiplicatively into a universal sampling-layer pattern and an empirically characterised angular amplitude with twelve-fold azimuthal modulation. Three further effects admit closed-form analytic descriptions: a sampling-fraction correction of a few percent, an azimuthal dipole proportional to |cos⁡θi|, and a dodecagonal imprint on the dipole amplitude. We validate the factorised operator directly, without training any model: applying it to near-normal-incidence Geant4 showers reproduces independent oblique-incidence targets, and each correction acts exclusively on the observable it targets: an orthogonality exact by construction and confirmed numerically. The dominant shower physics therefore depends solely on the incident energy, with all angular dependence captured by analytic closed-form corrections and a compact empirical tabulation applied as a post-processing step. Generative models for this detector can accordingly be designed with a one-dimensional conditioning space, eliminating the need to learn angular interpolation over the full acceptance.