CaloHadronic: a diffusion model for the generation of hadronic showers

TL;DR

CaloHadronic advances calorimeter shower simulation by unifying hadronic shower generation across ECal and HCal using a diffusion-transformer framework. It combines a continuous normalizing flow (PointCountFM) for layer-wise point counts with two EDM-diffusion blocks that incorporate transformer attention to model complex shower substructure, including track-like patterns. The approach yields high-fidelity distributions, correlations, and post-PandoraPFA reconstructions that closely match Geant4, while delivering substantial GPU-speedups and enabling efficient surrogates for HL-LHC and future collider studies. The work also introduces techniques such as monotonic EDM weighting and Fourier feature mappings, offering a flexible path toward faster, scalable, and more accurate hadronic-shower simulations in highly granular detectors.

Abstract

Simulating showers of particles in highly-granular calorimeters is a key frontier in the application of machine learning to particle physics. Achieving high accuracy and speed with generative machine learning models can enable them to augment traditional simulations and alleviate a major computing constraint. Recent developments have shown how diffusion based generative shower simulation approaches that do not rely on a fixed structure, but instead generate geometry-independent point clouds, are very efficient. We present a transformer-based extension to previous architectures which were developed for simulating electromagnetic showers in the highly granular electromagnetic calorimeter of the International Large Detector, ILD. The attention mechanism now allows us to generate complex hadronic showers with more pronounced substructure across both the electromagnetic and hadronic calorimeters. This is the first time that machine learning methods are used to holistically generate showers across the electromagnetic and hadronic calorimeter in highly granular imaging calorimeter systems.

Figures (22)

• Figure 1: Illustration of the structure of the ECal edm-diffusion (left) and HCal edm-diffusion (right). $time$ is the time of the diffusion model used to add noise to the data and $cond.$ are the incident energy and the number of points per layer. The electromagnetic calorimeter (ECal) consists of 30 layers, and the hadronic calorimeter (HCal) consists of 48 layers. input (ECal or HCal) is the input shower to be learned during training.
• Figure 2: Illustration of the generation process of CaloHadronic. The PointCountFM generates the number of points per layer. The ECal points per layer distributions along with E are fed into the ECal edm-diffusion block. Its output, E and the HCal points per layer distributions are passed into the HCal edm-diffusion block. By concatenating the output of the two blocks a new pion shower is generated.
• Figure 3: 3D view of a 50 GeV $\pi^{+}$ shower simulated with Geant4 (left) and a 50 GeV shower generated with CaloHadronic (right). The color represents the energy deposition in the cells. The red plane represents the division between ECal and HCal at layer 30.
• Figure 4: Histogram of the cell energies (left), longitudinal shower profile (center), and radial shower profile (right) for Geant4 and CaloHadronic. All distributions are calculated with $50 000$ events sampled with a uniform distribution of incident particle energies between 10 and 90 GeV. The bottom panel provides the ratio to Geant4. The error band corresponds to the statistical uncertainty in each bin.
• Figure 5: Position of the center of gravity of showers along the $x$ (left), $z$ (center), and $y$ (right) directions. All distributions are calculated for $50 000$ showers with a uniform distribution of incident particle energies between 10 and 90 GeV. The error band corresponds to the statistical uncertainty in each bin.