Probabilistic modeling of Cherenkov emission from particle showers

TL;DR

The paper tackles accurate yet efficient modeling of Cherenkov light from high-energy particle showers by treating shower-to-shower fluctuations probabilistically. It develops a two-part model: a gamma-like longitudinal profile with event-dependent shape $a,b$ and an independently modeled amplitude $\hat{\ell}_{tot}$, whose distributions are learned from extensive FLUKA simulations in ice. Shape fluctuations are captured by a joint density $f(a',b';E)$ on transformed variables $a'$, $b'$ modeled with penalized B-splines, while amplitudes use a skew normal or normal-inverse Gaussian distribution with energy-dependent parameters fitted across primaries. The resulting framework offers faster, more realistic simulations of Cherenkov signals for current and next-generation neutrino telescopes, enabling improved discrimination of signal and background. A validated implementation demonstrates substantial improvements over previous average-profile approaches, with practical applicability to neutrino DIS events via PYTHIA8 and a publicly available Python package.

Abstract

Subatomic particles can interact with target nuclei in matter or decay in flight, and an individual high-energy particle can induce a particle shower composed of numerous, lower-energy secondaries. These particle showers broadly exhibit universality across diverse media, including air, water, ice, and other materials, with their development governed by the Standard Model. Full Monte Carlo simulation of particle showers, where each secondary is individually tracked and propagated, can be a computational challenge to perform at scale. Experiments thus resort to parametrized approximations when efficient simulation becomes necessary. Here, we construct distributions of parameters capable of describing the Cherenkov light yield from particle showers in ice, and extensible to other, similar media. Sampling from the distributions allows for a much improved description of event-to-event fluctuations, in amplitude and shape, along the shower axis. Including these effects is essential for a more accurate simulation of signal and background events in current and next-generation neutrino telescopes.

Figures (10)

• Figure 1: The left panel shows four randomly chosen $\text{d}\hat{\ell}/\text{d}x$ distributions for a 1T $e^-$ from FLUKA in solid, colored lines. The parametrization for $\text{d}\hat{\ell}/\text{d}x$ from ref. Radel:2012ij is shown as the dashed, black curve. The right panel shows the $\hat{\ell}_\text{tot}$ distribution for a 1T $e^-$ ($\pi^+$) from FLUKA in blue (orange). The corresponding $\hat{\ell}_\text{tot}$ distribution as parametrized in refs. Raedel2012Radel:2012ij is shown as a dashed (dotted) black curves. Note that the parametrized $\hat{\ell}_\text{tot}$ curve for the 1T $e^-$ has a very narrow width, and its mean is assumed for the $\text{d}\hat{\ell}/\text{d}x$ parametrization in the left panel.
• Figure 2: The left panel shows 100 $\hat{\ell}_\text{tot}^{-1} \text{d}\hat{\ell}/\text{d}x$ distributions resulting from a 1T $e^-$ (blue) and $\pi^+$ (orange), chosen at random. The right panel shows the distribution of $\hat{\ell}_\text{tot} /E$ for $e^-$ (blue) and $\pi^+$ (orange) at three different energies as indicated in the legend. As is evident, $e^-$-initiated showers exhibits lower shower-to-shower fluctuations in both shape and amplitude compared to hadronic showers.
• Figure 3: The left panel shows fraction of simulation runs resulting in single-peak shower profiles for different primary particles, plotted as a function of its initial kinetic energy. Electromagnetic showers initiated by $e^-$ and $\gamma$ are shown in blue, mesons in orange, and baryons in green and red. All trend towards one as energy increases, but exhibit differences at lower energies between EM and hadronic showers, with smaller differences visible between the mesons and baryons. The right panel shows Spearman's rank coefficient between $\hat{\ell}_\text{tot}$, $a'$ and $b'$ (c.f. \ref{['eq:ab_transforms']}), also as a function of energy. The same line style and color scheme, corresponding to different primary particles, is used as the left panel. Values of $\varrho(a', b')$ lie close to one, indicating strong correlation. Values of $\varrho(\hat{\ell}_\text{tot}, \{a', b'\})$ for $e^-$ and $\gamma$ (blue) lie near zero, while hadronic shower amplitudes become more anticorrelated with $a'$ and $b'$ as energy increases. Note $e^-$ and $\gamma$ were simulated down to 1G ; hadrons 10G .
• Figure 4: The left panel shows fitted $\hat{\ell}_\text{tot}^{-1} \text{d}\hat{\ell}/\text{d}x$ distributions for the same set of simulations shown in the left panel of \ref{['fig:emvspi']}. Each shape is modeled by a two-parameter gamma distribution, which captures the per-shower longitudinal fluctuations. The right panel shows the fitted parameters $(a', b')$ for the distributions in the left panel, after applying \ref{['eq:ab_transforms']}. A clear distinction between $e^-$ and $\pi^+$ is visible with the denser $e^-$ cluster corresponding to its higher degree of similarity across different simulation runs.
• Figure 5: The top row shows normalized histograms of $(a', b')$ from FLUKA $\pi^+$ simulations at three different energies, 10G (left), 1T (middle) and 100T (right), binned as described in the text. The bottom row shows the probability density function $f(a', b' ; E)$ as generated from our model, for the same primary and energies. The same color scale is used for all six panels. Furthermore, the middle panels can be compared to the orange scatter in the left panel of \ref{['fig:emvspifits']}.