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TauPolaris: reconstructing tau lepton polarimetric vectors with conditional normalizing flows

The paper introduces TauPolaris, a tool utilizing conditional normalizing flows to reconstruct tau lepton polarimetric vectors by modeling the full kinematic density of undetected neutrinos, thereby improving spin observable resolution and enhancing sensitivity to quantum entanglement and CP violation in Higgs boson decays at the LHC.

Original authors: Daniel Winterbottom, Lucas Russell

Published 2026-08-12

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CC BY 4.0

Original authors: Daniel Winterbottom, Lucas Russell

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are a detective trying to solve a crime that happened inside a tiny, invisible box. The only clues you have are the footprints left behind by the suspects after they ran away, but the suspects themselves vanished into thin air before you could see them. This is the daily reality for physicists studying subatomic particles called tau leptons. These particles are like cosmic ghosts: they are born in high-energy collisions, spin around, and then instantly decay into other particles, including neutrinos. Neutrinos are the ultimate escape artists; they zip through the Earth and our detectors without leaving a single trace. Because we can't see the neutrinos, we can't directly measure the "spin" of the tau lepton—its internal rotation, which is like a tiny compass needle pointing in a specific direction.

Why does this invisible compass matter? Because the direction of that spin holds secrets about the fundamental laws of the universe. It can tell us if the universe treats matter and antimatter differently (a mystery called CP violation), if particles are "entangled" in a spooky quantum way that Einstein once doubted, or if there are new, hidden forces at play. For decades, physicists have been trying to guess where those invisible neutrinos went based on the footprints they left behind, but their guesses were often like trying to hit a moving target while blindfolded. They needed a better way to reconstruct the invisible path of the neutrinos to see the true spin of the tau lepton.

Enter TauPolaris, a new digital tool created by researchers Daniel Winterbottom and Lucas Russell. Think of TauPolaris not as a simple guesser, but as a masterful "probability wizard." Instead of just picking one single guess for where the neutrino went (like a standard computer program might do), TauPolaris imagines every possible place the neutrino could have gone, creating a complete map of probabilities. It uses a clever mathematical technique called a "conditional normalizing flow" to learn the complex rules of how these invisible particles behave based on the visible clues.

The researchers tested this tool using massive computer simulations of particle collisions, similar to those happening at the Large Hadron Collider (LHC). They found that TauPolaris is significantly better at reconstructing the tau lepton's spin than previous methods. In fact, when they used it to look for evidence of quantum entanglement (where two particles are linked across space) in Higgs boson decays, they showed that future experiments at the High-Luminosity LHC could distinguish between entangled and non-entangled particles with a confidence of at least 4.3σ (a statistical term meaning the result is extremely unlikely to be a fluke).

Furthermore, the tool improved the ability to measure CP violation (a type of symmetry breaking) in Higgs boson decays by 18%. It also helped create new "filters" to separate Higgs boson events from the noisy background of Z boson events, which is crucial for finding rare new physics. The key to this success is that TauPolaris doesn't just give a single answer; it provides the most likely configuration for each event and tells you how uncertain that guess is. By respecting the complex relationships between different parts of the collision, it avoids the common trap of previous methods that would accidentally average out the truth, effectively blurring the very spin information physicists are trying to see. In short, TauPolaris turns a blurry, guesswork-heavy puzzle into a sharp, high-definition picture of the quantum world.

Technical Summary: TauPolaris

Problem Statement
The spin orientation of tau leptons ( $\tau$ ) is a critical observable for probing the fundamental properties of the Standard Model and beyond, including searches for CP violation in Higgs boson decays ( $H \to \tau\tau$ ), tests of quantum entanglement, and measurements of effective operators. The optimal observable for determining $\tau$ spin is the polarimetric vector, which points along the most likely direction of the $\tau$ spin. However, reconstructing this vector is exceptionally challenging because $\tau$ decays invariably produce neutrinos that escape detection.

Current experimental methods rely on approximate constructions for spin-sensitive observables (e.g., using impact parameters or decay planes) because the neutrino momenta are unknown. These approximations limit experimental precision and are difficult to transfer between different processes. Furthermore, standard machine learning approaches that perform point regression (minimizing mean-squared error) often fail to capture the complex, multi-modal correlations between neutrino momentum components. When kinematics allow for two distinct solutions, a point-regression model tends to predict a mean value between them, biasing the neutrino direction toward the visible $\tau$ momentum and degrading the angular correlations essential for spin measurements.

Methodology
The authors present TauPolaris, a tool that reconstructs tau polarimetric vectors by estimating undetected neutrino momenta using conditional normalizing flows. Unlike point-regression networks, this method models the full conditional probability density, $p(\nu | c)$ , of the neutrino kinematics given the reconstructed event context $c$ .

Key architectural and methodological features include:

Conditional Normalizing Flow: The model maps the neutrino kinematics $\nu$ to a latent variable $z$ following a standard Gaussian distribution via a series of invertible transformations. This allows for both the sampling of the full density (to reproduce generator-level distributions) and the determination of the Maximum A Posteriori (MAP) estimate (the most probable configuration).
Coordinate System: Neutrino momenta are parametrized in an orthonormal basis $\{n, r, k\}$ defined relative to the visible $\tau$ momentum. This aligns the model's output with the basis used to express spin correlations, improving resolution.
Input Features: The model utilizes a transformer-based conditioning network to process event inputs, including the four-momenta of visible decay products (charged pions, leptons, and reconstructed $\pi^0$ strips), impact parameters, secondary vertex positions, and missing transverse momentum.
Gradient-Based MAP Estimation: To obtain the most probable neutrino configuration for a specific event without relying on stochastic sampling, the authors employ a gradient-based maximization routine in the latent space. This approach is more stable and reliable than maximizing directly over the neutrino components.
Training: The models are trained on simulated LHC proton-proton collision data (Pythia 8.312) with detector effects modeled via Delphes 3.5 and custom smearing. Separate models are trained for fully hadronic ( $\tau_h\tau_h$ ) and semileptonic ( $\tau_\ell\tau_h$ ) channels.

Key Contributions

Full Density Modeling: By modeling the joint density of neutrino components rather than independent point estimates, TauPolaris preserves the physical correlations between momentum components. This prevents the "collinearity bias" seen in mean-squared-error regressors, where the predicted neutrino is artificially aligned with the visible $\tau$ .
Uncertainty Quantification: The method provides not only a point estimate (MAP) but also a per-event uncertainty estimate derived from the spread of the sampled density.
Extension to All Topologies: The approach extends the use of polarimetric vectors to all $\tau$ decay topologies, including those previously difficult to analyze (e.g., $3\pi^\pm 1\pi^0$ and leptonic decays), by explicitly reconstructing the neutrino kinematics.
New Background Suppression Variables: The reconstructed polarimetric vectors are used to define new observables that exploit spin correlations to distinguish $H \to \tau\tau$ from $Z \to \tau\tau$ backgrounds without assuming a specific CP mixing angle.

Results
The performance of TauPolaris was evaluated on three benchmark applications using simulated High-Luminosity LHC (HL-LHC) data:

Quantum Entanglement: The authors demonstrate that the presence of quantum entanglement in $H \to \tau\tau$ decays can be distinguished from a separable (non-entangled) state with a significance of at least 4.3 $\sigma$ at the HL-LHC. This is achieved by reconstructing the spin density matrix elements ( $C_{ij}$ ) and calculating the concurrence ( $C$ ) and Bell non-locality witness ( $m_{12}$ ).
CP Violation Sensitivity: The method improves the sensitivity to CP violation in $H \to \tau\tau$ decays by 18% compared to previous approximate reconstruction methods. This is quantified by the asymmetry in the acoplanarity angle ( $\phi_{CP}$ ) distributions. The improvement is most significant in decay modes involving three charged hadrons (e.g., $3\pi^\pm 0\pi^0$ ), where the asymmetry increases by 88% relative to approximate methods.
Background Suppression: New variables constructed from the polarimetric vectors (specifically combinations of $\cos \theta_n \cos \theta_n$ and $\cos \theta_r \cos \theta_r$ ) effectively separate spin-0 Higgs events from spin-1 $Z$ boson events. These variables are independent of the Higgs CP state, providing a robust handle for background suppression.
Resolution: The resolution of the reconstructed spin observables (defined as the interquartile range of the prediction error) improves by approximately 40% compared to a stand-alone transformer regressor trained with mean-squared error.

Significance
The paper claims that TauPolaris represents a significant advancement in tau lepton polarimetry by moving beyond approximate kinematic constructions to a rigorous, density-based reconstruction of neutrino momenta. By accurately modeling the conditional density of neutrino kinematics, the tool recovers the optimal spin-analyzing observables with higher precision and provides event-level uncertainty estimates. This capability enables more sensitive tests of fundamental physics, including the first robust demonstration of distinguishing quantum entanglement from classical correlations in $H \to \tau\tau$ at the HL-LHC and a substantial reduction in the uncertainty of CP violation measurements. The authors note that the ability to estimate per-event uncertainties on the polarimetric reconstruction offers further potential for optimizing event selection in future analyses.

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