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Multivariate amplitude analysis of the cascade particle decays based on the Nearest Neighbors fitting

This paper introduces a Nearest Neighbors-based fitting method for multivariate amplitude analysis of particle decays that utilizes weighted Monte Carlo events to construct probability density functions, thereby eliminating the need for analytic formulas or intensive re-generation while ensuring accurate reconstruction modeling and efficient multi-threaded performance.

Original authors: I. V. Yeletskikh, A. O. Vasyukov

Published 2026-08-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: I. V. Yeletskikh, A. O. Vasyukov

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

In the subatomic world, particles do not simply vanish; they transform. When a heavy, unstable particle decays, it often breaks apart into a cascade of lighter particles, which may themselves break apart again before finally reaching a detector. Physicists study these cascades to understand the fundamental forces that govern matter. To make sense of the billions of collision events recorded in modern experiments, scientists use a technique called amplitude analysis. This method attempts to reconstruct the invisible history of a decay by comparing the observed patterns of particles against a theoretical model. The challenge lies in the sheer complexity of the data. A single decay event is described by many different measurements at once—speeds, angles, and energies—all of which are linked together. When these measurements are tangled in complex ways, traditional mathematical formulas often fail to describe the full picture, especially when the detectors themselves introduce small errors or distortions.

Researchers at the Joint Institute for Nuclear Research in Dubna, Russia, have developed a new way to tackle this problem. Instead of relying on rigid mathematical equations to describe the behavior of every possible decay, they created a method that learns directly from computer simulations. Their approach, known as the Nearest Neighbors fit, treats the problem like finding a familiar face in a crowd. When a real particle event is observed, the computer looks at its specific combination of measurements and finds the most similar events from a vast library of simulated data. It does not try to force the data into a pre-written formula. Instead, it assumes that the real event behaves much like its closest simulated cousins. By weighing these similar simulated events based on how well they match the current theoretical model, the researchers can build a complete picture of the decay without needing to calculate complex, multi-dimensional shapes at every step.

This new method offers a significant advantage in speed and accuracy. Traditional approaches often require the computer to regenerate massive, multi-dimensional maps of probability every time a parameter in the model changes, a process that can be incredibly slow and demand enormous computing power. The Nearest Neighbors method avoids this bottleneck. Because the computer only needs to look at the pre-simulated events that are closest to the real data, it can update the model quickly. The researchers demonstrated this by testing their technique on a complex scenario involving the decay of a B-meson, a heavy particle that breaks down into a J/psi meson, a kaon, and a pion. In this specific decay, the particles can travel through different intermediate states, creating a tangled web of possibilities that is difficult to untangle with standard tools.

The team generated a large set of fake data, or pseudodata, to mimic what a real experiment would see, including the messy effects of a particle detector. They then applied their new fitting method to this data. The results were precise. The algorithm successfully recovered the hidden properties of the decay, such as the mass and width of a short-lived intermediate state called a tetraquark, which the researchers had set to specific values in their simulation. The method also correctly identified the strength and phase of the different ways the particles could interact. The fit was able to distinguish between the various contributing signals and the background noise, even when the particles were mixed together in a way that defied simple separation.

What makes this work particularly notable is its efficiency. The researchers found that their method could run on a standard eight-core computer processor, taking only about ten seconds for each iteration of the fit. This is a stark contrast to other high-dimensional analysis techniques that might require days of computing time or massive supercomputers to achieve similar results. The method also handles the imperfections of real-world detectors naturally. Instead of ignoring the fact that detectors blur measurements or miss some particles, the simulation library includes these effects. When the computer compares a real event to the simulated ones, it is comparing apples to apples, including the same blurring and errors. This allows the model to account for reconstruction effects with high accuracy without needing to add extra, often inaccurate, assumptions about how the detector works.

The study confirms that this approach is a viable alternative for analyzing complex particle decays. It removes the need for scientists to guess the exact mathematical shape of the data distribution in advance. Instead, the data itself, guided by the simulation, reveals the underlying physics. The researchers showed that even with a high number of variables—six dimensions in their test case—the method could find the correct parameters with confidence. The ability to run these fits quickly and accurately opens the door to analyzing even more complex decay chains that were previously too difficult to study in detail. By relying on the proximity of simulated events rather than abstract formulas, this technique provides a robust and flexible tool for peering into the intricate dance of subatomic particles, ensuring that the subtle signals of new physics are not lost in the noise of the data.

Technical Summary: Multivariate Amplitude Analysis of Cascade Particle Decays Based on Nearest Neighbors Fitting

Problem Statement
Advanced scientific data processing in high-energy physics often requires Multivariate Analysis (MVA) to handle complex correlations and high-dimensional data. While methods like Principal Component Analysis or Machine Learning reduce dimensionality, they rely on a priori known data properties. In scenarios where model components (signals and backgrounds) are entangled via complex interference effects and cannot be factorized, full $N$ -dimensional ( $ND$ ) fits are necessary.

Current approaches face significant limitations:

Analytic $p.d.f.$ Fits: When an analytic probability distribution function (p.d.f.) is available, unbinned likelihood fits are possible. However, for complex data involving reconstruction effects (detector uncertainties), precise analytic models are rarely available. Approximations, such as folding analytic p.d.f.s with Gaussian error distributions, often ignore or inadequately model raw data uncertainties.
Kernel Density Estimation (KDE): Methods like RooNDKeysPdf generate $ND$ p.d.f.s from discrete Monte Carlo (MC) simulated events. However, if signal shapes (masses, widths) or interference patterns change during the fit, the p.d.f. must be regenerated at every step. This "regeneration" is computationally prohibitive for high-dimensional fits.
Binned $\chi^2$ Fits: These are limited to low dimensions (typically 2–3) due to memory and CPU constraints. Achieving sufficient statistics per bin requires non-uniform binning, which introduces its own optimization problems and can reduce sensitivity or increase uncertainties.

Methodology: Nearest Neighbors Likelihood Fit
The authors propose a method to overcome these limitations by estimating the goodness-of-fit using the "distance to nearest neighbor" concept, avoiding the need for explicit analytic p.d.f. definitions or computationally expensive p.d.f. regeneration.

Core Concept: The value of the p.d.f. for a specific data event is estimated based on the kinematic proximity of simulated MC events. Each MC event contributes to the total p.d.f. at a data event's position based on the distance between them.
Mathematical Formulation: The p.d.f. value $P_i$ at the $i$ -th data point (with coordinates $\vec{\theta}_i$ ) is calculated as a weighted sum of Gaussian functions centered on MC events ( $\vec{\theta}_j$ ):
$P_i = \frac{1}{\sum_k w_k} \sum_j w_j \times \text{Gauss}_{ND}(\vec{\theta}_i - \vec{\theta}_j, \vec{\sigma})$
Here, $w_j$ represents the weight of the $j$ -th MC event, derived from theoretical model parameters (e.g., helicity amplitudes) and detector efficiency factors. $\vec{\sigma}$ represents the covariance vector reflecting reconstruction uncertainties.
Optimization:
- Pre-computation: Since the kinematic positions of MC events are fixed during the fit, the Gaussian distance terms ( $\text{Gauss}_{ND}$ ) are pre-computed and stored. Only the weights $w_j$ are recalculated at each fit step as model parameters change.
- Local Search: To optimize CPU usage, the summation is restricted to MC events within a $4\text{--}5\sigma$ proximity of the data event, rather than summing over the entire MC sample.
- Parallelization: The calculation of weights and the resummation of contributions are highly parallelizable, allowing efficient execution on multi-core CPUs or GPUs.
Likelihood Function: The final log-likelihood is computed as $\log L = \sum_i \log(P_i) \times v_i$ , where $v_i$ are data event weights.

Key Contributions and Implementation
The paper demonstrates the implementation of this method through two specific tests:

5-Dimensional Pseudodata Fit:
- A synthetic dataset was generated using a simple analytic p.d.f. involving five variables with specific correlations.
- Six sets of MC events were generated to describe the data components.
- Result: The fit successfully recovered all input parameters with adequate accuracy, demonstrating that the method works for $ND$ fits without analytic p.d.f.s or per-step p.d.f. regeneration. No tuning of implicit parameters (like binning) was required.
Amplitude Analysis of $B^0$ Meson Decays:
- Physics Model: The authors simulated the decay chains $B^0 \to J/\psi K^+ \pi^-$ and $\bar{B}^0 \to J/\psi K^- \pi^+$ . These decays involve complex interference between different resonant states (e.g., $K^*$ and tetraquark $T_{c\bar{c}}$ states) and unknown decay constants.
- Setup: 137k pseudodata events were generated using Pythia 8.1 with specific intermediate resonances ( $K^*(1600)$ , $K^*(1950)$ , $T_{c\bar{c}}(4200)$ ). The fit model utilized 1.9 million MC events.
- Variables: The fit was performed in a 6-dimensional space using invariant masses ( $m(J/\psi\pi^\pm)$ , $m(J/\psi K^\pm)$ , $m(K^\pm \pi^\mp)$ ).
- Performance: The fitting code ran in multi-threaded mode on an 8-core CPU, taking approximately 10 seconds per iteration.
- Results: The fit accurately reconstructed the input parameters, including the mass and width of the $T_{c\bar{c}}$ state and the complex decay constants (amplitudes and phases) for the various resonances. The results closely matched the values used to generate the pseudodata.

Significance and Claims
The authors claim that the Nearest Neighbors likelihood fit offers a robust alternative to existing $ND$ fitting techniques with the following advantages:

Computational Efficiency: It demands modest CPU resources compared to alternative $ND$ approaches because it avoids the regeneration of $ND$ p.d.f.s at every fit step.
Accuracy: It allows for the accurate accounting of reconstruction effects and detector uncertainties by utilizing the full power of MC methods without relying on analytic approximations.
Scalability: The number of dimensions is limited only by the statistics of the simulated events required to keep model statistical uncertainties small, rather than by the complexity of defining binning or analytic functions.
Flexibility: The method is effective for complex models where signals and backgrounds are entangled via interference, a scenario where analytic p.d.f.s are often unavailable or inaccurate.

The paper concludes that this method is effectively multi-threaded and suitable for realistic amplitude analyses of particle cascades, as demonstrated by the successful reproduction of parameters in the $B^0 \to J/\psi K \pi$ decay model.

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