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Non-parametric and continuous extraction of amplitudes in realistic electroweak penguin decays

This paper demonstrates that a novel, model-independent method for extracting continuous decay amplitudes in $B\to V(\to M_1M_2)\ell^+\ell^-$ processes remains robust and effective when applied to realistic experimental data containing non-factorising detection efficiencies and backgrounds, thereby enabling broader kinematic studies and improved sensitivity to new physics.

Original authors: Tingzhen Xiao, Anja Beck, Eluned Smith

Published 2026-10-01

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CC BY 4.0

Original authors: Tingzhen Xiao, Anja Beck, Eluned Smith

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 called B mesons are unstable travelers that decay into lighter particles, often leaving behind a trail of clues about the fundamental forces of nature. Among the most intriguing of these trails are decays where a B meson transforms into a pair of muons—particles similar to electrons but much heavier—and a pair of other particles, such as a kaon and a pion. Physicists study these events, known as electroweak penguin decays, because they act as a sensitive microscope for searching for new physics beyond our current understanding. The Standard Model, our best theory of how particles interact, makes very specific predictions about how often these decays happen and how the resulting particles are arranged. When measurements deviate from these predictions, it hints at the presence of unknown particles or forces. However, interpreting these deviations is difficult because the process is clouded by complex interactions between quarks, the building blocks of the particles involved. To see clearly, scientists must separate the signal of the new physics from the noise of these known, but messy, strong-force interactions.

For years, researchers have struggled to map out the exact shape of these decays across the full range of energies involved. Traditional methods often force scientists to slice the data into large, coarse chunks, much like looking at a landscape through a grid of thick bars. This approach loses the fine details of how the decay changes from one energy level to the next. Furthermore, real-world experiments are never perfect; detectors have blind spots where they miss particles, and the data is often mixed with background events that look like the signal but are not. These imperfections can distort the picture, making it hard to tell if a deviation is a sign of new physics or just a flaw in how the data was collected. A team of physicists at MIT has now proposed a new way to untangle this mess, offering a method to extract the true shape of these decays without relying on rigid assumptions about how the particles behave.

The researchers focused on a technique called the sPlot method, which allows scientists to statistically separate signal events from background noise. In their previous work, they showed that this method could work in an ideal world where detectors are perfect and there is no background noise. However, the real world is far from ideal. In this new study, the team demonstrated that their approach remains robust even when faced with the messy reality of a real experiment. They tested their method using simulated data that mimicked the conditions of the LHCb experiment at CERN, including the specific ways detectors miss particles and the complex correlations between different types of background noise. They found that by making a small adjustment to how the statistical weights are calculated, the method could successfully account for detection efficiencies that vary in complicated ways. Similarly, they showed that even when background noise is strongly linked to the energy of the particles, the method could still isolate the true signal by expanding the description of the background into simpler, manageable parts.

The results of these simulations are promising. The team generated millions of fake decay events to test their approach, mixing in realistic background noise and detector imperfections. They found that the method could recover the true shapes of the decay amplitudes—the mathematical descriptions of how the particles are produced—with high precision. Unlike older methods that required binning the data into fixed ranges, this new approach allows for a continuous, smooth view of the decay across a wide range of energies, extending up to 19 GeV²/c⁴. This is a significant improvement, as it allows scientists to see subtle variations in the data that might otherwise be hidden. The study also confirmed that the method works well for different types of particle configurations, including those where the particles are produced in a scalar state, which is particularly difficult to study with previous techniques.

One of the most important findings is that this technique does not require scientists to assume a specific model for how the particles interact. In the past, researchers had to guess the shape of the underlying physics to make sense of their data, which could introduce bias if their guess was wrong. This new method is model-independent, meaning it lets the data speak for itself. By extracting the decay shapes directly, without forcing them into a pre-defined box, the method provides a clearer, more direct comparison with theoretical predictions. This is crucial for understanding the hadronic form factors, which describe how the strong force binds quarks together, and for identifying any genuine deviations that might point to new physics. The researchers showed that even with a realistic amount of background noise and imperfect detector performance, the statistical uncertainty on the results remains small enough to be useful for future experiments.

The implications of this work extend beyond just one type of decay. The method is flexible enough to be applied to other decays involving different combinations of particles, provided the mathematical description of the angles is adjusted accordingly. This opens the door for more precise studies of rare decays, such as those involving Lambda b baryons, which are harder to produce and study than B mesons. By providing a way to map the full energy spectrum of these decays without the need for coarse binning or heavy theoretical assumptions, the technique offers a powerful new tool for the field. It suggests that with the right statistical tools, the noise of the real world can be tamed, allowing physicists to see the subtle fingerprints of the universe's deepest secrets. The study concludes that this approach is ready to be applied to real experimental data, potentially improving the sensitivity of future searches for new physics and deepening our understanding of the strong force.

Technical Summary: Non-parametric and continuous extraction of amplitudes in realistic electroweak penguin decays

Problem Statement
Electroweak penguin decays, specifically $b \to s\ell^+\ell^-$ transitions, offer a critical avenue for searching for physics beyond the Standard Model (SM). While global analyses indicate tensions in Wilson coefficients (notably $C_9$ ), definitive interpretation is hindered by uncertainties in hadronic physics, particularly non-local charm-loop effects and the modeling of hadronic form factors. Current measurements often rely on binned observables or specific parametric models for decay amplitudes, which limits the ability to perform model-independent comparisons with theoretical predictions. Furthermore, existing methods to isolate specific partial waves (such as the $S$ -wave contribution in $B^0 \to K^+\pi^-\mu^+\mu^-$ ) often suffer from large systematic uncertainties due to the unknown shape of underlying contributions. A previous proof-of-principle study (Ref. [40]) demonstrated a method to extract decay amplitudes continuously and model-independently using the $sPlot$ technique but assumed an idealized detector with no background and uniform efficiency. This paper addresses the gap between that idealized scenario and the reality of experimental data, where detection efficiencies are non-uniform and non-factorizing, and backgrounds exhibit complex correlations.

Methodology
The paper proposes an extension of a non-conventional rearrangement of the two-dimensional angular decay rate for $B^0 \to K^+\pi^-\mu^+\mu^-$ (and similar $B \to V(\to M_1M_2)\ell^+\ell^-$ processes). The method involves:

Angular Decomposition: The decay rate is expressed in terms of helicity angles ( $\theta_h, \theta_\ell$ ) and control variables (dihadron mass $m(K\pi)$ and dimuon mass squared $q^2$ ). The rate is decomposed into angular functions multiplied by coefficients $(n)'$ which represent bilinear combinations of transversity amplitudes for $S$ -wave and $P$ -wave contributions.
Handling Non-Factorizing Efficiency: Realistic detection efficiencies often depend on both the fit variables (angles) and control variables (masses) in a non-factorizing way. The authors demonstrate that by weighting the data with the inverse of the efficiency ( $\epsilon^{-1}$ ) during the fit, the correct shapes can be recovered. Crucially, they modify the calculation of $sPlot$ weights to account for non-factorizing efficiencies by applying importance sampling (weighting integration terms by $\epsilon^{-1}$ ) rather than using standard libraries that assume factorization.
Handling Correlated Backgrounds: To separate signal from background, the four-body invariant mass ( $m$ ) is included in the fit. While random combinatorial backgrounds often factorize, specific sources (e.g., partially reconstructed decays) can create strong correlations between $m$ and $q^2$ . The authors propose expanding the background distribution into a sum of individually factorizing terms (using a polynomial basis for the mass dependence) to allow the $sPlot$ technique to extract the $q^2$ dependence of the background components individually.
Continuous Extraction: Once the fit to the angular distribution is performed, $sPlot$ weights are calculated to extract the continuous dependence of the amplitude coefficients on $m(K\pi)$ and $q^2$ without binning the control variables.

Key Contributions and Results
The paper validates this extended methodology using simulated toy data for $B^0 \to K^+\pi^-\mu^+\mu^-$ , incorporating:

Both $S$ -wave and $P$ -wave contributions.
Non-factorizing detection efficiencies.
Background candidates with correlations between the four-body and dimuon invariant masses.
A realistic $q^2$ range extending up to $19 \text{ GeV}^2/c^4$ .

Results include:

Validation of Efficiency Handling: The modified $sPlot$ weight calculation successfully recovers the true amplitude shapes in the presence of non-factorizing efficiencies.
Background Separation: The expansion of the background distribution into factorizing terms allows for the successful separation of signal and background even when correlations exist between the fit and control variables.
Comparison with Method of Moments: When compared to the traditional method of moments (calculated in $q^2$ bins), the continuous $sPlot$ method demonstrates reduced statistical scatter. This is attributed to the fact that every data point in the $sPlot$ method depends on the full dataset, whereas binned methods constrain each bin independently. The advantage is particularly pronounced at high $q^2$ , where data is sparse.
Massive Leptons: The study confirms the method works for massive muons, though it notes that generating reference samples for validation is non-trivial due to interference terms ( $b$ -terms) that vanish for massless leptons. However, specific combinations of coefficients allow for bias checks, and the bias is shown to be negligible for realistic dataset sizes.
Extension to D-wave: The formalism is extended to include $D$ -wave contributions, showing that the method can extract six observables (including $D$ -wave shapes) in a model-independent fashion.

Significance and Claims
The authors claim that this work demonstrates the feasibility of applying the continuous, model-independent amplitude extraction method to real experimental data. The significance of the work lies in:

Model Independence: It provides decay-amplitude shapes (specifically for $S$ - and $P$ -waves) that do not rely on specific hadronic form factor models, allowing for direct comparison with theoretical predictions.
Systematic Reduction: By providing the exact shape of the $S$ -wave contribution, the method can significantly reduce systematic uncertainties in future measurements of $b \to s\ell^+\ell^-$ transitions.
Enhanced Sensitivity: The continuous extraction over a wide $q^2$ range (up to $19 \text{ GeV}^2/c^4$ ) improves sensitivity to potential new-physics effects and aids in understanding the resonant structure above the $\psi(2S)$ region.
Applicability: The method is presented as applicable to any $B \to V(\to M_1M_2)\ell^+\ell^-$ decay and can be adapted to other spin configurations (e.g., $\Lambda_b^0 \to pK^-\mu^+\mu^-$ ) by establishing the appropriate angular decay rate expression.

The paper concludes that while the method extracts amplitudes squared (losing complex phase information accessible via full amplitude fits), it offers a simpler, more robust alternative to binned fits for extracting the $q^2$ and $m(K\pi)$ dependence of decay amplitudes, particularly in regimes with limited statistics or complex backgrounds.

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