Unbinned extraction of $\gamma$ from $B\to DK$ with normalizing flows

This paper introduces and validates an unbinned method using normalizing flows to extract the CKM angle $\gamma$ from $B^\pm \to (D \to K_S \pi^+ \pi^-) K^\pm$ decays, demonstrating its ability to accurately recover $\gamma$ and other parameters from Monte Carlo data while propagating statistical uncertainties through ensemble training.

The Gist

Imagine you are trying to solve a complex puzzle to find a hidden number, let's call it $\gamma$ (gamma). This number is a fundamental piece of the universe's rulebook, specifically related to why the universe is made of matter rather than antimatter.

Physicists usually try to find this number by watching specific particles, called B-mesons, decay (break apart) into other particles. The process is like watching a magician's trick: a B-meson splits, and one of its children is a D-meson, which then immediately splits again into a mix of pions and a kaon.

The Old Way: Looking Through a Grid

For decades, scientists have analyzed these particle breaks using a method called the BPGGSZ method. Imagine the possible outcomes of the D-meson breaking apart are mapped onto a square piece of graph paper (called a Dalitz plot).

In the traditional approach, scientists draw a grid over this paper, dividing it into 8 big boxes. They count how many particles land in each box and calculate an "average" for that box.

• The Problem: This is like trying to describe a detailed painting by only looking at it through a coarse window screen. You get the general idea, but you lose all the fine details and sharp edges inside the boxes. This "blurring" makes it harder to pinpoint the exact value of $\gamma$.

The New Way: The "Normalizing Flow" Camera

This paper introduces a new, sharper way to look at the data using a type of Artificial Intelligence (AI) called Normalizing Flows (NFs).

Think of a Normalizing Flow not as a grid, but as a high-definition, flexible camera that learns to take a perfect picture of the particle data.

1. Learning the Shape: The AI is fed millions of examples of how the D-meson breaks apart. Instead of counting boxes, the AI learns the exact, continuous shape of where the particles go. It captures every tiny ripple, peak, and valley in the data, just like a high-res photo captures every brushstroke.
2. The Tricky Part (The Constraint): There is a mathematical rule in physics that says these particle patterns must fit together perfectly, like three pieces of a puzzle that must form a circle. If you guess the shape of one piece, the others are locked in place.
   • The Challenge: If you use two separate AI models to guess the shapes, they might accidentally disagree with this rule (like two puzzle pieces that don't quite fit).
   • The Solution: The authors built two versions of their AI:
     • Version A (The "H-Network"): This AI is built with the rule hard-coded into its brain. It is physically impossible for it to make a mistake; it always produces shapes that fit the puzzle perfectly.
     • Version B (The "3-Flow"): This AI uses three separate models that learn independently. Sometimes they make tiny mistakes where the pieces don't fit. The authors fix this by smoothing out the errors, like gently sanding down a rough puzzle piece until it fits.

The Results: A Perfect Test

The authors tested this new method using computer simulations (a "closure test"). They created fake data with a known value for $\gamma$ and asked their AI to find it.

• The Outcome: Both versions of the AI successfully found the hidden number $\gamma$ with high precision.
• The Winner: The "H-Network" (the one with the rule hard-coded) was slightly more stable and precise, likely because it didn't have to waste time fixing its own mistakes.

Why This Matters

The paper claims that this method allows physicists to use all the information in the data, rather than throwing away the fine details by averaging them into boxes.

• The Benefit: As more data is collected from experiments (like those at CERN or Belle II), this AI method gets better and better, systematically improving the precision of the measurement.
• The Caveat: This is currently a "proof of concept" using simulated data. The authors note that before using this on real-world data, they will need to account for real-world messiness (like detector errors) and ensure the AI doesn't develop any subtle biases. They also suggest that in the future, using "Bayesian" versions of this AI could automatically calculate how uncertain the result is, without needing to run the simulation hundreds of times.

In short: The authors replaced a blurry, grid-based way of measuring a fundamental universe constant with a sharp, AI-driven method that learns the exact shape of the data, proving it can find the answer accurately in simulations.

Technical Summary

Technical Summary: Unbinned extraction of $\gamma$ from $B \to DK$ with normalizing flows

Problem Statement
The determination of the CKM unitarity triangle angle $\gamma$ (or $\phi_3$) from $B^\pm \to DK^\pm$ decays, where $D \to K_S \pi^+ \pi^-$, is theoretically clean but experimentally limited by statistical precision. The current state-of-the-art model-independent approach, the BPGGSZ method, partitions the $D$ decay Dalitz plot into bins. While this avoids model-dependent assumptions about the $D$ decay amplitude, it introduces a "coarse graining" of phase space. The averaging of the strong phase difference over finite bins washes out local interference effects, leading to a loss of sensitivity to $\gamma$. Previous unbinned alternatives (Fourier expansions, Legendre polynomials, nonparametric tests) have been proposed but often lack the flexibility to capture complex, multi-modal structures inherent in Dalitz plots without introducing irreducible modeling errors.

Methodology
The authors propose an unbinned extraction method utilizing Normalizing Flows (NFs) to learn a continuous, data-driven representation of the $D$ decay amplitude and strong phase variation across the square Dalitz plot. The method replaces the bin-averaged observables ($K_i, c_i, s_i$) of the BPGGSZ method with continuous functions ($K, C, S$) learned directly from $D$ decay data.

The core technical challenge addressed is the trigonometric constraint linking the learned functions: $C^2 + S^2 = K \bar{K}$. The paper explores two distinct architectures to handle this:

1. Constraint-Aware Architecture ($h$-network):

   • A flavor-tagged NF is trained to learn the density $K(m', \theta')$.
   • A separate neural network, $h(m', \theta')$, is trained on CP-tagged data to learn $\cos \Delta \delta$.
   • The network $h$ is constrained to the range $[-1, 1]$ via a $\tanh$ activation.
   • The interference observables are constructed as $C = \sqrt{K\bar{K}}h$ and $|S| = \sqrt{K\bar{K}}\sqrt{1-h^2}$. This construction guarantees the trigonometric constraint is satisfied by definition at every point.
   • To capture rapid oscillations near resonances, the $h$-network utilizes a SIREN (Sinusoidal Representation Network) architecture, which is better suited for high-frequency features than standard ReLU-based MLPs.

2. Deferred Constraint Enforcement (3-flow):

   • Three independent NFs are trained: one on flavor-tagged data (learning $K$) and two on CP-even and CP-odd tagged data (learning densities $p_+$ and $p_-$).
   • The observable $C$ is extracted algebraically from the CP densities.
   • Since the flows are independent, statistical fluctuations can lead to unphysical regions where $C^2 > K\bar{K}$ (resulting in negative $S^2$).
   • A post-processing step involving local neighbor averaging is applied to enforce the constraint $C^2 \leq K\bar{K}$.

The extracted continuous functions are then used in an unbinned maximum-likelihood fit to $B^\pm \to DK^\pm$ data to determine $r_B$, $\delta_B$, and $\gamma$.

Key Contributions

• Unbinned Density Estimation: The paper introduces the first application of Normalizing Flows to the extraction of $\gamma$, providing a flexible, non-parametric representation of the Dalitz plot that improves systematically with larger $D$ decay datasets.
• Constraint Handling: It demonstrates two strategies for incorporating the non-linear trigonometric constraint between amplitude and phase observables into machine learning architectures, comparing a "hard" constraint (via architecture design) against a "soft" constraint (via post-processing).
• SIREN for High-Frequency Physics: The work highlights the utility of SIREN architectures for modeling the rapid phase variations found in Dalitz plots, which standard neural networks often struggle to resolve.
• Bias and Uncertainty Analysis: The study provides a rigorous closure test using ensembles of simulated datasets to quantify statistical uncertainties and search for systematic biases introduced by the NF approximation.

Results
The methods were tested on Monte Carlo generated data using benchmark parameters ($r_B=0.1, \delta_B=130^\circ, \gamma=70^\circ$).

• Fidelity: The NFs successfully reproduce the input isobar model density, including narrow resonance structures. The ensemble mean of the learned densities matches the truth model with high fidelity (pointwise $1\sigma$ coverage of $\sim95\%$ for flavor flows and $\sim96-97\%$ for CP flows).
• Parameter Extraction: Both methods successfully recover the injected value of $\gamma$ within statistical uncertainties.
• Comparison of Methods:
  • The $h$-network approach yields smaller variances in the extracted parameters across different datasets compared to the 3-flow method. This is attributed to the built-in constraint preventing unphysical values that require numerical repair in the 3-flow approach.
  • The 3-flow method exhibits slightly larger variability in uncertainties, particularly when neighbor-averaging is required to fix unphysical $S^2$ values near resonance peaks.
• Bias: No statistically significant systematic biases were found. The estimated upper bounds on potential biases are small ($\sim 1^\circ$ for $\gamma$ and $\delta_B$, $\sim 0.002$ for $r_B$), comparable to or smaller than the statistical uncertainties of the simulated datasets.

Significance and Claims
The paper presents this work as a proof of principle for an unbinned extraction of $\gamma$. The authors claim that:

1. The NF-based approach offers a systematic improvement over binned methods as $D$ decay sample sizes increase, analogous to how the precision of $c_i$ and $s_i$ parameters improves in the binned method.
2. The method avoids the irreducible modeling errors associated with isobar models while retaining more phase-space information than binned approaches.
3. The proposed architectures do not introduce significant biases, making them viable candidates for future experimental implementation at LHCb and Belle II.

The authors explicitly note that the current implementation propagates statistical uncertainties via an ensemble of independently trained flows and does not yet capture systematic experimental errors (e.g., detector resolution, backgrounds). They suggest that future work should explore Bayesian Normalizing Flows to provide direct uncertainty estimates on the learned densities without requiring external ensembles, and that the method must be extended to include detector effects for real-data application. The paper advocates for pursuing both the traditional binned and the proposed unbinned NF methods in parallel to cross-check results and identify potential systematic effects.