Measurement of the CKM angle $\gamma$ in $B^{\pm} \rightarrow D(\rightarrow K^{0}_{\rm S} h^{\prime+}h^{\prime-})h^{\pm}$ decays with a novel approach

This paper presents the most precise single measurement of the CKM angle $\gamma$ to date, determining it to be $(71.3\pm 5.0)^{\circ}$ by applying a novel, model-independent approach that simultaneously analyzes joint datasets from the BESIII and LHCb experiments.

The Gist

Imagine the universe as a giant, complex clockwork machine. For decades, physicists have been trying to understand why this machine sometimes runs slightly differently when you look at it in a mirror. This phenomenon is called CP violation (Charge-Parity violation), and it's the reason our universe is made of matter instead of being an empty void of equal parts matter and antimatter.

To explain this, scientists use a special map called the CKM matrix. Think of this map as a compass with three main directions (angles). One of these directions, named Gamma ($\gamma$), has been the hardest to pin down. Knowing the exact value of Gamma is like finding the missing piece of a puzzle that could reveal if there are "ghosts" in the machine—new, unknown particles or forces that the Standard Model of physics doesn't predict.

The Problem: A Blurry Picture

Previously, scientists tried to measure Gamma by looking at specific particle collisions. Imagine trying to take a photo of a fast-moving race car, but your camera is a bit out of focus. You can see the car, but the details are blurry.

The old method was like taking a photo and then cutting the picture into a grid of squares (bins). You would count how many cars were in each square. While this worked, it threw away a lot of information. It was like trying to guess the speed of the car just by counting how many wheels were in the left half of the photo versus the right half, ignoring the fact that the car was leaning or speeding up in specific spots.

The Solution: A High-Definition, Smart Lens

This paper introduces a novel approach that acts like a high-definition, AI-powered lens. Instead of cutting the photo into squares, this new method looks at every single particle individually.

Here is how they did it, using a creative analogy:

1. The Two Labs: The Factory and the Race Track

• BESIII (The Factory): Located in Beijing, this lab acts like a pristine factory. They smash electrons and positrons together to create pairs of "charm" particles (called $D$ mesons) that are perfectly entangled twins. Because they are created in a controlled environment, scientists can measure the "strong phase" (a tricky internal setting of the particles) with extreme precision. Think of this as calibrating the lens.
• LHCb (The Race Track): Located at CERN in Switzerland, this lab is a high-speed race track. They smash protons together at nearly the speed of light, creating billions of "beauty" particles ($B$ mesons). These are messy, chaotic, and hard to catch, but they provide the massive amount of data needed to see the rare events.

2. The "Weight" Trick
The genius of this paper is the use of per-event weights.
Imagine you are a judge at a talent show. In the old method, you gave every contestant a score of 10, then averaged them.
In this new method, you look at each contestant and say, "You were amazing, so your score counts double!" or "You were a bit off, so your score counts half."
The scientists applied these "weights" to every single particle collision. They gave more importance to the collisions that happened in the most sensitive spots of the data and less importance to the noisy ones. This allowed them to extract much more information from the same amount of data.

3. The Result: Sharper Focus
By combining the precise calibration from the BESIII "factory" with the massive data from the LHCb "race track," and using this smart weighting system, they managed to sharpen the picture of Gamma.

• The Old Measurement: Was like seeing a blurry circle.
• The New Measurement: Is a sharp, clear point.

They found that Gamma is approximately 71.3 degrees, with an uncertainty of only 5.0 degrees. This is the most precise single measurement of this angle ever made.

Why Does This Matter?

Think of the Standard Model as a recipe for a cake. We know the ingredients (quarks, electrons, etc.) and the steps. But if the cake tastes slightly different than the recipe predicts, it means there's a secret ingredient we haven't found yet.

By measuring Gamma with such high precision, scientists are checking the recipe with a magnifying glass.

• If the value matches the predictions from other parts of the recipe perfectly, the Standard Model is solid.
• If there is a mismatch, it's a smoking gun for New Physics—perhaps a new particle or force that could explain dark matter or why the universe exists.

The Bottom Line

This paper is a triumph of teamwork between two massive experiments (BESIII and LHCb) and a clever mathematical trick (the optimal Fourier method). They took a blurry, noisy signal and turned it into a crystal-clear measurement. It's not just a number; it's a sharper window into the fundamental laws of our universe, bringing us one step closer to understanding the "ghosts" that might be hiding in the shadows of physics.

Technical Summary

1. Problem and Motivation

The Cabibbo-Kobayashi-Maskawa (CKM) angle $\gamma$ (also known as $\phi_3$) is a fundamental parameter in the Standard Model (SM) responsible for CP violation in the quark sector. It is defined as $\gamma = \arg(-V_{ud}V_{ub}^*/V_{cd}V_{cb}^*)$.

• Current Status: While global fits to CKM parameters provide indirect constraints on $\gamma$, direct measurements are crucial for testing the SM and searching for New Physics (NP) in loop-mediated processes. Existing direct measurements, primarily from the LHCb experiment using the binned phase-space approach, are statistically limited.
• Limitations of Previous Methods: The standard "binned" approach divides the $D$-meson Dalitz plot into discrete bins. This method discards intra-bin information, utilizing only about 85% of the available sensitivity to $\gamma$. Furthermore, it relies on strong-phase parameters measured in bins, which introduces systematic uncertainties due to model assumptions and statistical limitations in the input data.
• Goal: To achieve a more precise, single measurement of $\gamma$ by employing a novel, model-independent, unbinned approach that utilizes the full phase-space information and combines datasets from two complementary experiments: BESIII (providing strong-phase inputs) and LHCb (providing the $B$-decay statistics).

2. Methodology

The analysis employs a joint fit to data from the BESIII experiment ($e^+e^-$ collisions at the $\psi(3770)$ resonance) and the LHCb experiment ($pp$ collisions).

A. The Novel Approach: Optimal Fourier Method

Instead of binning the Dalitz plot, the authors apply per-event weights to the data to maximize sensitivity to CP-violating observables.

• Weighting Functions: The method combines two types of weights:
  1. Optimal Weights ($w_{opt}$): Derived from amplitude models (Belle/BaBar) and LHCb efficiency/background profiles. These weights enhance regions of the Dalitz plot with high signal purity and sensitivity, suppressing background-dominated regions.
  2. Fourier Weights: Based on the strong-phase difference $\phi(z) = \delta_D(z) - \bar{\delta}_D(z)$ across the Dalitz plot coordinates $z = (m^2_+, m^2_-)$. The weights are $\cos(k\phi)$ and $\sin(k\phi)$.
• Formalism: The decay rates for $B^\pm \to Dh^\pm$ are weighted by these functions. The resulting observables are linear combinations of the CP parameters ($x^\pm, y^\pm$) and strong-phase parameters ($C_n, S_n$).
• Fourier Order: The analysis uses a maximum Fourier order of $M_\pi = 2$ for $D \to K^0_S \pi^+ \pi^-$ and $M_K = 1$ for $D \to K^0_S K^+ K^-$, chosen to balance sensitivity with statistical power.

B. Data Samples

• BESIII: $8 \text{ fb}^{-1}$ of integrated luminosity collected at the $\psi(3770)$ resonance (2010–2011, 2021–2022). This dataset provides quantum-correlated $D^0\bar{D}^0$ pairs.
  • Role: Measures the strong-phase parameters ($C_n, S_n$) and flavor/CP-tag yields required as inputs for the LHCb analysis.
  • Channels: $D \to K^0_S \pi^+ \pi^-$, $K^0_L \pi^+ \pi^-$, $K^0_S K^+ K^-$, and $K^0_L K^+ K^-$.
• LHCb: $9 \text{ fb}^{-1}$ of integrated luminosity collected during Run 1 and Run 2 (2011–2018) at $\sqrt{s} = 7, 8, 13$ TeV.
  • Role: Measures the CP-violating observables in $B^\pm \to D h^\pm$ decays.
  • Channels: $B^\pm \to DK^\pm$ and $B^\pm \to D\pi^\pm$, with $D \to K^0_S \pi^+ \pi^-$ and $D \to K^0_S K^+ K^-$.

C. Analysis Strategy

1. Signal Extraction:
   • BESIII: Uses unbinned maximum-likelihood fits to mass distributions (beam-constrained mass $M_{BC}$ or missing mass squared $M^2_{miss}$) to extract signal yields for flavor and CP tags.
   • LHCb: Uses a two-stage fit to the $Dh^\pm$ mass spectrum. The sPlot technique is used to extract per-event weights ($f_i$) for signal events, correcting for background contamination.
2. Efficiency Correction: Relative efficiency distributions across the Dalitz plot are derived from simulation and corrected using control samples (e.g., $K^0_S$ tracking efficiency).
3. Joint Fit: A simultaneous $\chi^2$ minimization is performed on the weighted observables from both experiments.
   • Shared Parameters: The strong-phase parameters ($C_n, S_n$) are shared between the BESIII and LHCb fits.
   • Free Parameters: CP observables ($x^\pm, y^\pm$), hadronic parameters ($r_B, \delta_B$), and normalization factors.

3. Key Contributions

• First Application of Optimal Fourier Method: This is the first physics measurement to apply the "optimal Fourier" unbinned approach to a joint BESIII-LHCb dataset, demonstrating its superiority over the traditional binned method.
• Improved Sensitivity: By utilizing the intra-bin variation of the strong phase and optimal weighting, the method extracts more information from the same dataset size.
• Joint Strong-Phase Determination: The paper provides a new set of strong-phase parameters ($C_n, S_n$) measured specifically for this unbinned method (using higher Fourier orders $M_\pi=3, M_K=2$ for standalone BESIII results), which will serve as inputs for future $\gamma$ measurements.
• Systematic Control: The analysis rigorously evaluates systematic uncertainties arising from amplitude model imperfections, detector resolution, and background modeling, showing that the new method's systematics are comparable to the binned approach despite the increased complexity.

4. Results

The joint analysis yields the following results:

• CKM Angle $\gamma$:
  $$\gamma = (71.3 \pm 5.0)^\circ$$

  • This is the most precise single measurement of $\gamma$ to date.
  • The uncertainty is dominated by statistics ($4.9^\circ$ from LHCb, $0.6^\circ$ from BESIII inputs), with systematic uncertainties being negligible ($\sim 0.8^\circ$).
  • The result is consistent with world averages and previous LHCb binned measurements ($\gamma \approx 68.7^\circ$) but with improved precision.

• CP Observables ($B^\pm \to DK^\pm$):

  • $x_{DK}^- = (5.84 \pm 0.98 \pm 0.20) \times 10^{-2}$
  • $y_{DK}^- = (6.74 \pm 1.39 \pm 0.25) \times 10^{-2}$
  • $x_{DK}^+ = (-9.68 \pm 1.02 \pm 0.26) \times 10^{-2}$
  • $y_{DK}^+ = (-2.42 \pm 1.40 \pm 0.31) \times 10^{-2}$
  • The non-zero opening angle between the vectors $(x_{DK}^\pm, y_{DK}^\pm)$ confirms CP violation.

• Hadronic Parameters:

  • $r_B^{DK} = 0.0949^{+0.0086}_{-0.0085}$
  • $\delta_B^{DK} = (121.6^{+5.6}_{-5.9})^\circ$
  • $r_B^{D\pi} = 0.0064^{+0.0021}_{-0.0019}$
  • $\delta_B^{D\pi} = (311^{+17}_{-20})^\circ$

• Strong-Phase Parameters:

  • A comprehensive table of $C_n$ and $S_n$ parameters for $D \to K^0_S \pi^+ \pi^-$ and $K^0_S K^+ K^-$ is provided (Table 3), measured with higher Fourier orders than the baseline fit to ensure no truncation bias for future use.

5. Significance

• Precision Benchmark: The result $\gamma = (71.3 \pm 5.0)^\circ$ sets a new standard for single-experiment precision, reducing the uncertainty by approximately 15-20% compared to the previous best single measurement (LHCb binned).
• Validation of Unbinned Techniques: The paper successfully validates the "optimal Fourier" method, proving that unbinned approaches can outperform binned ones even when strong-phase inputs have finite statistical uncertainties. This paves the way for future analyses at LHCb (Run 3 and beyond) and Belle II.
• New Physics Sensitivity: The improved precision tightens the constraints on the Unitarity Triangle. Any future discrepancy between this direct measurement and indirect constraints (from global fits) would be a stronger indicator of New Physics entering loop processes.
• Community Resource: The strong-phase parameters and optimal weights provided in the paper (via HEPData) are essential inputs for the global community to perform future $\gamma$ measurements with even higher precision.

In conclusion, this paper represents a significant methodological advancement in flavor physics, leveraging the synergy between a $B$-factory (BESIII) and a hadron collider (LHCb) to extract the CKM angle $\gamma$ with unprecedented precision using a model-independent, unbinned analysis strategy.