Precise measurement of the CKM angle $\gamma$ with a novel approach

This paper presents the most precise measurement of the CKM angle $\gamma$ to date, yielding a value of $(71.3\pm 5.0)^{\circ}$ by performing a joint, model-independent analysis of quantum-correlated $D$ meson data from BESIII and $B^{\pm}$ decay data from LHCb.

The Gist

Imagine the universe as a giant, complex puzzle. For decades, physicists have been trying to solve a specific piece of that puzzle: Why is there more matter than antimatter? If the Big Bang had created equal amounts of both, they would have annihilated each other, and we wouldn't be here. Something must have tipped the scales.

This paper is about measuring a specific "tilt" in the laws of physics that explains this imbalance. The scientists call this tilt Angle $\gamma$ (gamma).

Here is the story of how the BESIII and LHCb collaborations (two massive teams of scientists working at different particle accelerators) solved this puzzle using a brand-new, clever trick.

1. The Problem: The "Blindfolded" Measurement

To measure this angle, scientists look at how certain particles (called B-mesons) decay. Think of a B-meson as a spinning top that eventually falls over and breaks into smaller pieces (a D-meson and a pion/kaon).

The tricky part is that the "D-meson" can break down in many different ways, creating a chaotic pattern of particles. To understand the angle $\gamma$, scientists need to know the "phase" of the waves involved in this breakup.

The Old Way (The Binned Approach):
Previously, scientists tried to measure this by dividing the chaotic pattern into a grid, like a chessboard. They would count how many particles landed in each square (or "bin") and take an average.

• The Flaw: It's like trying to describe the flavor of a complex soup by only tasting the average of a spoonful from each quadrant of the bowl. You lose the subtle, specific flavors happening inside the spoonful. This method threw away about 15% of the useful information.

2. The New Solution: The "Smart Filter"

In this new paper, the teams introduced a novel, unbinned approach. Instead of a rigid chessboard, they used a smart, adjustable filter.

Imagine you are trying to hear a specific instrument in a loud orchestra.

• The Old Way: You put on noise-canceling headphones that block out 15% of the music, hoping the rest is clear enough.
• The New Way: You use a high-tech AI ear that listens to the entire orchestra at once. It knows exactly which frequencies (parts of the sound) contain the most information about the instrument you are looking for. It amplifies the important parts and ignores the noise, without ever cutting out a chunk of the music.

In physics terms, they created a mathematical "weight function" (a filter) that assigns more importance to the parts of the particle decay where the "CP violation" (the matter-antimatter tilt) is strongest. This allowed them to use every single piece of data without throwing anything away.

3. The Teamwork: The "Chef" and the "Taster"

This measurement required a perfect partnership between two very different experiments, like a master chef and a professional taster.

• The Chef (BESIII Experiment in China):
  The BESIII experiment creates pairs of D-mesons in a very controlled environment (electron-positron collisions). Because these pairs are "quantum-correlated" (like two magic dice that always show opposite numbers), BESIII can measure the strong-phase parameters (the internal "flavor" of the D-meson decay) with extreme precision. They provide the "recipe" or the baseline map.
• The Taster (LHCb Experiment in Europe):
  The LHCb experiment smashes protons together at incredible speeds. This creates a massive amount of B-mesons. They are the "tasters" who observe the final decay and look for the tilt (Angle $\gamma$). However, they can't do it accurately without the recipe from BESIII.

The Magic: By combining the "recipe" from BESIII with the "tasting notes" from LHCb in a single, giant calculation, they got a result that is much sharper than either could achieve alone.

4. The Result: A Sharper Picture

The result of this new method is a measurement of the angle $\gamma$ as $71.3 \pm 5.0$ degrees.

• Why it matters: This is the most precise single direct measurement of this angle ever made.
• The Analogy: If the old method was like taking a photo with a slightly blurry lens, this new method is like switching to a 4K camera with a perfect lens. The picture of the universe's rules is now much clearer.
• Consistency: The result matches what we expected based on previous measurements, which is good news. It means the Standard Model (our current best theory of physics) is still holding up, but now we have a much more precise ruler to test it against.

Summary

The scientists didn't just count particles in boxes anymore. They built a smart, continuous filter that listened to the entire symphony of particle decays. By combining the precise "map" from the Chinese BESIII experiment with the massive "data stream" from the European LHCb experiment, they measured a fundamental angle of the universe with record-breaking precision.

This is a major step forward in understanding why we exist, proving that sometimes, the best way to see the whole picture is to stop looking at it in pieces.

Technical Summary

1. Problem Statement

The Cabibbo–Kobayashi–Maskawa (CKM) angle $\gamma$ (also denoted as $\phi_3$) is a fundamental parameter in the Standard Model (SM) describing CP violation in quark mixing.

• Current Status: While indirect measurements of $\gamma$ (via loop-mediated processes) are highly precise, they are sensitive to potential New Physics (NP). Direct measurements (via tree-level $b \to c\bar{u}s$ and $b \to u\bar{c}s$ interference in $B^\pm \to D h^\pm$ decays) are theoretically clean but currently less precise (uncertainty $\sim 3\times$ larger than indirect constraints).
• The Bottleneck: The dominant direct measurement method uses a binned phase-space approach (Dalitz plot analysis). In this method, the $D \to K_S^0 h^+ h^-$ decay phase space is divided into discrete bins. The strong-phase difference between $D^0$ and $\bar{D}^0$ decays is measured in each bin using quantum-correlated $D\bar{D}$ pairs from $e^+e^-$ collisions (BESIII).
• Limitation: The binned approach averages strong-phase information within each bin, discarding intra-bin variations. This results in a statistical sensitivity loss of approximately 15% compared to the theoretical maximum.

2. Methodology: The Novel Unbinned Approach

The paper introduces a novel, unbinned, model-independent approach that combines data from the BESIII experiment (quantum-correlated $D\bar{D}$ pairs) and the LHCb experiment ($B^\pm \to D h^\pm$ decays).

Key Technical Components:

1. Optimal Weight Functions ($w_{opt}$):
   Instead of using step functions (bins), the authors define continuous weight functions $w_n(z)$ over the Dalitz plot variables $z = (m^2_{K_S^0 h^+}, m^2_{K_S^0 h^-})$.

   • The weights are constructed using a Fourier expansion of the strong-phase difference $\phi(z)$:
     $$w_n(z) \propto w_{opt}(z) \times \begin{cases} \cos[k\phi(z)] & n=2k \\ \sin[k\phi(z)] & n=2k-1 \end{cases}$$
   • $w_{opt}(z)$: An optimal weight factor that incorporates the magnitude ratio $r_D(z)$, detection efficiency, and background distributions to maximize statistical sensitivity.
   • Fourier Order ($M$): The expansion is truncated at order $M$. The baseline analysis uses $M_\pi = 2$ and $M_K = 1$.

2. Joint Fit Strategy:

   • BESIII Data: Provides the strong-phase parameters $\{C_n, S_n\}$ via quantum-correlated $D\bar{D}$ decays. The joint decay rate of the $D\bar{D}$ pair is sensitive to the strong-phase difference.
   • LHCb Data: Provides the CP-violating observables $N^\pm_n$ from $B^\pm \to D h^\pm$ decays. These observables are calculated by summing signal events weighted by the functions $w_n(z)$.
   • Simultaneous Determination: A global $\chi^2$ minimization is performed on both datasets simultaneously to extract the CP-violating parameters ($x_{B}^{\pm}, y_{B}^{\pm}$) and the strong-phase parameters $\{C_n, S_n\}$ without relying on external amplitude models for the final $\gamma$ extraction.

3. Model Independence:
   Although amplitude models (e.g., from BaBar/Belle) are used to construct the weight functions (specifically to estimate $\phi(z)$ and $r_D(z)$), the measurement of $\gamma$ itself remains model-independent. The models only define the integration weights; the data determines the final observables.

3. Datasets and Experimental Setup

• BESIII: Collected at the $\psi(3770)$ resonance ($e^+e^-$ collisions).
  • Integrated Luminosity: 8 fb$^{-1}$.
  • Role: Measures strong-phase parameters in $D \to K_S^0 \pi^+ \pi^-$ and $D \to K_S^0 K^+ K^-$.
  • Signal Yield (Double-tagged): $564 \pm 27$.
• LHCb: Collected during Run 1 and Run 2 ($pp$ collisions at $\sqrt{s} = 7, 8, 13$ TeV).
  • Integrated Luminosity: 9 fb$^{-1}$.
  • Role: Measures CP observables in $B^\pm \to D K^\pm$ and $B^\pm \to D \pi^\pm$.
  • Signal Yields: $13,450 \pm 130$ ($D \to K_S^0 \pi^+ \pi^-$) and $2,005 \pm 50$ ($D \to K_S^0 K^+ K^-$).

4. Key Results

The joint fit yields the following results for the CKM angle $\gamma$ and associated hadronic parameters:

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

  • The uncertainty includes both statistical and systematic components.
  • Precision Improvement: Compared to a refitted binned analysis using the same datasets, the unbinned approach reduces the statistical uncertainty by 5% (from $\pm 5.1^\circ$ to $\pm 4.9^\circ$ for LHCb-only stats).
  • Consistency: The result is consistent with previous LHCb measurements, world averages, and indirect constraints.

• Hadronic Parameters:

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

• Systematic Uncertainties:

  • Systematic uncertainties are subleading to statistical uncertainties (approx. 10% of the statistical error from BESIII; comparable to binned methods for LHCb).
  • Sources include background modeling, efficiency corrections, amplitude model biases (for weights), and resolution effects.

5. Significance and Impact

• Breaking the Binning Limit: This work demonstrates that moving from a binned to an unbinned, optimal-weight approach can recover statistical sensitivity lost in traditional methods. While it does not yet fully recover the theoretical 15% sensitivity loss (due to the necessary truncation of the Fourier series), it establishes a robust framework for future improvements.
• Complementarity: The result provides a complementary determination of $\gamma$ that is consistent with existing measurements, validating the robustness of the CKM unitarity triangle.
• Future Prospects:
  • With increased luminosity (e.g., BESIII's projected 20 fb$^{-1}$ and LHCb Run 3/4), higher-order Fourier expansions ($M$) can be utilized to further reduce uncertainties.
  • The methodology is generalizable to other multi-body $D$ decays (e.g., $D \to K^+ K^- \pi^+ \pi^-$), paving the way for even more precise tests of the Standard Model and searches for New Physics in the flavor sector.
• Methodological Blueprint: The paper sets a "blueprint" for future precision measurements, showing how to leverage amplitude variations across phase space without introducing model-dependent biases in the final physics result.

In conclusion, this paper presents the most precise single direct measurement of $\gamma$ to date, achieved through a sophisticated joint analysis of quantum-correlated and $B$-decay data using a novel, unbinned, model-independent technique.