Updated model-independent measurement of the strong-phase differences between $D^0$ and $\bar{D}^0 \to K^{0}_{S/L}π^+π^-$ decays

Using quantum-correlated $D^0$-$\bar{D}^0$ decays from 7.93 fb$^{-1}$ of data collected by the BESIII experiment, this paper presents the most precise model-independent measurement of strong-phase differences for $D^0 \to K_{S/L}^0\pi^+\pi^-$ decays, which are critical inputs for reducing uncertainties in determining the $CP$-violating angle $\gamma$ and charm mixing parameters.

The Gist

The Big Picture: Measuring the "Twist" of the Universe

Imagine the universe has a secret rulebook called the Standard Model. This rulebook explains how particles interact, but it has a mysterious loophole: CP Violation. This is a fancy way of saying that nature sometimes treats "matter" and "anti-matter" slightly differently, like a left-handed glove that doesn't quite fit a right-handed hand.

Physicists want to measure a specific angle in this rulebook, called Gamma ($\gamma$). Knowing the exact value of $\gamma$ is crucial. If the measured value doesn't match the prediction, it means there is "New Physics" hiding in the shadows—something beyond our current understanding.

The Problem: The Foggy Mirror

To measure $\gamma$, scientists use a specific decay process involving a particle called a $B$ meson turning into a $D$ meson and a $K$ meson.

Think of the $D$ meson as a shapeshifting mirror. It can flip between being a particle ($D^0$) and an anti-particle ($\bar{D}^0$). When it decays into pions ($\pi^+\pi^-$), it creates a complex interference pattern, like ripples in a pond.

To read the value of $\gamma$ from this pattern, you need to know exactly how the "ripples" from the particle side differ from the "ripples" from the anti-particle side. This difference is called the Strong-Phase Difference.

The Analogy: Imagine trying to tune a radio to a specific station (measuring $\gamma$). But the radio dial is covered in a thick layer of fog (the strong-phase difference). If you don't know exactly how thick the fog is, you can't be sure which station you are listening to. Previous measurements of this "fog" were a bit blurry, leaving a lot of uncertainty in the final result.

The Solution: The "Quantum Twin" Experiment

The BESIII Collaboration (a team of scientists using a giant detector in Beijing) decided to clear up the fog. They used a special trick involving Quantum Entanglement.

1. The Setup: They smashed electrons and positrons together at a specific energy ($\psi(3770)$). This creates pairs of $D$ mesons that are "quantum twins."
2. The Twin Rule: Because they are twins, if one is a particle, the other must be an anti-particle, and vice versa. They are perfectly correlated.
3. The Tagging: They catch one twin (the "Tag") and see exactly what it is (e.g., "This is definitely a particle"). Because of the twin rule, they instantly know the other twin (the "Signal") is an anti-particle.
4. The Measurement: They watch how the "Signal" twin decays. Since they know exactly what it started as, they can measure the interference pattern (the ripples) with incredible precision, effectively measuring the "fog" thickness.

What They Did This Time

In the past, scientists used a smaller amount of data and had to rely on theoretical guesses (models) to fill in the gaps. It was like trying to guess the shape of a cloud by looking at a few pixels.

This paper reports on a massive upgrade:

• More Data: They collected 7.93 fb$^{-1}$ of data. That's about 2.7 times more data than their previous record. Imagine taking a blurry photo and then taking a high-definition 4K photo of the same scene.
• Model-Independent: Instead of guessing the shape of the cloud, they measured every single part of it directly. They didn't need to assume a specific model; the data spoke for itself.
• Two Types of Decay: They measured the "fog" for two different types of twins: those decaying into a Short-lived Kaon ($K_S$) and those decaying into a Long-lived Kaon ($K_L$).

The Results: Sharper Focus

The team calculated a set of numbers (called $c_i$ and $s_i$) that describe the "fog" in different regions of the decay.

• Precision: These are the most precise measurements of these numbers ever made.
• Consistency: The results matched their previous, less precise measurements, but with much smaller error bars.
• The "Constraint" Check: They also checked if using a theoretical model to help the measurement made things better. They found that while the model helped slightly, the raw data was so good that the model wasn't strictly necessary anymore. This is a huge win for "model-independent" science.

Why Does This Matter? (The Impact on $\gamma$)

Remember the radio station analogy? Now that the fog is cleared, the radio dial is much sharper.

• Before: The uncertainty in the "fog" measurement contributed about 1.5 degrees of error to the measurement of $\gamma$.
• Now: The uncertainty has dropped to about 0.9 degrees.

This might sound small, but in particle physics, it's a massive leap. It means that when experiments like LHCb (in Europe) and Belle II (in Japan) measure the angle $\gamma$ next, the "fog" from the BESIII experiment will no longer be the main thing holding them back. It clears the path to finding that elusive "New Physics."

Summary in One Sentence

The BESIII team used a massive amount of data from quantum-entangled particle twins to measure the "twist" between matter and anti-matter with record-breaking precision, effectively clearing the fog that was obscuring our view of the universe's fundamental rules.

Technical Summary

1. Problem Statement

The measurement of the Cabibbo-Kobayashi-Maskawa (CKM) unitary triangle angle $\gamma$ (also known as $\phi_3$) is a critical test of the Standard Model (SM) and a potential window into New Physics. The most precise direct measurement of $\gamma$ comes from the decay chain $B^\pm \to DK^\pm$, where the $D$ meson decays to a common final state such as $K^0_S \pi^+ \pi^-$.

The amplitude of this decay depends on the interference between $D^0$ and $\bar{D}^0$ amplitudes. To extract $\gamma$ without theoretical bias, one requires the strong-phase difference ($\Delta \delta_D$) between the $D^0$ and $\bar{D}^0$ decay amplitudes across the entire Dalitz plot phase space.

• The Challenge: Previous measurements relied on amplitude models to describe the $D$ decay, introducing model-dependent systematic uncertainties.
• The Goal: To perform a model-independent measurement of the strong-phase parameters ($c_i, s_i$) using quantum-correlated $D\bar{D}$ pairs produced at the $\psi(3770)$ resonance, utilizing a significantly larger dataset than previous analyses to reduce uncertainties and potentially remove model-dependent constraints entirely.

2. Methodology

Data Sample and Detector

• Experiment: BESIII detector at the BEPCII collider.
• Dataset: $7.93 \text{ fb}^{-1}$ of $e^+e^-$ annihilation data collected at $\sqrt{s} = 3.773 \text{ GeV}$, corresponding to the $\psi(3770)$ resonance where $D^0\bar{D}^0$ pairs are produced in a quantum-coherent $C$-odd state.
• Technique: The Double-Tag (DT) method. One $D$ meson is reconstructed in a specific "tag" mode (flavor-specific, CP-even, CP-odd, or self-conjugate), while the other is reconstructed in the signal mode ($K^0_{S/L}\pi^+\pi^-$).

Formalism and Binning

The Dalitz plot of the $K^0_{S/L}\pi^+\pi^-$ phase space is divided into bins ($i = \pm 1, \dots, \pm 8$) symmetric about the $m^2_+ = m^2_-$ axis. Three binning schemes are used:

1. Equal binning: Uniform phase space division.
2. Optimal binning: Maximizes sensitivity to $\gamma$.
3. Modified optimal binning: A variation of the optimal scheme.

The strong-phase differences are parameterized by amplitude-weighted averages of $\cos(\Delta \delta_D)$ and $\sin(\Delta \delta_D)$ in each bin, denoted as $c_i$ and $s_i$ for $K^0_S$ and $c'_i, s'_i$ for $K^0_L$.

Event Selection and Yields

• Signal Modes: $D \to K^0_S \pi^+ \pi^-$ and $D \to K^0_L \pi^+ \pi^-$.
• Tag Modes: Includes flavor tags ($K^+\pi^-$, etc.), CP eigenstates ($K^+K^-$, $\pi^+\pi^-$, etc.), and self-conjugate tags ($K^0_S \pi^+ \pi^-$ vs $K^0_S \pi^+ \pi^-$).
• Backgrounds: Peaking backgrounds (e.g., $D^0 \to \pi^+\pi^-\pi^+\pi^-$) and combinatorial backgrounds are estimated using inclusive Monte Carlo (MC) samples and subtracted or constrained in the fit.
• Efficiency Correction: Single-tag (ST) yields are used to normalize Double-tag (DT) yields, canceling many systematic uncertainties related to tracking and particle identification (PID).

Fitting Procedure

A maximum likelihood fit is performed to determine the parameters $c_i, s_i, c'_i, s'_i$.

• Unconstrained Fit: Determines parameters purely from data without external model inputs.
• Constrained Fit: Incorporates model-predicted differences ($\Delta c_i, \Delta s_i$) between $K^0_S$ and $K^0_L$ decays as Gaussian constraints. This utilizes the latest amplitude models (Belle and BESIII) and U-spin breaking parameters.
• Systematic Uncertainties: Evaluated using pseudo-experiments (1000 sets) varying inputs such as $K_i$ parameters, ST yields, MC statistics, and background fractions.

3. Key Contributions

1. Largest Dataset to Date: Utilization of $7.93 \text{ fb}^{-1}$ of $\psi(3770)$ data, more than doubling the previous BESIII dataset ($2.93 \text{ fb}^{-1}$) and significantly surpassing the CLEO dataset ($0.818 \text{ fb}^{-1}$).
2. Model-Independent Precision: The analysis provides the most precise model-independent determination of strong-phase parameters for $D \to K^0_{S/L}\pi^+\pi^-$ decays.
3. Removal of Model Dependence: The analysis demonstrates that with the current dataset size, the results are consistent with model predictions within $2\sigma$ even without constraints. This allows for the possibility of removing model-dependent constraints entirely in future $\gamma$ measurements, reducing systematic uncertainties.
4. Dual Decay Channel Analysis: Simultaneous measurement of both $K^0_S \pi^+ \pi^-$ and $K^0_L \pi^+ \pi^-$ modes, improving the constraints on the $K^0_L$ parameters which are historically more difficult to measure.

4. Key Results

• Parameters: The paper reports values for $c_i, s_i$ (for $K^0_S$) and $c'_i, s'_i$ (for $K^0_L$) across 8 bins for three different binning schemes.
  • Example (Equal Binning, Bin 1): $c_1 = 0.682 \pm 0.017 (\text{stat}) \pm 0.008 (\text{syst})$ (unconstrained).
• Comparison with Constraints: The unconstrained results are consistent with the model-predicted constraints within $2\sigma$. The constrained results show significantly reduced uncertainties (e.g., statistical uncertainty on $c_1$ reduced from $0.017$ to $0.015$).
• Systematic Uncertainties: Dominant sources include statistical fluctuations in the $K_i$ parameters and inputs for flavor tags. The total systematic uncertainties are generally smaller than the statistical ones.
• Impact on $\gamma$ Measurement:
  • Using the constrained parameters with the optimal binning scheme, the uncertainty contribution to the measurement of $\gamma$ is $0.9^\circ$.
  • Using the unconstrained parameters, the uncertainty is $1.5^\circ$.
  • The difference in the central value of $\gamma$ between constrained and unconstrained inputs is small ($\sim 0.4^\circ$).
  • Crucially, the uncertainty from these strong-phase inputs ($0.9^\circ$) is now sub-dominant compared to the statistical uncertainty of current $\gamma$ measurements at LHCb and Belle ($\sim 5^\circ$).

5. Significance

• Standard Model Test: These results provide the essential, high-precision inputs required for the next generation of $\gamma$ measurements at LHCb, Belle II, and future B-factories.
• Reduction of Systematics: By demonstrating that model-independent measurements can achieve precision comparable to or better than model-dependent constraints, this work paves the way for eliminating model bias in the determination of the CKM angle $\gamma$.
• Charm Physics: The results improve the understanding of charm mixing and indirect CP violation, providing inputs for other hadronic $D$ decay studies.
• Future Readiness: The precision achieved ensures that strong-phase uncertainties will not become the limiting factor in future high-luminosity experiments, allowing the full statistical potential of upcoming datasets to be realized.

In conclusion, the BESIII Collaboration has set a new benchmark for model-independent strong-phase measurements, significantly reducing the uncertainty in the determination of the CP-violating angle $\gamma$ and strengthening the foundation for future searches for New Physics.