Improved measurements of the coherence factors and strong-phase differences in $D\to K^-π^+π^+π^-$ and $D\to K^-π^+π^0$ with quantum-correlated $D\bar{D}$ decays

Using a 7.93 fb$^{-1}$ dataset from the BESIII experiment, this study reports improved measurements of coherence factors and strong-phase differences for $D\to K^-\pi^+\pi^+\pi^-$ and $D\to K^-\pi^+\pi^0$ decays via quantum-correlated $D\bar{D}$ pairs, providing essential inputs to reduce the uncertainty on the CKM angle $\gamma$ in future LHCb and Belle II analyses.

The Gist

Imagine you are trying to solve a giant, three-dimensional puzzle where the pieces are constantly shifting and changing shape. In the world of particle physics, this puzzle is the Standard Model, our best theory of how the universe works. One of the most mysterious pieces of this puzzle is matter-antimatter asymmetry: Why does the universe exist at all? Why is there more matter (us) than antimatter (which should have annihilated us long ago)?

To solve this, physicists are hunting for a specific number called the angle $\gamma$ (gamma). Think of $\gamma$ as the "secret code" that explains why matter won the battle against antimatter.

The Problem: A Blurry Lens

To find this secret code, scientists look at a specific type of particle decay involving a $B$ meson turning into a $D$ meson and a kaon. However, the $D$ meson is a tricky character. It doesn't just sit still; it exists as a quantum superposition of two states ($D^0$ and $\bar{D}^0$) that can interfere with each other, like two waves in a pond crashing together.

When these waves crash, they create a pattern of constructive interference (waves adding up) and destructive interference (waves canceling out). To read the secret code ($\gamma$) correctly, physicists need to know exactly how these waves are interfering. They need to measure two things:

1. Coherence Factor ($R$): How "in sync" are the waves? (Are they perfectly aligned, or is the pattern messy?)
2. Strong-Phase Difference ($\delta$): What is the exact timing difference between the two waves?

If you try to measure the secret code with a blurry lens (uncertain values for $R$ and $\delta$), your result will be fuzzy. Previous measurements were a bit like looking at the waves through a foggy window.

The Solution: The BESIII Experiment

The BESIII Collaboration (a team of hundreds of scientists working at a particle accelerator in Beijing) decided to clear up the fog. They used a machine called the BEPCII collider, which smashes electrons and positrons together to create pairs of $D$ and $\bar{D}$ mesons.

Here is the clever trick they used, which we can call the "Quantum Twin" method:

1. The Twins: When the machine creates a $D$ meson, it always creates its twin, the $\bar{D}$ meson, at the exact same time. They are "quantum-correlated," meaning they are linked like a pair of magic dice. If you know the state of one, you instantly know the state of the other.
2. The Tag: The scientists let one twin decay into a simple, easy-to-spot pattern (the "Tag"). This acts like a label telling them exactly what the other twin was doing at that moment.
3. The Signal: They then watch the other twin decay into a complex, messy pattern (the "Signal"), specifically looking at decays involving a Kaon and three pions ($K^-\pi^+\pi^+\pi^-$) or a Kaon, a pion, and a neutral pion ($K^-\pi^+\pi^0$).

By comparing the "Tag" twin with the "Signal" twin, they can measure exactly how the waves in the Signal twin are interfering.

The New Results: Sharper Vision

This paper reports a massive upgrade in their data. They didn't just look at a few events; they analyzed a dataset equivalent to 7.93 inverse femtobarns of collisions (a huge amount of data, including new data collected in 2022).

They found:

• For the 3-pion decay: The waves are about 51% coherent (not perfectly in sync, but not random either). The timing difference (phase) is about 182 degrees.
• For the 2-pion decay: The waves are 75% coherent (very in sync). The timing difference is about 209 degrees.

The Analogy: Imagine trying to tune a radio. Before, the signal was staticky, and you could only guess the station. Now, with these new measurements, the static is gone, and the music is crystal clear.

Why Binning Matters

The scientists didn't just look at the whole picture; they sliced the data into four different "bins" (like cutting a cake into four slices). In some slices of the decay, the waves interfere differently than in others. By measuring each slice separately, they got a much more detailed map of the interference pattern. This is crucial because it allows for a "model-independent" measurement of the secret code $\gamma$, meaning they don't have to rely on theoretical guesses about how the particles behave.

The Big Picture: What Does This Mean?

These new, precise numbers are like handing the LHCb (at CERN) and Belle II (in Japan) experiments a high-definition map.

• Before: The uncertainty in the secret code $\gamma$ was limited by how blurry the map was.
• Now: With these new measurements, the "blur" is reduced. The paper estimates that future measurements of $\gamma$ using these new inputs will have an uncertainty of only 3.5 degrees.

This is a huge step forward. It brings us closer to understanding why the universe is made of matter. If the value of $\gamma$ turns out to be different from what the Standard Model predicts, it would be a smoking gun for New Physics—perhaps a new force or particle that we haven't discovered yet.

In short: The BESIII team used a quantum twin trick to take a super-clear photo of how subatomic particles interfere. This photo removes the fog from our understanding of the universe's fundamental secrets, bringing us one step closer to solving the mystery of why we exist.

Technical Summary

1. Problem and Motivation

The precise determination of the Cabibbo-Kobayashi-Maskawa (CKM) unitarity triangle angle $\gamma$ (also known as $\phi_3$) is a primary goal of heavy flavor physics, as it provides a stringent test of the Standard Model's mechanism for CP violation. The most theoretically clean method to measure $\gamma$ involves the interference between $B^- \to D K^-$ and $B^- \to \bar{D} K^-$ decays, where the $D$ meson decays to a final state common to both $D^0$ and $\bar{D}^0$.

For multi-body $D$ decays (such as $D \to K^-\pi^+\pi^+\pi^-$ and $D \to K^-\pi^+\pi^0$), the sensitivity to $\gamma$ depends on three hadronic parameters:

1. Coherence factor ($R_S$): Quantifies the degree of interference between Cabibbo-favored (CF) and doubly Cabibbo-suppressed (DCS) amplitudes across the phase space.
2. Amplitude ratio ($r_S^D$): The ratio of DCS to CF amplitudes.
3. Strong-phase difference ($\delta_S^D$): The CP-conserving phase difference between CF and DCS amplitudes.

Previous measurements of these parameters, particularly for the four-body mode $D \to K^-\pi^+\pi^+\pi^-$, suffered from large uncertainties, which limited the precision of $\gamma$ measurements at LHCb and Belle II. Furthermore, the phase space of multi-body decays exhibits significant variations in strong phases; treating the entire phase space as a single bin dilutes the sensitivity. A binned analysis, where the phase space is divided into regions with distinct strong-phase properties, is required to maximize sensitivity.

2. Methodology

The analysis utilizes quantum-correlated $D\bar{D}$ pairs produced in $e^+e^-$ annihilation at the $\psi(3770)$ resonance ($\sqrt{s} = 3.773$ GeV). At this energy, the $D\bar{D}$ pair is produced in a $C$-odd eigenstate, meaning the two mesons are in an antisymmetric quantum state. This allows for the simultaneous reconstruction of one $D$ meson in a "signal" mode ($S$) and the other in a "tag" mode ($T$), where the quantum correlation modifies the joint decay rate based on the hadronic parameters.

Data Sample:

• Experiment: BESIII detector at the BEPCII collider.
• Integrated Luminosity: $7.93 \text{ fb}^{-1}$ (collected in 2010, 2011, and 2022). This represents a significant increase over the previous BESIII dataset ($2.93 \text{ fb}^{-1}$).

Analysis Strategy:
The study employs a "double-tag" technique with three categories of tag modes:

1. CP Tags: Decays to CP eigenstates (e.g., $K^+K^-$, $\pi^+\pi^-$, $K_S^0 \pi^0$).
2. Like-Sign Flavour Tags: Decays to $K^-n\pi$ where the kaon charge matches the signal mode (e.g., $K^- \pi^+$, $K^- \pi^+ \pi^0$).
3. Self-Conjugate Tags: Decays to $K_S^0 \pi^+ \pi^-$ and $K_L^0 \pi^+ \pi^-$. These are crucial for determining the strong-phase variation across the Dalitz plot.

Key Improvements over Previous Work:

• Increased Statistics: Nearly triple the integrated luminosity.
• New Tag Mode: Inclusion of $D \to K_L^0 \pi^+ \pi^-$ as a tag mode, which was not used in the previous global analysis.
• Refined Background Treatment: Improved modeling of peaking backgrounds, particularly $D \to K_S^0 K^- \pi^+$ in the like-sign samples.
• Binned Analysis: The phase space of $D \to K^- \pi^+ \pi^+ \pi^-$ is partitioned into four bins based on an amplitude model (LHCb) to optimize $\gamma$ sensitivity.

Observables:
The analysis measures ratios of efficiency-corrected yields ($\rho$) for different tag combinations. These observables are sensitive to $R_S$, $\delta_S^D$, and $r_S^D$. A global $\chi^2$ fit is performed to extract these parameters, incorporating external inputs for mixing parameters ($x, y$) and branching fractions.

3. Key Contributions

1. Global Measurements: First high-precision determination of the coherence factors and average strong-phase differences for $D \to K^- \pi^+ \pi^+ \pi^-$ and $D \to K^- \pi^+ \pi^0$ using the full BESIII dataset.
2. Binned Parameters: First measurement of the hadronic parameters for $D \to K^- \pi^+ \pi^+ \pi^-$ in four distinct phase-space bins, enabling model-independent $\gamma$ extraction.
3. Systematic Control: Detailed evaluation of systematic uncertainties, including particle identification, reconstruction efficiency, background modeling, and external parameter constraints.
4. Impact Assessment: Quantitative projection of how these new inputs reduce the uncertainty on future $\gamma$ measurements at LHCb.

4. Results

The paper reports the following improved measurements (uncertainties include both statistical and systematic contributions):

Global (Unbinned) Results:

• Coherence Factor ($R_{K3\pi}$): $0.51 \pm 0.04$ (Improved precision compared to previous $0.52^{+0.12}_{-0.10}$).
• Strong-Phase Difference ($\delta_{K3\pi}^D$): $(182^{+14}_{-13})^\circ$.
• Coherence Factor ($R_{K\pi\pi^0}$): $0.75 \pm 0.03$.
• Strong-Phase Difference ($\delta_{K\pi\pi^0}^D$): $(209^{+7}_{-8})^\circ$.

Binned Results ($D \to K^- \pi^+ \pi^+ \pi^-$):
The phase space was divided into four bins. The fit yielded coherence factors and strong-phase differences for each bin (e.g., Bin 1: $R \approx 0.72$, $\delta \approx 124^\circ$; Bin 4: $R \approx 0.54$, $\delta \approx 260^\circ$). These values are consistent with LHCb amplitude model predictions within uncertainties.

Uncertainty Reduction:

• The uncertainty on the coherence factor for $D \to K^- \pi^+ \pi^+ \pi^-$ was reduced by a factor of 2–3 compared to the previous BESIII result and the CLEO-c analysis.
• The statistical uncertainty remains the dominant source of error, indicating that further data (e.g., the full $20 \text{ fb}^{-1}$ BESIII dataset) will yield further improvements.

5. Significance

The results have immediate and profound implications for the measurement of the CKM angle $\gamma$:

• Reduction of Systematic Uncertainty: In the $B^- \to DK^-$ analysis at LHCb, the uncertainty on $\gamma$ is currently limited by the precision of the input charm hadronic parameters. The new BESIII measurements are projected to reduce the contribution of these inputs to the total $\gamma$ uncertainty to approximately $3.5^\circ$.
• Competitive Precision: This projected uncertainty ($3.5^\circ$) is comparable to the statistical uncertainty expected from the current LHCb $B$-meson sample size. This means the BESIII inputs are no longer the bottleneck; the precision of $\gamma$ is now limited by the size of the $B$-physics datasets.
• Model Independence: The binned results allow for a model-independent determination of $\gamma$ using $D \to K^- \pi^+ \pi^+ \pi^-$, avoiding reliance on specific amplitude models for the $D$ decay dynamics.
• Charm Physics: These measurements also provide essential inputs for studying $D^0-\bar{D}^0$ mixing and CP violation in the charm sector, refining the understanding of resonance structures in multi-body $D$ decays.

In conclusion, this paper represents a major step forward in the precision of charm hadronic parameters, directly enabling the next generation of CP violation measurements in $B$ decays and solidifying the Standard Model's description of quark mixing.