Study of $\phi\to K\bar{K}$ in the amplitude analysis of $D^{+}\to K_{S}^{0}K_{L}^{0}\pi^{+}$

Using 20.3 fb⁻¹ of data from the BESIII detector, this study presents the first amplitude analysis of the $D^{+} \to K_{S}^{0}K_{L}^{0}\pi^{+}$ decay, measuring its branching fraction and determining a relative branching fraction for $\phi \to K_{S}^{0}K_{L}^{0}$ versus $\phi \to K^{+}K^{-}$ that is significantly lower than previous world averages but consistent with isospin expectations.

The Gist

The Big Picture: Solving a 50-Year-Old Mystery

Imagine you have a mysterious machine that spits out pairs of shoes. Sometimes it makes a pair of left shoes (charged kaons), and sometimes it makes a pair of right shoes (neutral kaons).

For decades, physicists have been trying to figure out the ratio of left shoes to right shoes this machine produces. According to the basic rules of the universe (called "isospin symmetry"), the machine should make them in almost equal numbers. However, every time scientists have looked at the data from previous experiments, the machine seemed to be cheating—it was making far fewer right shoes than expected. This has been a confusing puzzle for 50 years.

This paper, written by the BESIII Collaboration, is like a new team of detectives who decided to look at the machine from a completely different angle. Instead of watching the machine run on its own, they watched it inside a specific type of "factory" (a decaying charm meson) to see if they could get a clearer, more honest count.

The Experiment: The "Tagging" Game

To solve this, the researchers used a massive dataset from the BESIII detector in China. They studied a specific event: a particle called a $D^+$ meson breaking apart into three pieces: a positive pion, a short-lived neutral kaon ($K_S$), and a long-lived neutral kaon ($K_L$).

To make sure they were counting the right events, they used a clever trick called "Tagging."

• The Analogy: Imagine a ballroom dance where every dancer has a partner. If you want to study the dance moves of the female dancers, you first find the male dancers.
• How it works: In this experiment, they first identified the "partner" particle (a $D^-$ meson) using known, easy-to-spot decay patterns. Once they found the partner, they knew exactly where to look for the "signal" ($D^+$) in the remaining debris. This ensured they had a very clean sample of the events they wanted to study, filtering out the noise.

The Detective Work: Amplitude Analysis

Once they had their clean sample, they didn't just count the shoes; they analyzed how the shoes were made. They used a technique called Amplitude Analysis.

• The Analogy: Imagine listening to a song that sounds like a mix of a guitar, a drum, and a violin. You can hear the whole song, but you want to know exactly how much of the sound comes from the guitar versus the drums.
• The Process: The researchers broke down the decay into its "ingredients." They found that the $D^+$ meson didn't just fall apart randomly. It mostly went through two main "paths" (intermediate steps):
  1. It briefly formed a $\phi$ meson (a specific type of particle) before breaking into the two neutral kaons.
  2. It formed other particles called $K^*$ resonances (like a temporary, unstable version of a kaon) before breaking apart.

By mathematically separating these paths, they could calculate exactly how often the $\phi$ meson was involved.

The Big Discovery: The Ratio is Different

The main goal was to measure the ratio of Neutral Kaons ($K_S K_L$) to Charged Kaons ($K^+ K^-$) produced by the $\phi$ meson.

• The Old View: Previous experiments suggested the ratio was around 0.74. This meant the $\phi$ meson was heavily biased against making neutral pairs, which broke the rules of symmetry.
• The New View: This new study found the ratio to be 0.628.

Why is this important?
This new number is significantly lower than the old average. In fact, it is much closer to 0.66 (or 2/3), which is what the rules of symmetry actually predict once you account for tiny differences in the mass of the particles.

Think of it like this: The old measurements were like looking at a blurry photo where the neutral shoes looked smaller than they really were. This new study took a high-definition photo and realized the neutral shoes were actually the right size all along. The "cheating" machine was just an illusion caused by how previous experiments were analyzed.

What They Also Found

While solving the shoe mystery, the team also measured:

1. The Branching Fraction: They calculated the exact probability of the $D^+$ meson turning into this specific trio of particles. It happens about 0.58% of the time.
2. The Phase Difference: They measured the "timing" or "phase" between the different paths the particles took. They found that the two main paths (involving $K_S$ and $K_L$) were almost perfectly out of step with each other (a difference of $\pi$ radians). This destructive interference (like noise-canceling headphones) explains why the total number of events is slightly less than the sum of the parts.

The Conclusion

The paper concludes that the long-standing puzzle of the $\phi$ meson's "broken symmetry" might not be broken at all. The new data from charm meson decays suggests that the $\phi$ meson behaves exactly as the laws of physics predict.

The authors suggest that the Particle Data Group (the organization that keeps the official record of all particle physics numbers) should update its global average to include these new findings. If they do, the "anomaly" that has confused physicists for decades might finally disappear, and the universe will look a little more symmetrical than we thought.

Technical Summary

Technical Summary: Study of $\phi \to K\bar{K}$ in the Amplitude Analysis of $D^+ \to K^0_S K^0_L \pi^+$

Problem and Motivation
The ratio of neutral-to-charged kaon-pair production rates from the $\phi$ meson, defined as $R^\phi_{KK} \equiv \mathcal{B}(\phi \to K^0_S K^0_L)/\mathcal{B}(\phi \to K^+ K^-)$, presents a long-standing puzzle in particle physics. Under isospin symmetry, this ratio is expected to be unity (after correcting for mass differences), yet experimental measurements have historically indicated significant isospin-symmetry breaking. Previous measurements, primarily derived from $e^+e^-$ annihilation experiments near the $\phi$ resonance, suffer from uncertainties related to cross-section parameterization, complex background interference (e.g., from $\omega$ and $\rho$ resonances), and sensitivity to the $\phi \to e^+e^-$ partial width. Furthermore, recent amplitude-model analyses consistent with unitarity have yielded values significantly lower than the current world average, highlighting a need for alternative measurement approaches to resolve this discrepancy.

Methodology
This study utilizes a dataset corresponding to an integrated luminosity of $20.3 \text{ fb}^{-1}$ collected by the BESIII detector at a center-of-mass energy of $\sqrt{s} = 3.773 \text{ GeV}$, where $D\bar{D}$ pairs are produced. The analysis employs a "tagging" technique to measure absolute branching fractions and select high-purity samples.

• Event Selection: Single-tag (ST) events are reconstructed where a $D^-$ meson decays into specific hadronic modes ($K^+\pi^-\pi^-$, $K^0_S\pi^-$, or $K^0_S\pi^+\pi^-\pi^-$). Double-tag (DT) events are formed by reconstructing the signal $D^+ \to K^0_S K^0_L \pi^+$ decay in association with the tagged $D^-$.
• Kinematic Constraints: The analysis utilizes beam-constrained mass ($M_{BC}$) and energy difference ($\Delta E$) variables for tag selection. For the signal, a kinematic fit constrains the total four-momentum to the initial $e^+e^-$ state and the invariant masses of the $D^\pm$ and $K^0_S$ candidates to their known values. The $K^0_L$ is identified via missing mass squared ($M^2_{miss}$).
• Background Suppression: Peaking backgrounds from $D^+ \to K^0_S \pi^+ \eta$ and $D^+ \to K^0_S K^0_S \pi^+$ are suppressed by vetoing reconstructed $\pi^0$ and $\eta$ candidates.
• Amplitude Analysis: An isobar model formulated with covariant tensors is used to describe the decay amplitude. The total amplitude is a coherent sum of intermediate processes. The fit includes the reference process $D^+ \to \phi \pi^+$ and other significant resonances such as $K^0_L K^{*}(892)^+$ and $K^0_S K^{*}(892)^+$. A maximum-likelihood fit determines the relative amplitudes and phases.
• Branching Fraction Measurement: The absolute branching fraction is calculated using the ratio of DT yields to ST yields, corrected for detection efficiencies and sub-decay branching fractions.

Key Contributions and Results

1. First Amplitude Analysis: This work presents the first amplitude analysis of the $D^+ \to K^0_S K^0_L \pi^+$ decay. The analysis identifies three significant intermediate contributions:

   • $D^+ \to \phi \pi^+$
   • $D^+ \to K^0_L K^{*}(892)^+$
   • $D^+ \to K^0_S K^{*}(892)^+$
     The phase difference between the $K^0_S K^{*}(892)^+$ and $K^0_L K^{*}(892)^+$ amplitudes is measured to be nearly $\pi$ radians, indicating destructive interference. The total fit fraction is $(107.7 \pm 1.5)\%$.

2. Branching Fraction Measurement: The absolute branching fraction for the decay is measured as:
   $$\mathcal{B}(D^+ \to K^0_S K^0_L \pi^+) = (5.780 \pm 0.085 \text{ (stat)} \pm 0.052 \text{ (syst)}) \times 10^{-3}$$

3. Determination of $R^\phi_{KK}$: By combining the measured branching fraction for the $\phi \pi^+$ component of the signal with the known value of $\mathcal{B}(D^+ \to \phi \pi^+, \phi \to K^+ K^-)$, the ratio $R^\phi_{KK}$ is determined:
   $$R^\phi_{KK} = 0.628 \pm 0.022 \text{ (stat)} \pm 0.015 \text{ (syst)} \pm 0.017 \text{ (external)}$$
   This result is significantly lower than the current world average of direct measurements but is consistent with the isospin expectation for the $\phi$ meson's coupling to charged and neutral kaon pairs.

4. Implications for $\phi$ Couplings: The measured $R^\phi_{KK}$ implies a coupling ratio $g_+/g_0$ (charged to neutral) consistent with unity, in contrast to the value significantly less than one implied by the PDG average of previous measurements. When combined with previous BESIII results on $D_s^+$ decays, the average $R^\phi_{KK}$ from charm decays is $0.611 \pm 0.023$.

Significance
The paper claims that this measurement provides a crucial alternative determination of $R^\phi_{KK}$ that is independent of the systematic limitations inherent in $e^+e^-$ annihilation studies. The result, which aligns with theoretical expectations of isospin symmetry (after mass corrections) and previous amplitude-model analyses, challenges the current world average derived from direct measurements. The authors suggest that the discrepancy warrants a comprehensive re-evaluation of $\phi$ decay branching fractions by the Particle Data Group, incorporating results from charm decays and recent $e^+e^-$ measurements (CMD2, CMD3) that are consistent with the charm-based average. Additionally, the study provides constraints on dynamical models of charmed meson decays through the measurement of singly Cabibbo-suppressed $D \to V P$ decay amplitudes.