Amplitude analysis and branching fraction measurement of the decay $D^0 \to K^+K^-\pi^0\pi^0$

Using 20.3 fb⁻¹ of $e^+e^-$ collision data collected by the BESIII detector, this paper presents the first amplitude analysis of the singly Cabibbo-suppressed decay $D^0 \to K^+K^-\pi^0\pi^0$, measuring its absolute branching fraction and revealing that the dominant intermediate process $D^0 \to K^{*}(892)^+K^{*}(892)^-$ is S-wave dominant with a longitudinal polarization fraction of approximately 0.47.

The Gist

Imagine the subatomic world as a bustling, chaotic dance floor. In this paper, the BESIII Collaboration acts like a team of high-speed photographers and detectives, trying to figure out exactly how a specific dancer, the $D^0$ meson (a type of charm particle), breaks apart into four smaller dancers: two kaons ($K^+$ and $K^-$) and two neutral pions ($\pi^0$).

Here is the story of their investigation, broken down into simple concepts:

1. The Setting: The Perfect Dance Hall

The scientists used the BESIII detector, a massive, high-tech camera located at the BEPCII particle accelerator in China. Think of the accelerator as a racetrack where they smash electrons and positrons (matter and antimatter) together at a very specific speed.

At this specific speed (3.773 GeV), the collision creates a special state called the $\psi(3770)$. This state is like a "factory" that almost exclusively produces pairs of $D^0$ and $\bar{D}^0$ mesons. It's the perfect environment because it's a clean factory floor with very little "trash" (background noise) to confuse the cameras. They collected data equivalent to 20.3 inverse femtobarns—a fancy way of saying they watched billions of these collisions.

2. The Mystery: How Does the Breakup Happen?

When the $D^0$ meson decays, it doesn't just explode into four pieces randomly. It usually goes through an intermediate step, like a relay race.

• The Race: The $D^0$ first splits into two temporary "middlemen" particles, which then decay into the final four particles.
• The Suspects: The scientists wanted to know: Which middlemen are doing the work? Is it the $K^*(892)$? The $\eta(1475)$? Or maybe the $f_1(1420)$?

This is where Amplitude Analysis comes in. Imagine you are trying to reconstruct a song just by listening to the final chord. You have to figure out which notes (intermediate particles) were played, how loud they were (magnitude), and when they started relative to each other (phase).

3. The Investigation: Sorting the Clues

The team used a technique called Double-Tagging.

• The Tag: They look at one side of the collision and say, "Okay, we definitely have a $D^0$ here."
• The Signal: Then they look at the other side to see how that partner $D^0$ decayed.
  By knowing exactly what the "tag" side is, they can be very sure about what happened on the "signal" side, filtering out the background noise.

They found 791 clean events where the $D^0$ decayed into $K^+K^-\pi^0\pi^0$.

4. The Big Discovery: The Dominant Dance Move

After running complex mathematical models (the "fit"), they found the most common way this decay happens:

• The Winner: The dominant process is $D^0 \to K^*(892)^+ K^*(892)^-$.
  • Analogy: Imagine the $D^0$ splits into two spinning tops ($K^*$ particles), which then wobble and break apart into the final four pieces.
• The Surprise: Theoretical models predicted that these spinning tops would mostly spin "sideways" (transverse polarization). However, the data showed they spin "up and down" (longitudinal polarization) about 47% of the time.
  • Why it matters: This is like predicting a coin toss will land on tails 90% of the time, but it actually lands on heads nearly half the time. This suggests our current understanding of how these particles interact (specifically "Final State Interactions") needs an update.

They also found other, less common dance moves involving particles like the $\eta(1475)$ and $K_1(1400)$, but the $K^*K^*$ pair was the star of the show.

5. The Result: How Often Does This Happen?

The scientists calculated the Branching Fraction, which is simply the percentage of the time this specific breakup happens out of all possible ways a $D^0$ can decay.

• The Number: It happens about 0.073% of the time.
• The Precision: They measured this with much higher precision than before (improving the accuracy by a factor of 2.5).

6. Why Should We Care?

You might ask, "Who cares about a charm meson breaking into pions?"

• Testing the Rules: The Standard Model of physics is our rulebook for how the universe works. But the "strong force" (which holds quarks together) is notoriously difficult to calculate.
• The Puzzle: By measuring exactly how these particles decay and how often, scientists can test their theories. If the theory says "Spin sideways" and the experiment says "Spin up and down," the theory needs to be rewritten.
• New Physics: Sometimes, these tiny discrepancies hint at "New Physics"—particles or forces we haven't discovered yet. While this paper confirms the Standard Model's general framework, the unexpected polarization results are a challenge for theorists to solve.

Summary

In short, the BESIII team took a snapshot of billions of particle collisions, filtered out the noise, and reconstructed the "movie" of how a $D^0$ meson breaks apart. They discovered that the most common way it breaks involves two specific intermediate particles, and surprisingly, these particles spin in a way that previous theories didn't fully predict. This helps refine our understanding of the fundamental forces that hold matter together.

Technical Summary

1. Problem and Motivation

The hadronic decays of charm mesons present a significant theoretical challenge because the charm quark mass is in an intermediate regime: it is too heavy for chiral perturbation theory to be reliable, yet too light for a robust heavy quark expansion. Consequently, non-perturbative methods are required, which rely heavily on precise experimental inputs to constrain model parameters.

Specifically, the decay $D^0 \to K^+K^-\pi^0\pi^0$ is a singly Cabibbo-suppressed (SCS) process. Theoretical models (such as Non-Relativistic Constituent Quark Models, Flavor SU(3) symmetry, and Factorization approaches) predict varying branching fractions and polarization states for the dominant intermediate process, $D^0 \to K^*(892)^+K^*(892)^-$.

• Polarization Puzzle: In $D \to VV$ (Vector-Vector) decays, the longitudinal polarization fraction ($F_L$) is a sensitive probe for Final State Interactions (FSIs) and potential new physics. Existing models predict transverse polarization dominance (low $F_L$) for this specific decay, but this had not been experimentally verified.
• Intermediate States: The decay involves complex intermediate resonances (e.g., $K_1$, $f_1$, $\eta$, $a_0$), and a comprehensive amplitude analysis is needed to disentangle these contributions and measure their relative magnitudes and phases.

2. Methodology

The analysis utilizes $e^+e^-$ collision data collected by the BESIII detector at the BEPCII storage ring.

• Data Sample: 20.3 fb$^{-1}$ of integrated luminosity collected at a center-of-mass energy of $\sqrt{s} = 3.773$ GeV, corresponding to the $\psi(3770)$ resonance where $D\bar{D}$ pairs are produced.
• Tagging Technique: The Double-Tag (DT) method is employed. One $D$ meson ($\bar{D}^0$) is reconstructed in specific "tag" modes ($\bar{D}^0 \to K^+\pi^-$, $K^+\pi^-\pi^0$, $K^+\pi^-\pi^-\pi^+$), while the other ($D^0$) is reconstructed in the signal mode $D^0 \to K^+K^-\pi^0\pi^0$. This technique suppresses background and allows for absolute branching fraction measurements.
• Event Selection:
  • Charged tracks and photons are selected with strict kinematic and particle identification (PID) criteria.
  • $\pi^0$ candidates are reconstructed from photon pairs with invariant mass constraints.
  • A kinematic fit constrains the four-momenta to the initial $e^+e^-$ system and known particle masses.
  • A $K_S^0$ mass veto is applied to suppress the background from $D^0 \to K_S^0 K^+K^-$.
  • The opening angle between the two $D$ mesons is required to be $>167^\circ$ to reject non-$D\bar{D}$ backgrounds.
• Signal Extraction: An unbinned 2D maximum likelihood fit to the beam-constrained mass ($M_{BC}$) distributions of the signal and tag sides yields a signal purity of 94.2%.
• Amplitude Analysis:
  • An Isobar model is used, where the total amplitude is a coherent sum of intermediate resonant processes: $\mathcal{M} = \sum c_n \mathcal{A}_n$.
  • Fit Method: An unbinned maximum likelihood fit is performed on the phase space variables. The signal PDF includes detection efficiency ($\epsilon$) and phase space factors.
  • Background Modeling: Background shapes are modeled using XGBoost machine learning techniques trained on inclusive Monte Carlo (MC) samples to predict background probabilities and efficiency corrections.
  • Resonances: The fit includes $K^*(892)$, $K_1(1400)$, $f_1(1420)$, $\eta(1475)$, $\eta(1405)$, and $a_0(980)$. Blatt-Weisskopf barrier factors and relativistic Breit-Wigner propagators are used.
  • Polarization: Helicity amplitudes ($H_0, H_\pm$) are constructed to calculate the longitudinal polarization fraction ($F_L$).

3. Key Contributions

• First Amplitude Analysis: This is the first amplitude analysis of the $D^0 \to K^+K^-\pi^0\pi^0$ decay, providing a complete map of the intermediate resonant structure.
• Polarization Measurement: It provides the first experimental measurement of the longitudinal polarization fraction for the $D^0 \to K^*(892)^+K^*(892)^-$ channel.
• Branching Fraction Precision: It presents a significantly more precise measurement of the absolute branching fraction compared to previous BESIII results.
• Model Testing: The results directly test theoretical predictions regarding polarization and branching fractions in SCS decays.

4. Key Results

A. Branching Fraction

The absolute branching fraction for the decay is measured as:
$$\mathcal{B}(D^0 \to K^+K^-\pi^0\pi^0) = (0.73 \pm 0.03_{\text{stat}} \pm 0.01_{\text{syst}}) \times 10^{-3}$$
This result is consistent with previous measurements but improves the precision by a factor of 2.5.

B. Dominant Intermediate Process

The dominant intermediate process is identified as $D^0 \to K^*(892)^+K^*(892)^-$.

• Branching Fraction: $\mathcal{B}(D^0 \to K^*(892)^+K^*(892)^-) = (2.61 \pm 0.31_{\text{stat}} \pm 0.15_{\text{syst}}) \times 10^{-3}$.
• Comparison with Theory: This value is significantly lower than predictions from Non-Relativistic Constituent Quark Models (NRCQM) and Factorization approaches (which predict $\sim 5.9 - 9.9 \times 10^{-3}$) but aligns well with the Flavor SU(3) symmetry model prediction ($2.4 \times 10^{-3}$).

C. Polarization and Wave Structure

• S-Wave Dominance: The amplitude analysis reveals that the $D^0 \to K^*(892)^+K^*(892)^-$ decay is S-wave dominant.
• Longitudinal Polarization Fraction ($F_L$):
  $$F_L = 0.468 \pm 0.046_{\text{stat}} \pm 0.011_{\text{syst}}$$
  This result is significantly higher than the theoretical prediction of $\sim 0.31-0.32$ (from NRCQM), representing a discrepancy of more than 3 standard deviations. This suggests that current models underestimate the longitudinal component or misrepresent the Final State Interaction dynamics.

D. Other Intermediate States

Significant contributions were also found from:

• $D^0 \to \eta(1475)\pi^0$ (with $\eta(1475) \to K^*K$)
• $D^0 \to \eta(1405)\pi^0$ (with $\eta(1405) \to a_0(980)\pi^0$)
• $D^0 \to K_1(1400)^+K^-$
• $D^0 \to f_1(1420)\pi^0$

5. Significance

• Theoretical Impact: The measurement of $F_L \approx 0.47$ challenges existing theoretical frameworks (specifically NRCQM) that predict transverse dominance in $D \to VV$ decays. This discrepancy highlights the need to refine models of Final State Interactions (FSIs) and non-perturbative QCD effects in charm decays.
• CP Violation Studies: Determining the CP-even fraction ($F_+ = 0.89 \pm 0.02$) and the detailed amplitude structure is crucial for future studies of CP violation in $D$ meson decays, as these parameters are inputs for time-dependent analyses.
• Standard Model Constraints: By providing precise branching fractions and polarization data, this work constrains the parameters of phenomenological models used to describe hadronic weak decays, aiding in the search for physics beyond the Standard Model in the charm sector.

In summary, this paper establishes a new benchmark for the understanding of four-body charm decays, resolving the polarization puzzle in the $K^*K^*$ channel and providing essential data for refining theoretical descriptions of strong interactions in the charm sector.