First Observation of \boldmath{$D^+ \to a_0(980)\rho$ and $D^+ \to a_0(980)^+ f_0(500)$} in \boldmath{$D^+ \to \pi^+\pi^+\pi^-\eta$ and $D^+ \to \pi^+\pi^0\pi^0\eta$} Decays

Using 20.3 fb⁻¹ of $e^+e^-$ collision data collected by the BESIII detector, this study presents the first amplitude analysis of singly Cabibbo-suppressed $D^+ \to \pi^+\pi^+\pi^-\eta$ and $D^+ \to \pi^+\pi^0\pi^0\eta$ decays, resulting in precise measurements of their absolute branching fractions and the first observation of the $D^+ \to a_0(980)^+ f_0(500)$ decay alongside the $D^+ \to a_0(980)\rho$ modes.

The Gist

Imagine the universe as a giant, high-speed particle collider, a cosmic "smash-up" zone where scientists crash particles together to see what happens. In this paper, the BESIII Collaboration (a team of hundreds of scientists from around the world) acted like master detectives at a crime scene, but instead of a murder, they were investigating the mysterious breakup of a specific particle called the $D^+$ meson.

Here is the story of their discovery, broken down into simple concepts and everyday analogies.

1. The Crime Scene: A Particle Breakup

Think of the $D^+$ meson as a heavy, unstable suitcase. When it breaks apart, it doesn't just fall apart randomly; it usually splits into smaller, recognizable pieces (like pions and an eta particle).

The scientists wanted to know: How does this suitcase break apart? Does it split into two big chunks first, which then break down further? Or does it just shatter into dust?

They looked at two specific ways this suitcase broke:

1. Case A: Into two positive pions, one negative pion, and an eta particle.
2. Case B: Into one positive pion, two neutral pions, and an eta particle.

2. The Mystery: The "Ghost" Particles

For decades, physicists have been puzzled by a group of particles called scalar mesons (specifically the $a_0(980)$ and the $f_0(500)$).

• The Analogy: Imagine you see a shadow in a room. You know something is casting it, but you can't quite tell what the object looks like. Is it a simple ball? A complex machine? Or is it actually two balls stuck together?
• The Problem: These scalar mesons are the "shadows." Standard physics theories (the "rulebook" of how particles are built) say they should be simple pairs of quarks (like a basic Lego brick). But their behavior is weird. They seem too heavy, or they decay in ways that shouldn't happen if they were simple bricks.
• The Theory: Some scientists think these aren't simple bricks at all. They might be "tetraquarks" (four quarks stuck together) or "molecules" (two particles holding hands loosely).

3. The Investigation: Amplitude Analysis

To solve this, the scientists didn't just count how many times the suitcase broke; they performed a 3D reconstruction of the crash. This is called "Amplitude Analysis."

Imagine a car crash. A simple count tells you "10 cars crashed." But an amplitude analysis is like having a video camera that shows exactly how the cars hit each other, which parts flew off first, and what the debris looked like.

They used a massive dataset (20.3 "inverse femtobarns" of data—think of this as a library containing millions of these crash videos) to map out every possible path the particles could take.

4. The Big Discoveries

Discovery A: The "Impossible" Double-Decker

The team found a specific path where the $D^+$ meson broke into two heavy, complex particles at once: the $a_0(980)$ and the $f_0(500)$.

• Why it's a big deal: According to the old "simple brick" theory, this should be extremely rare, like finding a four-leaf clover in a field of three-leaf clovers.
• The Result: They found it happening all the time. The frequency was surprisingly huge.
• The Metaphor: It's like expecting a car to break into two small wheels, but instead, it breaks into two giant engines. The fact that this happens so often suggests the "engines" (the scalar mesons) are built differently than we thought—likely as complex "tetraquarks" rather than simple pairs.

Discovery B: The "Dance Partner" Ratio

They also looked at how the $D^+$ meson paired up with a particle called the $\rho$ (rho) meson.

• The Puzzle: In the old theory, one specific dance move (pairing with a neutral $\rho$) should be much more common than another (pairing with a charged $\rho$).
• The Result: They measured the ratio and found it was 0.55. This is the opposite of what the simple theory predicted!
• The Meaning: This suggests that after the initial crash, the particles are "dancing" around each other and interacting strongly (a phenomenon called Final State Interaction) before settling down. It's like two dancers bumping into each other and changing their steps mid-dance.

Discovery C: The Missing Axial-Vector

They also looked for a particle called the $a_1(1260)$, which usually shows up in similar crashes.

• The Result: It was missing.
• The Metaphor: It's like going to a party where you always see a specific DJ, but tonight, the DJ is nowhere to be found. This "missing person" suggests that this specific type of particle breakup has unique, weird rules that we don't fully understand yet.

5. The Conclusion: Rewriting the Rulebook

The scientists concluded that the universe is playing by different rules than the "simple brick" model suggests.

• The Takeaway: The light scalar mesons ($a_0$ and $f_0$) are likely exotic structures (tetraquarks or molecules). The fact that the $D^+$ meson breaks into them so easily is the "smoking gun" evidence.
• Why it matters: Just as finding a new type of animal changes our understanding of biology, finding these exotic structures changes our understanding of the Strong Force (the glue that holds the universe together). It tells us that nature is more creative and complex than our current textbooks admit.

In short: The BESIII team looked at a billion particle crashes, found that the debris was forming complex, "exotic" shapes much more often than anyone expected, and proved that the building blocks of matter are stranger and more interconnected than we previously believed.

Technical Summary

1. Problem and Motivation

The internal structure of light scalar mesons, particularly the $f_0(500)$ (also known as $\sigma$), $f_0(980)$, and $a_0(980)$, remains a subject of intense debate in particle physics. While their existence is established, their classification within the standard constituent quark model (as conventional $q\bar{q}$ states) faces significant theoretical challenges. These challenges include:

• The unexpectedly large mass of the $a_0(980)$.
• The observation of OZI-suppressed decays like $\phi \to \gamma a_0(980)^0$.
• Anomalously large branching fractions (BFs) in decays involving scalar mesons that contradict predictions based on simple $q\bar{q}$ topological diagrams (specifically external $W$-emission).

Alternative hypotheses suggest these states may be compact tetraquarks, two-meson molecules, or mixed states. The production of such exotic structures is often governed by Final State Interactions (FSIs). Charmed meson decays ($D$ mesons) provide an exceptional laboratory to study these dynamics because they are dominated by non-perturbative QCD effects.

Specifically, this paper addresses two key gaps:

1. $D \to SV$ Decays: The ratio of $D^+ \to a_0(980)^+ \rho^0$ to $D^+ \to a_0(980)^0 \rho^+$ is predicted to be very small in the $q\bar{q}$ model due to suppression, but large in tetraquark models.
2. $D \to SS$ Decays: The channel $D^+ \to a_0(980)^+ f_0(500)$ (a scalar-scalar decay) has been rarely explored. In the $q\bar{q}$ model, this should be highly suppressed due to the small decay constant of $a_0(980)^+$. However, if $a_0(980)$ and $f_0(500)$ are tetraquarks, the hadronization process could significantly enhance the branching fraction.

2. Methodology

The study utilizes $e^+e^-$ collision data collected by the BESIII detector at the center-of-mass energy $\sqrt{s} = 3.773$ GeV, corresponding to the $\psi(3770)$ resonance. The dataset corresponds to an integrated luminosity of 20.3 fb$^{-1}$.

Key Experimental Techniques:

• Double-Tag (DT) Method: To measure absolute branching fractions with high precision, the collaboration uses the DT technique. One $D$ meson ($D^-$) is reconstructed in a "tag" mode (five specific hadronic decay modes), and the signal $D^+$ is reconstructed in the decay channels of interest within the same event. This method eliminates dependence on integrated luminosity and the $D\bar{D}$ production cross-section.
• Signal Channels:
  • $D^+ \to \pi^+ \pi^+ \pi^- \eta$
  • $D^+ \to \pi^+ \pi^0 \pi^0 \eta$
• Amplitude Analysis: An unbinned maximum likelihood fit is performed on the four-body phase space. The total amplitude is modeled as a coherent sum of intermediate resonant processes.
  • Resonance Modeling: The $a_0(980)$ is described using a coupled-channel Flatté formula; $\rho(770)$ uses the Gounaris-Sakurai lineshape; $f_0(500)$ follows a specific parameterization; other resonances ($\eta(1405)$, $f_1(1285)$, etc.) use relativistic Breit-Wigner functions.
  • Symmetry: Amplitudes are symmetrized to satisfy Bose-Einstein statistics for identical pions. Isospin constraints are applied via Clebsch-Gordan coefficients.
• Background Handling:
  • Background probability density functions (PDFs) are modeled using an XGBoost machine learning classifier trained on inclusive Monte Carlo (MC) samples to distinguish background from signal phase space.
  • A 2D fit on the beam-constrained mass ($M_{BC}$) of the tag and signal sides is used to extract yields.

3. Key Contributions and Results

A. First Observation of $D^+ \to a_0(980)^+ f_0(500)$

The most significant finding is the first observation of the $D^+ \to a_0(980)^+ f_0(500)$ decay in both $D^+ \to \pi^+ \pi^+ \pi^- \eta$ and $D^+ \to \pi^+ \pi^0 \pi^0 \eta$ channels.

• Significance: The statistical significance exceeds 10$\sigma$ in both channels.
• Branching Fraction: The measured BF is unexpectedly large, reaching the order of $10^{-3}$ (specifically $7.5 \times 10^{-4}$ for the charged mode and $8.4 \times 10^{-4}$ for the neutral mode).
• Implication: This large BF strongly contradicts the prediction for a conventional $q\bar{q}$ $a_0(980)$ meson (which would be suppressed) and provides compelling evidence supporting the tetraquark interpretation of light scalar mesons.

B. First Observation of $D^+ \to a_0(980)\rho$ and Ratio Measurement

The paper reports the first observation of both $D^+ \to a_0(980)^+ \rho(770)^0$ and $D^+ \to a_0(980)^0 \rho(770)^+$ decays.

• Significance: Both modes are observed with significance $> 10\sigma$.
• Ratio Measurement: The ratio $r_{+/0} \equiv \frac{\mathcal{B}(D^+ \to a_0(980)^+ \rho^0)}{\mathcal{B}(D^+ \to a_0(980)^0 \rho^+)}$ is measured for the first time:
  $$r_{+/0} = 0.55 \pm 0.08 (\text{stat}) \pm 0.05 (\text{syst})$$
• Implication: This ratio deviates significantly from the small value predicted by the $q\bar{q}$ model (where external $W$-emission is suppressed). The result implies substantial Final State Interaction (FSI) effects and supports the exotic nature of the $a_0(980)$.

C. Precision Branching Fraction Measurements

The absolute branching fractions for the parent decays are measured with unprecedented precision (threefold improvement over previous measurements):

• $\mathcal{B}(D^+ \to \pi^+ \pi^+ \pi^- \eta) = (3.20 \pm 0.06_{\text{stat}} \pm 0.03_{\text{syst}}) \times 10^{-3}$
• $\mathcal{B}(D^+ \to \pi^+ \pi^0 \pi^0 \eta) = (2.43 \pm 0.11_{\text{stat}} \pm 0.04_{\text{syst}}) \times 10^{-3}$

D. Negative Result on $a_1(1260)$

The study searched for $D^+ \to a_1(1260)^+ \eta$ but found no significant signal. This contradicts the abundant $a_1(1260)$ signals seen in other $D \to P3\pi$ decays, suggesting anomalous dynamics specific to this final state.

4. Significance

This work provides a critical breakthrough in understanding the nature of light scalar mesons:

1. Validation of Exotic Models: The unexpectedly large branching fractions for $D \to SS$ and $D \to SV$ decays involving $a_0(980)$ and $f_0(500)$ strongly favor models where these mesons are tetraquark states or have significant tetraquark components, rather than simple $q\bar{q}$ mesons.
2. Role of FSIs: The results highlight the crucial role of Final State Interactions in charmed meson decays, which can enhance or suppress decay rates in ways not predicted by simple topological diagrams.
3. New Window into QCD: By probing the $D \to SS$ channel, the study opens a new window into the fundamental dynamics of charmed meson decays and the hadronization of scalar particles.

In summary, the BESIII Collaboration has utilized a high-statistics dataset and advanced amplitude analysis techniques to overturn previous theoretical expectations regarding the suppression of scalar meson production in $D$ decays, offering strong empirical support for the exotic structure of the $a_0(980)$ and $f_0(500)$.