Partial wave analyses of $ψ(3686)\to p\bar{p}π^0$ and $ψ(3686)\to p\bar{p}η$

Using a large sample of $\psi(3686)$ events collected by the BESIII detector, this study performs partial wave analyses of the decays $\psi(3686)\to p\bar{p}\pi^0$ and $\psi(3686)\to p\bar{p}\eta$ to determine their branching fractions, observe several well-established $N^*$ resonances, and measure the decay width ratio of the $N(1535)$ baryon to $N\eta$ and $N\pi$ final states.

The Gist

Imagine the subatomic world as a giant, chaotic dance floor. In this paper, scientists from the BESIII collaboration (a team of physicists working at a particle accelerator in China) are trying to figure out exactly who is dancing with whom, and what steps they are taking.

Here is a simple breakdown of what they did, why it matters, and what they found.

1. The Setting: The Particle Dance Floor

The scientists used a machine called a collider to smash electrons and positrons (anti-electrons) together. When these particles collide, they sometimes create a heavy, short-lived particle called $\psi(3686)$. You can think of this particle as a "super-dancer" that exists for a split second before it immediately breaks apart into smaller pieces.

In this study, they watched what happened when the $\psi(3686)$ broke apart into three specific dancers:

• A proton (a piece of normal matter, like in your body).
• An anti-proton (the "evil twin" of the proton).
• A neutral pion ($\pi^0$) or an eta meson ($\eta$) (tiny packets of energy that act like the "music" or "glue" holding the dance together).

They had a massive dataset: about 2.7 billion of these collisions. That's like watching a dance floor for a very long time to catch every possible move.

2. The Mystery: The "Ghost" Dancers ($N^*$)

When the $\psi(3686)$ breaks apart, the proton and the pion (or eta) don't just fly away randomly. Sometimes, they briefly stick together to form a "ghost" dancer called an $N^*$ resonance before falling apart again.

Think of it like this: Two people (the proton and the pion) meet, do a quick, fancy spin together (forming a temporary $N^*$), and then separate. The problem is, there are many different "ghost" dancers with different names (like $N(1440)$, $N(1535)$, etc.), and they all look very similar. It's hard to tell which one is actually there just by looking at the final crowd.

To solve this, the scientists used a technique called Partial Wave Analysis (PWA).

• The Analogy: Imagine you are in a dark room with a loud band. You can't see the instruments, but you can hear the sound waves. By analyzing the specific frequencies and how the sound waves interfere with each other, you can figure out exactly which instruments are playing and how they are being played.
• In physics, they analyzed the "angles" and "energies" of the particles to mathematically reconstruct which "ghost" dancers were present and how often they appeared.

3. The Big Puzzle: The $N(1535)$ Mystery

The main character of this story is a specific ghost dancer named $N(1535)$.

• The Problem: For decades, physicists have been confused by this particle. According to the standard "rulebook" (the Quark Model), this dancer should be lighter than another famous dancer named $N(1440)$. But experiments show $N(1535)$ is actually heavier.
• The Weird Behavior: This dancer also has a strange habit. It loves to decay (break apart) into a proton and an eta particle almost as much as it decays into a proton and a pion. Standard theory predicted it should barely touch the eta particle at all.

Why does this matter?
If $N(1535)$ is so heavy and loves eta particles so much, it suggests it might not be a standard "three-quark" particle (like a normal proton). It might be a five-quark particle or a "dynamically generated" blob of energy. It's like finding a dancer who moves in a way that suggests they have an extra limb hidden under their costume.

4. The Interference: The "Constructive" vs. "Destructive" Dance

One of the most clever parts of this paper is how they handled interference.

• The Analogy: Imagine two speakers playing the same song. If they are perfectly in sync, the sound gets louder (Constructive Interference). If one is slightly out of phase, the sound cancels out and gets quieter (Destructive Interference).
• In the particle world, the "ghost" dancers ($N^*$) can be created in two ways: directly from the heavy $\psi(3686)$, or from a "background" process (like a faint echo). These two paths can interfere with each other.
• The scientists found that depending on how these waves interfere, the total number of particles they see changes. They calculated two possible solutions:
  1. Constructive: The waves add up, making the signal stronger.
  2. Destructive: The waves cancel out, making the signal weaker.
     Both solutions fit the data, so they reported both possibilities.

5. The Results: What Did They Find?

After crunching the numbers on their 2.7 billion events, they found:

1. Confirmed the Ghosts: They successfully identified several well-known "ghost" dancers ($N^*$ states) in the data, measuring exactly how often they appear.

2. The Ratio: They measured the ratio of how often $N(1535)$ decays into an eta vs. a pion.

   • Old Prediction: 0.17 (It should rarely touch the eta).
   • Old Experiments: ~1.00 (It touches them equally, but with big errors).
   • This New Result: 0.99.
   • The Meaning: This confirms that $N(1535)$ loves the eta particle just as much as the pion. This strongly supports the idea that $N(1535)$ has a special structure (possibly containing "strange" quarks) that makes it behave differently than standard theory predicts.

3. The "12% Rule": There is a famous rule in physics that says certain decay rates should be related by a factor of 12.

   • For the pion ($\pi^0$) channel, the rule holds up.
   • For the eta ($\eta$) channel, the rule is violated. This is another clue that the eta channel is doing something exotic and special.

Summary

In simple terms, this paper is a massive, high-definition investigation into the "family tree" of subatomic particles. By watching billions of collisions, the scientists proved that a specific particle, $N(1535)$, is indeed a weirdo that breaks the standard rules of physics. It is heavier than expected and interacts with specific particles in a way that suggests it might be made of more than just the usual three building blocks.

This isn't just about counting particles; it's about rewriting the rulebook of how matter is built, suggesting that the universe might be even more complex and "exotic" than we thought.

Technical Summary

1. Problem Statement and Motivation

The primary physics goal of this study is to investigate the nature of the lowest-lying negative-parity nucleon resonance, $N(1535)$.

• The Mass Puzzle: According to the constituent quark model, the first orbital excitation $N(1535)$ should be lighter than the first radial excitation $N(1440)$ (Roper resonance). However, experiments show $N(1535)$ is $\sim 100$ MeV heavier.
• The Coupling Puzzle: The partial decay width ratio $\Gamma_{N(1535)\to N\eta} / \Gamma_{N(1535)\to N\pi}$ is experimentally observed to be $\approx 1.0$, whereas flavor symmetry predictions suggest a value of $\sim 0.17$.
• Theoretical Implications: These anomalies suggest $N(1535)$ may not be a simple three-quark state but could contain significant $s\bar{s}$ components (as a dynamically generated quasi-bound state in $K\Lambda/K\Sigma$ channels) or a pentaquark component.
• Previous Limitations: Previous measurements by BESIII (using $\sim 10^8$ events) and CLEO yielded large systematic uncertainties and inconclusive results regarding the interference between the $\psi(3686)$ resonance and the continuum process.

2. Methodology

The analysis utilizes a significantly larger dataset and advanced amplitude analysis techniques compared to previous studies.

• Data Sample: The BESIII detector collected $(2712 \pm 14) \times 10^6$ $\psi(3686)$ events.
• Decay Channels: The study focuses on:
  • $\psi(3686) \to p\bar{p}\pi^0$ (with $\pi^0 \to \gamma\gamma$)
  • $\psi(3686) \to p\bar{p}\eta$ (with $\eta \to \gamma\gamma$)
• Event Selection:
  • Requires two charged tracks (proton/antiproton) and at least two photon candidates.
  • Applies kinematic constraints (5C fit for $\psi(3686) \to p\bar{p}\gamma\gamma$) and vetoes on $J/\psi \to p\bar{p}$ background.
  • Backgrounds are estimated using inclusive Monte Carlo (MC) samples and continuum data taken at $\sqrt{s} = 3.773$ GeV.
• Partial Wave Analysis (PWA):
  • Amplitude Construction: Uses relativistic covariant tensor formalism. The total amplitude includes contributions from intermediate $N^*$ resonances (in the $p\pi^0$ or $p\eta$ system) and $p\bar{p}$ structures (modeled as $\rho$, $\omega$, $\phi$ states).
  • Continuum Interference: A critical innovation is the explicit modeling of the interference between the $\psi(3686)$ resonance amplitude and the non-resonant continuum amplitude ($e^+e^- \to p\bar{p}\pi^0/\eta$). The interference phase angle is treated as a free parameter.
  • Fit Method: An unbinned maximum likelihood fit is performed to determine complex coupling constants.
  • Continuum Subtraction: To isolate the $\psi(3686)$ contribution, the continuum yield measured at 3.773 GeV is scaled to the $\psi(3686)$ peak energy (3.686 GeV) using luminosity, Born cross-sections, and radiative corrections.

3. Key Contributions

• High-Precision Dataset: The analysis uses the full $\psi(3686)$ sample collected by BESIII, offering a statistical improvement over previous BESIII results by a factor of $\sim 25$.
• Interference Treatment: The paper rigorously addresses the ambiguity caused by the interference between the resonance and continuum processes. It presents two distinct solutions (constructive and destructive interference) for the branching fractions, acknowledging the phase angle ambiguity.
• Comprehensive Resonance Search: The analysis includes a wide range of established and candidate $N^*$ states ($N(1440)$ through $N(2570)$) and vector meson structures ($\rho$, $\omega$, $\phi$) in the $p\bar{p}$ system.
• Cross-Section Measurement: The observed cross-sections for $e^+e^- \to p\bar{p}\pi^0$ and $e^+e^- \to p\bar{p}\eta$ are measured at nine energy points around the $\psi(3686)$ peak to extract the branching fractions.

4. Key Results

A. Branching Fractions of $\psi(3686)$ Decays

Due to the interference ambiguity, two solutions are reported:

• $\psi(3686) \to p\bar{p}\pi^0$:
  • Constructive: $(133.9 \pm 11.2 \pm 2.3) \times 10^{-6}$
  • Destructive: $(183.7 \pm 13.7 \pm 3.2) \times 10^{-6}$
• $\psi(3686) \to p\bar{p}\eta$:
  • Constructive: $(61.5 \pm 6.5 \pm 1.1) \times 10^{-6}$
  • Destructive: $(84.4 \pm 6.9 \pm 1.4) \times 10^{-6}$

B. Observation of $N^*$ States

Several well-established $N^*$ resonances were observed with high significance ($>5\sigma$) in the $p\pi^0$ and $p\eta$ systems. The branching fractions for these intermediate states were measured, including:

• $N(1440)$, $N(1520)$, $N(1535)$, $N(1650)$, $N(1710)$, $N(1720)$, $N(2100)$, $N(2300)$, and $N(2570)$.
• Notably, the $N(940)$ (virtual proton pole) was found significant in the $\pi^0$ channel but negligible in the $\eta$ channel.

C. The Ratio $\Gamma_{N(1535)\to N\eta} / \Gamma_{N(1535)\to N\pi}$

This is the most significant physics result. By combining the measured yields of $N(1535)$ in both channels and correcting for isospin factors:
$$\frac{\Gamma_{N(1535)\to N\eta}}{\Gamma_{N(1535)\to N\pi}} = 0.99 \pm 0.05 \text{ (stat)} \pm 0.19 \text{ (syst)}$$

• This result is consistent with the PDG average of $1.00 \pm 0.40$ but with significantly improved precision.
• It strongly supports the hypothesis that $N(1535)$ has a large $s\bar{s}$ component (or pentaquark nature), explaining the enhanced coupling to $\eta$ mesons.

D. The "12% Rule"

The ratio of branching fractions $B(\psi(3686) \to p\bar{p}M) / B(J/\psi \to p\bar{p}M)$ was tested against the "12% rule" (which predicts a ratio of $\approx 12\%$ for hadronic decays):

• $p\bar{p}\pi^0$: The ratio is $(11.3 \pm 1.2)\%$ (constructive) or $(15.4 \pm 1.6)\%$ (destructive). This confirms the 12% rule.
• $p\bar{p}\eta$: The ratio is $(3.1 \pm 0.4)\%$ (constructive) or $(4.2 \pm 0.4)\%$ (destructive). This significantly violates the 12% rule, further indicating the exotic nature of the $N(1535)$ and its specific coupling dynamics.

5. Significance

• Resolution of the $N(1535)$ Puzzle: The precise measurement of the decay width ratio provides strong experimental evidence for the non-standard quark structure of the $N(1535)$, favoring models with significant $s\bar{s}$ admixture or pentaquark components over simple three-quark models.
• Methodological Advancement: The study demonstrates the critical importance of accounting for resonance-continuum interference in $e^+e^-$ annihilation decays. Ignoring this effect (as in previous BESIII analyses) leads to ambiguous or incorrect branching fraction determinations.
• Benchmark for Theory: The high-precision branching fractions for various $N^*$ states and the violation of the 12% rule in the $\eta$ channel provide stringent constraints for theoretical models of baryon spectroscopy and QCD dynamics.

In conclusion, this paper represents a major step forward in understanding nucleon resonances, utilizing the world's largest $\psi(3686)$ dataset to resolve long-standing questions about the nature of the $N(1535)$ resonance.