Observation of a threshold enhancement in the $\pi^+\pi^-$ spectrum in $\psi(3686) \rightarrow \pi^{+}\pi^{-}J/\psi$ decays

Using a large sample of $\psi(3686)$ events collected by the BESIII detector, researchers discovered a statistically significant resonance-like structure near the $\pi^+\pi^-$ mass threshold that is well-described by a QCD multipole expansion model incorporating S-D wave mixing, while standard chiral perturbation theory fails to reproduce this enhancement.

The Gist

Imagine the subatomic world as a giant, chaotic dance floor. In this paper, the scientists from the BESIII collaboration are acting like high-tech bouncers and detectives, trying to figure out exactly what happens when two specific dancers, a $\psi(3686)$ (a heavy, excited "charmonium" particle) and a $J/\psi$ (a slightly calmer, heavier partner), interact.

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

1. The Setup: A Massive Dance Party

The researchers used a giant machine called the BESIII detector (located in Beijing) to watch billions of collisions. Think of this as setting up a super-fast camera to record a dance party that happens trillions of times a second.

They focused on a specific move: The heavy $\psi(3686)$ particle decays (breaks apart) into a $J/\psi$ particle and a pair of pions (tiny particles called $\pi^+$ and $\pi^-$).

• The Goal: They wanted to see the "mass spectrum" of these two pions. In plain English, they wanted to see how much energy the pions had when they were born together.

2. The Surprise: A "Bump" at the Door

Usually, when particles are created, their energy distribution looks like a smooth hill or a gentle curve. But when the scientists looked at the data for these pions, they saw something weird right at the very beginning (the "threshold").

• The Analogy: Imagine you are watching a crowd of people exit a stadium. You expect the crowd to flow out smoothly. Instead, you see a massive, sudden surge of people rushing out the very first second, creating a huge pile-up right at the exit door, before the flow settles down.
• The Discovery: They found a distinct "bump" or "enhancement" right at the lowest possible energy where these pions can exist. It was so clear and strong that the statistical chance of it being a fluke was less than 1 in a trillion (over 10 sigma).

3. The Mystery: What is this Bump?

The scientists had to ask: Is this a new particle? Is it a glitch in the machine? Or is it just the way the universe works?

• Ruling out "Pionium": One idea was that the pions were sticking together to form a temporary molecule called "pionium" (like a hydrogen atom made of pions). However, the math showed this new bump was too "wide" (it existed for a tiny fraction of a second) to be pionium. It was too short-lived to be a stable molecule.
• The "Adler Zero" Puzzle: There is a famous rule in physics (the Adler zero) that predicts the probability of this specific dance move should drop to zero if the pions are very slow. The data showed a dip near that zero point, which is a clue that the rules of "Chiral Symmetry" (a fundamental rule about how particles behave at low speeds) are at play.

4. The Investigation: Two Theories vs. The Data

To explain the bump, the team tested two different "rulebooks" (theoretical models) to see which one could predict what they saw.

• Theory A: Chiral Perturbation Theory (ChPT)

  • The Metaphor: This is like trying to predict the weather using a simple formula based on temperature and humidity. It works great for the "middle of the day" (higher energies), but it completely fails to predict the sudden storm at the "dawn" (the low-energy threshold).
  • Result: It couldn't explain the bump.

• Theory B: QCD Multipole Expansion (QCDME)

  • The Metaphor: This is a more complex model. It suggests that the heavy dancer ($\psi(3686)$) isn't just one type of dancer, but a mix of two different dance styles (S-wave and D-wave). When these two styles mix, they create a unique interference pattern.
  • The Twist: When they added "Final State Interactions" (FSI)—which is like accounting for how the pions bump into each other after they are born but before they fly away—this model matched the data almost perfectly.
  • Result: This model successfully reproduced the "bump" at the door and the "dip" nearby.

5. The Conclusion: Why It Matters

The paper concludes that:

1. We found something new: There is a real, physical enhancement of pions right at the energy threshold.
2. The "Mix" is real: The data strongly supports the idea that the $\psi(3686)$ particle is a mixture of different internal states (S-wave and D-wave).
3. The Rules are Complex: The interaction between heavy quarks (charm) and light particles (pions) is governed by a delicate balance of forces. The "dip" near 0.3 GeV suggests that the fundamental symmetry of the universe (Chiral Symmetry) is doing something very specific here.

In a nutshell:
The scientists watched billions of particle collisions and found a strange "traffic jam" of pions right at the starting line. By using a complex model that treats the heavy particle as a mix of two different states, they figured out that this jam is caused by the intricate dance of the strong nuclear force. It's a small bump in the data, but it tells us a huge story about how the building blocks of our universe stick together.

Technical Summary

1. Problem and Motivation

The paper addresses the dynamics of di-pion transitions in heavy quarkonium decays, specifically the process $\psi(3686) \to \pi^+\pi^-J/\psi$. This decay is a critical laboratory for testing Quantum Chromodynamics (QCD) in the non-perturbative regime.

• Theoretical Context: The transition is modeled as the emission of two soft gluons from the heavy quark pair, which subsequently hadronize into pions. Theoretical frameworks like Chiral Perturbation Theory (ChPT) and QCD Multipole Expansion (QCDME) are used to describe the $\pi\pi$ mass spectrum.
• The Puzzle: Previous measurements lacked the precision to resolve structures near the $\pi\pi$ threshold. Furthermore, the $\psi(3686)$ and $\psi(3770)$ states are hypothesized to be mixtures of $2^3S_1$ and $1^3D_1$ charmonium states, a mixing that should be evident in the decay dynamics.
• Specific Goal: To perform a high-precision measurement of the $\pi^+\pi^-$ invariant mass spectrum to search for threshold enhancements, test the "Adler zero" condition (which predicts the amplitude vanishes when one pion momentum approaches zero), and distinguish between competing theoretical models.

2. Methodology

The study utilizes a massive dataset collected by the BESIII detector at the BEPCII storage ring.

• Data Sample: The analysis is based on $(2712.4 \pm 14.4) \times 10^6$ $\psi(3686)$ events. From this, $3.7 \times 10^7$ signal candidates for $\psi(3686) \to \pi^+\pi^-J/\psi$ (with $J/\psi \to \ell^+\ell^-$) were selected.
• Event Selection:
  • Events with exactly four charged tracks and zero net charge.
  • Two low-momentum tracks ($< 1$ GeV/c) identified as pions; two high-momentum tracks ($> 1$ GeV/c) identified as leptons ($e$ or $\mu$).
  • Strict kinematic constraints: 5-constraint (5C) kinematic fit enforcing energy-momentum conservation and constraining the lepton pair mass to the nominal $J/\psi$ mass ($\chi^2_{5C} < 100$).
  • Angular cuts to suppress background from direct $e^+e^-$ annihilation.
• Background Suppression:
  • Extensive Monte Carlo (MC) simulations (2.7 billion $\psi(3686)$ decays) and data samples at $\sqrt{s} = 3.65$ GeV were used to estimate backgrounds.
  • The dominant background ($\psi(3686) \to \pi^+\pi^-J/\psi, J/\psi \to \pi^+\pi^-$) and non-resonant continuum were found to be negligible ($\sim 0.02\%$) and did not produce peaking structures near the threshold.
• Analysis Techniques:
  • Unfolding: To correct for detector resolution and efficiency distortions, the binned spectrum was unfolded using the iterative Bayesian method (RooUnfold).
  • Fitting:
    1. Resonance Hypothesis: An unbinned maximum likelihood fit using a Breit-Wigner (BW) function convolved with a Gaussian resolution function.
    2. Theoretical Models: A 2D binned maximum likelihood fit was performed on the unfolded spectrum (variables: $M_{\pi\pi}$ and $\cos\theta_{\pi^+}$) using:
       • ChPT: Including Final State Interactions (FSI) via the chiral unitary approach (CHUA).
       • QCDME: Modeling $\psi(3686)$ as an admixture of $S$- and $D$-wave states, also incorporating FSI.

3. Key Contributions and Results

A. Observation of Threshold Enhancement

The most significant finding is the first clear observation of a resonance-like structure near the $\pi^+\pi^-$ mass threshold.

• Breit-Wigner Fit: The fit to the threshold region ($[0.279, 0.325]$ GeV/$c^2$) yields:
  • Mass: $285.6 \pm 2.5$ MeV/$c^2$ (statistical) / $282.6 \pm 0.4 \pm 2.5$ MeV/$c^2$ (including systematics).
  • Width: $16.3 \pm 0.9$ MeV (statistical) / $17.3 \pm 0.8 \pm 0.4$ MeV (including systematics).
  • Significance: Exceeds 10$\sigma$.
• Nature of the Structure: The width corresponds to a lifetime orders of magnitude shorter than that of "pionium" ($\pi^+\pi^-$ Coulomb bound state), ruling out a conventional pionium interpretation. The structure suggests a strong interaction mechanism or exotic dynamics near the threshold.

B. Theoretical Model Comparison

The unfolded data was compared against theoretical predictions:

1. ChPT (without FSI): Failed to reproduce the threshold enhancement.
2. ChPT (with FSI): Successfully described the spectrum above 0.3 GeV/$c^2$ but failed to reproduce the threshold enhancement.
3. QCDME (without FSI): Reproduced the threshold enhancement but overshoot the spectrum at higher masses ($\sim 0.55$ GeV/$c^2$).
4. QCDME (with FSI): Provided the best overall description.
   • It successfully reproduced both the threshold enhancement and the high-mass region.
   • The fit required a mixing angle of $\theta_{mix} = 12^\circ$ between the $S$-wave ($2^3S_1$) and $D$-wave ($1^3D_1$) components of the $\psi(3686)$, supporting the hypothesis that $\psi(3686)$ is an $S$-$D$ wave mixture.
   • Discrepancy: A noticeable discrepancy remains in the low-mass region ($\chi^2/ndf \approx 33.7$ in the range $[0.279, 0.325]$ GeV/$c^2$) even with QCDME+FSI, suggesting missing physics in current models.

C. Systematic Uncertainties

A comprehensive error analysis was performed covering tracking efficiency, kinematic fits, resolution, unfolding iterations, and fit ranges. The total systematic uncertainties on the mass and width were determined to be approximately $2.5$ MeV/$c^2$ and $0.4$ MeV, respectively.

4. Significance

• QCD Dynamics: The observation provides a new, high-precision probe of low-energy QCD and the interplay between chiral symmetry and heavy-quark dynamics.
• Charmonium Structure: The results strongly support the hypothesis that $\psi(3686)$ is a mixture of $S$- and $D$-wave charmonium states, a conclusion derived from the success of the QCDME model with a specific mixing angle.
• Adler Zero and Chiral Symmetry: The pronounced dip observed near $0.3$ GeV/$c^2$ and the failure of standard ChPT to describe the threshold region offer new insights into the Adler zero condition and chiral symmetry breaking. The data suggests that the amplitude does not vanish as simply as predicted in the soft-pion limit, or that additional mechanisms are at play.
• Exotic Physics: The narrow, threshold-enhanced structure with a width inconsistent with pionium hints at potential exotic dynamics or strong interaction effects that are not yet fully understood, warranting further theoretical investigation.

In conclusion, this paper presents a landmark measurement that challenges existing theoretical descriptions of di-pion transitions in charmonium decays, highlighting the need for refined models of final-state interactions and the internal structure of excited charmonium states.