Search for the charmonium weak decay $ψ(2S)\to D_s^-π^+ + c.c.$ and $ψ(2S)\to D_s^-ρ^+ + c.c.$

Using a dataset of approximately 2.71 million $\psi(2S)$ events collected with the BESIII detector, this study reports the first search for the weak decays $\psi(2S)\to D_s^-\pi^+$ and $\psi(2S)\to D_s^-\rho^+$, finding no signal and establishing 90% confidence level upper limits on their branching fractions of $1.4\times 10^{-6}$ and $7.0\times 10^{-6}$, respectively.

The Gist

The Big Picture: Hunting for a Ghost in the Machine

Imagine the $\psi(2S)$ particle as a very heavy, very unstable "super-ball" made of two charm quarks holding hands. Usually, when this ball breaks apart, it does so with a massive explosion of energy, following the strict rules of the Strong Force (like a rubber band snapping). It breaks into other heavy particles, but it cannot break into a single heavy particle and a light one because it doesn't have enough energy to do so without breaking the laws of physics.

However, there is a tiny, almost magical possibility: the Weak Force. This is a much weaker, slower force (like a gentle breeze compared to the rubber band snap). The paper asks: Could this heavy super-ball ever decay via this gentle breeze into a specific, rare combination of particles?

Specifically, the scientists are looking for the ball to turn into a $D_s^-$ meson (a heavy particle) and either a $\pi^+$ (a light pion) or a $\rho^+$ (a slightly heavier pion).

Why does this matter?
According to the Standard Model (our current rulebook for the universe), this should happen so rarely that it's like finding a specific grain of sand on all the beaches of Earth. The predicted rate is about 1 in 10 billion.

• The Catch: If the scientists find more of these events than the rulebook predicts, it means the rulebook is wrong. It would be a smoking gun for New Physics—perhaps hidden dimensions, new particles, or forces we haven't discovered yet.

The Experiment: The Great Filter

The BESIII collaboration (a team of scientists using a giant detector in China) decided to play "Find the Needle in the Haystack."

1. The Haystack: They collected data from 2.7 billion $\psi(2S)$ events. Imagine a stadium filled with 2.7 billion people, and they are looking for two specific people wearing a red hat and a blue shoe.
2. The Filter (Event Selection): They built a complex set of rules to filter out the crowd.
   • They looked for specific "fingerprints": a specific mix of charged particles (electrons, pions, kaons) and photons (light particles).
   • They used a Blind Analysis method. This is like a chef tasting a soup before adding the secret ingredient. They set all their rules using computer simulations first, then looked at a small "taste test" sample of real data to make sure their rules worked, and only then did they look at the full 2.7 billion events. This prevents them from accidentally "seeing" what they want to see.

The Search: Looking for the Invisible

The tricky part is that one of the particles in the decay is a neutrino.

• The Analogy: Imagine you are watching a magic trick where a rabbit disappears. You see the hat, you see the wand, but the rabbit is gone. You know the rabbit must have gone somewhere because of conservation of energy, but you can't see it.
• In this experiment, the neutrino is the invisible rabbit. The scientists couldn't see it directly. Instead, they calculated where it should be by measuring everything else that was visible and seeing what was "missing." If the missing energy and momentum matched the profile of a neutrino, it was a potential candidate.

The Results: The Great Silence

After running their filters through the 2.7 billion events, what did they find?

• For the $\pi^+$ channel: They saw 9 events.
• For the $\rho^+$ channel: They saw 19 events.

But here is the twist: When they looked at the background noise (the "static" of the universe), they realized that these 9 and 19 events were likely just random accidents. They were indistinguishable from the background noise. It was like hearing a whisper in a crowded room and realizing it was just someone clearing their throat, not a secret message.

The Conclusion:
They found no signal. The universe remained silent.
Because they didn't find the "ghost," they set a new Upper Limit. This is like saying: "We looked everywhere, and if this ghost exists, it must be so rare that it happens less than 1.4 times in a million tries (for the pion) and 7 times in a million tries (for the rho)."

Why This is Still a Win

Even though they didn't find the "New Physics" they were hoping for, this is a huge success for science:

1. Ruling Out Possibilities: They have proven that certain "New Physics" theories (which predicted these decays would happen 100 or 1,000 times more often) are likely wrong. They have narrowed the search space.
2. Setting the Bar: They have established the strictest limits ever on these specific decays.
3. The Future: The paper concludes that while the current data isn't sensitive enough to find the real Standard Model prediction (which is even rarer), the experiment proved the method works. They need more data (a bigger haystack) to keep looking.

In short: The scientists built a super-sensitive metal detector, scanned a massive field, and found no gold. But by proving there is no gold in this specific spot, they have helped map out where the gold might be hidden elsewhere in the universe.

Technical Summary

1. Problem Statement and Motivation

• Kinematic Constraints: The mass of the $\psi(2S)$ resonance lies below the threshold for producing a $D\bar{D}$ pair. Consequently, strong decays into open charm are kinematically forbidden.
• Theoretical Context: While strong decays are forbidden, weak decays of the $\psi(2S)$ into a single charmed meson (e.g., $D_s^-$) are allowed within the Standard Model (SM). However, these processes are extremely rare, with SM predictions for branching fractions (BFs) on the order of $10^{-10}$ or lower.
• New Physics Potential: Several Beyond the Standard Model (BSM) scenarios, such as Top-color models, Minimal Supersymmetric SM (with/without R-parity), and Two-Higgs-Doublet models, predict that the branching fractions for these weak decays could be enhanced by 2–3 orders of magnitude compared to SM expectations.
• Objective: The primary goal is to perform the first experimental search for the specific weak decay modes $\psi(2S) \to D_s^- \pi^+$ and $\psi(2S) \to D_s^- \rho^+$ (and their charge conjugates) to test SM predictions and constrain new physics models.

2. Methodology

The analysis utilizes a "stepwise blind analysis" approach to avoid bias, ensuring that selection criteria are finalized before examining the full signal region in the data.

Data Sample and Detector

• Dataset: The analysis is based on $(2712.4 \pm 14.3) \times 10^6$ $\psi(2S)$ events collected at a center-of-mass energy $\sqrt{s} = 3.686$ GeV.
• Detector: The BESIII detector at the BEPCII storage ring. Key subsystems include a Multi-Drift Chamber (MDC), Time-of-Flight (TOF), and a CsI(Tl) Electromagnetic Calorimeter (EMC).

Event Reconstruction Strategy

• Decay Chain: The analysis reconstructs the $D_s^-$ via its semileptonic decay $D_s^- \to \phi e^- \bar{\nu}_e$, where $\phi \to K^+ K^-$.
  • Rationale: Semileptonic decays are chosen over hadronic ones because the $\psi(2S)$ predominantly decays via the strong interaction into hadrons. The presence of an electron and a neutrino (missing energy) significantly suppresses the background from strong decays.
• Final State Particles:
  • For $\psi(2S) \to D_s^- \pi^+$: Final state is $K^+ K^- \pi^+ e^-$.
  • For $\psi(2S) \to D_s^- \rho^+$: The $\rho^+$ is reconstructed via $\rho^+ \to \pi^+ \pi^0$ ($\pi^0 \to \gamma\gamma$). Final state is $K^+ K^- \pi^+ e^- \gamma \gamma$.
• Selection Criteria:
  • Tracking & PID: Charged tracks must satisfy vertex requirements ($|V_{xy}| < 1$ cm, $|V_z| < 10$ cm). Particle Identification (PID) combines $dE/dx$ and TOF measurements. Electrons are further constrained by $E/p$ ratios in the EMC.
  • Kinematic Fits:
    • $\phi$ candidates: $K^+K^-$ invariant mass within $(1.005, 1.035)$ GeV/$c^2$.
    • $\pi^0$ candidates: $\gamma\gamma$ mass constrained to the nominal $\pi^0$ mass.
    • $\rho^+$ candidates: $\pi^+\pi^0$ mass within $(0.61, 0.93)$ GeV/$c^2$.
  • Missing Energy/Momentum: To identify the neutrino and suppress backgrounds without missing particles, variables $\vec{p}_{miss}$ and $U_{miss}$ (related to missing energy and momentum) are calculated. Cuts are applied to ensure consistency with a single neutrino hypothesis.
  • Recoil Method: Since the neutrino prevents full reconstruction of the $D_s^-$, the $D_s^-$ candidate is identified using the recoil mass technique against the reconstructed $\pi^+$ or $\rho^+$. The recoil mass window is set to $(1.91, 2.02)$ GeV/$c^2$ for the $\pi^+$ mode and $(1.89, 2.08)$ GeV/$c^2$ for the $\rho^+$ mode.

Background Estimation

• Sources: Backgrounds are categorized into:
  1. $\psi(2S)$ decays: Estimated using an inclusive Monte Carlo (MC) sample of 2.7 billion events.
  2. Continuum backgrounds: Estimated using data collected at off-peak energies ($\sqrt{s} = 3.650, 3.682, 3.773$ GeV).
• Expected Backgrounds: The estimated background yields are $5.6 \pm 2.3$ events for the $\pi^+$ mode and $12.4 \pm 3.2$ events for the $\rho^+$ mode.

Systematic Uncertainties

Major sources of systematic uncertainty include:

• MC generator models (specifically for angular distributions in the $\rho^+$ mode).
• Tracking and PID efficiencies (estimated using control samples like $\psi(3770) \to D\bar{D}$).
• Photon detection efficiency.
• Intermediate branching fractions ($D_s^- \to \phi e^- \bar{\nu}_e$, $\phi \to K^+K^-$, etc.).
• Total number of $\psi(2S)$ events.
• Total systematic uncertainties are calculated to be 10.1% for the $\pi^+$ mode and 14.2% for the $\rho^+$ mode.

3. Key Contributions

• First Search: This represents the first experimental search for these specific weak decay channels of the $\psi(2S)$.
• Blind Analysis: The implementation of a rigorous blind analysis protocol ensures the integrity of the result by preventing tuning of cuts based on the signal region data.
• Comprehensive Background Modeling: The study provides a detailed breakdown of background contributions from both inclusive $\psi(2S)$ decays and continuum processes, utilizing off-peak data for validation.

4. Results

• Observation: After applying all selection criteria to the full dataset:
  • 9 events were observed in the signal region for $\psi(2S) \to D_s^- \pi^+$.
  • 19 events were observed in the signal region for $\psi(2S) \to D_s^- \rho^+$.
• Significance: In both cases, the number of observed events is consistent with the expected background yields within statistical uncertainties. No significant signal excess was found.
• Upper Limits: Using the Profile Likelihood method, 90% Confidence Level (C.L.) upper limits on the branching fractions were set:
  • $\mathcal{B}(\psi(2S) \to D_s^- \pi^+) < 1.4 \times 10^{-6}$
  • $\mathcal{B}(\psi(2S) \to D_s^- \rho^+) < 7.0 \times 10^{-6}$

5. Significance and Conclusion

• Comparison with SM: The obtained upper limits are consistent with Standard Model predictions, which place the branching fractions at the level of $10^{-10}$ or lower.
• Implications for New Physics: While the results do not rule out new physics models that predict enhancements of 2–3 orders of magnitude, the current limits suggest that the existing dataset lacks the sensitivity to probe these specific enhancements definitively.
• Future Outlook: The authors conclude that further data acquisition is highly motivated to improve sensitivity and potentially observe these rare weak decays or place tighter constraints on BSM theories.

In summary, this paper establishes a robust experimental framework for searching for weak charmonium decays, sets the first world-leading limits on these specific channels, and confirms that current data aligns with Standard Model expectations, leaving the door open for future high-statistics studies.