Searching for dark photons in $J/\psi$ decays

This paper utilizes non-relativistic QCD to analyze $J/\psi$ decays mediated by a light dark photon, presenting numerical predictions for event yields and statistical significances at BESIII across various mass ranges and decay channels (visible and invisible) to assess the feasibility of detecting dark photons and dark matter.

The Gist

Imagine the universe is like a giant, bustling city. We know about the "visible" citizens: the people, cars, and buildings that make up the Standard Model of physics. But we also know there's a massive, invisible "ghost district" called Dark Matter. We can't see it, but we know it's there because its gravity holds the city together. The problem? We've never caught a ghost in the act.

This paper is a proposal for a new "sting operation" to catch these ghosts, specifically looking for a new particle called the Dark Photon.

The Characters in Our Story

1. The J/ψ Particle (The Heavyweight Boxer): Think of the J/ψ as a heavy, unstable box of energy. It's like a giant balloon filled with helium that wants to pop. When it pops (decays), it usually releases standard particles like electrons or muons.
2. The Dark Photon (The Secret Messenger): This is the star of the show. It's a hypothetical particle that acts like a bridge between our visible world and the ghostly dark sector. It's "dark" because it doesn't interact with light, but it has a tiny, secret connection (called "kinetic mixing") to our regular photons (light).
3. Dark Matter (The Ghosts): These are the invisible particles the Dark Photon might be carrying. They could be fermions (ghostly particles with spin) or scalars (ghostly blobs).

The Plan: How They Catch the Ghosts

The scientists are using the BESIII experiment (a giant particle detector in China) as their hunting ground. They have a massive collection of J/ψ particles (over 87 billion of them!). They are waiting for the J/ψ to "pop" in a very specific, rare way.

There are two main scenarios they are looking for, depending on how heavy the Dark Photon is:

Scenario A: The "Visible" Trap (When the Dark Photon is Light)

Imagine the Dark Photon is very light (lighter than two dark matter particles). In this case, the Dark Photon is too weak to carry a heavy ghost. Instead, it immediately turns back into something we can see.

• The Process: The J/ψ decays into a Dark Photon, which instantly transforms into a pair of regular particles (like an electron and a positron, or a muon and an antimuon).
• The Clue: The scientists look for a specific "bump" in the data. If they see a sudden spike in the number of electron pairs with a specific energy, it's like hearing a ghost whisper. It means a Dark Photon was there, even if it didn't last long enough to be seen directly.
• The Result: The paper predicts they might see a few dozen of these events. However, the signal is very faint (like trying to hear a whisper in a hurricane), so they need extremely precise instruments to distinguish it from background noise.

Scenario B: The "Invisible" Trap (When the Dark Photon is Heavy)

Now, imagine the Dark Photon is heavy enough to carry a pair of dark matter ghosts.

• The Process: The J/ψ decays into a Dark Photon, which then splits into two invisible dark matter particles.
• The Clue: This is the "Missing Energy" hunt. The J/ψ decays, and the scientists see a photon (light) flying out, but the rest of the energy just... vanishes. It's like a magician pulling a rabbit out of a hat, but the rabbit is invisible. If the energy balance doesn't add up, it suggests the Dark Photon took the energy and hid it in the dark sector.
• The Result: The paper predicts they might see up to a dozen of these "missing energy" events. Interestingly, the "invisible" signal is actually easier to spot than the "visible" one because the background noise (other things that look like missing energy) is much quieter.

The Four-Body Hunt (The Complex Puzzle)

The scientists also looked at a more complex scenario where the J/ψ decays into four particles at once (two pairs of particles).

• The Analogy: Imagine the J/ψ doesn't just pop; it explodes into a shower of confetti. If the Dark Photon is very light, it might create a shower of four particles (like two pairs of electrons).
• The Result: In the very low mass range, they predict seeing about 100 to 170 of these complex events. It's like finding a specific pattern in a pile of confetti.

Why Does This Matter?

Think of the universe as a locked room. We know there's a secret door (Dark Matter), but we don't have the key. The Dark Photon is the keyhole.

• If they find these events, it proves the Dark Photon exists.
• It proves that Dark Matter interacts with our world, even if just a tiny bit.
• It opens a new window into the "ghost district" of the universe.

The Bottom Line

The authors did the math (using complex equations called NRQCD) to predict exactly how many times this "ghost hunting" should happen at the BESIII experiment.

• Good News: There are enough J/ψ particles collected to potentially see these events.
• Bad News: The signal is incredibly weak. It's like trying to find a single specific grain of sand on a beach, and that grain of sand is also trying to hide.
• The Verdict: While the numbers of expected events are small (ranging from 0 to about 170 depending on the scenario), the "significance" (how sure we can be it's not a fluke) is currently low. However, if the future Super Tau-Charm Facility (STCF) comes online with 100 times more data, this "sting operation" could finally succeed, potentially revealing the first direct evidence of how Dark Matter talks to our world.

In short: They are building a very sensitive net to catch a ghost that might be hiding in a pile of confetti, and they've calculated exactly how many ghosts they might catch if the ghost exists.

Technical Summary

1. Problem Statement

Dark Matter (DM) constitutes approximately 25% of the universe's energy density but remains unexplained by the Standard Model (SM). A leading theoretical framework proposes a "dark sector" governed by a new $U(1)_D$ gauge symmetry, mediated by a dark photon ($A'$ or $U$). This dark photon kinetically mixes with the SM hypercharge boson, creating a feeble interaction portal.

While extensive searches exist for dark photons at high-energy colliders (e.g., LHC, future $e^+e^-$ colliders) and fixed-target experiments, there is a need to explore the low-mass region ($m_{A'} < 3.0$ GeV) using heavy quarkonium systems. The $J/\psi$ meson offers a unique environment due to its large production cross-section at the BESIII experiment ($8.774 \times 10^{10}$ events) and its clean theoretical description via Non-Relativistic QCD (NRQCD). The paper aims to calculate the sensitivity of $J/\psi$ decays to dark photons in both visible (decaying to SM fermions) and invisible (decaying to dark matter) channels.

2. Methodology

The authors employ a theoretical framework combining NRQCD factorization with effective field theory for the dark sector.

• Theoretical Framework:

  • NRQCD Factorization: The decay rate is factorized into short-distance coefficients (calculable perturbatively) and long-distance matrix elements (non-perturbative). The $J/\psi$ is treated as a $c\bar{c}$ bound state.
  • Wavefunction: The long-distance matrix element is determined by the squared radial wavefunction at the origin, $|R_{J/\psi}(0)|^2$, extracted from experimental branching ratios of $J/\psi \to e^+e^-$ and $\mu^+\mu^-$.
  • Lagrangian: The interaction involves kinetic mixing ($\epsilon$) between the dark photon and the SM photon, and a dark gauge coupling ($g_\chi$) between the dark photon and dark matter ($\chi$).
  • Dark Matter Candidates: The study considers three types of dark matter: Dirac fermions ($\chi_D$), Majorana fermions ($\chi_M$), and complex scalars ($\phi$).

• Decay Channels Analyzed:

  1. Two-body decays: $J/\psi \to \gamma A' \to \gamma X$, where $X$ is either visible ($e^+e^-, \mu^+\mu^-, \text{hadrons}$) or invisible ($\bar{\chi}\chi$).
  2. Four-body decays: $J/\psi \to l^+l^- A' \to l^+l^- X$, where the dark photon is produced off-shell or on-shell and decays subsequently.

• Computational Tools:

  • FeynArts: Used to generate Feynman diagrams and amplitudes.
  • FeynCalc: Used for Dirac trace evaluation and Lorentz tensor algebra.
  • Parameters: Kinetic mixing $\epsilon = 10^{-3}$, dark fine-structure constant $\alpha_\chi = 0.05$, and mass ratio $m_U/m_\chi = 3.0$. The dark photon mass $m_U$ is scanned from 0.3 GeV to 3.0 GeV.

3. Key Contributions

• Comprehensive Decay Calculation: The paper provides the first detailed calculation of both two-body and four-body $J/\psi$ decay rates mediated by a light dark photon, covering the full mass range up to the $J/\psi$ mass.
• Invisible vs. Visible Comparison: It explicitly contrasts the phenomenology of dark photon decays into SM particles versus dark sector particles, highlighting the suppression of visible modes when $m_U \geq 2m_\chi$.
• Event Yield Predictions for BESIII: The authors translate theoretical decay widths into expected event numbers based on the current BESIII dataset, providing concrete targets for experimentalists.
• Significance and Kinematics: The study calculates the statistical significance ($S/\sqrt{B}$) and transverse momentum ($p_T$) distributions, offering guidance on how to distinguish signal from background.

4. Key Results

A. Two-Body Final States ($J/\psi \to \gamma X$)

• Visible Decays ($m_U < 2m_\chi$):
  • The decay channels $J/\psi \to \gamma e^+e^-$, $\gamma \mu^+\mu^-$, and $\gamma \text{hadrons}$ show event yields ranging from 0 to 17 and 0 to 37, respectively, for $m_U \in [0.3, 3.0]$ GeV.
  • The significance is extremely low ($S/\sqrt{B} \sim 10^{-5} - 10^{-6}$), indicating that current BESIII statistics are insufficient to claim a discovery without significant background suppression or higher luminosity.
• Invisible Decays ($m_U \geq 2m_\chi$):
  • Visible channels are severely suppressed (by a factor of $\sim 10^4$).
  • Invisible channels ($J/\psi \to \gamma + \text{missing energy}$) yield 0 to 12 events.
  • Significance improves slightly to $S/\sqrt{B} \sim 10^{-1} - 10^{-3}$, but remains challenging. The signal would appear as a resonance in the missing mass distribution.

B. Four-Body Final States ($J/\psi \to l^+l^- X$)

• Visible Decays ($m_U < 2m_\chi$):
  • In the very low mass region ($m_U < 0.2$ GeV), where the dark photon can only decay to $e^+e^-$, the four-body channels ($J/\psi \to e^+e^-e^+e^-\gamma$ and $J/\psi \to e^+e^-\mu^+\mu^-\gamma$) yield 94 to 172 events and 90 to 161 events, respectively.
  • For $m_U > 0.2$ GeV, yields drop significantly (below 100 events).
• Invisible Decays ($m_U \geq 2m_\chi$):
  • The invisible four-body channel ($J/\psi \to l^+l^- + \text{missing energy}$) yields up to 129 events.
  • Visible four-body decays are negligible ($\ll 1$ event).

C. Kinematic Distributions

• The transverse momentum ($p_T$) distributions of the decay products rise sharply as the dark photon mass $m_U$ increases. This kinematic feature can be used to separate signal from background in experimental analyses.

5. Significance and Conclusion

• Experimental Feasibility: The study concludes that while the predicted event yields at current BESIII luminosity are generally low (often $<100$ events) and significances are modest, the four-body visible channels at very low masses ($m_U < 0.2$ GeV) offer the most promising immediate search window with $\sim 100$ expected events.
• Future Prospects: The authors emphasize that the future Super Tau-Charm Facility (STCF), with an expected $J/\psi$ yield of $3.4 \times 10^{12}$ per year (two orders of magnitude higher than BESIII), will increase event yields and significances by roughly one order of magnitude, making these channels viable for discovery.
• Theoretical Utility: The provided analytical formulas and numerical tables serve as a baseline for experimental collaborations to set limits on the kinetic mixing parameter $\epsilon$ and dark gauge coupling $g_\chi$ in the sub-GeV mass range.

In summary, this work establishes a rigorous theoretical baseline for searching for dark photons in $J/\psi$ decays, identifying specific kinematic regions and decay topologies that offer the best sensitivity for current and future high-luminosity charm factories.