Two photon decay width of the fully charmed tetraquarks: revisiting prospects for ultraperipheral collisions

This paper investigates the production of fully charmed tetraquarks in ultraperipheral heavy-ion collisions by calculating their two-photon decay widths and comparing resonant contributions to continuum backgrounds, revealing that while resonant effects dominate the $J/\psi J/\psi$ channel, they are subdominant in the diphoton channel, contradicting predictions from a naive vector dominance model.

The Gist

Imagine the universe as a giant, bustling construction site. For a long time, physicists have known about the standard "bricks" of matter: protons, neutrons, and electrons. But in recent years, they've started finding strange, exotic structures built from these bricks in ways that shouldn't be possible according to the old blueprints.

One of the most exciting discoveries is the tetraquark. Think of a normal particle (like a proton) as a house made of three bricks stuck together. A tetraquark is a house made of four bricks. Even more strangely, the ones this paper focuses on are made entirely of "heavy" bricks called charm quarks. It's like finding a house built exclusively out of lead bricks, which is very rare and heavy.

Here is what the authors of this paper did, explained simply:

1. The Mystery of the "Ghost House"

Scientists at the Large Hadron Collider (LHC) have spotted these heavy four-brick houses (called $X(6900)$ and others) when smashing protons together. But it's like trying to study a ghost in a crowded, noisy party. The proton collisions are chaotic, with debris flying everywhere, making it hard to see the true shape and nature of these new particles.

The authors wanted to find a "clean room" to study them. They proposed using Ultraperipheral Collisions (UPCs). Imagine two massive trains (lead nuclei) speeding past each other on parallel tracks without actually crashing. Because they are so electrically charged, they throw off a shower of invisible "light bullets" (photons) that collide with each other. This creates a very quiet, clean environment where these heavy tetraquarks can be born without the messy debris of a full crash.

2. The Two Ways to Listen

Once these heavy particles are born in this clean environment, they don't stay alive for long. They immediately fall apart. The authors asked: How do they fall apart, and what does that tell us about what they are made of?

They looked at two specific ways these particles decay (fall apart):

• The "Double J/ψ" Exit: The particle splits into two smaller, well-known heavy particles (called $J/\psi$ mesons). This is like a heavy box opening up to reveal two smaller, identical boxes inside.
• The "Double Photon" Exit: The particle splits into two flashes of pure light (photons). This is like the heavy box vanishing and turning into two beams of light.

3. The Calculation: Weighing the Options

The authors used a sophisticated mathematical toolkit (called NRQCD) and a model of how these four bricks are arranged inside the particle (like a 3D map of the house's interior).

They calculated how likely it is for these particles to take the "Double Photon" exit versus the "Double J/ψ" exit.

• The Surprise: They found that for the "Double J/ψ" exit, the signal from these new tetraquarks is loud and clear. It stands out strongly against the background noise.
• The Disappointment: For the "Double Photon" exit, the signal is extremely faint. It is so quiet that it gets completely drowned out by the natural background light (the "QED box continuum").

4. The Verdict

The paper concludes with a clear message for experimentalists:

• Don't look for these particles in the "Double Photon" channel. The authors show that previous ideas suggesting these particles might be responsible for a bright flash of light in these collisions were likely wrong. The signal is too weak to be seen with current technology.
• Do look for them in the "Double J/ψ" channel. This is the promising path. If you have enough data (which the future High-Luminosity LHC will provide), you should be able to see these heavy tetraquarks clearly by looking for pairs of $J/\psi$ particles.

The Analogy Summary

Imagine you are trying to hear a specific violin soloist in a concert hall.

• The Proton Collision is like a rock concert where the soloist is playing, but the drums and guitars are so loud you can't hear the violin.
• The Ultraperipheral Collision is like moving the soloist to a silent, soundproof room.
• The "Double J/ψ" channel is like asking the soloist to play a specific note that echoes clearly in the room. The authors say, "Yes, we can hear them perfectly here!"
• The "Double Photon" channel is like asking the soloist to whisper a secret. The authors say, "Even in the silent room, the whisper is too quiet to hear over the wind outside. Don't bother listening for it."

In short, the paper tells us: Stop looking for these heavy particles in the light-flash channel; they are too quiet there. Instead, look for them in the heavy-particle channel, where they are loud enough to be found.

Technical Summary

Technical Summary: Two Photon Decay Width of Fully Charmed Tetraquarks: Revisiting Prospects for Ultraperipheral Collisions

Problem Statement
The observation of resonant structures in the $J/\psi$-pair mass spectrum by LHCb, ATLAS, and CMS has provided evidence for fully heavy $cc\bar{c}\bar{c}$ tetraquark states, most notably $X(6900)$. While hadroproduction in $pp$ collisions initiated these discoveries, the complex hadronic environment and potential contributions from double parton scattering complicate the interpretation of the tetraquark's intrinsic properties. A cleaner experimental channel is required to probe the internal structure directly. The exclusive process $\gamma\gamma \to X(6900)$ in ultraperipheral heavy-ion collisions (UPCs) offers such a channel, as its cross-section is directly proportional to the two-photon decay width, $\Gamma_{\gamma\gamma}$. This quantity is highly sensitive to the internal configuration (e.g., compact diquark-antidiquark vs. molecular) of the state. Previous literature has offered widely varying estimates for $\Gamma_{\gamma\gamma}$, ranging from $\sim 1$ eV to tens of keV, often relying on simplified models or vector meson dominance (VDM) assumptions that may not accurately reflect the four-body dynamics of fully heavy systems.

Methodology
The authors calculate the two-photon decay widths for scalar ($0^{++}$) and tensor ($2^{++}$) fully charmed tetraquarks using a two-step approach combining Non-Relativistic QCD (NRQCD) factorization with a specific quark model for the wave functions.

1. NRQCD Factorization: The amplitude for $\gamma\gamma \to T_{4c}$ is factorized into perturbative short-distance coefficients (SDCs) and non-perturbative long-distance matrix elements (LDMEs). The SDCs are calculated at leading order (LO) by matching full QCD to NRQCD, considering tree-level Feynman diagrams for the partonic subprocess $\gamma\gamma \to c\bar{c}c\bar{c}$. The calculation accounts for color configurations, specifically the mixing between color-antitriplet ($\bar{3} \otimes 3$) and color-sextet ($6 \otimes \bar{6}$) diquark-antidiquark states for scalar mesons, while tensor states are treated as pure $\bar{3} \otimes 3$ systems.
2. Wave Function Modeling: To evaluate the LDMEs, the authors utilize four-body wave functions obtained from an extended relativized quark model (Ref. [31]). This model solves the four-body Schrödinger equation for the $cc\bar{c}\bar{c}$ system using the Gaussian Expansion Method (GEM). The Hamiltonian includes relativistic kinetic energy and a potential comprising linear confinement and one-gluon-exchange (OGE) interactions.
3. Phenomenological Application: The calculated $\Gamma_{\gamma\gamma}$ values are used to estimate the production cross-sections for $T_{4c}$ states in $208\text{Pb} + 208\text{Pb}$ UPCs at $\sqrt{s_{NN}} = 5.5$ TeV. The authors evaluate two final states:
   • $J/\psi J/\psi$: Using simplified couplings derived from assumed branching fractions ($B_\psi$).
   • $\gamma\gamma$: Comparing resonant contributions against the QED box continuum.

Key Contributions and Results

• Decay Width Calculations: The paper provides a systematic calculation of $\Gamma_{\gamma\gamma}$ for ground ($1S$), first radial ($2S$), and second radial ($3S$) excitations of $0^{++}$ and $2^{++}$ tetraquarks.

  • The results range from approximately $0.1$ to $0.69$ keV.
  • For scalar states, the authors explicitly resolve the interference between color configurations ($\bar{3}\otimes3$ and $6\otimes\bar{6}$), showing that destructive interference can suppress or enhance the width depending on the mixing angle.
  • The tensor $2^{++}$ states are identified with the observed resonances: $X(6600)$ as $1S$, $X(6900)$ as $2S$, and $X(7100)$ as $3S$. Notably, the model predicts a vanishing two-photon width for the $3S$ state ($X(7100)$) due to the node in the wave function at the origin.
  • The calculated widths are roughly two orders of magnitude larger than early quark model estimates ($\sim 1$ eV) but generally consistent with or slightly smaller than recent NRQCD and VDM-based estimates, provided the branching fraction to $J/\psi J/\psi$ is of order unity.

• Production in UPCs ($J/\psi J/\psi$ Channel):

  • The resonant production cross-sections for $T_{4c} \to J/\psi J/\psi$ in Pb-Pb UPCs are calculated.
  • Assuming a conservative branching fraction $B_\psi \approx 3 \times 10^{-3}$, the resonant signal exceeds the continuum $\gamma\gamma \to J/\psi J/\psi$ background in the resonance mass region.
  • The tensor $2^{++}$ states show the largest production rates, with peak cross-sections reaching $\sim 2.2$ nb for the $X(6900)$ candidate. The total resonance production cross-section is on the order of microbarns.

• Production in UPCs ($\gamma\gamma$ Channel):

  • The paper evaluates the contribution of tetraquarks to the light-by-light (LbL) scattering process $\gamma\gamma \to \gamma\gamma$.
  • The resonant tetraquark contributions are found to be more than two orders of magnitude smaller than the standard QED box continuum.
  • This result stands in clear disagreement with previous claims (e.g., Ref. [21]) that suggested tetraquarks could explain discrepancies in ATLAS LbL data, which required much larger decay widths ($\sim 45-67$ keV).

Significance and Claims
The paper claims that the two-photon decay width is a powerful discriminant for the internal structure of fully heavy tetraquarks. By employing a realistic four-body wave function calculation within the NRQCD framework, the authors provide a robust benchmark for future experimental measurements at the LHC.

The primary significance lies in the differential prospects for the two channels:

1. $J/\psi J/\psi$ Channel: The authors conclude this channel is "substantially more promising" for tetraquark searches in UPCs, as the resonant signal dominates the continuum. However, they note that observing the four-muon final state ($\mu^+\mu^-\mu^+\mu^-$) requires high integrated luminosity (e.g., at the High-Luminosity LHC) due to the small expected event rates with current datasets and conservative branching fraction assumptions.
2. $\gamma\gamma$ Channel: The paper explicitly states that the $\gamma\gamma$ channel is "unpromising" for tetraquark searches with current UPC luminosities. The calculated signal-to-background ratio is too small to explain any potential excesses in LbL scattering data, refuting earlier hypotheses that relied on naive vector dominance or oversimplified point-like diquark models which overestimated the wave function at the origin.

In summary, the work re-evaluates the viability of UPCs for studying fully heavy tetraquarks, concluding that while the $J/\psi J/\psi$ channel offers a viable discovery path, the $\gamma\gamma$ channel is unlikely to yield significant signals given the realistic four-body dynamics of the $cc\bar{c}\bar{c}$ system.