Tensor molecule $J/\psi J/\psi$: A candidate to the resonance $X(6200)$

Using QCD sum rules, this study investigates the hadronic tensor molecule $J/\psi J/\psi$, calculating its mass and decay width to propose it as a viable candidate for the experimentally observed resonance $X(6200)$.

The Gist

Imagine the universe as a giant, chaotic construction site where tiny building blocks called quarks are constantly snapping together to form larger structures called particles. Usually, these blocks come in pairs (like a proton and an antiproton) or triplets (like a proton made of three quarks). But recently, scientists at giant particle colliders (like the LHC) have spotted some very strange, heavy structures made of four charm-quarks stuck together. They call these "exotic mesons."

One of these mysterious structures is named X(6200). It's like a ghost in the machine: we know it's there because we see a "bump" in the data, but we don't know exactly what it's made of or how it behaves.

This paper is like a team of theoretical detectives trying to solve the mystery of X(6200) by building a theoretical model of what it might be. Here is their investigation, broken down simply:

1. The Suspect: A "Molecular" Marriage

The authors propose that X(6200) isn't a single, tight knot of four quarks. Instead, they suggest it's a molecule.

• The Analogy: Imagine two heavy, glowing marbles (called J/ψ particles) that are holding hands loosely. They aren't fused into one solid rock; they are two distinct objects orbiting each other, held together by a force field.
• The Composition: This "molecule" is made of two J/ψ particles, which themselves are made of charm quarks. So, the whole thing is a "J/ψ-J/ψ" molecule.
• The Shape: The authors specifically look at a version of this molecule that has a "tensor" shape. Think of this as the molecule having a specific, rigid orientation in space, like a dumbbell spinning on its axis, rather than just a fuzzy cloud.

2. The Investigation: Weighing the Ghost

To see if this "molecule" could be the real X(6200), the authors used a mathematical tool called QCD Sum Rules.

• The Analogy: Imagine you can't see a ghost, but you can measure the temperature of the room and the sound of the floorboards creaking. By crunching these numbers, you can calculate exactly how heavy the ghost must be to cause those specific effects.
• The Result: They calculated the mass (weight) of their theoretical molecule. They found it weighs about 6,290 MeV (a unit of energy used for mass in particle physics).
• The Match: The real X(6200) observed by experiments weighs about 6,220 MeV. The numbers are very close (within the margin of error). This suggests the "molecule" theory is a strong candidate for what X(6200) actually is.

3. The Breakup: How the Molecule Falls Apart

A key part of identifying a particle is knowing how it dies (decays). The authors asked: "If this molecule exists, how does it break apart?"

• The Main Event (Dominant Decay): The easiest way for this molecule to break is for the two J/ψ marbles to simply let go of each other and fly apart. The authors calculated that this happens quite often.
• The Secret Handshake (Subdominant Decays): But there's a twist. Inside the molecule, the charm quarks can sometimes "annihilate" (destroy each other) and turn into lighter quarks.
  • The Analogy: Imagine the two heavy marbles holding hands suddenly explode into a cloud of smaller, lighter marbles (like D-mesons).
  • The authors calculated that the molecule can also break apart into pairs of these lighter particles (like $D^*D^*$ or $D_s D_s$). They did the math to see how likely each of these breakups is.

4. The Verdict: Does it Fit?

The authors added up all the ways their theoretical molecule could break apart to get its total "lifespan" (or decay width).

• The Calculation: They predicted the molecule should last for a very short time, corresponding to a width of about 149 MeV.
• The Comparison: The real X(6200) observed in experiments has a width of about 310 MeV, but with a huge margin of error (it could be anywhere between 110 and 480 MeV).
• The Conclusion: The authors' prediction (149 MeV) falls right inside the experimental "safe zone."

Summary

The paper argues that the mysterious X(6200) particle is likely a tensor molecule made of two J/ψ particles holding hands.

• Its calculated weight matches the experimental data.
• Its calculated lifespan (how fast it breaks apart) also fits within the experimental data.

The authors conclude that while X(6200) might not be purely this molecule (it could be a mix of different things), this "J/ψ-J/ψ molecule" is a very important piece of the puzzle that helps explain what we are seeing in the particle collider data. They didn't find a cure for a disease or a new engine; they simply solved a riddle about the fundamental building blocks of our universe.

Technical Summary

Technical Summary: Tensor Molecule $J/\psi J/\psi$ as a Candidate for the Resonance $X(6200)$

Problem Statement
The paper addresses the nature of the four-X resonances ($X(6200)$, $X(6600)$, $X(6900)$, and $X(7300)$) observed by the LHCb, ATLAS, and CMS collaborations in the di-$J/\psi$ and $J/\psi\psi'$ mass distributions. While these structures are presumed to be fully charmed ($cccc$) exotic mesons, their specific internal configurations—whether diquark-antidiquark states, hadronic molecules, or superpositions—remain under investigation. Recent experimental data suggest that the $X(6200)$ resonance possesses quantum numbers $J^{PC} = 2^{++}$, distinguishing it from scalar ($0^{++}$) or vector ($1^{+-}$) interpretations proposed in earlier theoretical works. The authors aim to investigate the hadronic tensor molecule $M = J/\psi J/\psi$ to determine if its spectroscopic parameters align with the experimental data for $X(6200)$.

Methodology
The study employs the Quantum Chromodynamics (QCD) sum rule (SR) method to calculate the spectroscopic parameters and decay properties of the tensor molecule $M$.

• Mass and Current Coupling: The mass ($m$) and current coupling ($\Lambda$) are evaluated using a two-point sum rule approach. The analysis utilizes the correlation function of the interpolating current $J_{\mu\nu}(x) = c_a(x)\gamma_\mu c_a(x) c_b(x)\gamma_\nu c_b(x)$, where $c$ represents the charm quark field. The Operator Product Expansion (OPE) is performed up to dimension-4 terms, specifically including the gluon condensate $\langle \alpha_s G^2/\pi \rangle$.
• Decay Widths and Couplings: To determine the decay width, the authors analyze both the dominant decay channel ($M \to J/\psi J/\psi$) and various subdominant channels involving light quark annihilation ($M \to D^{(*)}D^{(*)}$, $D_s^{(*)}D_s^{(*)}$, etc.). These processes are investigated using the three-point sum rule approach. This involves calculating correlation functions at the $M$-meson-meson vertices to extract form factors $G(q^2)$ and $g_i(q^2)$.
• Extrapolation: Since the sum rule method is valid in the deep Euclidean region ($Q^2 < 0$) while the physical decay occurs at the mass shell ($q^2 = m_{meson}^2 > 0$), the authors employ fitting functions (exponential forms) to extrapolate the sum rule data from the spacelike region to the timelike physical point.
• Parameters: The analysis uses standard input parameters for the charm quark mass ($m_c$), gluon condensate, and meson decay constants/masses. Borel and continuum subtraction parameters are constrained to ensure pole contribution dominance ($\geq 50\%$) and OPE convergence.

Key Contributions and Results

1. Mass Determination: The calculated mass of the tensor molecule $M$ is $m = (6290 \pm 50)$ MeV. This result is derived from the two-point sum rule analysis within the working window $M^2 \in [4.5, 5.5]$ GeV$^2$ and $s_0 \in [45, 46]$ GeV$^2$.
2. Dominant Decay Channel: The molecule $M$ is kinematically allowed to decay into a pair of $J/\psi$ mesons. The strong coupling at the $MJ/\psi J/\psi$ vertex is determined to be $G = (1.37 \pm 0.26)$ GeV$^{-1}$, yielding a partial width of $\Gamma[M \to J/\psi J/\psi] = (50.2 \pm 18.3)$ MeV.
3. Subdominant Decay Channels: The authors calculate partial widths for six non-strange subdominant modes ($M \to D^{(*)}D^{(*)}$, $DD_1$, etc.) and four charmed-strange modes ($M \to D_s^{(*)}D_s^{(*)}$, etc.). These decays proceed via the annihilation of constituent $c\bar{c}$ pairs into light $q\bar{q}$ or $s\bar{s}$ pairs.
4. Total Decay Width: Summing the partial widths of the dominant and subdominant channels, the total decay width is estimated as $\Gamma[M] = (149 \pm 21)$ MeV.

Significance and Claims
The paper posits that the tensor molecule $M = J/\psi J/\psi$ is a viable candidate for the experimentally observed resonance $X(6200)$.

• Mass Compatibility: The theoretical mass of $6290 \pm 50$ MeV is consistent with the ATLAS measurement of $X(6200)$ ($6220 \pm 50^{+40}_{-50}$ MeV).
• Width Comparison: The theoretical width of $149 \pm 21$ MeV is smaller than the central value of the experimental width ($310 \pm 120^{+70}_{-80}$ MeV). However, the authors note a significant overlap region ($110 - 170$ MeV) between the theoretical prediction and the experimental data.
• Interpretation: The authors conclude that while the physical state $X(6200)$ may not be a pure molecular state, the $J/\psi J/\psi$ molecular component is likely a crucial part of its structure.
• Excited States: The study suggests that the continuum threshold parameter used ($s_0 \approx 45.5$ GeV$^2$) implies that the first radial excitation $M(2S)$ would have a mass $m(2S) > 6708$ MeV. This excludes the $X(6600)$ resonance as a candidate for the $2S$ excitation but leaves open the possibility that the $X(6900)$ resonance could contain an excited molecular component.

The paper emphasizes that the interpretation of X resonances remains complex and that further precise experimental measurements and theoretical investigations (including other decay channels and excited states) are necessary to fully resolve the nature of these fully charmed exotic mesons.