Analysis of molecular state ${{\eta}_cD^*}$ and ${J/\psi D^*}$ in the effective Lagrangian approach

Using an effective Lagrangian approach and SU(3) flavor symmetry, this study investigates the production and decay of $J^P=1^+$ molecular states composed of $\eta_c D^*$ and $J/\psi D^*$, predicting sizable branching ratios from $B_c$ meson decays ($10^{-4}$ and $10^{-5}$, respectively) and narrow decay widths on the order of MeV.

The Gist

Imagine the subatomic world as a bustling, chaotic construction site. Usually, particles like protons and neutrons are built from three "bricks" (quarks) held together tightly. But recently, physicists have started finding strange, exotic structures made of four bricks. These are called "tetraquarks."

This paper is like a detective story where the authors try to figure out if a specific, very heavy, and very rare four-brick structure actually exists, how it gets built, and how long it lasts before falling apart.

Here is the breakdown of their investigation using simple analogies:

1. The Mystery: What are they looking for?

The scientists are hunting for a specific type of "exotic molecule" made of three heavy charm quarks and one light quark. Think of it like a heavy-duty truck (the charm quarks) towing a small trailer (the light quark).

They are focusing on two specific ways these trucks might be parked:

• The "ηcD" Configuration:* Imagine a heavy sedan (the ηc) parked right next to a heavy SUV (the D*). They are holding hands loosely, forming a "molecule."
• The "J/ψD" Configuration:* Imagine a different heavy sedan (the J/ψ) parked next to that same heavy SUV.

The big question is: Do these loose pairs actually stick together to form a stable new particle, or do they just bounce off each other?

2. The Factory: How are they made?

To find these particles, the scientists looked at a specific "factory" called the Bc meson. Think of the Bc meson as a giant, unstable delivery truck that is constantly breaking down.

When this truck breaks down (decays), it sometimes spits out a new particle. The authors calculated the odds of this happening.

• The Result: It turns out the factory is surprisingly good at making the "ηcD*" molecule. About 1 in 10,000 times the truck breaks down, it creates this specific molecule.
• The "J/ψD*" version is rarer, appearing about 1 in 100,000 times.

Why does this matter? In the world of particle physics, a 1-in-10,000 chance is huge! It means if we look hard enough at data from particle colliders (like the LHC), we should be able to spot these particles.

3. The Lifespan: How long do they last?

Once these molecules are made, they don't last forever. They are like a house of cards in a windy room; eventually, they collapse. The scientists calculated how fast they collapse (their "decay width").

• The Finding: These molecules are actually quite stable for such exotic things. They don't vanish instantly. They last long enough to be measured, with a "lifespan" corresponding to a width of just a few MeV (a tiny unit of energy).
• The Comparison: If these were "compact" particles (where all four bricks are glued into a single tight ball), they would fall apart much faster. The fact that they last this long supports the idea that they are indeed "molecules" (loose pairs) rather than tight balls.

4. The Method: How did they figure this out?

The authors didn't just guess; they used a mathematical toolkit called the Effective Lagrangian Approach.

• The Analogy: Imagine trying to predict the path of a ball thrown in a storm. You can't track every single air molecule, so you use a simplified map that accounts for wind and gravity.
• In their case, they used "maps" (mathematical formulas) based on Symmetry (rules that say nature looks the same in different flavors) and Triangle Diagrams.
• The Triangle Diagram: This is the most complex part. Imagine the Bc meson decaying, and the pieces don't just fly apart immediately. Instead, they swap a "messenger particle" (like a D* meson) back and forth in a triangular loop before settling into the final molecule. The authors calculated the energy of these loops to see if the math adds up to a real particle.

5. The Verdict

The paper concludes with a clear message for experimental physicists (the people who build the machines):

• Go look for these particles! The math says they should be there.
• Where to look? Specifically, look at the debris from Bc meson decays.
• What to expect? You should see a signal for the ηcD* molecule about 10 times more often than the J/ψD* one.
• How wide is the signal? It will be a narrow, distinct peak (a few MeV wide), not a blurry smear.

Summary in a Nutshell

This paper is a roadmap. It tells experimentalists: "We have done the math using advanced symmetry rules and loop calculations. We predict that a heavy, four-quark 'molecule' exists. It is made in about 1 out of every 10,000 Bc meson crashes, and it hangs around long enough to be seen. If you look in the right place, you will find it."

It's a bridge between the abstract math of theoretical physics and the concrete search for new matter in the universe.

Technical Summary

1. Problem Statement

The paper addresses the theoretical investigation of triply charmed four-quark states ($cc\bar{c}\bar{q}$), specifically focusing on those with quantum numbers $J^P = 1^+$. While experimental evidence for open-charm and doubly charmed tetraquarks (e.g., $X(2900)$, $T_{cc}(3875)$) has emerged, the existence and nature of fully or triply heavy four-quark states remain open questions.

• Core Question: Do stable or resonant triply charmed molecular states exist, and if so, are they compact tetraquarks or extended hadronic molecules?
• Specific Focus: The authors investigate the production and decay properties of two specific molecular configurations:
  1. *Vector-Pseudoscalar ($PV$):**$\eta_c D^$
  2. *Vector-Vector ($VV$):**$J/\psi D^$
• Challenge: Conventional analyses often disfavor molecular configurations for states with three heavy quarks due to the suppression of single long-range light meson exchange between charmonia. The paper aims to test if heavy meson exchange forces or two-light-meson exchange can sustain these molecular states and to provide concrete phenomenological predictions for experimental searches (e.g., at LHCb or Belle II).

2. Methodology

The authors employ a combination of phenomenological analysis and the effective Lagrangian approach within the framework of Quantum Chromodynamics (QCD) effective field theory.

• Theoretical Framework:

  • Effective Lagrangian: They construct gauge-invariant effective Lagrangians describing the coupling between the molecular states ($T_{\eta_c D^*}$ and $T_{J/\psi D^*}$) and their constituent mesons.
  • Coupling Constants: The coupling constants ($g_T$) are determined using the Weinberg compositeness condition ($Z_T = 0$), which ensures the state is a pure molecular bound state rather than a compact core. This involves calculating self-energy diagrams and renormalizing the mass operator.
  • Form Factors: To handle ultraviolet divergences and account for the finite size of hadrons, vertex form factors (monopole/dipole types) are introduced with a cutoff parameter $\Lambda$.

• Production Mechanism:

  • Source: Production via weak decays of the $B_c$ meson ($B_c \to K^* T_{cc\bar{c}\bar{q}}$).
  • Symmetry Analysis: An SU(3) flavor symmetry analysis is used to identify "golden channels" and relate different production modes. The weak transition $\bar{b} \to c\bar{c}\bar{s}/\bar{d}$ is modeled using an effective Hamiltonian with Wilson coefficients.
  • Dynamics: The production amplitude is calculated using triangle diagrams involving the weak decay of $B_c$ followed by the strong rescattering of intermediate mesons ($D^{(*)}$, $J/\psi$, $\eta_c$) to form the molecular state.

• Decay Mechanism:

  • Processes: Both two-body (e.g., $T \to J/\psi D$) and three-body (e.g., $T \to J/\psi D \pi$) strong decays are calculated.
  • Diagrams: The calculations include tree-level diagrams and one-loop triangle diagrams. Special attention is paid to processes involving $c\bar{c}$ annihilation, which are suppressed by the OZI rule and heavy quark mass but are included for completeness.
  • SU(4) Symmetry: Three-meson strong coupling constants are derived using SU(4) flavor symmetry, with uncertainties estimated by varying these constants by $\pm 10\%$.

3. Key Contributions

• Systematic Phenomenology: This is one of the first comprehensive studies applying the effective Lagrangian approach specifically to triply charmed molecular states ($cc\bar{c}\bar{q}$) with $J^P=1^+$.
• Golden Channels Identification: The authors identify specific production channels from $B_c$ decays that are most promising for experimental detection, accounting for CKM matrix elements and phase space.
• Quantitative Predictions: They provide precise numerical predictions for branching ratios and decay widths, including the dependence on binding energy ($\epsilon$) and the cutoff parameter ($\alpha_m$).
• Comparison of Configurations: The paper explicitly compares the $J/\psi D^*$ and $\eta_c D^*$ configurations, highlighting differences in their production rates and decay widths due to phase space and dynamical effects.

4. Key Results

A. Production Branching Ratios

• The production of these molecular states from $B_c$ meson decays is found to be sizable.
• $\eta_c D^*$ Configuration: The branching ratios for Cabibbo-allowed processes (e.g., $B_c^+ \to K^{*0} T_{\eta_c D^*}^+$) reach the order of $10^{-4}$.
  • Example: $Br(B_c^+ \to K^{*0} T_{\eta_c D^*}^+) \approx 1.13 \times 10^{-4}$.
• $J/\psi D^*$ Configuration: The branching ratios are smaller, reaching the order of $10^{-5}$.
  • Example: $Br(B_c^+ \to K^{*0} T_{J/\psi D^*}^+) \approx 8.16 \times 10^{-6}$.
• SU(3) Breaking: The calculated ratios between different channels (e.g., involving $K^*$ vs. $\rho$) deviate significantly from pure SU(3) symmetry predictions (which predict ratios $\approx 19.9$). The authors attribute this to significant SU(3) flavor breaking effects arising from phase space differences and loop dynamics (masses and decay constants of $D$ vs. $D_s$).

B. Decay Widths

• The total decay widths for both configurations are narrow, lying in the MeV scale.
  • $\eta_c D^*$: $\Gamma(T_{\eta_c D^*}^+) \approx 6.8$ MeV (for binding energy $\epsilon = 5$ MeV).
  • $J/\psi D^*$: $\Gamma(T_{J/\psi D^*}^+) \approx 0.16$ MeV (for $\epsilon = 5$ MeV).
• Dominant Modes: The dominant decay channels are two-body decays (e.g., $T \to J/\psi D$). Three-body decays and $c\bar{c}$ annihilation modes (e.g., to $\rho D$ or $D^* \pi$) are suppressed by orders of magnitude.
• Comparison with Models:
  • The narrow width of the $\eta_c D^*$ state is consistent with predictions from the Gaussian Expansion Method (GEM) for compact tetraquarks ($\sim 3.0$ MeV).
  • However, it contrasts sharply with Constituent Quark Model predictions for compact tetraquarks, which suggest much broader widths ($\sim 240$ MeV). This supports the molecular hypothesis where the state is loosely bound and decays via constituent rearrangement rather than internal quark rearrangement.

C. Sensitivity Analysis

• The results are relatively stable with respect to the cutoff parameter $\alpha_m$ (varying $100-450$ MeV).
• Uncertainties in strong coupling constants (due to SU(4) breaking) lead to a $\sim 20\%$ variation in production branching ratios and a $\sim 32\%$ variation in decay widths.

5. Significance

• Experimental Guidance: The paper provides specific "golden channels" ($B_c \to K^* T$) and expected branching ratios ($10^{-4}$ to $10^{-5}$) that experimentalists at LHCb and future facilities can target to search for triply charmed tetraquarks.
• Nature of Exotic States: The calculated narrow widths ($\mathcal{O}(\text{MeV})$) strongly suggest that if these states are observed, they likely possess a molecular structure rather than a compact tetraquark structure, as compact models typically predict much broader resonances.
• Theoretical Validation: The work validates the applicability of the effective Lagrangian approach and SU(3) flavor symmetry (with breaking corrections) to heavy-heavy-light systems, extending previous success in open-charm systems to the triply charmed sector.

In conclusion, the authors demonstrate that triply charmed molecular states $\eta_c D^*$ and $J/\psi D^*$ are viable candidates with observable production rates in $B_c$ decays and narrow decay widths, offering a clear pathway for experimental verification of the existence of such exotic hadrons.