Molecular states $J/\psi B_{c}^{+}$ and $\eta_{c}B_{c}^{\ast +}$

This paper investigates the hadronic molecule $J/\psi B_{c}^{+}$ using QCD sum rules, predicting its mass to be $(9740 \pm 70)$ MeV and its total width to be $(121 \pm 17)$ MeV, which indicates that the state decays strongly into ordinary mesons via dominant fall-apart and subdominant annihilation mechanisms.

The Gist

Imagine the universe is a giant, bustling 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 electron) or triplets (like a proton made of three quarks). But sometimes, they form exotic, four-block structures called tetraquarks.

This paper is like a theoretical blueprint for two very specific, heavy-duty constructions made of four quarks: three charm quarks and one bottom quark. The authors are trying to figure out what these structures look like, how heavy they are, and how long they last before falling apart.

Here is the breakdown of their findings using simple analogies:

1. The Two Blueprints: "The Twin Structures"

The scientists looked at two potential ways to arrange these four quarks:

• Structure A: A "J/ψ" particle (a heavy charm-anticharm pair) holding hands with a "B+" particle (a bottom-anticharm pair).
• Structure B: An "ηc" particle (a different type of charm pair) holding hands with a "B*" particle (a slightly excited bottom pair).

Think of these like two different ways to stack the same four Lego bricks. The authors calculated the weight and stability of both stacks. They found that, mathematically, these two stacks are almost identical in weight and stability. Because the difference is so small (like the difference between two grains of sand), the paper decides to focus on just one of them (Structure A) to save time, treating them as effectively the same for their calculations.

2. The Weight: "Too Heavy to Stand Still"

The team calculated the mass (weight) of this particle to be roughly 9,740 MeV (a unit of energy used in particle physics).

To understand what this means, imagine a heavy box sitting on a scale. The authors compared this weight to the combined weight of the two smaller boxes that make it up (the J/ψ and the B+).

• The Result: The big box is heavier than the two smaller boxes combined.
• The Analogy: Imagine you try to glue two heavy suitcases together to make a super-suitcase. If the super-suitcase ends up weighing more than the two suitcases added together, it's unstable. It's like a wobbly tower that wants to collapse immediately.
• The Conclusion: Because it is heavier than its parts, this particle cannot sit still as a stable "bound state." Instead, it is a resonance—a fleeting, unstable structure that immediately falls apart into its two constituent pieces.

3. The Breakup: "Two Ways to Fall Apart"

Since this particle is unstable, the authors asked: How does it break apart? They identified two main mechanisms, like two different ways a house of cards might collapse:

Mechanism 1: The "Snap" (Dominant Decay)
This is the most common way it breaks. The molecule simply falls apart into its two original components: the J/ψ and the B+ (or the ηc and B*).

• Analogy: Imagine a magnet holding two metal balls. If the magnet is weak, the balls just pop apart and fly away. This happens about 64% of the time.

Mechanism 2: The "Explosion" (Subdominant Decay)
This is a more complex process. Inside the molecule, the two charm quarks annihilate each other (they destroy one another), releasing energy that instantly creates new particles.

• Analogy: Imagine the two metal balls inside the magnet suddenly turn into a flash of light, which then instantly reshapes into four different balls (like a B-meson and a D-meson). This is like a chemical reaction where ingredients are swapped for something entirely new.
• The Result: This happens about 36% of the time, creating various combinations of B and D mesons.

4. The Lifespan: "A Very Short Flash"

The authors calculated the total "width" of the particle, which in physics is a measure of how quickly it decays (how short its life is).

• They found the particle lives for a very brief moment, with a width of 121 ± 17 MeV.
• The Analogy: If a stable particle is like a stone that sits on the ground for years, this particle is like a firework spark. It exists for a split second and then vanishes. Because it decays so quickly, it is considered a "broad" resonance, meaning it's hard to pin down exactly.

5. Why This Matters

The authors aren't just guessing; they used a rigorous mathematical tool called QCD Sum Rules (think of it as a high-powered calculator that uses the fundamental laws of the strong nuclear force).

• The Goal: They want to help experimentalists (the people with the giant particle colliders like the LHC) know what to look for.
• The Prediction: If scientists scan the data for a "bump" or a "peak" in the mass of particles around 9,740 MeV, they might find this exotic molecule.
• The Caveat: The authors note that a different type of structure (a "diquark-antidiquark" arrangement) could also exist at a similar weight. Distinguishing between a "molecule" (two particles holding hands) and a "tetraquark" (four particles fused into one blob) is tricky and requires comparing their predicted decay patterns with real-world data.

Summary

In short, this paper predicts the existence of a heavy, exotic particle made of four quarks. It is unstable, heavier than its parts, and falls apart very quickly (in about 121 MeV of "time"). It mostly breaks back into the two heavy particles it was made of, but sometimes it explodes into a different set of lighter particles. The authors hope this blueprint helps experimentalists spot this fleeting ghost in the data.

Technical Summary

1. Problem Statement

The paper investigates the spectroscopic properties and decay mechanisms of fully heavy hadronic molecules composed of four heavy quarks ($c c \bar{c} \bar{b}$). Specifically, the authors focus on two axial-vector ($J^P = 1^+$) molecular states:

• $M = J/\psi B^+_c$ (composed of a vector charmonium and a pseudoscalar $B_c$ meson).
• $\tilde{M} = \eta_c B^{*+}_c$ (composed of a pseudoscalar charmonium and a vector $B_c$ meson).

While fully heavy tetraquarks (diquark-antidiquark structures) have been studied extensively, the nature of these specific molecular configurations remains theoretically under-explored. The primary goals are to:

1. Determine the masses and current couplings of these states.
2. Determine whether they form bound states or resonant states above the two-meson thresholds.
3. Calculate their total decay widths and branching fractions via two distinct mechanisms: "fall-apart" decays and annihilation-induced decays.

2. Methodology

The authors employ the QCD Sum Rule (SR) method, a non-perturbative approach that connects hadronic properties to QCD parameters via the Operator Product Expansion (OPE).

• Two-Point Sum Rules (Spectroscopy):

  • Interpolating currents $I_\mu(x)$ were constructed for the $J/\psi B^+_c$ and $\eta_c B^{*+}_c$ molecules.
  • Correlation functions were calculated on the "phenomenological side" (using hadron parameters like mass $m$ and coupling $\Lambda$) and the "QCD side" (using heavy quark propagators and vacuum condensates).
  • The Borel transformation and continuum subtraction were applied to suppress higher resonance contributions.
  • Parameters such as the Borel mass $M^2$ and continuum threshold $s_0$ were optimized to ensure pole contribution (PC) $\geq 0.5$ and OPE convergence.

• Three-Point Sum Rules (Decay Widths):

  • To calculate decay widths, the authors evaluated strong coupling constants ($g_i$) at the vertices where the molecule decays into two mesons.
  • Dominant Channels: Calculated couplings for $M \to J/\psi B^+_c$ and $M \to \eta_c B^{*+}_c$.
  • Subdominant Channels (Annihilation): Calculated couplings for processes where the $c\bar{c}$ pair annihilates into light quarks ($u\bar{u}, d\bar{d}, s\bar{s}$), leading to final states like $B^* D$, $B D^*$, and strange meson pairs.
  • Condensate Relation: For annihilation channels involving light quarks, the authors utilized the relation between the heavy quark condensate $\langle \bar{c}c \rangle$ and the gluon condensate $\langle \alpha_s G^2/\pi \rangle$ to close the calculation without introducing new free parameters.
  • Extrapolation: Since SRs are valid in the Euclidean region ($q^2 < 0$), the authors used extrapolating functions (e.g., $Z(Q^2)$) to estimate the couplings at the physical mass shells ($q^2 = m^2_{meson}$).

3. Key Contributions

• Mass Prediction: The paper provides the first detailed mass prediction for the $J/\psi B^+_c$ and $\eta_c B^{*+}_c$ molecular states using QCD SRs.
• Distinction of Mechanisms: It explicitly separates and calculates contributions from two decay mechanisms:
  1. Fall-apart mechanism: The molecule dissociates directly into its constituent mesons.
  2. Annihilation mechanism: The $c\bar{c}$ pair annihilates, transforming the molecule into conventional mesons containing light quarks ($B, D, B_s, D_s$).
• Comparison of Structures: The authors demonstrate that while $M$ and $\tilde{M}$ have different internal color configurations, their masses and couplings are numerically indistinguishable within the uncertainties of the SR method.
• Stability Analysis: The study clarifies that these states are resonant (unstable) rather than bound, as their predicted masses lie significantly above the relevant two-meson thresholds.

4. Key Results

Spectroscopic Parameters

• Mass of $M$ ($J/\psi B^+_c$): $m = (9740 \pm 70)$ MeV.
• Mass of $\tilde{M}$ ($\eta_c B^{*+}_c$): $\tilde{m} = (9737 \pm 71)$ MeV.
• Current Couplings: $\Lambda \approx 5.19 \times 10^{-1}$ GeV$^5$ and $\tilde{\Lambda} \approx 5.23 \times 10^{-1}$ GeV$^5$.
• Conclusion on Stability: The mass is well above the $J/\psi B^+_c$ threshold ($\approx 9372$ MeV). Even at the lower bound of the error margin ($9660$ MeV), the state is unbound and will decay strongly.

Decay Widths

The total width is dominated by fall-apart decays, with significant contributions from annihilation channels.

Decay Channel	Mechanism	Partial Width (MeV)
$M \to J/\psi B^+_c$	Fall-apart	$40.6 \pm 12.4$
$M \to \eta_c B^{*+}_c$	Fall-apart	$36.8 \pm 9.5$
$M \to B^{*+} D^0, B^{*0} D^+$	Annihilation ($c\bar{c} \to u\bar{u}/d\bar{d}$)	$9.7 \pm 3.0$ (each)
$M \to B^+ D^{*0}, B^0 D^{*+}$	Annihilation ($c\bar{c} \to u\bar{u}/d\bar{d}$)	$6.6 \pm 1.7$ (each)
$M \to B^{*0}_s D^+_s$	Annihilation ($c\bar{c} \to s\bar{s}$)	$6.8 \pm 1.8$
$M \to B^0_s D^{*+}_s$	Annihilation ($c\bar{c} \to s\bar{s}$)	$3.9 \pm 1.1$
Total Width	Sum	$121 \pm 17$

• Branching Fractions:
  • Dominant (Fall-apart) modes constitute ~64% of the total width.
  • Subdominant (Annihilation) modes constitute ~36% of the total width.

5. Significance and Discussion

• Experimental Guidance: The predicted mass ($\sim 9.74$ GeV) and broad width ($\sim 121$ MeV) provide specific targets for experiments at LHCb, CMS, and ATLAS. The authors suggest searching for a broad peak in the $J/\psi B^+_c$ or $\eta_c B^{*+}_c$ invariant mass distributions.
• Structural Differentiation: The paper notes that while a diquark-antidiquark tetraquark with the same quark content ($cc\bar{c}\bar{b}$) might have a similar mass (predicted elsewhere as $\sim 9611-9706$ MeV), the decay patterns and widths could help distinguish between a molecular structure and a compact tetraquark.
• Comparison with $bb$ Systems: The authors contrast these $cc\bar{c}\bar{b}$ states with $bb\bar{c}\bar{c}$ systems (like $\Upsilon B^-_c$). Unlike the $bb$-based molecules which can be stable bound states below the threshold, the $cc\bar{c}\bar{b}$ molecules are inherently unstable resonances due to the lighter charm quark mass.
• Theoretical Consistency: The results align with previous studies suggesting that fully heavy asymmetric molecules are generally resonant states, though the specific calculation of the annihilation-induced partial widths adds necessary detail for experimental phenomenology.

In conclusion, this work establishes the $J/\psi B^+_c$ and $\eta_c B^{*+}_c$ as broad, unstable hadronic molecules with a total width of approximately 121 MeV, decaying primarily into their constituent mesons but with a substantial fraction decaying via $c\bar{c}$ annihilation into open-charm/beauty meson pairs.