Tensor states $ΥB_{c}^{\ast -}$ and $J/ψB_{c}^{\ast +}$

Using the QCD sum rule method, this paper investigates the properties of two heavy-quark hadronic molecules, $\Upsilon B_{c}^{*-}$ and $J/\psi B_{c}^{*+}$, predicting their masses and relatively broad decay widths to demonstrate their instability against dissociation.

The Gist

The Cosmic Lego Set: A Guide to "Heavy" Mystery Particles

Imagine you are playing with a high-tech Lego set. Most of the bricks you have are standard—small, light, and easy to snap together. These represent the "ordinary" particles that make up everything we see, like atoms.

But physicists have discovered a "VIP" set of bricks. These are much heavier, much more powerful, and much harder to handle. These are the heavy quarks (specifically the bottom and charm quarks). This paper is a mathematical blueprint exploring what happens when you try to build something incredibly complex using only these heavy, "VIP" bricks.

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1. The Main Characters: The "Tensor" Molecules

The researchers are looking at two specific, theoretical structures they call Tensor States ($M^b_T$ and $M^c_T$).

Think of these not as single solid bricks, but as "Hadronic Molecules."

• The Analogy: Imagine two heavy bowling balls held together by a very strong magnetic force. They aren't fused into one single object, but they are stuck together so tightly that they move as a single unit.

These molecules are "asymmetric," meaning they aren't perfectly balanced. One is made of a specific mix of heavy quarks ($bbbc$), and the other is a different mix ($cccb$). Because they are so heavy and "unbalanced," they are incredibly unstable.

2. The Problem: The "Exploding" Molecule

The paper's main goal is to figure out two things: How much do they weigh? and How long do they last before they fall apart?

Because these molecules are so heavy, they are like a spinning top made of glass. They want to stay together, but the sheer energy inside them makes them want to shatter. The researchers used a complex mathematical tool called QCD Sum Rules (think of this as a high-powered digital scale and a stopwatch combined) to predict their properties.

3. The Two Ways They Break

The researchers found that these "bowling ball molecules" can break apart in two distinct ways:

• The "Fall-Apart" Method (Leading Decays): This is the easy way. The two bowling balls simply let go of each other and roll away as two separate, smaller objects. This is the most common way they "die."
• The "Annihilation" Method (Subleading Decays): This is the dramatic way. Imagine if, while the two bowling balls were spinning, the atoms inside them suddenly collided and vanished in a flash of light, transforming into a completely different set of lighter particles (like marbles or ping-pong balls). This is much rarer, but it happens.

4. The Verdict: "Broad and Unstable"

The researchers concluded that these particles are "relatively broad."

In particle physics, "broad" is a polite way of saying "extremely short-lived." If a particle is "narrow," it lives a long time (like a sturdy building). If it is "broad," it exists for only a tiny fraction of a blink of an eye before exploding into other things (like a firework).

Why does this matter?

Right now, these particles are mostly theoretical—they are "ghosts" predicted by math. However, massive experiments like the LHC (Large Hadron Collider) are constantly smashing things together to see if these "ghosts" actually appear in real life.

By providing these precise "weights" and "lifespans," this paper gives experimental scientists a treasure map. It tells them exactly what to look for in the wreckage of high-speed collisions to prove that these heavy, exotic molecules actually exist in our universe.

Technical Summary

Technical Summary: Tensor States $\Upsilon_{c}^{b\bar{b}\bar{c}}$ and $J/\psi_{c}^{c\bar{c}\bar{b}}$

1. Problem Statement

The study of exotic hadrons, particularly tetraquarks composed entirely of heavy quarks ($b$ and $c$), is a frontier in high-energy physics. While recent experimental observations by LHCb, CMS, and ATLAS have provided evidence for fully heavy $cccc$ tetraquarks, the properties of asymmetric fully heavy tetraquarks—those with non-identical quark contents such as $bb\bar{b}\bar{c}$ and $cc\bar{c}\bar{b}$—remain theoretically under-explored. The paper specifically investigates the existence, mass spectra, and decay properties of two particular tensor hadronic molecules ($J^P = 2^+$):

• $M_T^b = \Upsilon B_c^{*-}$ (quark content $bb\bar{b}\bar{c}$)
• $M_T^c = J/\psi B_c^{*+}$ (quark content $cc\bar{c}\bar{b}$)

2. Methodology

The authors employ the QCD Sum Rule (QCDSR) method, a non-perturbative approach based on the Operator Product Expansion (OPE) and quark-hadron duality. The technical workflow is as follows:

• Spectroscopic Analysis: Two-point correlation functions are constructed using interpolating currents specifically designed for tensor states. The masses ($m$) and current couplings ($\Lambda$) are extracted by matching the phenomenological side (representing the physical state) with the OPE side (representing the QCD vacuum contributions, including perturbative terms and gluon condensates).
• Decay Width Calculations: To determine the stability and lifetimes of these molecules, the authors utilize three-point QCD Sum Rules. This allows for the calculation of strong coupling constants at the molecule-meson-meson vertices.
• Decay Channel Classification:
  • Leading Channels: "Fall-apart" decays where the constituent mesons are preserved (e.g., $M_T^b \to \Upsilon B_c^{*-}$).
  • Subleading Channels: Processes involving the annihilation of heavy quark pairs ($bb \to q\bar{q}$ or $cc \to q\bar{q}$), resulting in the production of $B^{(*)}D^{(*)}$ meson pairs.
• Extrapolation: Since sum rules are valid in the deep Euclidean region ($q^2 < 0$), the authors use specific fit functions to extrapolate the form factors to the physical mass shell ($q^2 = m_{meson}^2$).

3. Key Contributions

• First Calculation of Tensor Asymmetric Molecules: This paper provides the first theoretical estimation of the masses and decay widths for these specific tensor structures using the molecular model.
• Comprehensive Decay Mapping: The study distinguishes between the dominant "constituent" decays and the subleading "annihilation" decays, providing a complete picture of the particle's instability.
• Stability Assessment: By comparing calculated masses to the thresholds of constituent mesons, the authors determine whether these states are bound states or resonances.

4. Results

• Mass Predictions:
  • $m(M_T^b) = (15864 \pm 85) \text{ MeV}$
  • $m(M_T^c) = (9870 \pm 82) \text{ MeV}$
    The results indicate that these states are above the thresholds of their constituent mesons, meaning they are unstable against dissociation.
• Decay Widths ($\Gamma$):
  • $\Gamma[M_T^b] = 120^{+17}_{-12} \text{ MeV}$
  • $\Gamma[M_T^c] = (71 \pm 9) \text{ MeV}$
    These relatively large widths characterize these molecules as broad structures.
• Dominant Decay Modes:
  • For $M_T^b$: $\Upsilon B_c^{*-}$ and $\eta_b B_c^-$.
  • For $M_T^c$: $J/\psi B_c^{*+}$ and $\eta_c B_c^+$.

5. Significance

This research contributes significantly to the theoretical landscape of exotic spectroscopy. By predicting the masses and widths of these asymmetric tetraquarks, the paper provides a roadmap for experimentalists at facilities like the LHC. If these states are observed with the predicted widths and decay patterns, it would validate the hadronic molecule model for fully heavy systems and deepen our understanding of the color-force organization in multi-quark states.