Scalar molecules $\eta _{b}B_{c}^{-}$ and $\eta _{c}B_{c}^{+}$ with asymmetric quark contents

Using the QCD sum rule method, this paper investigates the masses, decay widths, and dominant decay channels of the hypothetical scalar molecules $\mathcal{M}_{b}$ ($bb\overline{b}\overline{c}$) and $\mathcal{M}_{c}$ ($cc\overline{c}\overline{b}$), predicting them to be strong-interaction unstable particles with masses of approximately 15.7 GeV and 9.7 GeV, respectively, to guide future experimental searches.

The Gist

Imagine the universe as a giant, bustling construction site. Most of the buildings we see are made of standard bricks: protons and neutrons. But physicists have long suspected that under certain conditions, these bricks can stick together in weird, temporary ways to form "exotic" structures that don't follow the usual rules.

This paper is like a theoretical blueprint for two very specific, very heavy, and very unstable "ghost buildings" made of four quarks (the fundamental particles that make up protons and neutrons). The authors, using a sophisticated mathematical tool called the QCD Sum Rule method (think of it as a high-powered calculator that predicts how particles behave based on the laws of the strong force), investigated two specific molecular structures:

1. $M_b$: A molecule made of three bottom quarks and one charm quark ($bbbc$).
2. $M_c$: A molecule made of three charm quarks and one bottom quark ($cccb$).

Here is the breakdown of their findings in plain language:

1. What are these molecules?

Usually, particles are like single Lego bricks (a quark and an antiquark). Sometimes, they form "tetraquarks," which are like two bricks glued tightly together. But the authors are looking at hadronic molecules.

Think of a hadronic molecule not as a single glued brick, but as two separate Lego structures (ordinary mesons) that are holding hands loosely.

• $M_b$ is imagined as a loose partnership between an $\eta_b$ particle and a $B_c^-$ particle.
• $M_c$ is a loose partnership between an $\eta_c$ particle and a $B_c^+$ particle.

Because they are "asymmetric" (they have three of one type of heavy quark and only one of another), they are unique and have never been clearly seen in experiments yet.

2. How heavy are they?

The authors calculated the "weight" (mass) of these ghost buildings:

• $M_b$ weighs about 15,728 MeV. This is incredibly heavy—about 16 times the mass of a proton. Interestingly, this weight is just barely heavy enough to fall apart into its two component parts ($\eta_b$ and $B_c^-$). It's like a tower that is so tall it's teetering on the edge of collapsing.
• $M_c$ weighs about 9,712 MeV. This is also very heavy, but it sits comfortably above the weight needed to break apart. It's a tower that is definitely ready to collapse.

3. How long do they last? (The Decay)

These molecules are not stable. They are like soap bubbles that pop almost instantly. The authors calculated how fast they pop (their "width" or decay rate):

• $M_b$ lasts for a tiny fraction of a second, with a decay width of about 93 MeV.
• $M_c$ is slightly more stable but still fleeting, with a width of about 70 MeV.

How do they pop?
They don't just vanish; they transform into other, more common particles.

• The Main Event: The most likely way they break is by simply separating into their two component parts (like a couple breaking up and walking away).
  • $M_b$ splits into $\eta_b$ and $B_c^-$.
  • $M_c$ splits into $\eta_c$ and $B_c^+$, or sometimes into a $J/\psi$ and a $B_c^*$.
• The "Annihilation" Side Effect: Sometimes, the heavy quarks inside the molecule (like the three bottom quarks in $M_b$) can crash into each other and annihilate (disappear), turning their energy into new pairs of lighter particles (like $B$ and $D$ mesons). The authors found that while this happens less often than the main breakup, it still contributes significantly to how fast the molecule disappears.

4. Why does this matter?

The authors compared their "loose molecule" models to "tight tetraquark" models (where the four quarks are glued together in a tight cluster).

• They found that their loose molecules are slightly heavier than the tight clusters.
• They also found that the loose molecules are broader (they decay faster) than the tight clusters.

The Bottom Line for Experimenters:
The paper serves as a "Wanted Poster" for experimental physicists working at facilities like the LHC. It says: "If you look for a particle with a mass around 15,728 MeV or 9,712 MeV that decays into these specific pairs of particles, you might find these exotic molecules."

The authors conclude that while these particles are unstable and short-lived, their specific masses and decay patterns provide a clear target for scientists to hunt down in future experiments. They are essentially saying, "We've done the math; now go look for them there."

Technical Summary

Technical Summary: Scalar Molecules $\eta_b B^-_c$ and $\eta_c B^+_c$ with Asymmetric Quark Contents

Problem Statement
This work investigates the spectroscopic properties and decay dynamics of two scalar heavy hadronic molecules with asymmetric quark contents: $M_b$ (quark content $bb\bar{b}\bar{c}$) and $M_c$ (quark content $cc\bar{c}\bar{b}$). These structures are proposed as molecular analogues to asymmetric tetraquarks ($T_b$ and $T_c$) previously studied in a diquark-antidiquark framework. The primary objective is to determine whether these scalar states exist as bound states or resonances, calculate their masses and current couplings, and evaluate their full decay widths by analyzing both dominant decay channels (where constituent quarks rearrange) and subleading channels (generated by heavy quark annihilation).

Methodology
The authors employ the Quantum Chromodynamics (QCD) sum rule (SR) method to analyze these four-quark systems. The study is divided into two main computational phases:

1. Two-Point Sum Rules (Spectroscopy):

   • Interpolating currents are constructed for the scalar molecules $M_b = \eta_b B^-_c$ and $M_c = \eta_c B^+_c$ using products of pseudoscalar meson currents.
   • Two-point correlation functions are calculated in both the physical representation (pole approximation) and the Operator Product Expansion (OPE) representation.
   • The OPE includes perturbative contributions and non-perturbative condensates up to dimension-6 ($\langle \alpha_s G^2/\pi \rangle$ and $\langle g_s^3 G^3 \rangle$).
   • Borel transformations and continuum subtraction are applied to extract the masses ($m, \tilde{m}$) and current couplings ($\Lambda, \tilde{\Lambda}$). Stability windows for the Borel parameter $M^2$ and continuum threshold $s_0$ are established to ensure pole dominance and OPE convergence.

2. Three-Point Sum Rules (Decay Widths):

   • To calculate partial decay widths, the strong coupling constants at the molecule-meson-meson vertices are determined.
   • Three-point correlation functions are analyzed for vertices such as $M_b \eta_b B^-_c$, $M_b B^- D^0$, and $M_c \eta_c B^+_c$.
   • For decay channels involving heavy quark annihilation (e.g., $bb \to q\bar{q}$), the vacuum expectation value of the heavy quark pair $\langle b\bar{b} \rangle$ is related to the gluon condensate $\langle \alpha_s G^2/\pi \rangle$ within the SR framework.
   • Form factors are computed in the Euclidean region ($Q^2 < 0$) and extrapolated to the physical mass shell using fitting functions to extract the strong couplings.
   • Partial widths are calculated using standard kinematic formulas, and the total width is obtained by summing contributions from all allowed channels.

Key Results

• Masses and Stability:

  • The mass of the molecule $M_b$ is calculated as $m = (15728 \pm 90)$ MeV. This value is close to the $\eta_b B^-_c$ threshold (15673 MeV). The authors note that within the lower uncertainty limit ($m \approx 15638$ MeV), $M_b$ could be interpreted as a bound state of $\eta_b$ and $B^-_c$.
  • The mass of the molecule $M_c$ is calculated as $\tilde{m} = (9712 \pm 72)$ MeV. This places $M_c$ approximately 250 MeV above the $\eta_c B^+_c$ threshold, identifying it as a broad resonance rather than a bound state.
  • Current couplings are determined as $\Lambda = (3.09 \pm 0.32)$ GeV$^5$ for $M_b$ and $\tilde{\Lambda} = (5.11 \pm 0.48) \times 10^{-1}$ GeV$^5$ for $M_c$.

• Decay Channels and Widths:

  • $M_b$ Decay: The dominant channel is $M_b \to \eta_b B^-_c$. Additionally, subleading channels generated by $bb$ annihilation are considered, leading to pseudoscalar pairs ($B^- D^0, B^0 D^-, B^0_s D^-_s$) and vector pairs ($B^{*-} D^{*0}, B^{*0} D^{*-}, B^{*0}_s D^{*-}_s$).
    • Total width: $\Gamma[M_b] = (93 \pm 17)$ MeV.
    • If treated as a bound state (lower mass limit), the width is reduced to $\Gamma[M_b]|_{l.m.} = (49 \pm 6)$ MeV, driven solely by annihilation mechanisms.
  • $M_c$ Decay: The dominant channels are $M_c \to \eta_c B^+_c$ and $M_c \to J/\psi B^{*+}_c$. Subleading channels arise from $cc$ annihilation into $B^{(*)} D^{(*)}$ pairs.
    • Total width: $\Gamma[M_c] = (70 \pm 10)$ MeV.

Significance and Claims
The paper claims to provide the first detailed QCD sum rule analysis of these specific scalar molecules with asymmetric quark contents. The authors highlight several points of comparison and significance:

1. Molecular vs. Diquark-Antidiquark Structure: The calculated masses for the molecular states $M_b$ and $M_c$ are slightly higher than those of their diquark-antidiquark counterparts ($T_b$ and $T_c$) calculated in previous work [38]. The authors attribute this to the internal organization: molecules are composed of color-singlet mesons, whereas diquark models rely on tightly bound colored diquarks. Consequently, the molecular states are predicted to be broader (larger widths) than the diquark states.
2. Experimental Guidance: The predicted masses and widths offer specific targets for experimental searches at existing facilities (e.g., LHCb, ATLAS, CMS). The authors emphasize that the $M_b$ state, particularly near the lower mass limit, represents a potential bound state candidate, while $M_c$ is a distinct resonance.
3. Mechanism Validation: The study validates the application of the QCD sum rule method to subleading decay mechanisms involving heavy quark annihilation, demonstrating that these processes contribute significantly to the total width of such exotic states.

The authors conclude that while their findings are consistent with some theoretical expectations regarding fully heavy molecules lying above thresholds, the results require confirmation via alternative theoretical approaches. The work serves as a step toward understanding the internal structure of potential all-heavy four-quark mesons.