Hadronic Molecules with Four Charm or Beauty Quarks

Abstract

We apply extended local hidden-gauge formalism to study meson–meson interactions with the quark constituents $cc\overline{c}\overline{c}$, $cc\overline{c}\overline{b}/\overline{c}\overline{c}cb$, $cc\overline{b}\overline{b}/\overline{c}\overline{c}bb$, $bb\overline{c}\overline{b}/\overline{b}\overline{b}cb$, and $bb\overline{b}\overline{b}$, in which the exchanged mesons are the fully heavy vector mesons $J/\psi$, $B_c^{*}$ and $\mathrm{{\rm Y}}$. We solve the coupled-channel Bethe–Salpeter equation to derive two poles in the $bb\overline{c}\overline{b}$ system and two poles in the $cc\overline{c}\overline{b}$ system. There are also four charge-conjugated poles in the $\overline{b}\overline{b}cb$ and $\overline{c}\overline{c}cb$ systems. In the $bb\overline{c}\overline{b}$ system, one pole corresponds to a sub-threshold bound state when the cutoff momentum is set to $\mathrm{\Lambda }>850 \mathrm{MeV}$. The other pole in this system corresponds to a sub-threshold bound state when $\mathrm{\Lambda }>1100 \mathrm{MeV}$. In the $cc\overline{c}\overline{b}$ system, the two poles correspond to sub-threshold bound states only when $\mathrm{\Lambda }>1550 \mathrm{MeV}$ and $\mathrm{\Lambda }>2650 \mathrm{MeV}$. This makes them difficult to identify as deeply bound hadronic molecules. We propose investigating the two poles of the $bb\overline{c}\overline{b}$ system in the ${\mu }^{+}{\mu }^{-}{B}_c^{-}$ channel at the LHC.

1. Introduction

In recent decades, the study of exotic hadrons has gradually become a focal point in hadron physics. Some exotic hadrons exhibit multiquark compositions, such as the compact tetraquark states and the meson-meson molecular states [1,2,3,4,5,6,7]. The picture of hadronic molecules has achieved significant success in the light-quark sector [8,9,10,11,12,13,14], which can be used to explain many resonances, such as $f_0\left(980\right)$ and $a_0\left(980\right)$, etc. Additionally, many hidden-charm pentaquark states observed in the past decade can be interpreted as hadronic molecules that are dynamically generated through the meson–baryon interactions within the local hidden-gauge framework [15,16,17,18,19,20,21,22]. In recent years, several exotic structures in the di-$J/\psi$-invariant mass spectrum have been reported through the LHCb, CMS, and ATLAS collaborations [23,24,25], including $X\left(6200\right)$, $X\left(6600\right)$, $X\left(6900\right)$, and $X\left(7200\right)$. These structures are good candidates for fully charmed tetraquark states. Extensive theoretical investigations have been performed to elucidate their nature [26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68], some of which have attempted to explore their nature as molecular states [69,70,71,72,73,74,75], but a definitive and conclusive understanding of their nature remains elusive.

Previous theoretical studies on fully heavy tetraquark states mainly focus on the interpretation of compact tetraquark states, while there are not so many studies based on the interpretation of hadronic molecular states. This is because the exchanged hadrons of these systems have quite large masses, such as the fully heavy vector mesons $J/\psi$, $B_c^{*}$, and $\mathrm{{\rm Y}}$ exchanged in the $cb\overline{c}\overline{b}$ system within extended local hidden-gauge formalism, so their induced interactions are significantly suppressed. In Ref. [76], we studied the $cb\overline{c}\overline{b}$ system to explore the existence of fully heavy hadronic molecules $B_c^{(*)}{\overline{B}}_c^{(*)}$. Within the extended local hidden-gauge framework, we found that the two fully heavy mesons $B_c^{(*)}$ and ${\overline{B}}_c^{(*)}$ are able to form a bound state by exchanging the relatively lighter meson $J/\psi$.

In this paper, we apply extended local hidden-gauge formalism to further investigate the fully heavy hadronic molecules that exist in $cc\overline{c}\overline{c}$, $cc\overline{c}\overline{b}/\overline{c}\overline{c}cb$, $cc\overline{b}\overline{b}/\overline{c}\overline{c}bb$, $bb\overline{c}\overline{b}/\overline{b}\overline{b}cb$, and $bb\overline{b}\overline{b}$ systems. By solving the coupled-channel Bethe–Salpeter equation, we evaluate the hadronic molecules generated via the meson–meson interactions in these systems. Our results indicate the possible existence of two bound states in the $bb\overline{c}\overline{b}$ system, along with two charge-conjugated states in the $\overline{b}\overline{b}cb$ system. However, their manifestation depends on the cutoff momentum, as a result of which they may appear as threshold effects. Both structures share the same spin-parity quantum number $J^P=1^{+}$, and they can potentially be observed in the ${\mu }^{+}{\mu }^{-}{B}_c^{-}$ channel at LHC. Additionally, we find two poles in the $cc\overline{c}\overline{b}$ system (with two charge-conjugated poles in the $\overline{c}\overline{c}cb$ system), but it is difficult to identify them as deeply bound hadronic molecules.

This paper is organized as follows. In Section 2, we apply local hidden-gauge formalism to derive the potentials for the interactions of the $cc\overline{c}\overline{c}$, $cc\overline{c}\overline{b}/\overline{c}\overline{c}cb$, $cc\overline{b}\overline{b}/\overline{c}\overline{c}bb$, $bb\overline{c}\overline{b}/\overline{b}\overline{b}cb$, and $bb\overline{b}\overline{b}$ systems. Based on the obtained results, we solve the coupled-channel Bethe–Salpeter equation in Section 3 to extract the poles, some of which may qualify as fully heavy hadronic molecules. A brief summary is provided in Section 4.

2. Formalism

In Ref. [76], we applied extended local hidden-gauge formalism to study the interactions of the $cb\overline{c}\overline{b}$ system. In this section, we follow the same approach to study the $cc\overline{c}\overline{c}$, $cc\overline{c}\overline{b}/\overline{c}\overline{c}cb$, $cc\overline{b}\overline{b}/\overline{c}\overline{c}bb$, $bb\overline{c}\overline{b}/\overline{b}\overline{b}cb$, and $bb\overline{b}\overline{b}$ systems. Note that the validity of extended local hidden-gauge formalism in these systems is still questionable. Within the framework of local hidden-gauge formalism [77,78], the vector mesons are considered to act as gauge bosons, transmitting interactions. This mechanism successfully describes many low-energy interactions [15,79,80,81,82], especially in processes dominated by the exchange of light vector mesons. Unfortunately, it is not clear whether this mechanism remains effective in the heavy flavor region. An extension of this method has also been developed to study molecular state candidates containing heavy quarks [15,20,83]. In these cases, generally, the exchange of light vector mesons and heavy vector mesons is allowed simultaneously. When the vector meson–vector meson interaction in the $c\overline{c}s\overline{s}$ system is taken as an example, this system involves two coupling channels: $D_s^{*}{\overline{D}}_s^{*}$ and $J/\psi \varphi$. In the $D_s^{*}{\overline{D}}_s^{*}$ channel, the exchange of $\varphi$ and $J/\psi$ occurs simultaneously. Compared to the $\varphi$ exchange, the $J/\psi$ exchange is considered to be secondary due to its larger mass. However, things are somewhat different in the $J/\psi \varphi$ channel. The minimal constituents for the exchanged meson should be $c\overline{c}s\overline{s}$, which we do not consider in local hidden-gauge formalism. Such a type of exchange mechanism should also be suppressed due to its involvement in the exchange of four quarks. On the other hand, the $J/\psi \varphi$ channel can couple with the $D_s^{*}{\overline{D}}_s^{*}$ channel, which allows for the exchange of the heavy vector meson $D_s^{*}$ between these two channels. Although the exchange of heavy vector mesons should be relatively minimal, it plays a role in this scenario. Under this assumption, results consistent with experiments can be obtained [83]. So, the present study, as well as Ref. [76], serve as pioneering research investigating fully heavy hadronic molecules. Within this framework, the interactions are primarily contributed via the vector meson exchange, as depicted in Figure 1. The corresponding Lagrangians are written as follows:

$$\begin{array}{ccc}\hfill {\mathcal{L}}_{VPP}& =& -ig⟨[P,{\partial }_{\mu }P]V^{\mu }⟩,\hfill \\ \hfill {\mathcal{L}}_{VVV}& =& ig⟨(V^{\mu }{\partial }_{\nu }{V}_{\mu }-{\partial }_{\nu }{V}^{\mu }{V}_{\mu })V^{\nu }⟩,\hfill \\ \hfill {\mathcal{L}}_{VVVV}& =& {\displaystyle \frac{g^2}{2}}⟨V_{\mu }{V}_{\nu }{V}^{\mu }{V}^{\nu }-V_{\nu }{V}_{\mu }{V}^{\mu }{V}^{\nu }⟩,\hfill \end{array}$$

(1)

where

$$P=\left(\begin{array}{cc}{\eta }_c& B_c^{+}\\ B_c^{-}& {\eta }_b\end{array}\right),V=\left(\begin{array}{cc}J/\psi & B_c^{*+}\\ B_c^{*-}& \mathrm{{\rm Y}}\end{array}\right).$$

(2)

The coupling constant g is generally defined as $g=M_V/\left(2f_P\right)$, with $M_V$ being the mass of the exchanged vector meson and $f_P$ being the decay constant of its corresponding pseudoscalar meson. Since the charm and bottom quarks do not form a flavor symmetry, $SU\left(2\right)$, we cannot use an overall parameter. For the exchange of the $J/\psi$, $B_c^{*}$, and $\mathrm{{\rm Y}}$ mesons, we respectively use $M_{J/\psi }=3096.9\mathrm{MeV}$ [84], $f_{{\eta }_c}=387/\sqrt{2}\mathrm{MeV}$ [85], $M_{B_c^{*}}=6331\mathrm{MeV}$ [86], $f_{B_c}=427/\sqrt{2}\mathrm{MeV}$ [87], $M_{\mathrm{{\rm Y}}}=9460.4\mathrm{MeV}$ [84], $f_{{\eta }_b}=667/\sqrt{2}\mathrm{MeV}$ [87]. Additionally, we use $g^4=g_{V_1}{g}_{V_2}{g}_{V_3}{g}_{V_4}$ for the contact term, with $V_{1\cdots 4}$ denoting the four connected vector mesons.

Figure 1. The meson–meson interactions arising from the vector meson exchange: (a) between two pseudoscalar mesons, (b) between one vector meson and one pseudoscalar meson, and (c) between two vector mesons. The subfigure (d) describes the contact term connecting four vector mesons.

We derive the following interaction potential from the Lagrangians given in Equation (1):

$$\begin{array}{ccc}\hfill V_{PP}\left(s\right)& =& C_{PP}^t\times g^2(p_1+p_3)(p_2+p_4)\hfill \\ \hfill & +& C_{PP}^u\times g^2(p_1+p_4)(p_2+p_3),\hfill \end{array}$$

(3)

$$\begin{array}{ccc}\hfill V_{VP}\left(s\right)& =& C_{VP}^t\times g^2(p_1+p_3)(p_2+p_4){ϵ}_1·{ϵ}_3\hfill \\ \hfill & +& C_{VP}^u\times g^2(p_1+p_4)(p_2+p_3){ϵ}_1·{ϵ}_3,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}\hfill V_{VV}\left(s\right)& =& V_{VV}^{ex}\left(s\right)+V_{VV}^{co}\left(s\right),\hfill \end{array}$$

(5)

$$\begin{array}{ccc}\hfill V_{VV}^{ex}\left(s\right)& =& C_{VV}^t\times g^2(p_1+p_3)(p_2+p_4){ϵ}_1·{ϵ}_3{ϵ}_2·{ϵ}_4\hfill \\ \hfill & +& C_{VV}^u\times g^2(p_1+p_3)(p_2+p_4){ϵ}_1·{ϵ}_4{ϵ}_2·{ϵ}_3,\hfill \end{array}$$

(6)

where $p_1$ and $p_2$ are four-momenta of the incoming mesons, $p_3$ and $p_4$ are four-momenta of the outgoing mesons, ${ϵ}_1$ and ${ϵ}_2$ are polarization vectors of the incoming mesons, and ${ϵ}_3$ and ${ϵ}_4$ are polarization vectors of the outgoing mesons. We use the subscripts $PP$, $VP$, and $VV$ to denote the pseudoscalar–pseudoscalar, vector–pseudoscalar, and vector–vector sectors, respectively. We use the superscripts t, u, and $co$ to denote the vector meson exchange in the t and u channels, as well as the contact term, respectively. Note that the coefficient $C_{VP}^u$ is zero because we only consider the vector meson exchange in the present study. Actually, there also exists the pseudoscalar meson exchange. As evaluated in detail in Refs. [88,89], the contribution of the pseudoscalar meson exchange is negligible near the threshold compared to the vector meson exchange. This is because the amplitude of the vector meson exchange is proportional to the energy of the external meson, while the amplitude of the pseudoscalar meson exchange is proportional to the three-momentum of the external meson, which is always negligible near the threshold.

We further derive the scattering amplitudes from Equations (3)–(6) by solving the Bethe–Salpeter equation as

$$T_{PP/VP/VV}={\left(\mathbf{1}-V_{PP/VP/VV}•G\right)}^{-1}•V_{PP/VP/VV},$$

(7)

where $G\left(s\right)$ is the diagonal loop function, whose expression for the ith channel is

$$G_{ii}\left(s\right)=i\int {\displaystyle \frac{d^{4}q}{{\left(2\pi \right)}^4}}{\displaystyle \frac{1}{q^2-m_1^2+iϵ}}{\displaystyle \frac{1}{{(p-q)}^2-m_2^2+iϵ}}.$$

(8)

In the above expression, $m_{1,2}$ are the masses of the two mesons involved in this channel, and $s=p^2$ with p is the total four-momentum. We apply the cutoff method to regularize it as

$$G_{ii}\left(s\right)={\int }_0^{\mathrm{\Lambda }}{\displaystyle \frac{d^{3}q}{{\left(2\pi \right)}^3}}{\displaystyle \frac{{\omega }_1+{\omega }_2}{2{\omega }_1{\omega }_2}}{\displaystyle \frac{1}{s-{({\omega }_1+{\omega }_2)}^2+iϵ}},$$

(9)

where ${\omega }_1=\sqrt{m_1^2+{\overrightarrow{q}}^2}$, ${\omega }_2=\sqrt{m_2^2+{\overrightarrow{q}}^2}$, and $\mathrm{\Lambda }$ is the cutoff momentum.

We shall calculate the coefficient matrices $C_{PP/VP/VV}^{t/u/co}$ in the following subsections separately for the $cc\overline{c}\overline{c}$, $cc\overline{c}\overline{b}/\overline{c}\overline{c}cb$, $cc\overline{b}\overline{b}/\overline{c}\overline{c}bb$, $bb\overline{c}\overline{b}/\overline{b}\overline{b}cb$, and $bb\overline{b}\overline{b}$ systems. Before doing this, we summarize their relevant coupled channels in Table 1.

Table 1. Coupled channels considered for the $cc\overline{c}\overline{c}$, $cc\overline{c}\overline{b}/\overline{c}\overline{c}cb$, $cc\overline{b}\overline{b}/\overline{c}\overline{c}bb$, $bb\overline{c}\overline{b}/\overline{b}\overline{b}cb$, and $bb\overline{b}\overline{b}$ systems.

Constituent	$\mathit{PP}$ Sector	$\mathit{VP}$ Sector	$\mathit{VV}$ Sector
$cc\overline{c}\overline{c}$	${\eta }_c{\eta }_c$	$J/\psi {\eta }_c$	$J/\psi J/\psi$
$cc\overline{c}\overline{b}$	${\eta }_c{B}_c^{+}$	$J/\psi B_c^{+},B_c^{*+}{\eta }_c$	$J/\psi B_c^{*+}$
$cc\overline{b}\overline{b}$	$B_c^{+}{B}_c^{+}$	$B_c^{*+}{B}_c^{+}$	$B_c^{*+}{B}_c^{*+}$
$bb\overline{c}\overline{b}$	${\eta }_b{B}_c^{-}$	$\mathrm{{\rm Y}}{B}_c^{-},B_c^{*-}{\eta }_b$	$\mathrm{{\rm Y}}{B}_c^{*-}$
$bb\overline{b}\overline{b}$	${\eta }_b{\eta }_b$	$\mathrm{{\rm Y}}{\eta }_b$	$\mathrm{{\rm Y}}\mathrm{{\rm Y}}$

2.1. The $cc\overline{c}\overline{c}$ System

In the $cc\overline{c}\overline{c}$ system, the $PP$ interaction involves only one channel, ${\eta }_c{\eta }_c$, the $VP$ interaction involves only one channel, $J/\psi {\eta }_c$, and the $VV$ interaction also involves only one channel, $J/\psi J/\psi$. Additionally, the $bb\overline{b}\overline{b}$ system can be similarly investigated.

Since the two vertices are both zero, ${\mathcal{L}}_{{\eta }_c{\eta }_{c}J/\psi }=0$ and ${\mathcal{L}}_{J/\psi J/\psi J/\psi }=0$, the interactions in this system all vanish within the extended local hidden-gauge framework:

$$\begin{array}{cc}\hfill C_{PP}^t& =C_{PP}^u=0,\hfill \\ \hfill C_{VP}^t& =C_{VP}^u=0,\hfill \\ \hfill C_{VV}^t& =C_{VV}^u=0,\hfill \\ \hfill C_{VV}^{co}& =0.\hfill \end{array}$$

(10)

The $bb\overline{b}\overline{b}$ system and the $cc\overline{c}\overline{c}$ system exhibit very similar dynamic properties, with the only distinction being their mass differences. Consequently, they share the same coefficients listed in Equation (10), implying that the interactions within the $bb\overline{b}\overline{b}$ system also vanish.

Therefore, our results do not support the existence of hadronic molecules in the $cc\overline{c}\overline{c}$ and $bb\overline{b}\overline{b}$ systems. Moreover, our conclusions do not actually depend on the value of the coupling constant g. As long as the near-threshold interactions in these systems are dominated by vector meson exchange and can be described by the Lagrangian in Equation (1), the validity of the conclusions can be maintained. Let us examine the specific form of the t-channel exchange potential in Equation (3):

$$\begin{array}{c}\hfill -i{V}^t=i{g}_1{(p_1+p_3)}^{\mu }i{g}_2{(p_2+p_4)}^{\nu }{\displaystyle \frac{i(-g_{\mu \nu }+\frac{q_{\mu }{q}_{\nu }}{m_V^2})}{t-m_V^2}},\end{array}$$

(11)

where $p_1$ and $p_2$ are four-momenta of the incoming mesons, $p_3$ and $p_4$ are four-momenta of the outgoing mesons, q is a four-momentum of the exchanged meson and $g_1$ and $g_2$ are two coupling constants corresponding to the top and bottom vertices in Figure 1a. When the three-momenta of the particles are ignored and t is approximated as 0, the following simplification can be obtained:

$$\begin{array}{c}\hfill V^t\simeq {\displaystyle \frac{g_1{g}_2}{m_V^2}}(m_1+m_3)·(m_2+m_4).\end{array}$$

(12)

It can be found that, if the particles in the channels are identical, i.e., $g_1=g_2$, the interaction will be repulsive. Similar derivations also apply to the exchange potential $V^u$ in the u-channel. This indicates that the existence of near-threshold molecular states in channels similar to ${\eta }_c{\eta }_c$ is difficult to support. The experimentally observed states in these systems are, therefore, good candidates to be tetraquark states.

2.2. The $cc\overline{c}\overline{b}$ and $\overline{c}\overline{c}cb$ Systems

The results for the $cc\overline{c}\overline{b}$ and $\overline{c}\overline{c}cb$ systems are the same, so we only need to study the $cc\overline{c}\overline{b}$ system. Additionally, the $bb\overline{c}\overline{b}$ and $\overline{b}\overline{b}cb$ systems can be similarly investigated. In the $cc\overline{c}\overline{b}$ system, the $PP$ interaction involves only one channel, ${\eta }_c{B}_c^{+}$, whose coefficients are

$$\begin{array}{ccc}\hfill C_{PP}^t& =& 0,\hfill \\ \hfill C_{PP}^u& =& \lambda {\displaystyle \frac{1}{m_{B_c^{*}}^2}}.\hfill \end{array}$$

(13)

The reduction factor $\lambda$ existing in the u channel, is introduced to account for the large mass difference between the initial meson ${\eta }_c$ and the final meson $B_c^{+}$ (or between the initial meson $B_c^{+}$ and the final meson ${\eta }_c$). Following Ref. [90], numerically, we use

$${\lambda }_{{\eta }_c{B}_c^{+}\to B_c^{+}{\eta }_c}\approx {\displaystyle \frac{-m_{B_c^{*}}^2}{{(m_{{\eta }_c}-m_{B_c})}^2-m_{B_c^{*}}^2}}=1.37.$$

(14)

The coefficient $C_{PP}^u$ is positive, indicating that the interaction due to the exchange of the $B_c^{*}$ meson is repulsive; hence, hadronic molecules in the PP sector are not expected to exist.

The $VP$ interaction involves two coupled channels, $J/\psi B_c^{+}$ and $B_c^{*+}{\eta }_c$, whose coefficients are

$$\begin{array}{ccc}\hfill C_{VP}^t& =& \left(\begin{array}{ccc}J=1& J/\psi B_c^{+}& B_c^{*+}{\eta }_c\\ J/\psi B_c^{+}& 0& \lambda {\displaystyle \frac{1}{m_{B_c^{*}}^2}}\\ B_c^{*+}{\eta }_c& \lambda {\displaystyle \frac{1}{m_{B_c^{*}}^2}}& 0\end{array}\right)and\hfill \\ \hfill C_{VP}^u& =& {\mathbf{0}}_{2\times 2}.\hfill \end{array}$$

(15)

Diagonalizing this $2\times 2$ matrix, we obtain two decoupled channels:

$$\begin{array}{ccc}\hfill |V{P}^{+}⟩& =& {\displaystyle \frac{1}{\sqrt{2}}}\left(|J/\psi B_c^{+}⟩+|B_c^{*+}{\eta }_c⟩\right),\hfill \end{array}$$

(16)

$$\begin{array}{ccc}\hfill |V{P}^{-}⟩& =& {\displaystyle \frac{1}{\sqrt{2}}}\left(|J/\psi B_c^{+}⟩-|B_c^{*+}{\eta }_c⟩\right),\hfill \end{array}$$

(17)

whose coefficient is

$$\begin{array}{ccc}\hfill C_{VP}^{\prime t}& =& \left(\begin{array}{ccc}J=1& V{P}^{+}& V{P}^{-}\\ V{P}^{+}& \lambda {\displaystyle \frac{1}{m_{B_c^{*}}^2}}& 0\\ V{P}^{-}& 0& -\lambda {\displaystyle \frac{1}{m_{B_c^{*}}^2}}\end{array}\right).\hfill \end{array}$$

(18)

Hence, the interaction due to the exchange of the $B_c^{*}$ meson in the $V{P}^{-}$ channel turns out to be attractive, so there may exist a hadronic molecule of $J^P=1^{+}$ in the $VP$ sector.

The $VV$ interaction involves only one channel, $J/\psi B_c^{*+}$, whose coefficients are

$$\begin{array}{ccc}\hfill C_{VV}^t& =& 0and\hfill \\ \hfill C_{VV}^u& =& \lambda {\displaystyle \frac{1}{m_{B_c^{*}}^2}}.\hfill \end{array}$$

(19)

The relevant contact term is

$$\begin{array}{ccc}\hfill V_{J/\psi B_c^{*+}\to J/\psi B_c^{*+}}^{co}\left(s\right)& =& \left\{\begin{array}{cc}-2g^2& \mathrm{for} J=0,\\ 3g^2& \mathrm{for} J=1,\\ g^2& \mathrm{for} J=2.\end{array}\right.\hfill \end{array}$$

(20)

After performing the spin projection, we find the $J=1$ channel to be attractive, so there may exist a hadronic molecule of $J^P=1^{+}$ in the $VV$ sector.

2.3. The $cc\overline{b}\overline{b}$ and $\overline{c}\overline{c}bb$ Systems

The results for the $cc\overline{b}\overline{b}$ and $\overline{c}\overline{c}bb$ systems are the same, so we only need to study the $cc\overline{b}\overline{b}$ system. In this system, the $PP$ interaction involves only one channel, $B_c^{+}{B}_c^{+}$, whose coefficients are

$$\begin{array}{ccc}\hfill C_{PP}^t& =& {\displaystyle \frac{1}{m_{J/\psi }^2}}+{\displaystyle \frac{1}{m_{\mathrm{{\rm Y}}}^2}}and\hfill \\ \hfill C_{PP}^u& =& {\displaystyle \frac{1}{m_{J/\psi }^2}}+{\displaystyle \frac{1}{m_{\mathrm{{\rm Y}}}^2}}.\hfill \end{array}$$

(21)

The $VP$ interaction involves only one channel, $B_c^{*+}{B}_c^{+}$, whose coefficient are

$$\begin{array}{ccc}\hfill C_{VP}^t& =& {\displaystyle \frac{1}{m_{J/\psi }^2}}+{\displaystyle \frac{1}{m_{\mathrm{{\rm Y}}}^2}}and\hfill \\ \hfill C_{VP}^u& =& 0.\hfill \end{array}$$

(22)

The $VV$ interaction also involves only one channel, $B_c^{*+}{B}_c^{*+}$, whose coefficients are

$$\begin{array}{ccc}\hfill C_{VV}^t& =& {\displaystyle \frac{1}{m_{J/\psi }^2}}+{\displaystyle \frac{1}{m_{\mathrm{{\rm Y}}}^2}}and\hfill \\ \hfill C_{VV}^u& =& {\displaystyle \frac{1}{m_{J/\psi }^2}}+{\displaystyle \frac{1}{m_{\mathrm{{\rm Y}}}^2}}.\hfill \end{array}$$

(23)

The relevant contact term is

$$\begin{array}{ccc}\hfill V_{B_c^{*+}{B}_c^{*+}\to B_c^{*+}{B}_c^{*+}}^{co}\left(s\right)& =& \left\{\begin{array}{cc}-8g^2& \mathrm{for} J=0,\\ 0& \mathrm{for} J=1,\\ 4g^2& \mathrm{for} J=2.\end{array}\right.\hfill \end{array}$$

(24)

In sectors other than the ${|\left(VV\right)}_{cc\overline{b}\overline{b}};J^P=0^{+}⟩$ sector, the coefficients are positive, corresponding to repulsive interactions. As for the ${|\left(VV\right)}_{cc\overline{b}\overline{b}};J^P=0^{+}⟩$ sector, although the contact term provides an attractive potential of $-8g^2$, the repulsive potential generated via the exchange of vector mesons is approximately $+13g^2$ at the threshold, resulting in an overall repulsive interaction. Therefore, in these sectors, the exchange of vector mesons cannot bind mesons together; hence, the above coefficients do not support the existence of hadronic molecules in the $cc\overline{b}\overline{b}$ system.

3. Numerical Results

In the previous section, we studied the interactions of the $cc\overline{c}\overline{c}$, $cc\overline{c}\overline{b}/\overline{c}\overline{c}cb$, $cc\overline{b}\overline{b}/\overline{c}\overline{c}bb$, $bb\overline{c}\overline{b}/\overline{b}\overline{b}cb$, and $bb\overline{b}\overline{b}$ systems. In this section, we numerically study their properties. As shown in Equation (9), the loop function $G\left(s\right)$ is regularized using the cutoff method, with the cutoff momentum $\mathrm{\Lambda }$ describing the dynamical scale to be integrated out. Its value is quite uncertain for the exchange of fully heavy vector mesons, and we follow Ref. [76] in choosing a broad region, $\mathrm{\Lambda }=400\sim 1400\mathrm{MeV}$, to perform numerical analyses, since the authors of Refs. [91,92] have already found that the requirement of heavy quark symmetry demands the use of the same cutoff momentum in the charm and bottom sectors. We note that the value of this important parameter is quite uncertain for the exchange of fully heavy vector mesons, so the present study, as well as Ref. [76], serves as pioneering research investigating the fully heavy hadronic molecules, but there do exist large theoretical uncertainties.

Within the extended local hidden-gauge framework, the resonances are dynamically generated as poles of the scattering amplitudes $T_{PP/VP/VV}\left(s\right)$. We find the existence of eight poles that may lead to some singular structures on the invariant mass spectrum. There exist two poles in the $cc\overline{c}\overline{b}$ system and two poles in the $bb\overline{c}\overline{b}$ system, which we shall discuss in detail later. We summarize their positions in Table 2 with respect to the cutoff momentum $\mathrm{\Lambda }$. Additionally, there exist four charge-conjugated poles in the $\overline{c}\overline{c}cb$ and $\overline{b}\overline{b}cb$ systems.

Table 2. Pole positions with respect to the cutoff momentum $\mathrm{\Lambda }$ in units of MeV. We only list the poles that correspond to the sub-threshold bound states.

	$\mathbf{\Lambda }=400$	$\mathbf{\Lambda }=600$	$\mathbf{\Lambda }=800$	$\mathbf{\Lambda }=1000$	$\mathbf{\Lambda }=1200$	$\mathbf{\Lambda }=1400$
${|\left(VP\right)}_{cc\overline{c}\overline{b}};J^P=1^{+}⟩$	$--$	$--$	$--$	$--$	$--$	$--$
${|\left(VV\right)}_{cc\overline{c}\overline{b}};J^P=1^{+}⟩$	$--$	$--$	$--$	$--$	$--$	$--$
${|\left(VP\right)}_{bb\overline{c}\overline{b}};J^P=1^{+}⟩$	$--$	$--$	$--$	15,725.3 − $i0$	15,710.8 − $i0$	15,685.2 − $i0$
${|\left(VV\right)}_{bb\overline{c}\overline{b}};J^P=1^{+}⟩$	$--$	$--$	$--$	$--$	15,790.3 − $i0$	15,784.0 − $i0$

We find two poles in the $cc\overline{c}\overline{b}$ system: one pole in the $VP$ sector and the other in the $VV$ sector. However, both of them correspond to virtual states when the cutoff momentum is set to $\mathrm{\Lambda }=400\sim 1400\mathrm{MeV}$, so they can only result in some threshold effects. The pole in the $VP$ sector corresponds to the sub-threshold bound state with $\mathrm{\Lambda }>1550\mathrm{MeV}$, and the pole in the $VV$ sector corresponds to the sub-threshold bound state with $\mathrm{\Lambda }>2650\mathrm{MeV}$. We generally consider the cutoff momentum $\mathrm{\Lambda }$ to be consistent with the chiral unitary approach, which takes $\mathrm{\Lambda }\approx 4\pi f_{\pi }\simeq 1200$ MeV. This value reflects certain non-perturbative properties of QCD. However, it is important to note that the parameter $\mathrm{\Lambda }$ also functions as a free parameter, absorbing some implicitly considered interactions, and as a result, it may deviate from 1200 MeV in practical applications. Empirically, $\mathrm{\Lambda }$ is usually taken to fall within the range of 400 to 700 MeV. Given the uncertainties inherent to our work, we believe it is appropriate to extend this range to 400 to 1400 MeV. Therefore, our results do not support the existence of deeply bound hadronic molecules in the $cc\overline{c}\overline{b}$ system.

We also find two poles in the $bb\overline{c}\overline{b}$ system: one pole in the $VP$ sector and the other in the $VV$ sector. The pole in the $VP$ sector corresponds to the sub-threshold bound state when $\mathrm{\Lambda }>850\mathrm{MeV}$ is set, making it possible to be identified as a hadronic molecule. This pole transfers to a virtual state and results in the threshold effect when $\mathrm{\Lambda }<850\mathrm{MeV}$ is set. The pole in the $VV$ sector corresponds to the sub-threshold bound state with $\mathrm{\Lambda }>1100\mathrm{MeV}$. To illustrate these two poles, we present, in Figure 2, the transition amplitudes using several different values of the cutoff momentum $\mathrm{\Lambda }$.

Figure 2. Line shapes of the transition amplitudes ${\left|T\left(s\right)\right|}^2$ for the cutoff momentum $\mathrm{\Lambda }=$ (a) 750 MeV, (b) 850 MeV, and (c) 950 MeV in the $VP$ sector, as well as $\mathrm{\Lambda }=$ (d) 1000 MeV, (e) 1100 MeV, and (f) 1200 MeV in the ${VV|}_{J=1}$ sector of the $bb\overline{c}\overline{b}$ system. The relevant thresholds are indicated by dashed lines. In subfigures (a–c), the green line labeled $T_{11}$ and the red line labeled $T_{22}$ represent ${|T_{\mathrm{{\rm Y}}{B}_c^{-}\to \mathrm{{\rm Y}}{B}_c^{-}}(s)|}^2$ and ${|T_{B_c^{*-}{\eta }_b\to B_c^{*-}{\eta }_b}(s)|}^2$, respectively.

4. Conclusions

In this paper, we have studied fully heavy meson–meson interactions with the quark constituents $cc\overline{c}\overline{c}$, $cc\overline{c}\overline{b}/\overline{c}\overline{c}cb$, $cc\overline{b}\overline{b}/\overline{c}\overline{c}bb$, $bb\overline{c}\overline{b}/\overline{b}\overline{b}cb$, and $bb\overline{b}\overline{b}$ through extended local hidden-gauge formalism. After solving the coupled-channel Bethe–Salpeter equation, we searched for poles on both the first (physical) and second Riemann sheets. The obtained results are summarized in Table 2 with respect to the cutoff momentum $\mathrm{\Lambda }$. We found two poles in the $bb\overline{c}\overline{b}$ system (and two charge-conjugated poles in the $\overline{b}\overline{b}cb$ system): the pole generated in the $VP$ sector corresponds to the sub-threshold bound state when the cutoff momentum $\mathrm{\Lambda }>850\mathrm{MeV}$ is set, and the pole generated in the $VV$ sector corresponds to the sub-threshold bound state with $\mathrm{\Lambda }>1100\mathrm{MeV}$. These two poles are potential fully heavy hadronic molecules, and we propose to investigate them in the ${\mu }^{+}{\mu }^{-}{B}_c^{-}$ channel at LHC. However, our results do not support the existence of hadronic molecules in the $cc\overline{c}\overline{c}$, $cc\overline{c}\overline{b}/\overline{c}\overline{c}cb$, $cc\overline{b}\overline{b}/\overline{c}\overline{c}bb$, and $bb\overline{b}\overline{b}$ systems.

Additionally, the $cb\overline{c}\overline{b}$ system has already been investigated in our previous study [76], where we found the existence of the fully heavy hadronic molecules $|B_c^{+}{B}_c^{-};J^{PC}=0^{++}⟩$, $|B_c^{*+}{B}_c^{-}-c.c.;J^{PC}=1^{+-}⟩$, and $|B_c^{*+}{B}_c^{*-};J^{PC}=2^{++}⟩$, as well as the possible existence of $|B_c^{*+}{B}_c^{-}+c.c.;J^{PC}=1^{++}⟩$. We further proposed in Ref. [76] that a lower-mass fully heavy meson may be able to bind two higher-mass fully heavy hadrons. The results obtained in the present study are consistent with this proposal: the exchanged mesons of the $bb\overline{c}\overline{b}$ and $cc\overline{c}\overline{b}$ systems are both the $B_c^{*}$ meson, but the larger mass of the $bb\overline{c}\overline{b}$ system facilitates the formation of bound states. It is a topic of considerable interest whether the interaction of the heavy meson exchange is strong enough to form hadronic molecules. This question serves as a crucial test for the extensively investigated interaction of the light meson exchange. Therefore, the present study, as well as Ref. [76], both of which concentrate on the interaction of the fully heavy meson exchange, are of particular interest.