The past decade has been a golden era in hadron physics [1], and our understanding of the non-perturbative behavior of the strong interaction in the low-energy regime has been significantly improved [2–16]. This presents an ideal opportunity to compare it with the electromagnetic interaction through investigation of the hydrogen atom/molecule and the charmed meson/molecule.

A hydrogen atom consists of an electron bound to a proton, with a binding energy of 13.6 eV. In the ground state, the interaction between the electron and proton spins results in a slight increase in energy when the spins are parallel and a decrease when they are antiparallel. The energy difference between these two hyperfine states is approximately 5.87 × 10⁻⁶ eV. The situation becomes more complex in the hydrogen molecule, which consists of two hydrogen atoms bound together by a chemical covalent bond, with a binding energy of 4.73 eV. There are two spin configurations of the hydrogen molecule, i.e. ortho-hydrogen has parallel proton spins and a higher energy while para-hydrogen has antiparallel proton spins and a lower energy. The energy difference between these two spin isomers is approximately 0.0151 eV.

A comparable picture has been found in the strong interaction, namely, the charmed meson and its hadronic molecule. A charmed meson consists of a light up/down antiquark bound to a charm quark within the conventional quark model. There are two ground states: D with J^P = 0⁻ and D^∗ with J^P = 1⁻, with a mass difference of about 140 MeV. For brevity, we denote them collectively as D^(∗), and refer to $\bar{D}$ and ${\bar{D}}^{\ast }$ collectively as ${\bar{D}}^{(\ast )}$. Two ${\bar{D}}^{(\ast )}$ mesons may form a ${\bar{D}}^{(\ast )}{\bar{D}}^{(\ast )}$ hadronic molecule, but so far only T_cc(3875) has been observed in the LHCb experiment [17, 18], which can be interpreted as the $\bar{D}{\bar{D}}^{\ast }$ hadronic molecule of (I)J^P = (0)1⁺, with a binding energy of the order of 1 MeV. More ${\bar{D}}^{(\ast )}{\bar{D}}^{(\ast )}$ hadronic molecules, such as the ${\bar{D}}^{\ast }{\bar{D}}^{\ast }$ hadronic molecule of (I)J^P = (0)1⁺, have been predicted in the literature and are yet to be discovered, which would contribute to completing this picture [2–16].

In addition to the many similarities between the hydrogen atom/molecule and the charmed meson/molecule, there are two distinct differences:

•In the hydrogen atom and the hydrogen molecule, the contribution from the proton spin is small and negligible. In contrast, in the ${\bar{D}}^{(\ast )}$ meson and its hadronic molecule, the contribution from the spin of the charm antiquark is no longer negligible.

•The binding energy of the hydrogen molecule, resulting from the residual electromagnetic interaction between two hydrogen atoms, is of the same order as that of the hydrogen atom, which arises from the direct electromagnetic interaction between the electron and proton. In contrast, the binding energy of T_cc(3875), resulting from the residual strong interaction between $\bar{D}$ and ${\bar{D}}^{\ast }$ mesons, is significantly smaller than the ‘binding’ energy of the ${\bar{D}}^{(\ast )}$ meson, which arises from the direct strong interaction between the light up/down quark and the charm antiquark.

Another related observation is that a muonic Z atom is believed to behave as if it were a Z − 1 atom, although this will not be discussed in the present study.

Similar to a chemical covalent bond, we proposed in [19] a hadronic covalent bond to explain P_c/P_cs/T_cc [17, 18, 20–24] and the deuteron as possible hadronic covalent molecules. In the hydrogen molecule, the two electrons shared between two protons are antisymmetric so that they obey the Pauli principle. Similarly, we proposed in T_cc(3875) that the two light up/down quarks shared between two charm antiquarks are also antisymmetric so that they obey the Pauli principle, as illustrated in Figure 1(a). Since the binding energy of T_cc(3875) is significantly smaller than the ‘binding’ energy of the ${\bar{D}}^{(\ast )}$ meson, the two shared light up/down quarks need not be fully antisymmetric, as long as there is enough antisymmetric component to provide sufficient attraction.

However, the hadronic covalent bond cannot explain the hadronic molecule composed of one D^(∗) meson and one ${\bar{D}}^{(\ast )}$ meson. Specifically, Z_c(3900) [25, 26] and X(3872) [27] can be interpreted as such hadronic molecules [2–16]. An interesting question related to the electromagnetic interaction is whether a chemical molecule composed of one hydrogen atom and one anti-hydrogen atom exists. While the answer is not yet known to us, it is likely that additional binding mechanisms exist in the strong interaction.

In this paper we extend the hadronic covalent bond to explain Z_c(3900) as the $D{\bar{D}}^{* }/{D}^{* }\bar{D}$ hadronic molecule with I^GJ^PC = 1⁺1⁺⁻. As illustrated in Figure 1(b), we consider not only the light quark–antiquark pair from the D and ${\bar{D}}^{\ast }$ mesons but also two sea quark–antiquark pairs from the vacuum. Consequently, annihilation of the light quark–antiquark pair should also be considered. As illustrated in Figure 1(c), this effect has been studied in [28–31] to explain X(3872) as a mixture of a $c\bar{c}$ state and a $D{\bar{D}}^{* }/{D}^{* }\bar{D}$ component with I^GJ^PC = 0⁺1⁺⁺, and this analysis is closely related to the ³P₀ model studies on the hadronic decay of charmonium [32–34]. We refer to the two binding mechanisms above as the ‘creation’ and ‘annihilation’ bonds, which exist only in the strong interaction not in the electromagnetic interaction. These two bonds provide a quasi-static low-energy platform for studying quantum chromodynamic (QCD) confinement, so we refer to them collectively as the ‘confined’ bond, with the relevant molecule termed the hadronic ‘confined’ molecule.

This paper is organized as follows. In section 2 we improve the hadronic covalent bond proposed in [19] and apply it to study T_cc(3875) and the deuteron. In section 3 we propose the hadronic creation bond to explain Z_c(3900) through the shared light quark–antiquark pair along with sea quark–antiquark pairs from the vacuum. In section 4 we use the hadronic annihilation bond to explain X(3872) by further considering the annihilation of the light quark–antiquark pair, after which we conclude the paper.

We proposed in [19] a hadronic covalent bond induced by shared light quarks, as illustrated in Figure 2. In a chemical covalent bond the shared electrons are totally antisymmetric so that they obey the Pauli principle. Similarly, we found in the hadronic covalent bond that ‘the light-quark-exchange interaction is attractive when the shared light quarks are totally antisymmetric so that [they] obey the Pauli principle’. In this section we improve this mechanism and apply it to study several examples in the following subsections.

The hadronic covalent bond studied in the previous section cannot explain the hadronic molecule composed of one D^(∗) meson and one ${\bar{D}}^{(\ast )}$ meson, which will be explored in this section. To address this, we extend the covalent bond by incorporating not only the light quark–antiquark pair from the D^(∗) and ${\bar{D}}^{(\ast )}$ mesons but also two sea quark–antiquark pairs from the vacuum, as illustrated in Figure 1(b).

We use ${\bar{q}}_{1}{q}_{2}$ to denote the light quark–antiquark pair from the D^(∗) and ${\bar{D}}^{(\ast )}$ mesons, where ${\bar{q}}_{1}$ is from the D^(∗) meson and q₂ is from the ${\bar{D}}^{(\ast )}$ meson. We use ${\bar{q}}_{3}{q}_{4}$ and ${\bar{q}}_{5}{q}_{6}$ to denote the two sea quark–antiquark pairs from the vacuum. According to the Lagrangian ${ \mathcal L }=\bar{q}{\rm{i}}{\gamma }^{\mu }{D}_{\mu }q$ with D_μ = ∂_μ + ig_sA_μ, we assume their quantum numbers to both beand their total quantum numbers to be In addition, two ³P₀ quark–antiquark pairs are also possible, but they would result in a higher energy configuration. Moreover, we require that the three antiquarks ${\bar{q}}_{1,3,5}$, which orbit around the charm quark in the D^(∗) meson, have the same color and so a symmetric color structure; the three quarks q_2,4,6, which orbit around the charm antiquark in the ${\bar{D}}^{(\ast )}$ meson also have the same color and so a symmetric color structure; the two antiquarks ${\bar{q}}_{3}$ and ${\bar{q}}_{5}$ form a strong bond, while the two quarks q₄ and q₆ form another strong bond. There are numerous sea quark–antiquark pairs and sea gluons in the QCD vacuum, making it possible for these assumptions to be satisfied.

Based on the above assumptions, we find that the ${\bar{q}}_{1}{q}_{2}$ pair with the quantum numbers is capable of forming the following configuration:

We refer to this configuration as the hadronic creation bond due to the activity of sea quark–antiquark pairs from the vacuum, thus being closely related to QCD confinement. It contains four strong bonds (${\bar{q}}_{1}{\bar{q}}_{5}$, ${\bar{q}}_{3}{\bar{q}}_{5}$, q₂q₄ and q₄q₆) and two weak bonds (${\bar{q}}_{1}{\bar{q}}_{3}$ and q₂q₆), which attract the D^(∗) and ${\bar{D}}^{(\ast )}$ mesons to potentially form a hadronic confined molecule that behaves as a resonance lying above the ${D}^{(\ast )}{\bar{D}}^{(\ast )}$ threshold. Additionally, the two color-octet sea quark–antiquark pairs from the vacuum, ${\bar{q}}_{3}{q}_{4}$ and ${\bar{q}}_{5}{q}_{6}$, both with I = 0 and J = 1, couple strongly to sea gluons, so there is a tremendous number of them; moreover, they can be recombined into the two color-singlet quark–antiquark pairs, ${\bar{q}}_{3}{q}_{6}$ and ${\bar{q}}_{5}{q}_{4}$ both with I = 1 and J = 0, which couple strongly to the pseudoscalar mesons π and so are capable of providing long-range attraction.

Besides, the ${\bar{q}}_{1}{q}_{2}$ pair with the quantum numbers can form another similar configuration:

However, the ${\bar{q}}_{1}{q}_{2}$ pair with the quantum numbers cannot form this type of configuration, nor can the ${\bar{q}}_{1}{q}_{2}$ pair with the quantum numbers Besides the configurations mentioned above, there are other possible configurations that could form hadronic creation bonds, and thus hadronic confined molecules. We list these in Appendix A but shall not investigate them in the present study.

We apply the above binding mechanism to study several examples, as follows:

•$D[{\bar{q}}_{1}{c}_{7}]$–$\bar{D}[{q}_{2}{\bar{c}}_{8}]$. The D meson cannot employ the weak ${\bar{q}}_{1}{\bar{q}}_{3}/{\bar{q}}_{1}{\bar{q}}_{5}$ bond, and the $\bar{D}$ meson cannot employ the weak q₂q₄/q₂q₆ bond either. Accordingly, there is no $D\bar{D}$ confined molecule, neither with I = 0 nor with I = 1.

•$D[{\bar{q}}_{1}{c}_{7}]$–${\bar{D}}^{* }[{q}_{2}{\bar{c}}_{8}]\oplus {D}^{* }[{\bar{q}}_{1}{c}_{7}]$–$\bar{D}[{q}_{2}{\bar{c}}_{8}]$. The D meson itself cannot employ the weak ${\bar{q}}_{1}{\bar{q}}_{3}/{\bar{q}}_{1}{\bar{q}}_{5}$ bond, but the D and D^* mesons together can employ the weak ${\bar{q}}_{1}{\bar{q}}_{3}/{\bar{q}}_{1}{\bar{q}}_{5}$ bond, making it possible for the combination of $D{\bar{D}}^{* }$ and ${D}^{* }\bar{D}$ to form a hadronic confined molecule. According to heavy quark spin symmetry [35–41], we perform the spin decomposition:

There are three possibilities:

1. The state given in equation (10) contains the ${\bar{q}}_{1}{q}_{2}$ pair with ${s}_{{\bar{q}}_{1}{q}_{2}}=1$, so its isoscalar component can form a creation bond, making it possible to interpret X(3872) as the $D{\bar{D}}^{* }/{D}^{* }\bar{D}$ confined molecule with I^GJ^PC = 0⁺1⁺⁺.

2. The state given in equation (11) contains the ${\bar{q}}_{1}{q}_{2}$ pair with ${s}_{{\bar{q}}_{1}{q}_{2}}=0$, so its isovector component can form a creation bond, making it possible to interpret Z_c(3900) as the $D{\bar{D}}^{* }/{D}^{* }\bar{D}$ confined molecule with I^GJ^PC = 1⁺1⁺⁻.

3. The state given in equation (11) also contains the ${\bar{q}}_{1}{q}_{2}$ pair with ${s}_{{\bar{q}}_{1}{q}_{2}}=1$, suggesting the possible existence of the $D{\bar{D}}^{* }/{D}^{* }\bar{D}$ confined molecule with I^GJ^PC = 0⁻1⁺⁻. However, its corresponding ${\bar{c}}_{8}{c}_{7}$ pair has ${s}_{{\bar{c}}_{8}{c}_{7}}=0$, which could render this system potentially unstable.

•${D}^{* }[{\bar{q}}_{1}{c}_{7}]$–${\bar{D}}^{* }[{q}_{2}{\bar{c}}_{8}]$. According to heavy quark spin symmetry, we perform the spin decomposition

After requiring ${s}_{{\bar{c}}_{8}{c}_{7}}=1$, there are three possibilities remaining:

1. The state given in equation (12) contains the ${\bar{q}}_{1}{q}_{2}$ pair with ${s}_{{\bar{q}}_{1}{q}_{2}}=1$, so its isoscalar component can form a creation bond, suggesting the possible existence of the ${D}^{* }{\bar{D}}^{* }$ confined molecule with I^GJ^PC = 0⁺0⁺⁺.

2. The state given in equation (13) contains the ${\bar{q}}_{1}{q}_{2}$ pair with ${s}_{{\bar{q}}_{1}{q}_{2}}=0$, so its isovector component can form a creation bond, suggesting the possible existence of the ${D}^{* }{\bar{D}}^{* }$ confined molecule with I^GJ^PC = 1⁺1⁺⁻.

3. The state given in equation (14) contains the ${\bar{q}}_{1}{q}_{2}$ pair with ${s}_{{\bar{q}}_{1}{q}_{2}}=1$, so its isoscalar component can form a creation bond, suggesting the possible existence of the ${D}^{* }{\bar{D}}^{* }$ confined molecule with I^GJ^PC = 0⁺2⁺⁺.

Summarizing the above discussions, our results suggest the possible existence of the I^GJ^PC = 0⁺1⁺⁺/1⁺1⁺⁻$D{\bar{D}}^{* }/{D}^{* }\bar{D}$ confined molecules and the I^GJ^PC = 0⁺0⁺⁺/1⁺1⁺⁻/0⁺2⁺⁺${D}^{* }{\bar{D}}^{* }$ confined molecules, all of which contain the hadronic creation bond.

In this paper we systematically study the ${\bar{D}}^{(* )}{\bar{D}}^{(* )}$ and ${D}^{(* )}{\bar{D}}^{(* )}$ hadronic molecules. For the ${\bar{D}}^{(* )}{\bar{D}}^{(* )}$ hadronic molecules, we proposed in [19] a hadronic covalent bond induced by shared light quarks and used this binding mechanism to interpret T_cc(3875) as the $\bar{D}{\bar{D}}^{* }$ covalent molecule with (I)J^P = (0)1⁺. In this paper we improve upon this binding mechanism and estimate the binding energies of some possible hadronic covalent molecules. The obtained results are summarized in table 1, based on the hypothesis proposed in [19] that ‘the light-quark-exchange interaction is attractive when the shared light quarks are totally antisymmetric so that [they] obey the Pauli principle’.

The covalent bond cannot explain the ${D}^{(* )}{\bar{D}}^{(* )}$ hadronic molecules. In this paper we attempt to explain them by incorporating not only the light quark–antiquark pair from the D(*) and ${\bar{D}}^{(* )}$ mesons but also two sea quark–antiquark pairs from the vacuum ²

²

We also attempted to use only one sea quark–antiquark pair instead of two but were unable to obtain a reasonable result. However, we still consider this a potential option.

() ³

³https://en.wikipedia.org/wiki/Coherent_turbulent_structure.

(). These three quarks and three antiquarks together can form four strong covalent bonds and two weak covalent bonds, potentially attracting the D(*) and ${\bar{D}}^{(* )}$ mesons to form a hadronic confined molecule. Our results suggest the possible existence of the I^GJ^PC = 0⁺1⁺⁺/1⁺1⁺⁻$D{\bar{D}}^{* }/{D}^{* }\bar{D}$ and I^GJ^PC = 0⁺0⁺⁺/1⁺1⁺⁻/0⁺2⁺⁺${D}^{* }{\bar{D}}^{* }$ confined molecules.

Given that the creation of sea quark–antiquark pairs from the vacuum has been considered, their annihilation back into the vacuum should receive equal consideration:

•The annihilation bond does not affect the isovector ${D}^{(* )}{\bar{D}}^{(* )}$ molecule, but it could potentially destabilize the isoscalar ${D}^{(* )}{\bar{D}}^{(* )}$ molecule unless a relevant charmonium state is nearby (consequently, there would be mixing between the molecular state and the charmonium state, and the mass of this charmonium state would increase). Accordingly, the I^GJ^PC = 0⁺0⁺⁺/0⁺2⁺⁺${D}^{* }{\bar{D}}^{* }$ confined molecule might not exist.

•The annihilation bond could reduce the mass of the molecule due to its mixing with the relevant charmonium state. Accordingly, the mass of X(3872), interpreted as the $D{\bar{D}}^{* }/{D}^{* }\bar{D}$ confined molecule of I^GJ^PC = 0⁺1⁺⁺ mixed with the χ_c1(2P) state, becomes smaller than the mass of Z_c(3900), interpreted as the $D{\bar{D}}^{* }/{D}^{* }\bar{D}$ confined molecule of I^GJ^PC = 1⁺1⁺⁻.

The hadronic creation and annihilation bonds provide a quasi-static, low-energy platform for studying QCD confinement. In addition to the ${D}^{(* )}{\bar{D}}^{(* )}$ confined molecule, a brief application of these two bonds to the hidden-charm baryonium states is presented in appendix B. We will apply this framework to study additional properties of hadrons in the future, such as the electromagnetic structure of the deuteron.

To end this paper, we list several unique features of our framework, which is based on the hadronic covalent, creation and annihilation bonds:

•The hadronic covalent molecule formed by shared light quarks often behaves as a bound state. By contrast, the hadronic confined molecule formed by the shared light quark–antiquark pair often behaves as a resonance lying above the relevant threshold due to the activity of sea quark–antiquark pairs from the vacuum, which might be connected to the behavior of hadrons lying above the thresholds of current quarks.

•$DD/\bar{B}\bar{B}$ covalent molecules do not exist, neither for I = 0 nor for I = 1; $D\bar{D}/B\bar{B}$ confined molecules possibly do not exist, neither for I = 0 nor for I = 1; however, $\bar{D}{{\rm{\Sigma }}}_{c}/\bar{D}{{\rm{\Sigma }}}_{b}/B{{\rm{\Sigma }}}_{c}/B{{\rm{\Sigma }}}_{b}$ covalent molecules with (I)J^P = (1/2)1/2⁺ do exist. Note that if $D\bar{D}/B\bar{B}$ confined molecules exist we would need to consider the configurations provided in appendix A and significantly modify our framework.

•The binding energies of (I)J^P = (0)1⁺$D{\bar{B}}^{* }/{D}^{* }\bar{B}$ covalent molecules are much larger than those of (I)J^P = (0)1⁺$D{D}^{* }/\bar{B}{\bar{B}}^{* }$ covalent molecules, while the (I)J^P = (1/2)1/2⁺$\bar{D}{{\rm{\Sigma }}}_{{\rm{c}}}/\bar{D}{{\rm{\Sigma }}}_{{\rm{b}}}/B{{\rm{\Sigma }}}_{{\rm{c}}}/B{{\rm{\Sigma }}}_{{\rm{b}}}$ covalent molecules have similar binding energies.

Besides the configurations provided in section 3 there are other possible configurations that could form hadronic confined molecules. We list them as follows:

•The ${\bar{q}}_{1}{q}_{2}$ pair with the quantum numbers can form only one configuration, which has been provided in section 3.

•Besides the configuration provided in section 3, the ${\bar{q}}_{1}{q}_{2}$ pair with the quantum numbers can form another configuration:

•The ${\bar{q}}_{1}{q}_{2}$ pair with the quantum numbers is capable of forming two configurations, either

or

•The ${\bar{q}}_{1}{q}_{2}$ pair with the quantum numbers is capable of forming the following configuration:

The above configurations might form some hadronic confined molecules. Furthermore, we can incorporate the light strange quark to construct more configurations, which could potentially form additional hadronic confined molecules to explain the Z_cs states. All these possibilities will be explored in our future studies.

In this appendix we briefly investigate the hidden-charm baryonium state, i.e. the hadronic molecule composed of one charmed antibaryon and one charmed baryon, which have the quark contents ${\bar{q}}_{1}{\bar{q}}_{7}{\bar{c}}_{9}$ and q₂q₈c₁₀, respectively. We use ${\bar{q}}_{3}{q}_{4}$ and ${\bar{q}}_{5}{q}_{6}$ to denote the two sea quark–antiquark pairs from the vacuum. We require that the three antiquarks ${\bar{q}}_{1,3,5}$ have the same color, and the three quarks q_2,4,6 also have the same color. Utilizing the hadronic creation and annihilation bonds, we examine several examples, as follows:

•${\bar{{\rm{\Lambda }}}}_{{\rm{c}}}[{\bar{q}}_{1}{\bar{q}}_{7}{\bar{c}}_{9}]$–Λ_c[q₂q₈c₁₀]. Let us exchange the light quark q₂ from the Λ_c baryon with the sea quark q₄ from the vacuum. q₂ and q₈ inside the Λ_c baryon spin in opposite directions, q₄ and q₈ also need to spin in opposite directions in order to form another Λ_c baryon, so q₂ and q₄ spin in the same direction with a symmetric spin structure. Similarly, q₂ and q₄ have parallel isospin with a symmetric flavor structure. As a result, the two light quarks q₂ and q₄ are totally symmetric:  

Accordingly, there is no ${{\rm{\Lambda }}}_{c}{\bar{{\rm{\Lambda }}}}_{c}$ confined molecule.

•${\bar{{\rm{\Lambda }}}}_{{\rm{c}}}[{\bar{q}}_{1}{\bar{q}}_{7}{\bar{c}}_{9}]$–${{\rm{\Sigma }}}_{{\rm{c}}}[{q}_{2}{q}_{8}{c}_{10}]\oplus {\bar{{\rm{\Sigma }}}}_{{\rm{c}}}[{\bar{q}}_{1}{\bar{q}}_{7}{\bar{c}}_{9}]$–Λ_c[q₂q₈c₁₀]. As discussed above, the Λ_c baryon itself cannot employ a strong bond, but the Λ_c and Σ_c baryons together can, making it possible for the combination of ${\bar{{\rm{\Lambda }}}}_{{\rm{c}}}{{\rm{\Sigma }}}_{{\rm{c}}}$ and ${\bar{{\rm{\Sigma }}}}_{{\rm{c}}}{{\rm{\Lambda }}}_{{\rm{c}}}$ to form a hadronic confined molecule. This state has I = 1, so we can assume that the ${\bar{q}}_{1}{q}_{2}$ pair also has I = 1. Since there is no constraint on the spin of this quark–antiquark pair, the following configuration can potentially be formed:

This configuration is identical to the one formed by the quark–antiquark pair given in equation (6). Its corresponding hadronic creation bond attracts the combination of ${\bar{{\rm{\Lambda }}}}_{{\rm{c}}}{{\rm{\Sigma }}}_{{\rm{c}}}$ and ${\bar{{\rm{\Sigma }}}}_{{\rm{c}}}{{\rm{\Lambda }}}_{{\rm{c}}}$ to potentially form a hadronic confined molecule. However, the hadronic annihilation bond may annihilate the ${\bar{q}}_{7}{q}_{8}$ pair depending on its isospin, while leaving the isovector ${\bar{q}}_{1}{q}_{2}$ pair unaffected.

•${\bar{{\rm{\Sigma }}}}_{{\rm{c}}}[{\bar{q}}_{1}{\bar{q}}_{7}{\bar{c}}_{9}]$–Σ_c[q₂q₈c₁₀]. This molecule can have I = 0, 1 or 2. Specifically, both the ${\bar{q}}_{1}{q}_{2}$ and ${\bar{q}}_{7}{q}_{8}$ pairs inside the isotensor state have I = 1. The ${\bar{q}}_{1}{q}_{2}$ pair can form a configuration identical to the one described above, and the hadronic annihilation bond does not affect either pair, suggesting that the isotensor ${\bar{{\rm{\Sigma }}}}_{{\rm{c}}}{{\rm{\Sigma }}}_{{\rm{c}}}$ confined molecule is likely to exist.

•${\bar{{\rm{\Xi }}}}_{{\rm{c}}}^{{\prime} }[{\bar{q}}_{1}{\bar{s}}_{7}{\bar{c}}_{9}]$–${{\rm{\Xi }}}_{{\rm{c}}}^{{\prime} }[{q}_{2}{s}_{8}{c}_{10}]$. This molecule can have I = 0 or I = 1. Specifically, the ${\bar{q}}_{1}{q}_{2}$ pair inside the isoscalar state has I = 0. Since there is no constraint on its spin, the following configuration can potentially be formed:

This configuration is identical to the one formed by the quark–antiquark pair given in equation (7). Its corresponding hadronic creation bond attracts the ${\bar{{\rm{\Xi }}}}_{{\rm{c}}}^{{\prime} }$ and ${{\rm{\Xi }}}_{{\rm{c}}}^{{\prime} }$ baryons to potentially form a hadronic confined molecule. However, the hadronic annihilation bond can annihilate the ${\bar{q}}_{1}{q}_{2}$ pair, potentially destabilizing this molecule.

Summarizing the above discussions, our results suggest the possible existence of the ${\bar{{\rm{\Lambda }}}}_{{\rm{c}}}{{\rm{\Sigma }}}_{{\rm{c}}}/{\bar{{\rm{\Sigma }}}}_{{\rm{c}}}{{\rm{\Lambda }}}_{{\rm{c}}}$, ${\bar{{\rm{\Sigma }}}}_{{\rm{c}}}{{\rm{\Sigma }}}_{{\rm{c}}}$, and ${\bar{{\rm{\Xi }}}}_{{\rm{c}}}^{{\prime} }{{\rm{\Xi }}}_{{\rm{c}}}^{{\prime} }$ confined molecules, along with many others. These states may behave as resonances lying above the relevant thresholds due to the activity of sea quark–antiquark pairs from the vacuum, possibly linked to the behavior of hadrons lying above the thresholds of current quarks. These hadronic molecules are formed by the hadronic creation bond proposed in the present study, while the annihilation bond also proposed in the present study may destabilize them. A detailed analysis will be presented in our future work.