Systematics of the heavy flavor hadronic molecules

Abstract

With a quark level interaction, we give a unified description of the loosely bound molecular systems composed of the heavy flavor hadrons $({\bar{D}},{\bar{D}}^*)$, $(\Lambda _c, \Sigma _c, \Sigma _c^*)$, and $(\Xi _c, \Xi _c^\prime ,\Xi _c^*)$. Using the $P_c$ states as inputs to fix the interaction strength of light quark-quark pairs, we reproduce the observed $P_{cs}$ and $T_{cc}^+$ states and predict another narrow $T_{cc}^{\prime +}$ state with quantum numbers $[D^*D^*]_{J=1}^{I=0}$. If we require a satisfactory description of the $T_{cc}^+$ and $P_c$ states simultaneously, our framework prefers the assignments of the $P_{c}(4440)$ and $P_{c}(4457)$ as the $[\Sigma _c{\bar{D}}^*]_{J=1/2}^{I=1/2}$ and $[\Sigma _c{\bar{D}}^*]_{J=3/2}^{I=1/2}$ states, respectively. We propose the isospin criterion to explain naturally why the experimentally observed $T_{cc}$, $P_c$, and $P_{cs}$ molecular candidates prefer the lowest isospin numbers. We also predict the loosely bound states for the bottom di-hadrons.

1 Introduction

The conventional mesons ($q{\bar{q}}$) and baryons (qqq) have been extensively discussed at the birth of quark model [1,2,3], and have become the main part of hadron spectrum nowadays [4]. Besides conventional mesons and baryons, quantum chromodynamics (QCD) also allows the existence of hadrons with more complicated configurations, such as $qq{\bar{q}}{\bar{q}}$, ${\bar{q}}qqqq$, qqqqqq ($qqq{\bar{q}}{\bar{q}}{\bar{q}}$), etc. In the past decades, many XYZ states have been observed [5,6,7,8,9,10,11,12] since the discovery of X(3872) in 2003 [13]. Some of these states are below the thresholds of di-hadrons from several to several tens MeVs. The molecular explanations were widely proposed to understand their underlying structures.

In 2015, the LHCb Collaboration reported two structures $P_c(4380)$ and $P_c(4450)$ in the $J/\psi p$ mass spectrum [14]. Their masses are consistent with the predictions of the hidden-charm pentaquarks [15,16,17]. In 2019, the LHCb Collaboration updated their analyses with larger data samples and found that the $P_{c}(4450)$ consists of two $P_c$ states, i.e., $P_{c}(4440)$ and $P_{c}(4457)$ [18]. Besides, they further reported another near threshold hidden-charm pentaquark $P_c(4312)$. These important results from LHCb provide strong evidences for the existence of the hidden-charm molecular pentaquarks [19,20,21,22,23,24,25,26,27,28,29]. Thus, it is desirable to see whether there exist the hadronic molecules in other heavy flavor di-hadron systems.

If enlarging the flavor symmetry group to SU(3), one may expect that the $\Xi _c{\bar{D}}^{(*)}$, $\Xi ^\prime _c{\bar{D}}^{(*)}$, and $\Xi _c^*{\bar{D}}^{(*)}$ systems may also form molecular bound states. The $P_{cs}$ pentaquarks were investigated in Refs. [15, 30,31,32,33,34] and the most promising production channel $\Xi _b^-\rightarrow J/\psi \Lambda K$ was suggested in Refs. [31, 35]. Later, the LHCb reported the evidence of $P_{cs}(4459)$ in the $J/\psi \Lambda$ invariant mass spectrum [36], which agrees very well with the prediction from the chiral effective field theory in Ref. [34]. However, this state still needs further confirmation due to limited data samples at present [36]. Very probably, the $P_c$ and $P_{cs}$ pentaquarks share a very similar binding mechanism.

Very recently, the LHCb reported a very narrow structure $T_{cc}^+(3875)$ in the $D^0D^0\pi ^+$ spectrum [37, 38]. Its mass is slightly below the $D^{*+}D^0$ threshold about 300 keV. This signal tends to confirm the predictions of the $DD^*$ molecular state with quantum numbers $I(J^P)=0(1^+)$ [39]. The doubly heavy tetraquark states have been extensively studied and debated in the literatures [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]. One can refer to Ref. [6] for a review of the $QQ{\bar{q}}{\bar{q}}$ system. This inspiring discovery also stimulated a series of theoretical studies [66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84].

The minimal valence quark components for the $P_c$, $P_{cs}$, and $T_{cc}^+$ states are $c{\bar{c}}uud$, $c{\bar{c}}uds$, and $cc{\bar{u}}{\bar{d}}$, respectively. Such states can not be accommodated within the conventional quark model and thus give us a golden platform to study the structures and dynamics of the multiquark states.

We propose the following picture to understand the above heavy flavor di-hadron systems. The heavy quark (c or b) behaves like a static color triplet source in each hadron. In the heavy quark limit, its velocity v does not change with time. The non-relativistic property of heavy quarks is stimulative to stabilize a molecular system [85,86,87]. On the other hand, the type of possible exchanged light meson depends on the light quark components within the heavy flavor di-hadron system. Thus, if the particular combinations of the light quark components coincidentally allow the exchange of light mesons that can provide enough attractive force, then the residual strong interaction that mainly comes from their light degrees of freedom (d.o.f) may render this system to be bound. This picture is similar to a hydrogen molecule in QED, where the two electrons are shared by the two protons and the residual electromagnetic force binds this system. Therefore, the question of which heavy flavor di-hadron system can form a bound state becomes which kind of light quark combinations in the di-hadron system can exchange particular light mesons that can provide enough attractive force.

In this work, we adopt the quark level Lagrangian to address the above question. We relate the heavy flavor di-hadron effective potentials to their matrix elements in the flavor and spin spaces of their light d.o.f, so that we can qualitatively determine which heavy flavor di-hadron is possible to form a bound state.

This paper is organized as follows. In Sect. 2, we introduce our theoretical framework. Then we present our numerical results and discussions in Sect. 3. In Sect. 4, we conclude this work with a short summary.

2 Theoretical framework

We consider the two-body systems that are the combinations of the ground-state hadrons $({\bar{D}},{\bar{D}}^*)$, $(\Lambda _c, \Sigma _c, \Sigma _c^*)$, and $(\Xi _c, \Xi _c^\prime ,\Xi _c^*)$. We list the physical allowed heavy flavor meson-meson, meson-baryon, and baryon-baryon systems $[H_1H_2]_{J}^I$ in Table 1, where $H_1$ and $H_2$ denote the considered heavy flavor hadrons, while J and I are the total angular momentum and total isospin of the di-hadron system, respectively. If $H_1$ and $H_2$ are the general identical particles (e.g., ${\bar{D}}{\bar{D}}$, $\Sigma _c\Sigma _c\ldots$), then the quantum numbers of $[H_1H_2]_J^I$ system must satisfy the following selection rules

$$\begin{aligned} L+S_{\mathrm{{tot}}}+I_{\mathrm{{tot}}}+2i+2s=\left\{ \begin{aligned} \mathrm {Even},&\quad \mathrm {for\quad bosons},\\ \mathrm {Odd},&\quad \mathrm {for\quad fermions}. \end{aligned}\right. \end{aligned}$$

(1)

where L is the orbital angular momentum between $H_1$ and $H_2$ ($L=0$ for the S-wave case in our calculations), while $S_{\text {tot}}$ and $I_{\text {tot}}$ are the total spin and total isospin of the general identical di-hadron system, respectively. i denotes the isospin of $H_1$ or $H_2$, e.g., $i=1/2$ for the D meson, and $i=1$, $\frac{1}{2}$ for the $\Sigma _c$ and $\Xi _c$ baryons, respectively. Since the 2s (s is the spin of $H_1$ or $H_2$) must be an even or odd number for the two-body identical bosons or fermions systems, respectively, thus, we can collectively rewrite the selection rule in Eq. (1) as

$$\begin{aligned} L+S_{\text {tot}}+I_{\text {tot}}+2i=\text {Even}. \end{aligned}$$

(2)

As discussed in the introduction, we analyze the type of possible exchanged light mesons within a $[H_1H_2]_J^I$ system from their light quark components. The corrections from the exchange of heavy flavor mesons are neglected, this approximation is widely adopted to study the interactions of the systems that are composed of two heavy flavor hadrons [5,6,7,8,9,10,11,12]. Then the heavy flavor meson-meson, meson-baryon, and baryon-baryon can be studied simultaneously in the same formalism by checking the interactions with all possible quantum numbers.

We focus on the two-body S-wave interactions among the considered ground heavy flavor mesons/baryons. One can formulate the corresponding quark level Lagrangians as [34, 88, 89]

$$\begin{aligned} {\mathcal {L}}=g_s{\bar{q}}{\mathcal {S}}q+g_a{\bar{q}}\gamma _\mu \gamma ^5{\mathcal {A}}^\mu q, \end{aligned}$$

(3)

where $q=(u,d,s)$, $g_s$ and $g_a$ are two independent coupling constants. They encode the nonperturbative dynamics between light quarks of two color singlet hadrons and can be determined from the experimental data.

In this work, we focus on the di-hadron systems that can only allow the exchange of the isospin singlet and triplet fields. For the systems listed in Table 1, it is clear that the ${\bar{D}}^{(*)}{\bar{D}}^{(*)}$, $\Lambda _c{\bar{D}}^{*}$, $\Sigma _c^{(*)}{\bar{D}}^{(*)}$, $\Lambda _c\Lambda _c$, $\Lambda _c\Sigma _c^{(*)}$, $\Sigma _c^{(*)}\Sigma _c^{(*)}$, and $\Xi _c{\bar{D}}^{(*)}$ ($\Xi _c^{\prime (*)}{\bar{D}}^{(*)}$) cannot exchange isospin doublet strange mesons. In addition, we further calculate the matrix elements $\langle \lambda ^j\lambda ^j\rangle$ ($j=4,5,6,7$) for the $\Xi _c\Xi _c$ related systems listed in Table 1, they all have vanishing contributions. Thus, we also include these systems in this work. For the systems that can exchange isospin doublet fields, we need to take into account the SU(3) breaking effect when we construct their corresponding wave functions. They deserve a separate and detailed study, and we analyze these systems in a subsequent work [90]. Then the fictitious scalar field ${\mathcal {S}}$ and axial-vector field ${\mathcal {A}}^\mu$ reduce to the form

$$\begin{aligned} {\mathcal {S}}= & {} {\mathcal {S}}_3\lambda ^i+{\mathcal {S}}_1\lambda ^8, \end{aligned}$$

(4)

$$\begin{aligned} {\mathcal {A}}^\mu= & {} {\mathcal {A}}_3^\mu \lambda ^i+{\mathcal {A}}^\mu _1\lambda ^8, \end{aligned}$$

(5)

where $\lambda ^i$ ($i=1,2,3$) and $\lambda ^8$ are the generators of SU(3) group. ${\mathcal {S}}_3$ (${\mathcal {A}}_3^\mu$) and ${\mathcal {S}}_1$ (${\mathcal {A}}_1^\mu$) denote the isospin triplet and isospin singlet fields, respectively.

Table 1 The allowed heavy flavor di-hadron systems ($[H_1H_2]^{I}_{J}$) that are considered in this work, where the quantum numbers of the general identical systems are constrained by Eq. (2)

The effective potential of light quark-quark interactions can be deduced from Eq. (3), and we have

$$\begin{aligned} V_{l_1l_2}= & {} {\tilde{g}}_s\left( \lambda _1^8\lambda _2^8+\lambda _1^i\lambda _2^i\right) +{\tilde{g}}_a\left( \lambda _1^8\lambda _2^8+\lambda _1^i\lambda _2^i\right) \varvec{\sigma }_1\cdot \varvec{\sigma }_2.\nonumber \\ \end{aligned}$$

(6)

where the effective potential $V_{l_1l_2}$ is reduced to the local form when we integrate out the exchanged spurions (which is analogous to the resonance saturation model [91]). The redefined coupling constants are ${\tilde{g}}_s= g_s^2/m_{{\mathcal {S}}}^2$ and ${\tilde{g}}_a=g_a^2/m_{{\mathcal {A}}}^2$. Then the heavy flavor di-hadron effective potential from the interactions of their light quark components can be written as

$$\begin{aligned} V_{[H_1H_2]_J^I}=\left\langle [H_1H_2]_J^{I}\left| V_{l_1l_2}\right| [H_1H_2]_J^I\right\rangle , \end{aligned}$$

(7)

where $|[H_1H_2]_J^I\rangle$ denotes the quark-level spin-flavor wave function of $H_1H_2$ system with total isospin I and total angular momentum J, which is the direct product of spin and flavor wave functions

$$\begin{aligned} \left| [H_1H_2]_J^I\right\rangle= & {} \sum _{m_{I_1},m_{I_2}}C_{I_1,m_{I_1};I_2,m_{I_2}}^{I,I_z} \phi ^{H_{1f}}_{I_1,m_{I_1}}\phi ^{H_{2f}}_{I_2,m_{I_2}}\nonumber \\&\otimes \sum _{m_{l_1},m_{l_2}}C_{l_1,m_{l_1};l_2,m_{l_2}}^{l,l_z}\phi ^{H_{1s}}_{l_1,m_{l_1}}\phi ^{H_{2s}}_{l_2,m_{l_2}}. \end{aligned}$$

(8)

The $C_{I_1,m_{I_1};I_2,m_{I_2}}^{I,I_z}$ and $C_{l_1,m_{l_1};l_2,m_{l_2}}^{l,l_z}$ are the Clebsch-Gordan (CG) coefficients. ($\phi ^{H_{1f}}_{I_1,m_{I_1}}$, $\phi ^{H_{1s}}_{l_1,m_{l_1}}$), ($\phi ^{H_{2f}}_{I_2,m_{I_2}}$, $\phi ^{H_{2s}}_{l_2,m_{l_2}}$) are the quark-level (flavor, spin) wave functions for the $H_1$ and $H_2$ hadrons, respectively. l denotes the total light spin of this two-body system.

From Eqs. (6) and (7), we can see that the residual strong interaction of a specific $[H_1H_2]_J^I$ system can be divided into four parts, i.e., the scalar type ($\lambda ^8\lambda ^8$), the isospin related type ($\lambda ^i\lambda ^i$), the spin related type ($\lambda ^8\lambda ^8\varvec{\sigma }_1\cdot \varvec{\sigma }_2$) and the isospin-spin related type $(\lambda ^i\lambda ^i\varvec{\sigma }_1\cdot \varvec{\sigma }_2)$ interactions. In Table 2, we present the matrix elements of these four types of operators for the considered di-hadron systems in Table 1.

Table 2 The matrix elements of the operators ${\mathcal {O}}_1$ ($\lambda ^8_1\lambda ^8_2$), ${\mathcal {O}}_2$ ($\lambda ^i_1\lambda ^i_2$), ${\mathcal {O}}_3$ ($\lambda ^8_1\lambda ^8_2\varvec{\sigma }_1\cdot \varvec{\sigma }_2$), and ${\mathcal {O}}_4$ ($\lambda ^i_1\lambda ^i_2\varvec{\sigma }_1\cdot \varvec{\sigma }_2$) for the considered heavy flavor hadron-hadron systems ($[H_1H_2]_J^I$) listed in Table 1

After we obtain the effective potential of the $[H_1H_2]_J^I$ system, we need to check whether this system can form a bound state. This can be achieved by solving the following Lippmann-Schwinger equation (LSE),

$$\begin{aligned} T(p^\prime ,p)=V(p^\prime ,p)+\int \frac{d^3q}{\left( 2\pi \right) ^3}\frac{V\left( p^\prime ,q\right) T\left( q,p\right) }{E-\frac{ q^2}{2m_\mu }+i\epsilon }, \end{aligned}$$

(9)

where $m_\mu$ is the reduced mass of the $H_1$ and $H_2$. p and $p^\prime$ represent the momentum of the initial and final states in the center of mass frame, respectively.

Here, we introduce a hard regulator to exclude the contributions from higher momenta [73, 92]

$$\begin{aligned} V(p,p^\prime )=V_{[H_1H_2]_J^I}\Theta \left( \Lambda -p\right) \Theta \left( \Lambda -p^\prime \right) , \end{aligned}$$

(10)

where $\Theta$ is the step function. The amplitude $T(p^\prime ,p)$ is a function of $p^\prime$, p, and binding energy E with a separable form

$$\begin{aligned} T\left( p^\prime ,p\right) =\beta (E)\Theta (\Lambda -p^\prime )\Theta (\Lambda -p). \end{aligned}$$

(11)

Then the LSE can be reduced to an algebraic equation

$$\begin{aligned} \beta (E)=\frac{V_{[H_1H_2]_J^I}}{1-V_{[H_1H_2]_J^I}G}, \end{aligned}$$

(12)

with

$$\begin{aligned} G=\frac{m_\mu }{\pi ^2}\left[ -\Lambda +k \tan ^{-1}\left( \frac{\Lambda }{k}\right) \right] , k=\sqrt{-2m_\mu E}.\nonumber \\ \end{aligned}$$

(13)

We can search for the pole position of Eq. (12) to obtain the binding energy of the $[H_1H_2]_J^I$ system.

3 Numerical results

3.1 The results of the $P_c$, $P_{cs}$, and $T_{cc}$ states

We first use the masses of the $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$ in Ref. [18] to fix the parameters in our model. In our previous work [29], we suggested that the $P_c(4312)$, $P_{c}(4440)$, and $P_c(4457)$ have the assignments $[\Sigma _c{\bar{D}}]_{1/2}^{1/2}$, $[\Sigma _c{\bar{D}}^*]_{1/2}^{1/2}$, and $[\Sigma _c{\bar{D}}^*]_{3/2}^{1/2}$, respectively. With the matrix elements listed in Table 2, we can easily read out the effective potentials for these three $P_c$ states,

$$\begin{aligned} V_{P_{c}(4312)}= & {} -\frac{10}{3}{\tilde{g}}_s, \end{aligned}$$

(14)

$$\begin{aligned} V_{P_{c}(4440)}= & {} -\frac{10}{3}{\tilde{g}}_s+\frac{40{\tilde{g}}_a}{9},\end{aligned}$$

(15)

$$\begin{aligned} V_{P_{c}(4457)}= & {} -\frac{10}{3}{\tilde{g}}_s-\frac{20{\tilde{g}}_a}{9}. \end{aligned}$$

(16)

There exist three undetermined parameters in Eq. (12), the light quark-quark coupling constants ${\tilde{g}}_s$, ${\tilde{g}}_a$, and the momentum cutoff $\Lambda$. We use the experimental mass of the $P_c$ states [18] to precisely extract these three parameters. The solutions are ${\tilde{g}}_s=11.739$ $\hbox {GeV}^{-2}$, ${\tilde{g}}_a=-2.860$ $\hbox {GeV}^{-2}$, and $\Lambda =0.409$ GeV. To give a rough estimation on the theoretical errors of the masses of these three $P_c$ states, we fix the obtained parameters ${\tilde{g}}_s$ and ${\tilde{g}}_a$ since they provide a base line to produce bound states and a good description of the mass splitting of the $[\Sigma _c{\bar{D}}^*]_{\frac{1}{2}/\frac{3}{2}}^{\frac{1}{2}}$ spin multiplets, respectively. Then we adjust the parameter $\Lambda$ so that the obtained masses of these three $P_c$ states can cover the corresponding experimental error ranges. The corresponding range for the $\Lambda$ is at (0.33, 0.43) GeV. We use these two values to give the lower and upper limits of the considered heavy flavor di-hadron systems. In our convention, a positive (negative) $V_{[H_1H_2]_J^I}$ means a(n) repulsive (attractive) interaction. Once we determine the signs of the ${\tilde{g}}_s$ and ${\tilde{g}}_a$, we can directly find out whether the considered systems have repulsive or attractive forces from the values in Table 2.

From the point of view of the potential model, the spin-spin interaction is suppressed by a factor of $1/(m_{\Sigma _c^{(*)}}m_{{\bar{D}}^{(*)}})$, which is roughly consistent with our obtained ratio $|{\tilde{g}}_a|/|{\tilde{g}}_s|\approx 0.24$.

Table 3 The experimental data [14, 18, 36,37,38] and our results of the masses and binding energies (BE) for the $T_{cc}^+$, $P_{c}(4312)$, $P_{c}(4380)$, $P_{c}(4440)$, $P_{c}(4457)$, and $P_{cs}(4459)$. The tiny experimental error for the $T_{cc}(3875)$ [37, 38] is not presented. We use the values $\Lambda =0.33$ and 0.43 GeV as inputs to give the upper and lower limits of our theoretical results, respectively. We adopt the isospin averaged masses for the single-charm mesons and baryons [4]. The listed values are all in units of MeV

The cutoff $\Lambda$ is smaller than the masses of the ground scalar or axial-vector mesons, which are regarded as the hard scales and integrated out in the effective field theory. In principle, there may exist contributions from the pion-exchange interactions. Although the two ground heavy meson/baryons can easily couple to the pion field via the p-wave interactions, our calculations based on the chiral effective field theory [23, 29, 34, 89, 93] showed that the magnitude of the one-pion-exchange (OPE) interaction is comparable to that of the next-to-leading order two-pion-exchange (TPE) interaction. The OPE and TPE have considerable corrections to the binding energies of the bound states, but they are not the main driving force of the formation of the bound states. In this work, we do not include the pion exchange dynamics.

In Table 3, we list the masses of the experimentally observed molecular candidates and the results from our model. The center values of the $P_c(4312)$, $P_{c}(4440)$, and $P_c(4457)$ are used as inputs to determine the values of ${\tilde{g}}_s$, ${\tilde{g}}_a$, and $\Lambda$. We also predict a $[\Sigma _c^*{\bar{D}}]^{1/2}_{3/2}$ molecular state with the mass 4376.2 MeV, which may correspond to the observed $P_{c}(4380)$ state [14]. The bound state $[\Sigma _c^*{\bar{D}}]^{1/2}_{3/2}$ is also obtained in different models [19, 20, 22, 24,25,26, 28, 29].

We further adopt our picture to study the recently observed $T_{cc}^+$ state. We assign the $T_{cc}^+$ as the $[DD^*]_1^0$ molecular state and calculate its mass with the same parameters extracted from the $P_c$ states. As shown in Table 3, our approach gives a rather good description of the mass of $T_{cc}^+$. This nice agreement indicates that neglecting the corrections from the heavy degrees of freedom is a fairly good approximation in this case. The heavy flavor meson-meson and baryon-baryon systems share the same binding mechanism that is dominated by their light degrees of freedom.

The $P_{cs}(4459)$ is close to the threshold of $\Xi _c {\bar{D}}^*$, which can be assigned as a $[\Xi _c {\bar{D}}^*]^0_{1/2}$ or $[\Xi _c {\bar{D}}^*]_{3/2}^{0}$ molecular state [34]. The spin of the light diquark in the $\Xi _c$ baryon is 0, so the $\Xi _c {\bar{D}}^*$ system has the vanishing spin-spin interaction from its light d.o.f. The $[\Xi _c {\bar{D}}^*]^0_{1/2}$ and $[\Xi _c {\bar{D}}^*]_{3/2}^{0}$ states are degenerate in our formalism, as can be seen from Table 2. The inclusion of the spin-spin interaction from heavy degrees of freedom shall distinguish these two states. However, from a serious calculation within the framework of chiral effective field theory [34], the mass gaps induced from the spin-spin interactions are within several MeVs. In this sense, our prediction is consistent with the observed $P_{cs}(4459)$. Indeed, the LHCb collaboration also fitted the data using two resonances with masses $4454.9\pm 2.7$ MeV and $4467.8\pm 3.7$ MeV [36]. However, the limited data samples cannot confirm or refute the two-peak hypothesis. An updated analysis with more data samples is desired to clarify this issue.

We further swap the assignments of the $P_c(4440)$ and $P_{c}(4457)$, and regard them as the $[\Sigma _c{\bar{D}}^*]_{3/2}^{1/2}$ and $[\Sigma _c{\bar{D}}^{*}]_{1/2}^{1/2}$ molecular states, respectively. We can also find a set of solutions that can reproduce the masses of the three $P_c$ states. However, the cutoff $\Lambda$ is at 1.763 GeV, which is far away from the scale of light scalar mesons. Moreover, we can not reproduce the $T_{cc}^+$ state in this case. Thus, we rule out this set of assignments for the $P_c(4440)$ and $P_c(4457)$. In our framework, we can identify the quantum numbers of the $P_{c}(4440)$ and $P_{c}(4457)$ states if we require a satisfactory description of the $T_{cc}^+$ and $P_c$ states simultaneously.

3.2 $T_{cc}^\prime$ state and other heavy flavor molecular states

In the previous section, we have shown that our framework gives a nice description of the observed $T_{cc}$, $P_c$, and $P_{cs}$ states. In the following, we further adopt the fitted parameters ${\tilde{g}}_s$, ${\tilde{g}}_a$, and $\Lambda$ to calculate the other heavy flavor di-hadron systems listed in Table 1. The effective potentials for these systems can be easily read from Table 2. Their calculated masses and binding energies are listed in Table 4.

Table 4 The predicted masses and binding energies (BE) for the charmed di-hadrons ($[H_1H_2]_J^{I}$) in Table 1. We adopt the isospin averaged masses for the single-charm hadrons [4]. We use the values $\Lambda =0.33$ and 0.43 GeV as inputs to give the upper and lower limits of our theoretical results, respectively. The “$+*$" denotes that the corresponding molecular states no longer exist when we adopt $\Lambda =0.33$ GeV. The values are all in units of MeV

The effective potentials for the $[{\bar{D}}^{(*)}{\bar{D}}^{(*)}]_J^I$ systems can be collectively written as

$$\begin{aligned} V_{[{\bar{D}}^{(*)}{\bar{D}}^{(*)}]_J^I}={\tilde{g}}_s\left( {\mathcal {O}}_1+{\mathcal {O}}_2\right) +{\tilde{g}}_a\left( {\mathcal {O}}_3+{\mathcal {O}}_4\right) , \end{aligned}$$

(17)

where the values of matrix elements for the ${\mathcal {O}}_{1,\ldots ,4}$ are listed in Table 2. Note that the ${\tilde{g}}_s$ is much larger than ${\tilde{g}}_a$. Thus, the terms that are related to ${\tilde{g}}_s$ dominate the total effective potentials of the $[{\bar{D}}^{(*)}{\bar{D}}^{(*)}]_J^I$ systems. We can easily read out the total effective potentials of the $[{\bar{D}}{\bar{D}}^*]_1^0$ and $[{\bar{D}}^*{\bar{D}}^*]_1^0$ systems from Table 2 as

$$\begin{aligned} V_{[{\bar{D}}{\bar{D}}^*]_1^0}= & {} -\frac{8}{3}{\tilde{g}}_s, \end{aligned}$$

(18)

$$\begin{aligned} V_{[{\bar{D}}^*{\bar{D}}^*]_{1}^0}= & {} -\frac{8}{3}{\tilde{g}}_s+\frac{8}{3}{\tilde{g}}_a, \end{aligned}$$

(19)

these two potentials are negative due to the dominant ${\tilde{g}}_s$-related terms and can be easily checked numerically. The interactions of these two systems are attractive. Similarly, we can also obtain the effective potentials for the rest of the ${\bar{D}}^{(*)}{\bar{D}}^{(*)}$ systems, i.e., the $[{\bar{D}}{\bar{D}}]_0^1$, $[{\bar{D}}{\bar{D}}^*]_1^1$, $[{\bar{D}}^*{\bar{D}}^*]_{0}^1$, and $[{\bar{D}}^*{\bar{D}}^*]_{2}^1$ systems directly from Table 2. These four systems have repulsive interactions due to the positive ${\tilde{g}}_s$-related terms, and they cannot form bound states. We check all the physically allowed $D^{(*)}D^{(*)}$ systems and find that besides the $T_{cc}$ state with the assignment $[DD^*]_1^0$, the $[D^*D^*]_1^0$ system can also form a molecular state, and we assign it as the $T_{cc}^\prime$. Thus, the $T_{cc}$ and $T_{cc}^\prime$ are the only two molecular states in the $D^{(*)}D^{(*)}$ systems. The $T_{cc}^{\prime +}$ lies about 7 MeV below the $D^*D^*$ threshold. This state has no hidden-charm strong decay channels due to its $cc{\bar{u}}{\bar{d}}$ valence quark component, thus should decay into $D^0D^0\pi ^0\pi ^+$ or $D^0D^+\pi ^0\pi ^0$ final states. In addition, because of the small phase space for $D^{*+}\rightarrow D^+\pi ^0$ ($D^0\pi ^+$) and $D^{*0}\rightarrow D^0\pi ^0$, if the $T_{cc}^{\prime +}$ does exist, similar to the $T_{cc}^+$, the $T_{cc}^{\prime +}$ should also be a narrow state in a loosely bound molecular picture. We suggest the LHCb Collaboration to look for this state in the future.

Before the observation of the $T_{cc}$ state, the $D^{(*)}D^{(*)}$ interactions were studied in the framework of the one-boson-exchange (OBE) potential model [39], they found that the $[DD^*]_1^0$ and $[D^*D^*]_1^0$ systems are good molecular candidates. While the $D^{(*)}D^{(*)}$ systems with quantum numbers $I(J^P)=1(0^+)$, $1(1^+)$, $1(2^+)$ are not good molecular candidates due to the large cutoff. Thus, the results obtained from the OBE model are consistent with ours. The $[D^*D^*]_1^0$ molecular state was also suggested in Ref. [94], the authors introduced the vector-vector interaction within the hidden-gauge formalism to describe the interactions of the $D^{(*)}D^{(*)}$ systems. They also found that the $[DD^*]_1^0$ and $[D^*D^*]_1^0$ systems can form bound states. Within the Bethe–Salpeter (B–S) framework [95], the authors employed the OBE model to study the interaction kernels in the B-S equations. Their results supported the $[DD^{*}]_1^0$ assignment of the observed $T_{cc}$ state. However, they found that the $D^*D^*$ molecular state with quantum number $I(J^P)=1(0^+)$ may exist. Besides, the possible molecular states with the assignments $[DD^*]_1^0$, $[D^*D^*]_1^0$ were also suggested in Ref. [96] by adopting the heavy-quark spin symmetry. Their results also indicated that the isovector $D^{(*)}D^{*}$ molecular states may also exist.

The results for the charmed baryon-meson (baryon) systems are also presented in Table 4. We predict three more $P_c$ states in the $\Sigma _c^{(*)}{\bar{D}}^{(*)}$ systems, which are also suggested in Ref. [20, 22, 29]. For the meson-baryon systems composed of the ($\Xi _c$, $\Xi _c^\prime$, $\Xi _c^*$) baryons and (${\bar{D}}$, ${\bar{D}}^*$) mesons. We obtain 10 molecular candidates in the $\Xi _c^{\prime }{\bar{D}}^{(*)}$ and $\Xi _c^{\prime (*)}{\bar{D}}^{(*)}$ systems. They all have isospin 0 and their binding energies lie in the range (-30,0) MeV by including the theoretical uncertainties extracted from the experimental uncertainties of the three $P_c$ states. The systematical studies on the $\Xi _c{\bar{D}}^{(*)}$, $\Xi _c^{\prime (*)}{\bar{D}}^{(*)}$ systems based on the quasipotential Bethe-Salpeter equation approach [97], coupled channel unitary approach [33, 98], and chiral effective field theory [34] give very similar predictions in the isoscalar channels. Among them, five states with $1/2^-$, four states with $3/2^-$, the molecular candidate with $5/2^-$ has marginal attractive interaction and the existence of this state will depend on the detailed theoretical treatments from different models [33, 34, 97, 98]. Thus, the results obtained from our formalism are close to the above theoretical calculations.

We do not find any molecular states in the $\Lambda _c\Lambda _c$ system. The existence of $\Lambda _c\Lambda _c$ molecular states is controversial and has been discussed in many literatures [83, 85, 86, 99,100,101,102,103,104,105,106,107]. It was suggested that the single channel $\Lambda _c\Lambda _c$ system can not form the bound state [83, 85, 102, 103, 105], while the authors in Ref. [86, 99, 100, 107] suggested that the coupling to the $\Sigma _c^{(*)}\Sigma _c^{(*)}$ channels may render the $\Lambda _c\Lambda _c$ system to be bound. However, the authors in Ref. [104, 106] found that the $\Lambda _c\Lambda _c$ system can be bound by itself. Similarly, we do not find any bound states in the $\Lambda _c\Sigma _c^{(*)}$ systems due to the weak interaction between the spin-0 and spin-1 light diquarks. This result is consistent with Ref. [105] obtained from the framework of quark delocalozation color screening model. The studies from the OBE model [86], the dispersion relation technique [104] and effective theory [107] suggested that the $\Lambda _c\Sigma _c$ and $\Lambda _c\Sigma _c^{(*)}$ molecular states may exist, respectively. We predict two molecular states $[\Sigma _c\Sigma _c]_0^0$ and $[\Sigma _c\Sigma _c]_1^1$ in the $\Sigma _c\Sigma _c$ systems. The existence of the molecular states in the $\Sigma _c\Sigma _c$ system is solid and supported by many literatures [83, 85, 102, 104, 105, 108, 109]. These literatures all suggested the existence of the molecular state $[\Sigma _c\Sigma _c]_0^0$, while for the existence of $[\Sigma _c\Sigma _c]_1^1$ molecule, the conclusion still relied on the different treatments of theoretical models. For the di-baryon systems that are composed of two ($\Xi _c$, $\Xi _c^\prime$, $\Xi _c^*$) baryons, we also predict several molecular states in Table 4. The molecular candidates of the $\Xi _c\Xi _c$ and related systems were predicted in a realistic phenomenological nucleon-nucleon interaction models [108] and OBE model [85, 86], while the authors in Ref. [83] adopted the resonance saturation model and suggested that the interactions for the $\Xi _c\Xi _c$ related systems are two weak and thus cannot form bound states. Thus, the experimental exploration of the double-charm di-baryon systems would be crucial to distinguish different approaches in the future.

The determined ${\tilde{g}}_s$ is a positive value and about 3 times larger than the ${\tilde{g}}_a$. From Eq. (6) we can see that for the lowest isospin di-hadron systems, the isospin-isospin matrix elements are negative and dominate the whole effective potentials of the di-hadron systems. Thus, if a heavy flavor two-body system has a large negative $\lambda ^i_1\lambda _2^i$ eigenvalue, this two-body system will have an attractive force and may form a bound state. As shown in Tables 2 and 4, this feature is universal for all the studied heavy flavor meson-meson, meson-baryon, and baryon-baryon systems. The $\lambda _1^i\lambda _2^i$ reduces to $\varvec{\tau }_1\cdot \varvec{\tau }_2$ in the SU(2) case, and can be calculated with

$$\begin{aligned} \varvec{\tau }_1\cdot \varvec{\tau }_2=2\left[ I(I+1)-I_1(I_1+1)-I_2(I_2+1)\right] . \end{aligned}$$

(20)

As shown in Eq. (20), the lowest total isospin generally leads to a negative eigenvalue and corresponds to an attractive force. Our formalism gives a very practical criterion to understand why the currently observed $T_{cc}^+$, $P_c$, and $P_{cs}$ states all prefer the lowest isospins.

3.3 Implications for the bottom hadron-hadron systems

In our calculations, we neglect the corrections from the heavy quarks in the charmed two-body systems and obtain a good description of the observed $T_{cc}^+$, $P_c$, and $P_{cs}$ states. If we adopt the same approximation for the bottom di-hadrons, then the ($T_{cc}^+$, $P_{c}$, $P_{cs}$) and ($T_{bb}^-$, $P_{b}$, $P_{bs}$) molecular states share the identical effective potentials from their light d.o.f.

In Fig. 1, we present the variation of binding energies for some typical molecular states as their corresponding reduced masses gradually increase. In each system, there exists a critical reduced mass at $E_{\text {BE}}=0$, from which the system starts to form a bound state. Then the absolute values of binding energies increase as their reduced masses increase. The increased rate depends on the different types of light quark combinations in the two-body heavy flavor systems. We mark the $T_{cc}^+$, $T_{cc}^{\prime +}$, $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$, as well as their bottom partners in Fig. 1. For the rest of the considered bottom meson-meson, meson-baryon and baryon-baryon systems, we list our predictions in Table 5. As shown in Fig. 1, due to the large reduced masses of the bottom di-hadron systems, if there exist bound states in the charm di-hadrons, there should also exist the bottom partners with deeper binding energies as well.

Fig. 1

The variation of binding energies for the $T_{cc}^+$, $T_{cc}^{\prime +}$, $P_{c}(4312)$, $P_c(4440)$, and $P_{c}(4457)$ states as their reduced masses increase. At $m_Q=m_b$, we have their bottom partners $T_{bb}^-$, $T_{bb}^{\prime -}$, $[\Sigma _bB]_{1/2}^{1/2}$, $[\Sigma _bB^*]_{1/2}^{1/2}$, and $[\Sigma _bB^*]_{3/2}^{1/2}$, respectively

Table 5 The predicted masses and binding energies (BE) for the bottom di-hadrons ($[H_1H_2]_J^{I}$) in Table 1. We adopt the isospin averaged masses for the single-bottom hadrons [4]. We use the values $\Lambda =0.33$ and 0.43 GeV as inputs to give the upper and lower limits of our theoretical results, respectively. The “$+*$" denotes that the corresponding molecular states no longer exist when we adopt $\Lambda =0.33$ GeV. The values are all in units of MeV

4 Summary

In this work, we use a quark level Lagrangian to give a universal description of the heavy flavor hadronic molecules that are composed of the ground $({\bar{D}},{\bar{D}}^*)$, $(\Lambda _c, \Sigma _c, \Sigma _c^*)$, and $(\Xi _c, \Xi _c^\prime ,\Xi _c^*)$ hadrons. Based on this quark-level effective Lagrangian, we neglect the contributions from heavy quarks and relate the effective potentials of di-hadrons to their flavor and spin interaction operators of light degrees of freedom.

In our approach, we only introduce three parameters ${\tilde{g}}_s$, ${\tilde{g}}_a$, and $\Lambda$. They can be well extracted from the observed $P_{c}(4312)$, $P_c(4440)$, and $P_c(4457)$. We exclude the assignments of $P_{c}(4440)$ and $P_{c}(4457)$ as the $[\Sigma _c{\bar{D}}^*]_{3/2}^{1/2}$ and $[\Sigma _c{\bar{D}}^*]_{1/2}^{1/2}$ states, respectively, due to the poor description of $T_{cc}^+$ in this case. Our results strongly indicate a very similar binding mechanism between the heavy flavor meson-meson and meson-baryon systems, i.e., they are bound dominantly by the interactions of their light degrees of freedom. We further generalize this similarity to the heavy flavor baryon-baryon systems.

We predict another $T_{cc}^{\prime +}$ state with the assignment $[D^*D^*]_{1}^0$. From our calculations, the $T_{cc}^+$ and $T_{cc}^{\prime +}$ are the only two molecular states in the $D^{(*)}D^{(*)}$ systems. We suggest the LHCb to look for this state in the future. We also predict other possible heavy flavor hadronic molecules in the charmed and bottom sectors (e.g., see Tables 4 and 5).

In this work, we adopt the non-relativization/heavy quark symmetry approximations. Indeed, the complete potential for the considered heavy flavor di-hadron system at the quark-level can be parameterized as

$$\begin{aligned} V_{\mathrm {quark-level}}= & {} V_{\mathrm {quark-level}}^{\mathrm {HQS}}+V_{\mathrm {quark-level}}^{\mathrm {HQSB}}, \end{aligned}$$

(21)

where $V_{\mathrm {quark-level}}^{\mathrm {HQS}}$ and $V_{\mathrm {quark-level}}^{\mathrm {HQSB}}$ denote the heavy quark symmetry (HQS) keeping and breaking terms, and they can be written as, respectively,

$$\begin{aligned} V_{\mathrm {quark-level}}^{\mathrm {HQS}}= & {} V_a+{\tilde{V}}_a\varvec{\ell }_1\cdot \varvec{\ell }_2, \end{aligned}$$

(22)

$$\begin{aligned} V_{\mathrm {quark-level}}^{\mathrm {HQSB}}= & {} \frac{{\tilde{V}}_b}{m^2_{c{\bar{q}}}}\varvec{\ell }_1\cdot \varvec{h}_2+\frac{{\tilde{V}}_c}{m^2_{c{\bar{q}}}}\varvec{\ell }_2\cdot \varvec{h}_1+\frac{{\tilde{V}}_d}{m^2_{c{\bar{c}}}}\varvec{h}_1\cdot \varvec{h}_2,\nonumber \\ \end{aligned}$$

(23)

with $\varvec{\ell }_i(\varvec{h}_i)$ the light (heavy) spin in the ith hadrons. The $V_a,{\tilde{V}}_{a,\dots ,d}$ are the functions (LECs) used to parametrize the strength of the corresponding potentials. On the one hand, at tree level, to ensure that the final $[H_1H_2]_J^I$ state is unchanged at flavor space, the $[H_1H_2]_J^I$ system can only exchange light mesons or heavy charmoniums, while the exchange of the charmed mesons is forbidden. Thus, the ${\tilde{V}}_b$ and ${\tilde{V}}_c$ terms vanish. Then we need to determine the parameter ${\tilde{g}}_{c{\bar{c}}}\equiv {\tilde{V}}_d/m^2_{c{\bar{c}}}$ to estimate the effect induced from the non-relativization approximation. However, based on the present date, we do not have an independent relation to fix the ${\tilde{g}}_{c{\bar{c}}}$. On the other hand, the couplings from the exchange of $c{\bar{c}}$ light mesons and charmoniums are suppressed by $1/m^2_{{\mathcal {S}}/{\mathcal {A}}}$ and $1/m_{cc}^2$, respectively. Thus, the interaction induced from the exchange of heavy charmoniums are expected to be suppressed by a factor of about 1/10 compared to that of exchanging light mesons, and our results indicate that this is a fairly good approximation.

Although we do not explicitly include the OPE and TPE potentials, we still obtain a good description to the masses of the observed $P_c$, $P_{cs}$, and $T_{cc}$ states, as presented in Table 3. One reason is that unlike the contact terms, the pion-exchange interactions depend on the transferred momentum $\varvec{q}^2$, which is a small quantity especially for the near-threshold two body systems. From our calculations based on the chiral effective field theory, the magnitudes of the OPE potentials are comparable to that of the NLO TPE potentials, they provide small corrections to the total effective potential of the $[H_1H_2]_J^I$ system. The most important reason is that in our approach, we use the form of the leading order contact terms $V_{\mathrm{{Cont}}}$ to approximate the total effective potentials of the $[H_1H_2]_J^I$ system, and use this type of contact potential to fit the mass spectrum of the discussed $P_c$ states (here we assign this approach as the first scheme). If we calculate the effective potential of the $[H_1H_2]_J^I$ system in the framework of chiral effective field theory, i.e., we include the contact ($V^\prime _{\mathrm{{Cont}}}$), OPE ($V_{\mathrm{{OPE}}}$) and TPE ($V_{\mathrm{{TPE}}}$) potentials (we assign this approach as the second scheme). Then we have implicitly adopted the approximation

$$\begin{aligned} V_{\mathrm{{Cont}}}=V_{\mathrm{{Cont}}}^\prime +V_{\mathrm{{OPE}}}+V_{\mathrm{{TPE}}}. \end{aligned}$$

(24)

In both schemes, we need to fix the coupling parameters (${\tilde{g}}_s$, ${\tilde{g}}_a$) in $V_{\mathrm{{Cont}}}$ and $V_{\mathrm{{Cont}}}^\prime$ with the experimental masses. Thus, it is obvious that the obtained ${\tilde{g}}_s$ and ${\tilde{g}}_a$ in the first scheme is different from that of the second scheme. We could say that the effects induced from pion-exchange interactions are partly “smeared" and implicitly included in the coupling parameters ${\tilde{g}}_s$ and ${\tilde{g}}_a$ obtained from the first scheme. Thus, we can still give a good description to the masses of the observed $P_c$, $P_{cs}$, and $T_{cc}$ states.

As a primary application, we study the interactions of heavy flavor di-hadron systems in the single-channel case. We introduce a simple Lagrangian to describe the effective potential of the heavy flavor $[H_1H_2]_J^I$ system, to discuss the couple channel effect of this system, this framework allow us to include as many as possible channels that can couple to the considered $[H_1H_2]_J^I$ system, the number of included channels, the roles of these couple channels to the considered $[H_1H_2]_J^I$ system are also very interesting topics. There still remain rich contents and deserve a detailed and extended discussion in the future.

In this work, we only introduce the leading order contact interactions to describe the effective potentials of the considered heavy flavor di-hadron systems. The obtained results to the observed $P_c$, $P_{cs}$, and $T_{cc}$ states indicate that the interactions of heavy flavor di-hadron systems may have a heavy quark spin and flavor symmetry. Specifically, the spin multiplets in the same system obey the heavy quark spin symmetry, for example, the $P_{c}(4312)$, $P_{c}(4440)$, and $P_{c}(4457)$. While different heavy flavor di-hadron systems obey the heavy quark flavor symmetry, for example, the observed $P_{c}$, $P_{cs}$, and $T_{cc}$ states (for a detailed and extended discussion on this symmetry, we refer to Ref. [90]). This is the main task of our work. However, at present, we do not include the OPE and TPE interactions in the effective potentials of the considered heavy flavor di-hadron systems, which may provide considerable corrections to the binding energies of the discussed di-hadron systems, these effects still need to be elaborately studied with the chiral effective field theory in the future.