Search for singly charmed dibaryons in baryon-baryon scattering

The model has become one of the most common approaches to describe hadron spectra, hadron-hadron interactions and multiquark states ChQM(RPP) . The construction of the ChQM is based on the breaking of chiral symmetry dynamics chqm1 ; chqm2 . The model mainly uses one-gluon-exchange potential to describe the short-range interactions, a $\sigma$ meson exchange (only between u, d quarks) potential to provide the mid-range attractions, and Goldstone boson exchange potential for the long-range effects chqm3 . In addition to the Goldstone bosons exchange, there are additional $D$ meson that can be exchanged between u/d and c quarks, $D_{s}$ meson that can be exchanged between s and c quarks, and $\eta_{c}$ that can be exchanged between any two quarks of the u, d, s and c quarks. In order to incorporate the charm quark well and study the effect of the $D$, $D_{s},\eta_{c}$ meson exchange interaction, we extend the model to $SU(4)$, and add the interaction of these heavy mesons interactions. The extension is made in the spirit of the phenomenological approach of Refs.  Glozman ; Stancu . The detail of ChQM used in the present work can be found in the references ChQM1 ; ChQM2 ; ChQM3 . In the following, only the Hamiltonian and parameters are given.

Where $T_{c}$ is the kinetic energy of the center of mass; $S_{ij}$ is quark tensor operator. We only consider the $S-$wave systems at present, so the tensor force dose not work here; $Y(x)$ and $H(x)$ are standard Yukawa functions ChQM(RPP) ; $\alpha_{ch}$ is the chiral coupling constant, determined as usual from the $\pi$-nucleon coupling constant; $\alpha_{s}$ is the quark-gluon coupling constant ChQM1 . Here $m_{\chi}$ is the mass of the mesons, which are experimental value; $\Lambda_{\chi}$ is the cut-off parameters of different mesons, which can refer to Ref D . The coupling constant $g_{ch}$ for scalar chiral field is determined from the $NN\pi$ coupling constant through

All other symbols have their usual meanings.

All parameters were determined by fitting the masses of the baryons of light and heavy flavors. The model parameters and the fitting masses of baryons are shown in Table 1 and Table 2, respectively.

In this work, RGM RGM1 ; RGM2 is used to carry out a dynamical calculation. In the framework of RGM, which split the dibaryon system into two clusters, the main feature of RGM is that for a system consisting of two clusters, it can assume that the two clusters are frozen inside, and only consider the relative motion between the two clusters, so the conventional ansatz for the two-cluster wave function is:

where the symbol ${\cal A}$ is the anti-symmetrization operator. With the $SU(4)$ extension, both the light and heavy quarks are considered as identical particles. So ${\cal A}=1-9P_{36}$. $[\sigma]=[222]$ gives the total color symmetry and all other symbols have their usual meanings. $\phi_{B_{i}}$ is the $3-$quark cluster wave function. From the variational principle, after variation with respect to the relative motion wave function $\chi(\bm{\mathbf{R}})=\sum_{L}\chi_{L}(\bm{\mathbf{R}})$, one obtains the RGM equation

where $H(\bm{\mathbf{R}},\bm{\mathbf{R^{\prime}}})$ and $N(\bm{\mathbf{R}},\bm{\mathbf{R^{\prime}}})$ are Hamiltonian and norm kernels. The RGM can be written as

where

where $\mu$ is the approximate mass between the two quark clusters; $E_{rel}=E-E_{int}$ is the relative motion energy; $V_{rel}^{D}$ is the direct term in the interaction potential. By solving the RGM equation, we can get the energies $E$ and the wave functions. In fact, it is not convenient to work with the RGM expressions. Then, we expand the relative motion wave function $\chi(\bm{\mathbf{R}})$ by using a set of gaussians with different centers,

where $L$ is the orbital angular momentum between two clusters. Since the system we studied are all $S-$waves, $L=0$ in this work, and $\bm{S_{i}}$, $i=1,2,...,n$ are the generator coordinates, which are introduced to expand the relative motion wave function. By including the center of mass motion:

the ansatz Eq.(10) can be rewritten as

where $\chi_{I_{1}S_{1}}$ and $\chi_{I_{2}S_{2}}$ are the product of the flavor and spin wave functions, and $\chi_{c}$ is the color wave function. The flavor, spin, and color wave functions are constructed in two steps. First, constructing the wave functions for the baryon and baryon clusters; then, coupling the two wave functions of two clusters to form the wave function for the dibaryon system. The detail of constructing the wave functions are presented in Appendix. For the orbital wave functions, $\phi_{\alpha}(\bm{S}_{i})$ and $\phi_{\beta}(-\bm{S}_{i})$ are the single-particle orbital wave functions with different reference centers:

By expanding the relative motion wave function between two clusters in the RGM equation by gaussians, the integro-differential equation of RGM can be reduced to an algebraic equation, which is the generalized eigen-equation. With the reformulated ansatz, the RGM equation Eq.(11) becomes an algebraic eigenvalue equation:

where $H_{i,j}$ and $N_{i,j}$ are the Hamiltonian matrix elements and overlaps, respectively. Besides, to keep the matrix dimension manageably small, the baryon-baryon separation is taken to be less than 6 fm in the calculation. By solving the generalized energy problem, we can obtain the energy and the corresponding wave functions of the dibaryon system. On the basis of RGM, we can further calculate scattering problems to find resonance states.

For a scattering problem, the relative wave function of the baryon-baryon is expanded as

with

where

$C_{i}$ are the expansion coefficients, and $C_{i}$ satisfy $\sum_{i=1}^{n}C_{i}=1$. n is the number of Gaussion bases (which is determined by the stability of the results), and $j_{L}$ is the $L$th spherical Bessel function. $h_{L}^{\pm}$ are the $L$th spherical Hankel functions, $k$ is the momentum of the relative motion with $k=\sqrt{2\mu E_{cm}}$, $\mu$ is the reduced mass of two baryons of the open channel, $E_{cm}$ is the incident energy of the relevant open channels, and $R_{C}$ is a cutoff radius beyond which all of the strong interactions can be disregarded. $\alpha_{i}$ and $s_{i}$ are complex parameters that determined in terms of continuity conditions at $R=R_{C}$. After performing the variational procedure by the Kohn-Hulth$\acute{e}$n-Kato(KHK) variational method KHK , a $L$th partial-wave equation for the scattering problem can be reduced as

with

and

By solving Eq.(22) we obtain the expansion coefficients $C_{i}$. Then, the $S$ matrix element $S_{L}$ and the phase shifts $\delta_{L}$ are given by

Through the scattering process, not only can we better study the interaction between hadrons, but it can also help us research resonance states. The general scattering phase shift diagram should be a smooth curve, that is, the phase shift will change gently as the incident energy increases. But in some cases, the phase shift will be abrupt, the change will be more than 90 degrees, which is the resonance phenomena. The rapid phase change is a general feature of resonance phenomena, see Fig.1. The center of mass energy with phase shift $\frac{\pi}{2}$ gives the mass of the resonance ($M^{\prime}$ in Fig.1), and the difference of the energies with phase shift $\frac{3\pi}{4}$ and $\frac{\pi}{4}$ gives the partial decay width of the resonance ($\Gamma$ in Fig.1).

The effective potential between two baryons is shown as

where $S_{i}$ stands for the distance between two clusters and $E(\infty)$ stands for a sufficient large distance of two clusters, and the expression of $E(S_{i})$ is as follow.

$\Psi_{6q}(S_{i})$ represents the wave function of a certain channel. Besides, $\left\langle\Psi_{6q}(S_{i})\left|H\right|\Psi_{6q}(S_{i})\right\rangle$ and $\left\langle\Psi_{6q}(S_{i})|\Psi_{6q}(S_{i})\right\rangle$ are the Hamiltonian matrix and the overlap of the states. The effective potentials of all channels with different strange numbers are shown in Fig.2, Fig.3 and Fig.4 respectively.

For $S=-1$ system, as shown in Fig.2, all of the seven channels are attractive, the potentials for the four channels $\Sigma\Sigma_{c},\Sigma\Sigma_{c}^{*},\Sigma^{*}\Sigma_{c}$ and $\Sigma^{*}\Sigma_{c}^{*}$ are deeper than the other three channels $\Lambda\Lambda_{c},N\Xi_{c}$ and $N\Xi_{c}^{\prime}$, which indicates that the $\Sigma\Sigma_{c},\Sigma\Sigma_{c}^{*},\Sigma^{*}\Sigma_{c}$ and $\Sigma^{*}\Sigma_{c}^{*}$ are more likely to form bound states or resonance states.

For $S=-3$ system, from Fig.3 we can see that the potentials of the $\Xi\Xi_{c}^{*},\Xi^{*}\Xi_{c}^{*}$ and $\Lambda\Omega_{c}^{*}$ are attractive, while the potentials for the other six channels are repulsive. The attraction of $\Xi\Xi_{c}^{*}$ and $\Xi^{*}\Xi_{c}^{*}$ is much stronger than that of $\Lambda\Omega_{c}^{*}$, which implies that it is more possible for $\Xi\Xi_{c}^{*}$ and $\Xi^{*}\Xi_{c}^{*}$ to form bound states or resonance states. However, compared to $S=-1$, the attraction is much weaker.

For $S=-5$ system, see Fig.4, there are only two channels in this system, one of which is $\Omega\Omega_{c}$, a purely repulsive state; and the other is $\Omega\Omega_{c}^{*}$, which is weakly attractive. Therefore, it is difficult for these channels to form any bound state. However, we still need to confirm the existence of bound states or resonance states by performing the dynamic calculations.

In order to see whether there is any bound state, a dynamic calculation based on RGM RGM2 has been performed. The energies of each channel as well as the one with channel coupling calculation are listed in Table 3, Table 4 and Table 5. The first column is the state of every channel; the second column $E_{th}$ denotes the theoretical threshold of each corresponding state; the third column $E_{sc}$ represents the energy of every single channel; the fourth column $B_{sc}$ stands for the binding energy of every single channel, which is $B_{sc}=E_{sc}-E_{th}$; the fifth column $E_{cc}$ denotes the lowest energy of the system by channel coupling calculation; and the last column $B_{cc}$ represents the binding energy with all channels coupling, which is $B_{cc}=E_{cc}-E_{th}$. Here, we should notice that when the state is unbound, we label it as “ub”.

S=-1: The single channel calculation shows that the channels $\Sigma\Sigma_{c},\Sigma\Sigma_{c}^{*},\Sigma^{*}\Sigma_{c}$ and $\Sigma^{*}\Sigma_{c}^{*}$ are bound states with the binding energies -23 MeV, -25 MeV, -32 MeV and -25 MeV, respectively (see Table 3). This conclusion is consistent with the property that there is a strong effective attraction of these channels. However, for the $\Lambda\Lambda_{c},N\Xi_{c}$ and $N\Xi_{c}^{\prime}$ channels, which are unbound, the energies obtained by single channel calculations are above their corresponding thresholds due to the weak attraction of these channels. For the calculation of the channel coupling, the lowest energy is still above the lowest threshold ($\Lambda\Lambda_{c}$). Therefore, for this system, no bound states below the lowest threshold were found. For higher-energy single-channel bound states, they can be coupled to the open channels and the scattering process is needed to determine the existence of resonance states.

S=-3: From Table 4, the single channel calculation shows that all these nine channels are unbound. After the channel coupling calculation, the lowest energy of this system is 3790 MeV (still higher than the threshold of the lowest channel $\Xi\Xi_{c}$ ), which indicates that the singly charmed dibaryon system with $IJ=01,S=-3$ is unbound. This is reasonable. The attractions of the channels $\Xi\Xi_{c}^{*},\Xi^{*}\Xi_{c}^{*}$ and $\Lambda\Omega_{c}^{*}$ are not strong enough to form any bound state, and the interaction of the other channels are repulsive, as shown in Fig.3.

S=-5: The situation is similar to that of the $S=-3$ system. As shown in Table 5, both of the channels $\Omega\Omega_{c}$ and $\Omega\Omega_{c}^{*}$ are unbound. The lowest energy of the system is higher than the threshold of the $\Omega\Omega_{c}$ by the channel coupling calculation. So the system with $S=-5$ is unbound.

As mentioned above, some channels are bound due to the strong attractions of the system. However, these states will decay to the corresponding open channels by coupling with them and become resonance states. Besides, some states will become scattering state by the effect of coupling to both the open and closed channels. To further check the existence of the resonance states, we studied the scattering phase shifts of all possible open channels. Since no resonance states are obtained in the $S=-3$ and $S=-5$ systems, we only show the scattering phase shifts of the $S=-1$ system here.

In the $S=-1$ system, four singly bound states are obtained, which are $\Sigma\Sigma_{c}$, $\Sigma\Sigma_{c}^{*}$, $\Sigma^{*}\Sigma_{c}$ and $\Sigma^{*}\Sigma_{c}^{*}$, and there are three open channels, which are $\Lambda\Lambda_{c},N\Xi_{c}$ and $N\Xi_{c}^{\prime}$. We analyze two types of channel coupling in this work. The first is the two-channel coupling with a singly bound state and a related open channel, while the other is the five-channel coupling with four bound states and a corresponding open channel. The general features of the calculated results are as follows.

Here, we should note that the horizontal axis $E_{c.m.}$ in Fig.5 is the incident energy without the theoretical threshold of the corresponding open channel. So the resonance mass $M^{\prime}$ is obtained by adding $E_{c.m.}$ and the theoretical threshold of the corresponding open channel. In order to minimize the theoretical errors and compare our predictions with future experimental data, we shift the resonance mass by $M=M^{\prime}-E_{th}+E_{exp}$, where $E_{th}$ and $E_{exp}$ are the theoretical and experimental thresholds of the resonance state, respectively. Taking the resonance state $\Lambda\Lambda_{c}$ in the $\Sigma\Sigma_{c}$ channel as an example, the resonance mass shown in Fig.5(a) is $M^{\prime}=3597$ MeV, the theoretical threshold is $M_{th}=3595$ MeV, and the experimental threshold is $M_{exp}=3618$ MeV. Then the final resonance mass $M=3597-3595+3618=3620$ MeV. The estimated masses and widths of the resonances in different channels are listed in Table6, where $M$ is the resonance mass, $\Gamma_{i}$ is the partial decay width of the resonance state decaying to different open channels, and $\Gamma_{total}$ is the total decay width of the resonance state.

For the case of the two-channel coupling, in $\Lambda\Lambda_{c}$ scattering process, it is obvious that $\Sigma\Sigma_{c}$ and $\Sigma^{*}\Sigma_{c}^{*}$ appear as resonance states, as shown in Fig.5(a) and Fig.5(d), respectively. The resonance mass and decay width of every resonance state are obtained from the $\Lambda\Lambda_{c}$ scattering phase shifts. At the same time, $\Sigma\Sigma_{c}^{*}$ and $\Sigma^{*}\Sigma_{c}$ do not behave as resonance states in $\Lambda\Lambda_{c}$ scattering process, as shown in Fig.5(b) and Fig.5(c), respectively. There may be two reasons: the one is that stronger coupling between the two channels causes the bound state to be pushed above the threshold and become a scattering state; the other one is that the coupling between the two channels is so weak that the resonance state does not manifest during the scattering process. To clarify this issue, we calculate the cross matrix elements between the two channels ($\Lambda\Lambda_{c}$ and $\Sigma\Sigma_{c}^{*}$/$\Sigma^{*}\Sigma_{c}$), they are all close to zero, which means that the coupling between $\Lambda\Lambda_{c}$ and $\Sigma\Sigma_{c}^{*}$/$\Sigma^{*}\Sigma_{c}$ is very weak. Therefore, neither $\Sigma\Sigma_{c}^{*}$ nor $\Sigma^{*}\Sigma_{c}$ behaves as a resonance state in the $\Lambda\Lambda_{c}$ scattering phase shifts.

However, in the $N\Xi_{c}$ scattering process, the situation reversed. From Fig.6, both $\Sigma\Sigma_{c}^{*}$ and $\Sigma^{*}\Sigma_{c}$ appear as resonance states, while the other two channels $\Sigma\Sigma_{c}$ and $\Sigma^{*}\Sigma_{c}^{*}$ do not. The cross matrix elements between $N\Xi_{c}$ and $\Sigma\Sigma_{c}$/$\Sigma^{*}\Sigma_{c}^{*}$ show that the coupling between them is very weak, which results in the absence of resonance state $\Sigma\Sigma_{c}$/$\Sigma^{*}\Sigma_{c}^{*}$ in the $N\Xi_{c}$ scattering phase shift. In the $N\Xi_{c}^{\prime}$ scattering process, as shown in Fig.7, the conclusion is similar to the one in the $N\Xi_{c}$ scattering process. All the resonance masses and decay width are shown in Table6.

For the case of five-channel coupling, the scattering phase shifts are shown in Fig.8, and the resonance masses and decay widths are listed in Table6. There is only one resonance state $\Sigma\Sigma_{c}$ appears in the $\Lambda\Lambda_{c}$ phase shifts, as shown in Fig.8(a). From Table6, the resonance mass of the $\Sigma\Sigma_{c}$ in the five-channel coupling case is 3591 MeV, which is lower than the one in the two-channel coupling case (3620 MeV). This is because the coupling between closed channels will push down the channels with lower energy. At the same time, the channel coupling can also raises the energy of the higher state, even pushes the higher state above the threshold. Therefore, the resonance state $\Sigma^{*}\Sigma_{c}^{*}$ in two-channel coupling disappears in the five-channel coupling. Similarly, there is only one resonance state $\Sigma\Sigma_{c}^{*}$ appears in the $N\Xi_{c}$ phase shifts, which is shown in Fig.8(b). In the $N\Xi_{c}^{\prime}$ phase shifts, the situation is slightly different. Two resonance states $\Sigma\Sigma_{c}^{*}$ and $\Sigma^{*}\Sigma_{c}$ are shown in Fig.8(c). By comparing with the results in the two-channel coupling, the resonance mass of $\Sigma\Sigma_{c}^{*}$ is 30 MeV lower, while the one of $\Sigma^{*}\Sigma_{c}$ is 9 MeV higher. However, since the resonance $\Sigma^{*}\Sigma_{c}$ disappears in the $N\Xi_{c}$ scattering phase shifts, it will decay through the $N\Xi_{c}$ open channel. So the $\Sigma^{*}\Sigma_{c}$ cannot be identified as a resonance state. All these results show that the existence of the resonance states and the resonance energy are both affected by the multi-channel coupling. So the effect of the channel coupling cannot be ignored in the multi-quark system.