Masses and strong decays of open charm hexaquark states $\Sigma_{c}^{(\ast)}{\Sigma}_{c}^{(\ast)}$

Inspired by the recent discovery of the doubly charmed tetraquark state $T_{cc}^{+}$ by the LHCb Collaboration, we perform a systematic study of masses and strong decays of open charm hexaquark states ${\Sigma}_{c}^{(\ast)}\Sigma_{c}^{(\ast)}$. Taking into account heavy quark spin symmetry breaking, we predict several bound states of isospin $I=0$, $I=1$, and $I=2$ in the one boson exchange model. Moreover, we adopt the effective Lagrangian approach to estimate the decay widths of ${\Sigma}_{c}^{(\ast)}\Sigma_{c}^{(\ast)} \to \Lambda_{c}\Lambda_{c}$ and their relevant ratios via the triangle diagram mechanism, which range from a few MeV to a few tens of MeV. We strongly recommend future experimental searches for the ${\Sigma}_{c}^{(\ast)}\Sigma_{c}^{(\ast)}$ hexaquark states in the $\Lambda_c\Lambda_c$ invariant mass distributions.


I. INTRODUCTION
The quenched (or conventional) quark model can well describe the properties of traditional hadrons, especially the ground-state ones [1,2]. However, a large number of states that can not be easily explained by the quenched quark model have been accumulating since D * s0 (2317) and X(3872) were discovered by the BaBar [3] and Belle [4] collaborations in 2003. To unveil the nature of these exotic states, many pictures, such as multiquark states, hadronic molecules, and kinetic effects, have been proposed to understand them from different perspectives, including production mechanisms, mass spectra, and decay widths [5][6][7][8][9][10][11][12]. Among them, the hadronic molecular picture is probably the most popular one since most of these exotic states are located close to the mass thresholds of conventional hadron pairs. Two-body hadronic molecules are expected to exist in three configurations, i.e., meson-meson, meson-baryon, and baryon-baryon. With more and more XY Z states, P c states, and T cc discovered in the past two decades [11], heavy hadronic molecules have attracted a lot attention.
Experimental searches for dibaryons containing charm quarks is challenging because of the high energy required and the low production of charmed baryons. On the other hand, many theoretical studies have been performed. In Ref. [73], Liu et al. predicted one bound state generated by Λ c N and Σ c N couple channels in the OBE model, which was later explored in the chiral constituent quark model [74,75], lattice QCD simulations [76], and effective field theories [77]. Later, using the same approach, Li et al. systematically studied possible bound states made of two singly charmed baryons [17,78,79]. Some of these states were investigated in the quark delocalization color screening model [80,81]. After the discovery of the doubly charmed baryon Ξ cc by the LHCb Collaboration [82], Meng et al. investigated bound states made of two doubly charmed baryons [83]. Inspired by the discovery of the pentaquark states by the LHCb Collaboration [84], c system utilizing heavy antiquark diquark symmetry in the OBE model and effective field theory [85,86] and obtained a complete HQSS multiplet of hadronic molecules.
The manuscript is structured as follows. In Sec. II we present the details of the OBE model as applied to the Σ ( * ) c Σ ( * ) c system and the numerical results of their mass spectra. In Sec. III we explain the effective Lagrangian approach and calculate the decay widths of Σ Finally we conclude in Sec. IV In this work, we employ the OBE model to derive the Σ c interaction, which has been successfully used to investigate hadronic molecules composed of heavy hadrons [93][94][95][96]. The OBE interaction of two heavy hadrons is generated by the exchange of light mesons, π, σ, ρ, and ω. The vector mesons, ρ and ω, provide the short-range interaction, the scalar meson σ provides the medium-range interaction, and the π meson provides the long-range interaction. Given the exploratory nature of the present work, the contributions of other mesons are neglected [97]. To derive the OBE potential, the effective Lagrangians describing the interaction between a charmed baryon and a light meson are necessary. The relevant Lagrangians are the same as those of thē c interaction [97], which read where S c = ( 1 √ 3 Σ c σ + Σ * c ) denotes the superfield of Σ c and Σ * c dictated by HQSS. The coupling of the π meson to Σ c and Σ * c , g, was determined to be 0.84 in lattice QCD [98], which is smaller than the prediction of the quark model [73]. In this work we take g = 0.84. For the coupling to the σ meson, we estimate it using the quark model. From the nucleon and σ meson coupling of the liner sigma model, g σN N = 10.2, the corresponding coupling is determined to be g σ = 2 3 g σN N = 6.8 [99]. The couplings of the light vector mesons contain both electric-type(g v ) and magnetic-type(f v ) ones, which are related via f v = κ v g v . For the ρ and ω couplings, from SU(3)symmetry and the OZI rule, we obtain g ω = g ρ and f ω = f ρ . The electric-type coupling of the singly charmed baryons to the vector mesons is determined to be g v = 5.8 [73] as well as the corresponding magnetic-type coupling is estimated to be κ v = 1.7 [100]. All the couplings are given in Table I for easy reference.
With the above Lagrangians, the OBE potentials for the Σ c system in coordinate space can be obtained, which read where T 1 · T 2 is the isospin factor for the Σ ( * ) c Σ ( * ) c system, and a 1 · a 2 and S 12 (r) denote the spin-spin and tensor terms, respectively, and a denotes the spin operator of the Σ Since the charmed baryons and light mesons involved in our study are not point-like particles, we introduce a form factor in the interaction vertices. Here we use a monopolar form factor (for more details we refer to Refs. [45,97]) where m E and q are the mass and 4-momentum of the exchanged meson. Λ is an unknown cutoff, which is often fixed by reproducing some hadronic molecular candidates that can be related to hadronic molecules of interest via symmetries. Assuming X(3872) and P c (4312) as bound states ofDD * andDΣ c , we can determine the corresponding cutoff of the OBE model to be 1.04 GeV [45] and 1.12 GeV [97], respectively. In addition, the cutoff needed to reproduce the binding energy of the deuteron is 0.86 GeV [101]. All these results indicate that a reasonable value for the cutoff of the OBE model is about 1 GeV. In this work, assuming that the Σ c system is similar to the nucleon-nucleon system, we take a cutoff of Λ =0.86 GeV to calculate the binding energies of the Σ ( * ) c Σ ( * ) c dibaryon system, With the above form factor, the functions δ, W Y , and W T in Eqs. (5)(6)(7)(8) need to be changed with λ = Λ/m. The corresponding functions d, W Y , and W T read The generic wave function of a baryon-baryon system reads where |IM I denotes the isospin wave function and Ψ JM ( r) are the spin and spatial wave functions. The dynamics in isospin space is embodied in the isospin factor T 1 · T 2 . In this work the total isospin is either 0, 1, or 2 for the Σ c system, then the corresponding isospin factor are −2, −1, and 1, respectively.
The spin wave function can be written as a sum over partial wave functions, which reads (in the spectroscopic notation) where LM L SM S |JM are the Clebsch-Gordan coefficients, |SM S the spin wavefunction, and Y LM L (r) the spherical harmonics.
In the partial wave decomposition of the OBE potential, we encounter both spin-spin and tensor components In the present study, we consider both S and D waves. The relevant matrix elements are listed in Table IV.

B. Numerical results and discussions
For a pair of identical fermions, the total spin and isospin of Σ c have to fulfill the condition, I + S=even. The possible combinations of spin and isospin are shown in Table II. Additionally, when deriving the OBE potential, we have assumed HQSS. As is well known, the charm quark mass is not large enough to strictly satisfy HQSS. In the present work, we take into account a HQSS breaking of the order of ∆ = 15% in the following way [85]: With the above OBE potentials, the binding energies of the Σ c system for isospin I = 0, 1, and 2, which are given in Table II. After taking into account the HQSS breaking, we find four hadronic molecules with I = 0, i.e., J P = 0 + Σ c Σ c ,

III. STRONG DECAYS OF Σ
c molecules decaying into Λ c Λ c by exchanging a π meson.

A. Effective Lagrangian approach
In this section, we further investigate the strong decays of Σ where p E is the Euclidean momentum and Λ is the size parameter.
To estimate the couplings between bound states and their constituents, we employ the compositeness condition [107][108][109]. For dibaryon bound states with total angular momentum J = 0 the compositeness condition reads where Σ(m 2 ΣcΣc ) is the self energy of a Σ ( * ) c bound states as shown in Fig. 2. The self-energy for a vector state is expressed as Σ µν (p 2 ) with Lorentz indices µ and ν, which can be decomposed into two parts, longitudinal and transverse, Σ µν (p 2 ) =ĝ µν Σ T (p 2 ) + p µ p ν p 2 Σ L (p 2 ), whereĝ µν = g µν − p µ p ν /p 2 . Substituting the transverse term into Eq (25), we can also determine the couplings of a J = 1 dibaryon state to their constituents.
The mass operators for the Σ ( * ) c Σ ( * ) c dibaryons read as follows: M µν In Refs. [110,111], the above cutoff is found to be around 1 GeV. One should note, however, that the couplings of a molecular state to its components are related to its binding energy [112], and hence we take a cutoff of Λ = 0.86 GeV, the same as that used to study the Σ ( * ) c Σ ( * ) c dibaryons, to determine the couplings in this work. In Table III components. Because the binding energy of H * * cc is larger than those of H * cc and H cc , g H * * cc Σ * c Σ * c is larger than g H * cc Σ * c Σc and g HccΣcΣc as well. The other vertices of the triangle diagrams of Fig 1 can be classified into two categories, Σ c → Λ c π and Σ * c → Λ c π. The Lagrangians describing these interactions are given by where f π = 132 MeV and the couplings g πΛcΣc and g πΛcΣ * c can be determined by fitting to experimental data. From the decay widths of Γ(Σ c → Λ c π) = 1.89 MeV and Γ(Σ * c → Λ c π) = 15.0 MeV [113], we obtain the couplings g πΛcΣc = 0.55 and g πΛcΣ * c = 0.97, consistent with other works [73,114]. In addition, we find that the two couplings approximately satisfy the relationship g πΛcΣ * c = √ 3g πΛcΣc , given by the quark model [73].
With the above Lagrangians, the amplitudes of H ( * * ) cc → Λ c Λ c can be easily written down whereū Λc andū T Λc represent the spinors of the final-state Λ c Λ c pair, and F (q, m E , Λ) is the form factor which not only removes ultraviolet divergence of the loop diagram, but also takes into account the off-shell effects. The m E is the mass of the exchanged particle. The cutoff is expressed by where Λ QCD is around 200-300 MeV and the dimensionless parameter α is around 1 [115]. Thus we vary the cutoff from 0.4 GeV to 0.6 GeV to estimate the uncertainties induced. With the amplitudes of H ( * * ) cc → Λ c Λ c determined, one can obtain the corresponding partial decay widths as where J is the total angular momentum of the H their masses in Fig. 4, where the ranges of masses are obtained from the upper and lower limits of binding energies of Table II, and the cutoff is taken to be 0.5 GeV. One can see that the decay widths of H cc and H * cc only vary by several MeV, while the decay width of H * * cc varies by tens of MeV, which shows that the decay width of H ( * * ) cc molecules are not very sensitive to their masses.
All of the H ( * * ) cc molecules can decay into Λ c Λ c , which indicates that all of them could be detected in the Λ c Λ c mass distributions. In Fig .5 we show the ratios of the decay widths of the H * * cc and H * cc molecules to that of the H cc molecule, and the corresponding ratios are around 10 and 1, respectively, which tells that three peaks will appear in the Λ c Λ c invariant mass spectrum, two narrow structures and one rather wide structure. In addition, we find that these ratios are insensitive to the cutoff used.

IV. SUMMARY AND OUTLOOK
Inspired by the recent discovery of the doubly charmed tetraquark sate T cc , we performed a systematic study of the mass spectrum and strong decays of doubly charmed hexquark states c . We adopted the one-boson exchange model to calculate the binding energies of the Σ ( * ) c Σ ( * ) c system, where the cutoff is fixed by reproducing the binding energy of the deuteron. After considering breaking of HQSS, we found four bound states with isospin 0, i.e., J P = 0 + Σ c Σ c , J P = 1 + Σ * c Σ c , J P = 0 + Σ * c Σ * c , and J P = 2 + Σ * c Σ * c , four bound states with isospin 1, J P = 1 + Σ c Σ c , J P = 1 + Σ * c Σ c , J P = 2 + Σ * c Σ c , and J P = 1 + Σ * c Σ * c , and one bound state with isospin 2, J P = 2 + Σ * c Σ c . Among them, the J P = 0 + Σ c Σ c state could be regarded as the deuteron counterpart with double charm, which is much more bound than the deuteron. In addition, we      G H cc */G H cc G H cc **/G H cc