The $\Omega_{cc}$ resonances with negative parity in the chiral constituent quark model

Spectrum of the low-lying $\Omega_{cc}$ resonances with negative parity, which are assumed to be dominated by $sccq\bar{q}$ pentaquark components, is investigated using the chiral constituent quark model. Energies of the $\Omega_{cc}$ resonances are obtained by considering the hyperfine interaction between quarks by exchanging Goldstone boson. Possible $sccq\bar{q}$ configurations with spin-parity $1/2^{-}$, $3/2^{-}$ and $5/2^{-}$ are taken into account. Numerical results show that the lowest $\Omega_{cc}$ resonances with negative parity may lie at $4050 \pm 100$ MeV. In addition, the transitions of the $\Omega_{cc}$ resonance to a pseudoscalar meson and a ground baryon state are also investigated within the chiral Lagrangian approach. We expect that these $\Omega_{cc}$ resonances could be observed in the $\bar{D}\Xi_{c}$ channel by future experiments.

The observation of the Ξ cc states have brought new opportunities for us to study the doubly charmed baryons, since this finding suggests the potential of discovering more low-lying doubly charmed baryons in the near future, and thus one needs to have a solid theoretical calculations for the corresponding spectrum. In the present work, based on the chiral constituent quark model, we study the spectrum of the low-lying Ω cc resonances with negative parity. And the transitions of these Ω cc states to a pseudoscalar meson and a ground baryon state (M B) are also studied, employing the chiral Lagrangian approach which has been explicitly developed to study the strong decays of the N ss nucleon resonances as in Ref. [49].
The present manuscript is organized as following: in section II, we briefly present our theoretical formalism which includes the Hamiltonian and wave functions for the Ω cc pentaquark system, and the chiral Lagrangian approach for strong decays of a five-quark system, we show our explicit numerical results in section III, and section IV contains summary and conclusions.

II. THEORETICAL FRAME
We will briefly introduce the Hamiltonian and wave functions for the Ω cc resonances with negative parity as pentaquark states in Sec. II A, and the chiral Lagrangian approach for strong decays of the Ω cc states in Sec. II B.

A. Hamiltonian and wave functions
In present work, the constituent quark model is employed to study the spectrum of Ω cc resonances, within which the Hamiltonian for a five-quark system can be written as where H ij hyp represents the hyperfine interaction between the ith and jth quarks in the five-quark system, m i is the constituent mass of the ith quark, and H o is the Hamiltonian concerning orbital motions of the quarks, which should contain the kinetic term, the confinement potential of the quarks, and the flavor symmetry breaking term.
In general, the corresponding eigenvalue (1) should depend on the constituent masses of quarks and the model parameters in the quark confinement model, for instance, the confinement strength C and constant V 0 in the harmonic oscillator potential model [53]. In this work, we study the low-lying Ω cc resonances with negative parity as pentaquark states, which require all the quarks and antiquark to be in their ground states, accordingly, the eigenvalue E 0 should be the same one for different pentaquark configurations.
The parameter E 0 has been taken to be 2127 MeV for investigations on the intrinsic sea flavor content of nucleon in Ref. [55], with which value the data for light sea quark asymmetryd −ū in the proton can be well reproduced, while to fit the experimental data about Ω 0 c resonances, we took E 0 = 3132 MeV in Ref [13]. As discussed in details in Ref. [13], the resulted different values of E 0 by fitting the experimental data should be consistent with the chiral constituent quark model if all the model parameters are taken to be the empirical values.
In this work, the value of E 0 should be ∼ 1140 MeV higher than the one we took in Ref. [13], because of the different quark content in Ω cc and Ω c sates, while the SU (4) flavor symmetry breaking effects caused by two charm quarks in present case will lower E 0 by ∼ 170 MeV than those caused by one charm quark as in Ref. [13], if the Hamiltonian for symmetry breaking correction is taken to be the form similar as in Ref. [38]. Consequently, hereafter we will take E 0 = 4102 MeV, based on the investigations on the intrinsic sea content of nucleon and spectrum of low-lying Ω 0 c resonances, and the requirements of the chiral constituent quark model. Neverthe-less, we will investigate the dependency of the results on E 0 .
The hyperfine interaction between quarks is taken to be mediated by goldstone boson exchange and the corresponding H ij hyp is taken as following where V M (r ij ) denotes the coupling strength for a meson M exchanged between the ith and jth quarks. In this work, the π, K, η, D, D s and η c mesons are taken into account. For a five-quark system with the quark flavor as Ω cc resonances, namely, the sccqq system, a general wave function can be written as where [F i ] b,Y,Tz , [S i ] c,Sz and [211; C] a are the flavor, spin and color wave functions of the four-quark subsystem denoted by Young tableaux, the label i enumerates different pentaquark configurations. The ξ j is the Jacobi coordinates for a five quark system, which is defined as According to the SU (2) symmetry, the spin wave function of a four-quark system may be [4] S , [31] S or [22] S , the corresponding spin quantum numbers are 2, 1 and 0, respectively. While the coupling between spin of the fourquark subsystem and the antiquark leads to J = 1/2, 3/2 or 5/2. Given that all the quarks and antiquark are in their ground states, namely, the orbital wave function of the four-quark system is [4] respectively. One should note that the different spin symmetries of the four-quark subsystem result in vanishing coupling between different five-quark configurations, this is another reason for us to categorize the states in three groups by the four-quark spin wave functions.

B. The chiral Lagrangian approach
We consider the decays sccqq → M B, which mainly proceed through the process of qq → M , where the final baryon and meson are assumed to be composed of threequark and a quark-antiquark pair, respectively. We name this kind of decays as the annihilation transitions. The sccqq toKΞ cc and DΞ cc transitions are shown in Fig. 1.
To compute the transitions of the Ω cc →KΞ cc and Ω cc → DΞ c shown in Fig. 1, we use the chiral Lagrangian approach. Within this approach, the quark pseudoscalar (P) and vector (V) meson couplings are respectively, where the summation on j runs over the quark in the initial hadron. ψ j represents the quark field, and φ m and φ µ m are the pseudoscalar and vector meson fields. m j is the constituent mass of the jth quark, while k ν M denotes the four-momentum of the vector meson. a and b are the vector and tensor coupling constants, respectively.
In the non-relativistic approximation, Eqs. (8) and (9) lead to the operators for process involving q → q ′ M transitions as following where ω M and k M are the energy and three-momentum of the final meson. E i(f ) , M i(f ) and P i(f ) are the energy, mass and three-momentum of the initial (final) baryon, while p j , r j and m j are the momentum, coordinate and constituent mass of the quark which emits a meson. The µ q is the reduced mass of the jth quark before and after emitting the meson. For the vector meson emission, in Eqs. (11) and (12), the transition operators are denoted as T V qq d,T and T V qq d,L for the meson being transversely and longitudinally polarized, respectively. The b ′ in Eq. (11) is defined by b ′ = b − a. M V is the mass of the vector meson, and the polarization vectors of the final vector meson are taken to be with and E V is the energy of the final vector meson.
Finally, X j P and X j V are the operators in flavor space for a pseudoscalar and vector meson emission, which only depends on the quark-antiquark content of the emitted meson. For instance, X j P for a light pseudoscalar meson q s q emission in Eq. (10) can be defined as with λ j i and I the Gell-Mann matrix and unit matrix in flavor space. θ denotes the mixing angle for the mixing between η 1 and η 8 , leading to the physical states η and where the empirical value for the mixing angle is θ = −23°. The flavor operators for other pseudoscalar mesons or the vector mesons can be obtained straightforward.
Accordingly, the transition operators for a pseudoscalar meson emission T P qq a , a transversely polarized vector meson emission T V qq a,T and a longitudinally polarized vector meson emission can be obtained as where m j and mq are the constituent masses of the jth quark and the antiquark, respectively. C j XF SC denotes the operator to calculate the orbital, flavor, spin and color overlap factor between the residual wave function of the pentaquark configuration after the quark-antiquark annihilation and the wave function of the final baryon. χ † z I 2 χ j z is the spin operator for the quark-antiquark annihilation.

III. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we present our theoretical results for the mass spectrum of the low-lying sccqq states with J P = 1/2 − , 3/2 − and 5/2 − , and the decay behaviours of the obtained Ω cc pentaquark states.
A. The mass spectrum of the low-lying sccqq states With the Hamiltonian in Eq. (1) and wave function in Eq. (3), one can obtain the following nonzero H ij = i|H|j matrix elements between the pentaquark configurations in Eq. (5), and between the pentaquark configurations in Eq. (6), and between the pentaquark configurations in Eq. (7). In the above equations, C M are the corresponding matrix elements of the hyperfine interaction coupling strength V M (r ij ) between the S-wave orbital wave functions of the quarks in sccqq system, namely C cc is obtained from the last term in Eq. (2), and it contains the exchanges of the uū, dd, ss and cc pairs. The coupling strength constants C M are taken to be the empirical values [53] as shown in Table I.
With the above values for the model parameters and the diagonalization of the matrices obtained by Eqs. (25)(26)(27), one can get the physical states which are shown in Fig. 2, while the explicit probability amplitudes are shown in Table II. For instances, Eq. (25) leads to the following energy matrix Then one can directly obtain the eigenvalues and eigenvectors of matrix in Eq. (29). The three obtained eigenvalues are the energies of the physical states |i ′ with i = 1, 2, 3, respectively, and a obtained eigenvector just show the coefficients for the decoupling of a corresponding physical state |i ′ to the configurations |i listed in Eq. (5). The energies for the obtained Ω cc states in present work are at 4050 ± 100 MeV. Similar as the results for Ω 0 c obtained in Ref. [13], mixing between the pentaquark configurations |i caused by the goldstone boson exchange is strong, while the mass splitting for the obtained states |i ′ is not very large. On the other hand, the spectrum of the ten obtained states is not sensitive to the values of the coupling strength C M .
Up to now, there are no solid experimental data for the Ω cc resonances, while theoretical investigations on the doubly heavy baryon resonances have been intensively taken using various of approaches, such as the constituent quark model [9,10,[56][57][58][59][60], QCD sum rules [17], chiral perturbation theory [20][21][22], the unitarized coupled channel approach [61], and the lattice QCD calculations [25], etc. The corresponding obtained energies for the P -wave Ω cc in a three-quark picture are around 4000 − 4200 MeV in most of the literatures, and one may note that in Ref. [61], the S-wave interactions between pseudo-Nambu-Goldstone bosons (π, K and η) and the II: The ten physical pentaquark states obtained in present model, line three shows the energies for the states |i ′ (in MeV), and lines four to ten are the corresponding probability amplitudes. 3942  4053  4054  3979  4024  4069  4083  4146  4092  J P = 1/2 + ground state doubly charmed baryons in the energy region around the corresponding thresholds are investigated, two quasistable narrow J P = 1/2 − Ω cc are predicted to lie at the energy below 4200 MeV, and their strong decay mode is predicted to be only the Ω cc π 0 , which is isospin breaking channel. Therefore, the two obtained Ω cc resonances in a meson-baryon picture should be very narrow. One may also study the spectrum of low-lying Ω cc resonances with negative parity using the chiral constituent quark model in a three-quark picture [54]. As P -wave states whose parity are negative, there are three possible Ω cc configurations: whose spin-parity quantum number J P may be 1/2 − or 3/2 − for the first two configurations, and 1/2 − , 3/2 − or 5/2 − for the last one. Direct calculations employing the chiral constituent quark model as in Ref. [54] lead to the following values for the energies of the three Ω cc states, respectively. Consequently, the energies of low-lying Ω cc states in the five-quark picture are lower than those in the three-quark picture, this conclusion is the same as that for the Ω * resonances [37].
In Ref. [56], a relativistic quark model was applied to study the spectrum of doubly heavy baryons. Considering the Ω cc resonances to be dominated by threequark components, it was obtained that the low-lying Ω cc resonances with negative parity fall in the range of 4200 − 4300 MeV, which are consistent with the results obtained in Ref. [10] by employing a three-quark model. While in Ref. [9], a three-quark model was employed to investigate the spectrum of the doubly heavy baryons, in which model the two heavy quarks were treated as a diquark, and the resulting energies of the low-lying Ω cc were in the range of 4050 − 4150 MeV. Those results are about 100 MeV lower than the present rough estimation using a three-quark model, and the results in [10,56]. So one may expect that the diquark assumption for the two heavy quark in Ω cc resonances may reduce the energies.
In any case, we can conclude that the Ω cc resonances should lie at a energy below 4200 MeV in both the compact five-quark model (present) and the meson-baryon model [61].
Finally, we show the dependency of presently obtained spectrum on the model parameter E 0 . By taking E 0 = 3132 MeV as given in Ref. [13], one can get Obviously, the obtained energies are much lower than those predicted by using other approaches. Namely, the value E 0 = 4102 MeV employed in our calculations should be reasonable. We also present the numerical results with E 0 changed by 2%, then In fact, change of E 0 should lead to almost the same change for energy of each physical state |i ′ . In addition, the coefficients for decompositions of the physical states |i ′ are not sensitive to E 0 .
B. S-wave coupling of the sccqq to pseudoscalar meson and ground baryon states From Fig. 2 and Table II, one can find that most of the obtained physical sccqq states are above the threshold of the SU (2) isospin breaking πΩ cc channel, but, below the thresholds of the other pseudoscalar meson and ground state baryons channels. It is expected that the decay widths of the presently obtained Ω cc should not be very large. This is in consistent with these findings in Ref. [61]. Therefore, in present work, we only try to estimate the S-wave transitions of the obtained 1/2 − and 3/2 − Ω cc resonances to M B channels.
Using the transition operator in Eq. (22), and the wave functions obtained in Sec. III A, one can calculate the transition matrix elements of the obtained Ω cc resonances toKΞ cc ,KΞ * cc , DΞ c , and DΞ * c channels, respectively. It is found that all the transition amplitudes of Ω cc → M B processes share a common factor which involves the overlap between the orbital wave functions of the pentaquark configurations and the final meson-baryon, namely, z I 2 χ j z X j P in Eq. (22). In the three quark model, for Ξ c baryon, the light and strange quarks in its flavor wave function can be either symmetric or antisymmetric. We denote the former sate as Ξ c , while the latter one as Ξ ′ c . Therefore, we consider the transition processes of Ω cc with spin-parity 1/2 − tō KΞ cc , DΞ c , and DΞ ′ c channels, and the Ω * cc with spinparity 3/2 − toKΞ * cc and DΞ * c channels. Then, straightforward calculations on the transition matrix elements for the operator C j XF SCχ † z I 2 χ j z X j P in the processes of |i in Eqs. (5)(6) to the above mentioned M B channels lead to the results shown in Table III, where the first three rows of the numerical results are the orbitalflavor-spin-color overlap factors for the configurations |i with spin-parity quantum number 1/2 − to M B channels, while the last two rows are those for 3/2 − configurations.
Considering the probability amplitudes for the mixing of configurations |i presented in Table II, we obtain the corresponding overlap factors for the physical Ω cc resonances listed in Table IV. One may note that there are some zeros obtained for some configurations as shown in Table III, however, they become finite for the physical states. For instance, the overlap factor for the configuration |8 to the channelKΞ cc is 0, while that for |8 ′ →KΞ cc is −0.635. This is because of that the physical state |8 ′ decouples to the configurations |i as which is shown in Table II.
Compared to the overlap factor [49,50] for the strangeness five-quark configurations |uudss to ηp channels that is about ∼ 0.75, which may account for the strong coupling between S 11 (1535) and strangeness channels, one can expect that the presently obtained physical states |i ′ may couple strongly to the M B channels for which the overlap factors shown in Table IV are larger than 0.8.
It should be very interesting to compare the decay behaviours of the presently obtained Ω cc resonances with those in a three-quark model. In Ref. [9], five Ω cc resonances lying at 4208 − 4303 MeV were obtained, and the decay widths of these resonances toKΞ cc orKΞ * cc channels were estimated explicitly. It was found that some of the obtained decay widths should be larger than 100 MeV. This is very different from the conclusion that most of the obtained Ω cc resonances using a pentaquark picture can only decay to the isospin breaking channel πΩ cc .
In a three-quark picture, one can also estimate the flavor-spin-color overlap factor of the Ω cc resonances and the M B channels using Eq. (10). For instance, a straightforward calculation on the overlap factors of the threequark states given in Eq. (30), shows that coupling for Ω cc →KΞ cc may be comparable to that for Ω cc → DΞ c , since the obtained flavor-spin-color overlap factor for a given three-quark Ω cc resonance to theKΞ cc channel is √ 2 times of that for the DΞ c channel, this is determined by the flavor-spin structure of the Ω cc resonances and the effective chiral Lagrangian. However, as we can see in Table IV, the presently obtained numerical results for several states are very different from the three-quark results.

IV. SUMMARY
In present work, we investigate the spectrum of lowlying Ω cc resonances with negative parity as pentaquark states, using the chiral constituent quark model within a five-quark picture. We obtain ten pentaquark states III: Color-flavor-spin factors for the transitions J P = 1/2 − (shown in the 2 nd to 4 th rows) and 3/2 − (shown in the last two rows) sccqq configurations |i to M B channels.

|1
|2   with spin-parity J P = 1/2 − , 3/2 − , 5/2 − , which lie at 4050 ± 100 MeV. Most of the obtained states are above the isospin breaking decay channel πΩ cc , but below the other meson-baryon channels. So we just try to calculate the flavor, spin, orbital, and color overlap factor for the final M B states and the residual three-quark-meson configurations of the Ω cc states after the annihilation of the quark-antiquark qq → M . It is found that several ones of the presently obtained Ω cc may couple strongly to DΞ c orKΞ cc channels. One may expect that these calculations here could be compared with the future experimental measurements which are likely to be done by Belle II and/or LHCb.

Acknowledgments
We thank Yun-Xia Lang for her contributions at the very beginning of present work. This work is partly