Doubly Charmed Pentaquarks

The LHCb Collaboration, using its full data set from runs $1$ and $2$, announced in $2019$ a surprising update of the hidden-charm pentaquark states $P_c(4380)^+$ and $P_c(4450)^+$, observed in 2015. A new state, $P_c(4312)^+$, was clearly seen at lower energies; furthermore, the original $P_c(4450)$ resonance was resolved into two individual states, named the $P_c(4440)^+$ and the $P_c(4457)^+$. Motivated by the fact that these new hidden-charm pentaquark states were successfully predicted by our chiral quark model, we extend herein such study to the doubly charmed sector. The analyzed total spin and parity quantum numbers are $J^P=\frac12^-$, $\frac32^-$ and $\frac52^-$, in the $I=\frac12$ and $\frac32$ isospin channels. We find several possible narrow baryon-meson resonances (theoretical masses in parenthesis): $IJ^P = \frac12 \frac12^-$ $\Sigma_c D(4356)$, $\frac12 \frac32^-$ $\Sigma^*_c D(4449)$, $\frac32 \frac12^-$ $\Sigma_c D(4431)$, $\frac32 \frac12^-$ $\Sigma_c D(4446)$, $\frac32 \frac32^-$ $\Sigma_c D^*(4514)$ and $\frac32 \frac52^-$ $\Xi^*_{cc} \rho(4461)$ whose widths are $4.8$, $8.0$, $2.6$, $2.2$, $4.0$ and $3.0\,\text{MeV}$, respectively. Moreover, one shallow bound-state is found, too, with quantum numbers $IJ^P = \frac12 \frac32^-$ $\Xi^*_{cc} \pi(3757)$. These doubly charmed pentaquark states are expected to be identified in future experiments.


I. INTRODUCTION
During the past 15 years, more than two dozens of nontraditional charmonium-and bottomonium-like states, the so-called XYZ mesons, have been observed at Bfactories (BaBar, Belle and CLEO), τ -charm facilities (CLEO-c and BESIII) and also proton-(anti)proton colliders (CDF, D0, LHCb, ATLAS and CMS). Among all of them, one can highlight the new three hiddencharm pentaquark candidates observed in 2019 by the LHCb Collaboration [1] in the J/ψp invariant mass spectrum of Λ 0 b → J/ψK − p decays, they are signed as P c (4312) + , P c (4440) + and P c (4457) + , respectively. The story of hidden-charm pentaquark states can actually be dated back to 2015, when two exotic signals: P c (4380) + and P c (4450) + , were announced by the same collaboration [2]. Two striking features characterized these states: they appear quite close to baryon-meson thresholds and all are very narrow; this is believed to be an invaluable information towards discriminating between different explanations on how the quarks are arranged within the pentaquarks.
It is important to highlight here that, before the LHCb's announcement of the three new hidden-charm pentaquark states, their existence were predicted by some of the present authors in Ref. [22] (see Tables III and IV). The P + c (4312), P + c (4440) and P + c (4457) were described as baryon-meson molecular states of the form J P = 1 2 − Σ cD , 1 2 − Σ cD * and 3 2 − Σ cD * , respectively; belonging all of them to the isospin I = 1 2 sector. Moreover, these results are supported by other theoretical studies such as the ones reported in Refs. [3,4,6,19].
Apart from the hidden-charm pentaquark states, there are also other pentaquark configurations triggering theoretical interest. One heavy antiquark pentaquarks, Qqqqq, are analyzed within a constituent quark model and no bound-state is found [23]. Doubly heavy pentaquarks are systematically studied in a phenomenological potential model with the conclusion that either stable states or narrow resonances are possible [24,25]. In Ref. [26], light pseudoscalar meson and doubly charmed baryon scattering lengths are calculated by means of the heavy baryon chiral perturbation theory. Possible triply charmed molecular pentaquarks such as Ξ cc D 1 (D 1 ) and Ξ cc D * 2 (D * 2 ) are proposed using a one-boson-exchange model in Ref. [27]; and the mass splittings for the Swave triply heavy pentaquark states are systematically calculated [28]. Meanwhile, some interesting reviews dis-cussing the pentaquark issue but also collecting information about, e.g., tetraquark states can be found in Refs. [29,30]; moreover, potential prospects on the production of multiquark systems containing heavy quarks with the ALICE experiment at LHC are discussed in Ref. [31].
Within a chiral quark model formalism 1 , we systematically study herein the possibility of having either bound or resonance states in the doubly charmed pentaquark sector with quantum numbers J P = 1 2 − , 3 2 − and 5 2 − , and in the I = 1 2 and 3 2 isospin sectors. This 5-body bound state problem is solved by means of the Gaußian expansion method (GEM) [41], which has been demonstrated to be as accurate as a Faddeev calculation (see Figs. 15 and 16 of Ref. [41]). Note, too, that the same approach has been applied in previous studies of P c [22] and P b states [42]. In this work, a powerful technique named complex scaling method (CSM) is employed in order to disentangle bound, resonance, and continuum (scattering) states within the same calculation. As illustration, Fig. 1 shows a schematic distribution of the complex energy 2-body states obtained by the CSM, according to Ref. [43]. As one can see, the resonance states can be computed as an equivalent bound-state problem without resorting to the Lippmann-Schwinger equation formalism. Up to our knowledge, this is the first time that the CSM is applied to study pentaquark systems. During the past decades, CSM has been extensively applied to nuclear physics problems [43,44], and recently to the study of charmed di-baryon resonances [45] and doubly-heavy tetraquarks [46].
The structure of the present work is organized in the following way. In Sec. II the ChQM, pentaquark wavefunctions, GEM and CSM are briefly presented and discussed. Section III is devoted to the analysis and discussion of our theoretical results. We summarize and give some prospects in Sec. IV.

II. THEORETICAL FRAMEWORK
Lattice-QCD (LQCD) has made in the last decade or so an impressive progress on understanding multiquark systems [47,48] and meson-meson, meson-baryon and baryon-baryon interactions [49][50][51]; however, QCDinspired quark models are still the main tool to shed some light on the nature of the multiquark candidates observed by experimentalists.
The general form of our five-body Hamiltonian, within the CSM approach, is given by where each quark is considered nonrelativistic, T CM is the center-of-mass kinetic energy and the two-body potential includes color-confining, one-gluon-exchange and Goldstone-boson-exchange interactions.
Herein, the coordinates of relative motions between quarks are transformed with a complex rotation, r → re iθ . Therefore, in the framework of complex range, the five-body systems are solved in a complex scaled Schrödinger equation: According to the so-called ABC theorem [52,53], there are three types of complex eigenenergies of Eq. (3), as shown in Fig. 1: • Bound states below threshold are always located on the energy's negative real axis.
• Discretized continuum states are aligned along the cut line with a rotated angle of 2θ, related to the real axis.
• Resonance states are fixed poles under the complex scaling transformation, and they are located above the continuum cut line. The resonance's width is given by Γ = −2 Im(E).
Coming back to the quark-(anti-)quark interacting potentials shown in Eq. (2). Color confinement should be encoded in the non-Abelian character of QCD. LQCD studies have demonstrated that multi-gluon exchanges produce an attractive linearly rising potential proportional to the distance between infinite-heavy quarks [54]. However, the spontaneous creation of light-quark pairs from the QCD vacuum may give rise at the same scale to a breakup of the color flux-tube [54]. We have tried to mimic these two phenomenological observations by the following expression, in complex scaling: where a c and µ c are model parameters, and the SU(3) color Gell-Mann matrices are denoted as λ c . One can see in Eq. (4) that the potential is linear at short interquark distances with an effective confinement strength σ = −a c µ c ( λ c i · λ c j ), while it becomes constant at large distances.
The one-gluon-exchange potential contains central, tensor and spin-orbit contributions. We consider only the central term but also with a complex transformation, r → re iθ : where m i is the quark mass and the Pauli matrices are denoted by σ. The contact term of the central potential has been regularized as with r 0 (µ ij ) =r 0 /µ ij a regulator that depends on µ ij , the reduced mass of the quark-(anti-)quark pair.
The wide energy range needed to provide a consistent description of mesons and baryons from light to heavy quark sectors requires an effective scale-dependent strong coupling constant. We use the frozen coupling constant of, for instance, Ref. [55] in which α 0 , µ 0 and Λ 0 are parameters of the model.
The central terms of the chiral quark-(anti-)quark in-teraction with CSM can be written as where Y (x) is the standard Yukawa function defined by Y (x) = e −x /x. We consider the physical η meson instead of the octet one and so we introduce the angle θ p . The λ a are the SU(3) flavor Gell-Mann matrices. Taken from their experimental values, m π , m K and m η are the masses of the SU(3) Goldstone bosons. The value of m σ is determined through the PCAC relation m 2 σ m 2 π + 4m 2 u,d [56]. Finally, the chiral coupling constant, g ch , is determined from the πN N coupling constant through which assumes that flavor SU(3) is an exact symmetry, only broken by the different mass of the strange quark.
As it is well known, the quark model parameters are crucial. In our case, the model parameters have been taken from, e.g., Ref. [22] and, for completeness, they are listed in Table I. Note that the same set of model parameters was used in Refs. [22] and [42] to study, respectively, possible hidden-charm and -bottom pentaquark boundand resonance-states.
There are four sets of baryon-meson configurations for ccqqq (q = u or d) systems 2 , and they are shown in Figs. 2 to 5. Moreover, the anti-symmetry property in these

Quark masses
28.170 identical fermion systems is necessary; however, due to the asymmetry between light and heavy quarks, the two charmed quarks can be coupled first within a 3-quark cluster as shown in Fig. 2. Therefore, the antisymmetry operator for the ccqqq pentaquark system is Figure 3 shows a different arrangement in the 3-quark cluster with two heavy quarks included. In this case, the antisymmetry operator is given by The cases in which the two charm quarks are separated in different clusters are also considered and shown in Figs. 4 and 5. When the two light quarks are coupled as in Fig. 4, the antisymmetry operator is whereas the last configuration, shown in Fig. 5, has the same antisymmetry operator of Eq. (14); this is to say The pentaquark wave function is a product of four terms: color, flavor, spin and space wave functions. Concerning the color degrees-of-freedom, multiquark systems  have richer structure than the conventional mesons and baryons. For instance, the 5-quark wave function must be colorless but the way of reaching this condition can be done through either a color-singlet or a hidden-color channel, or both. The authors of Refs. [57,58] assert that it is enough to consider the color singlet channel when all possible excited states of a system are included. However, a more economical way of computing is considering both; the color singlet wave function: where n=1-4 is a label for each quark configuration shown in Figs. 2 to 5, respectively (it is of the same meaning for spin, flavor and space wave functions). In other words, they have a common form but with different quark sequence: 123;45, 132;45, 352;41 and 253;41. When computing matrix elements, one should switch the last three cases into the first one. The hidden-color channel is given by: is an index which stands for the symmetric (anti-symmetric) configuration of two quarks in the 3-quark cluster. All color configurations have been used herein, as in the case of the P + c (P + b ) hidden-charm (-bottom) pentaquarks studied in Refs. [22,42].
According to the SU(2) symmetry in isospin space, the flavor wave functions for the clusters mentioned above are The two heavy quarks are divided into two clusters with the light-and heavy-quark coupled firstly.
given by: where the superscript of the flavor wave functions of 3quark clusters stand for the number of each pentaquark configuration. Consequently, the flavor wave-functions for the 5-quark system with isospin I = 1/2 or 3/2 are where the third component of the isospin is set to be equal to the total one without loss of generality, because there is no interaction term in the Hamiltonian that can distinguish such component.
We consider herein 5-quark systems with total spin ranging from 1/2 to 5/2. Our Hamiltonian does not have any spin-orbital coupling dependent potential, and thus we can assume that the spin wave function has its third component equal to the total one, without loss of generality: for S = 1/2, and for S = 3/2, and for S = 5/2. These expressions can be obtained easily using SU(2) algebra and considering the 3-quark and quark-antiquark clusters separately. They were derived in Ref. [22] for the hidden-charm pentaquarks. The complex Schrödinger-like 5-body bound state equation is solved using the Rayleigh-Ritz variational principle, which is one of the most extended tools to solve eigenvalue problems due to its simplicity and flexibility. However, it is of great importance how to choose the basis on which to expand the wave function. The spatial wave function of a 5-quark system is written as follows: Taking the first pentaquark configuration shown in Fig. 2 as an example 3 , the internal Jacobi coordinates are de-fined as This choice is convenient because, on one hand, the center-of-mass kinetic term T CM can be completely eliminated for a nonrelativistic system and, in the other hand, the spatial wave functions related with the relative motions between quarks can be also extended to the complex scaling.
In order to make the calculation tractable, even for complicated interactions, we replace the orbital wave functions, φ's in Eq. (47), by a superposition of infinitesimally-shifted Gaussians (ISG) [41]: where the limit ε → 0 must be carried out after the matrix elements have been calculated analytically. This new set of basis functions makes the calculation of 5body matrix elements easier without the laborious Racah algebra. Moreover, all the advantages of using Gaußians remain with the new basis functions. Finally, in order to fulfill the Pauli principle, the complete antisymmetry complex wave-function is written as where A n is the antisymmetry operator of the 5-quark system and their expressions are shown in Eqs. (13) to (16). This is needed because we have constructed an antisymmetric wave function for only two quarks of the 3-quark cluster, the remaining (anti-)quarks of the system have been added to the wave function by simply considering the appropriate Clebsch-Gordan coefficients.

III. RESULTS
In the present calculation, we investigate the possible lowest-lying and resonance states of the ccqqq (q = u or d) pentaquark systems by taking into account the (ccq)(qq), (cqc)(qq), (qqc)(qc) and (cqq)(qc) configurations in which the considered baryons have always positive parity and mesons are either pseudoscalars (J P = 0 − ) or vectors (1 − ). This means that, in our approach, a pentaquark state with negative parity has L = 0. In this case, we assume that the angular momenta l 1 , l 2 , l 3 and l 4 , appearing in Eq. (47), are all equal to zero. Accordingly, the total angular momentum, J, coincides with the total spin, S, and can take values 1/2, 3/2 and 5/2. The possible baryon-meson channels which are under consideration in the computation are listed in Tables II and III shows the baryon-meson channel in which a bound state appears, it also indicates in parenthesis the experimental value of the noninteracting baryon-meson threshold; the second column refers to color-singlet (S), hidden-color (H) and coupled-channels (S+H) calculations; the third and fourth columns show the theoretical mass and binding energy of the pentaquark bound-state; and the fifth column presents the theoretical mass of the pentaquark state but re-scaled attending to the experimental baryonmeson threshold, this is in order to avoid theoretical uncertainties coming from the quark model prediction of the baryon and meson spectra. In addition to the study sketched briefly in the last paragraph, we use the mentioned complex scaling method (CSM) to investigate the nature of a given pentaquark state in coupled-channels calculation. There exist (resonance) poles for pentaquark states with quantum numbers I(J P ) = 1 No resonance state is found in the present work with total spin J P = 5 2 − and isospin I = 1 2 . As for those possible resonance states, their complex energies (masses and widths) are established in Figs. 6 to 11. Moreover, Table XIV summarized our theoretical findings of these We proceed now to describe in detail our theoretical findings: The I(J P ) = 1 2 ( 1 2 − ) channel: Among all the possible baryon-meson channels: Ξ cc η, Ξ cc ω, Ξ cc π, Ξ cc ρ, Ξ * cc ω, Ξ * cc ρ, Λ c D, Λ c D * , Σ c D, Σ c D * and Σ * c D * , only Λ c D * is possibly bound in real-range calculation with a binding energy E B = −2 MeV and its modified mass is 4291 MeV. One can clearly see in Table V that the coupling between color-singlet and hidden-color channels is quite weak. However, after a coupled-channels calculation for all of these possible channels in complex-range with a rotated angle θ varied from 0 • to 6 • , one possible Σ c D resonance state is obtained.
The distribution of complex energies with quantum numbers I(J P ) = 1 2 ( 1 2 − ) are shown in Fig. 6. The green dots on the positive real-axis are the masses of coupledchannels calculation with θ = 0 • . Meanwhile, black, red and blue dots are for those with θ = 2 • , 4 • and 6 • , respectively. Generally, they are aligned along the threshold lines with the same color. If we focus on, e.g., Ξ cc π channel, whose lowest theoretical mass is 3812 MeV, the nature of scattering state is clearly identified because the obtained poles always move along the cut lines when the scaling angle θ changes. This feature is also observed for the other channels: Λ c D, Λ c D * , Ξ * cc ρ, Σ c D * and Σ * c D * . Note, too, that the radial excited state of Ξ cc π is also obtained, as shown in Fig. 6.
An important feature to highlight here is the following. The bound state of Λ c D * , with a mass of 4291 MeV, is pushed above its threshold within the coupled-channels calculation. In Fig. 6, one can see that the pole of Λ c D * is always going down with larger values of θ. Since we are working with a finite Fock-space, some numerical noise is found in the high energy region, from 4.6 GeV. This issue can be settled with a large number Gaussian basis; however, such higher energies are not interesting for the scope of this work.
The top panel of Fig. 6 also shows a dense distribution of Ξ cc η, Ξ cc ω, Σ c D, Ξ * cc ω and Ξ cc ρ states in the energy region 4.35 − 4.46 GeV; for this reason, the bottom panel shows an enlarged version of it which concentrates on [4.35 − 4.46] GeV. One can see that the calculated complex energies fall mostly into the kind of continuum states, except a possible resonance pole whose mass and width are ∼ 4416 MeV and ∼ 4.8 MeV, respectively. In the same figure, there are three almost overlap-  [3,4,6,19,22]; moreover, its 5-quark configuration is identified with a molecular state of Σ cD with mass and width 4311.9 ± 0.7 +6.8 −0.6 MeV and 9.8 ± 2.7 +3.7 −4.5 MeV, respectively. Hence, this new resonance state is expected to be identified in near future high-energy physics experiments.  Table VI shows our findings with θ = 0 • . The definition of each column is the same as that in Table IV of the I(J P ) = 1 2 ( 1 2 − ) case. No bound state is found in these channels; however, a loosely bound one of Σ c D * with a binding energy of E = −1 MeV could be obtained, as shown in Table VII. For the possible bound state of Σ c D * (4461), hidden-color channel helps a little in forming the baryon-meson molecular state. When the rotated angle θ is varied from 0 • to 6 • in coupled-channels calculation, several interesting results are observed. In Fig. 7, the possible channels are mostly scattering states moving along their corresponding cut lines. Besides, there is a Ξ * cc π(3757) bound state circled with purple in the real axis. Its binding energy is E = −3 MeV when compared with the threshold's theoretical value, 3866 MeV in Table VI. Therefore, after a mass shift with respect to the experimental value 3760 MeV, the modified mass is 3757 MeV. Consequently, the coupled-channels calculation results in a Ξ * cc π(3757) bound state with I(J P ) = cc ω, Ξ cc η, Σ * c D and Ξ ( * ) cc ρ). The nature of Σ c D * scattering state can be identified clearly in Fig. 7 where the corresponding calculated poles (E ∼ 4.5 GeV in real axis) go always down when increasing the rotated angle, θ.
An enlarged figure for the energy region 4.4 − 4.6 GeV is shown in the bottom panel of Fig. 7. A resonance state is obtained and surrounded by a green circle (three calculated results of different θ are almost unchanged inside of it). The resonance's mass and width are about 4492 MeV and 8.0 MeV, respectively. Due to this pole is above two almost degenerate scattering states of Σ * c D and Ξ cc ρ whose theoretical thresholds are 4432 MeV and 4434 MeV, in present work, the obtained resonance state is preferred to be identified as a molecular state of Σ * c D. Hence, after a mass shift according to Σ * c D(4389) with ∆ threshold = 43 MeV, the obtained resonance state has a mass of E = 4449 MeV and a width of Γ = 8.0 MeV respectively. Note again that there is also a significant similarity between Σ * c D(4449) and the hidden-   Table VIII lists the masses of possible states in the channels Ξ * cc ω, Ξ * cc ρ and Σ * c D * , taking into account singlet-color, hidden-color and their coupling. The real-range calculation with rotated angle θ = 0 • does not provide bound states. In a further complex-scaling study within coupled-channels calculation, neither bound nor resonance states are obtained. In Fig. 8, the continuum states of Ξ * cc ω, Ξ * cc ρ and Σ * c D * are shown and they basically fall along the corresponding cut lines.
The I(J P ) =      Our results from the coupled-channels calculation within the CSM taking into account a range of rotated angle θ ∈ [0 • , 6 • ] is shown in Fig. 9. The distribution of Ξ cc π states is the same as that seen in IJ P = 1 On one hand, it is clear that the effects of coupledchannels lead to a scattering state of Σ * c D * whose original modified bound state mass is 4523 MeV and the corresponding pole (E = 4547 MeV in real axis of Fig. 9) descends gradually with a larger values of the rotated angle θ. On the other hand, an unchanged resonance pole with mass (E) and width (Γ) of 4491 MeV and 2.6 MeV, respectively, is circled with green. We identify this state as a baryon-meson molecule of nature Σ c D with a shifted mass of 4431 MeV due to the difference between our theoretical and the experimental values of the Σ c D threshold.
The bottom panel of Fig. 9 shows our results in the energy interval of 4.50 to 4.53 GeV. One can guess that another possible Σ c D resonance state is found, whose mass and width are 4506 MeV and 2.2 MeV, respectively. By a mass shift with respect to Σ c D, according to previous discussion, the obtained resonance state is Σ c D(4446) with a very small width of Γ = 2.2 MeV. As one can elucidate from our discussion until now is that the doubly-charmed pentaquark states present similar features than those  hidden-charm ones observed experimentally, P + c (4312), P + c (4440) and P + c (4457) [1], which are mainly explained as molecular states of Σ ( * ) cD We expect that, in the near future, the potential molecular candidates in the doubly-charm sector, Σ c D(4431) and Σ c D(4446), being confirmed experimentally.
The I(J P ) = With a rotational manipulation for the relative motions of five-quark systems in complex plane, the coupledchannels results are shown in Fig. 10. Again, the lowest and radial excited states of Ξ * cc π are both scattering ones with theoretical a mass of 3866 MeV and 4305 MeV, respectively.
A possible resonance state of Σ c D * is found in the bottom panel of Fig. 10 which is an enlarged part involving the energy interval 4.48 − 4.65 GeV. Clearly, there are three almost overlapped poles inside the green circle which is above the cut lines of Σ c D * , and the corresponding masses and widths can be cluster around 4555 MeV and 4.0 MeV respectively. This resonance can be identified as a Σ c D * (4514) molecular state whose modified mass E = 4514 MeV is obtained by a mass shift of ∆ = 41 MeV according to the calculated results of Σ c D * (4462) channel in Table XI   state of Σ * c D * (4524) turned into a scattering one with an unstable pole with a theoretical mass of 4548 MeV in Fig. 10.
I(J P ) = 3 2 ( 5 2 − ) channel: Only two baryon-meson channels contribute to this case: Ξ * cc ρ and Σ * c D * . Table XIII shows that we do not find any bound state in these two configurations. However, in coupled-channels calculation within complex-scaling, a possible Ξ * cc ρ resonance state with a small decay width is found. In Fig. 11,