Doubly charmed molecular pentaquarks

We perform a systematic exploration of the possible doubly charmed molecular pentaquarks composed of $\Sigma_c^{(*)}D^{(*)}$ with the one-boson-exchange potential model. After taking into account the $S-D$ wave mixing and the coupled channel effects, we predict several possible doubly charmed molecular pentaquarks, which include the $\Sigma_cD$ with $I(J^P) = 1/2(1/2^-)$, $\Sigma_c^*D$ with $1/2(3/2^-)$, and $\Sigma_cD^*$ with $1/2(1/2^-)$, $1/2(3/2^-)$. The $\Sigma_cD$ state with $3/2(1/2^-)$ and $\Sigma_cD^*$ state with $3/2(1/2^-)$ may also be suggested as candidates of doubly charmed molecular pentaquarks. The $\Sigma_cD$ and $\Sigma_c^*D$ states can be searched for by analyzing the $\Lambda_cD\pi$ invariant mass spectrum of the bottom baryon and $B$ meson decays. The $\Sigma_cD^*$ states can be searched for in the invariant mass spectrum of $\Lambda_cD^*\pi$, $\Lambda_cD\pi\pi$ and $\Lambda_cD\pi\gamma$. Since the width of $\Sigma_c^*$ is much larger than that of $D^*$, $\Sigma_c^*D\rightarrow \Lambda_cD\pi$ will be the dominant decay mode. We sincerely hope these candidates for the doubly charmed molecular pentaqurks will be searched by LHCb or BelleII collaboration in the near future.


I. INTRODUCTION
In the past decades, the observations of X/Y/Z/P c states have stimulated theorist's extensive interest in exploring the properties of exotic states. Among these possible configurations of exotic state, the hadronic molecular state is composed of the color-singlet hadrons, which is different from other exotic state configurations, like the hybrid, glueball, multiquarks. Since many observed X/Y/Z/P c states are near the threshold of one hadron pair, the molecular assignments have received extensive attentions [1][2][3][4]. The study of hadronic molecular state is an active and important research field in hadron physics.
Among these extensive studies of charmoniumlike XYZ states and P c state, we may find shaped integrated and clear venation. The charmoniumlike XYZ states inspired the discussion of the interaction between charmed meson and anticharmed meson [1][2][3]. Here, a typical example is that the DD * molecular explanation of the X(3872) was proposed [5][6][7][8][9][10][11], which has been viewed as a starting point of exploring the hadronic molecular tetraquark state since 2003. From these studies, the applicability and reliability of the involved phenomenological models like the one-boson exchanged model adopted in this work were also tested. The authors in Refs. [12][13][14][15][16][17] benefitted from their experience with the hadronic molecular tetraquark states and further investigated the interaction between the charmed meson and anti-charmed baryon and predicted the hidden-charm pentaquarks. In 2015, LHCb reported the observation of several P c states [18], which are consistent with the prediction of the hidden-charm pen- * Electronic address: chen rui@pku.edu.cn † Electronic address: lining59@mail.sysu.edu.cn ‡ Electronic address: sunzf@lzu.edu.cn § Electronic address: xiangliu@lzu.edu.cn ¶ Electronic address: zhusl@pku.edu.cn taquarks. In 2019, with more precise data, LHCb again analyzed the same process and found the characteristic mass spectrum of the P c states [19], which provides direct evidence of the existence of the hidden-charm molecular pentaquarks [20][21][22][23][24][25][26][27][28][29][30].
Very recently, the LHCb Collaboration analyzed the D 0 D 0 π + mass spectrum using the full Run1 plus Run2 data corresponding to 9 fb −1 , and observed a very narrow doubly charmed tetraquark T + cc as its minimal valence quark component is ccūd [31]. Its mass relative to the D 0 D * + mass threshold and decay width are δm = −273 ± 61 ± 5 +11 −14 keV/c 2 , Γ = 410 ± 165 ± 43 +18 −38 keV, respectively. The spin-parity is estimated as 1 + . The mass and spin-parity are well consistent with the prediction of the DD * doubly charmed molecular state [32][33][34]. In particular, after considering the isospin breaking effects, the newly T cc state can be assigned as the S −wave D 0 D * + molecular state. We also predict another doubly charmed D + D * 0 resonance with its mass around 3876 MeV [35]. The light quark pair within the Σ ( * ) c baryon shares the same color configuration with the light anti-quark within the D 0 meson. If the T cc is the doubly charmed molecule, there should also exist the doubly charmed molecular pentaquarks composed of the charmed baryon Σ ( * ) c and charmed meson D ( * ) as shown in Figure 1. The above argument is very similar to the underlying reasoning of predicting the hidden-charm molecular pentaquarks from the existence of the hidden-charm molecular states [12]. The only difference is that now we have two charm quarks instead of a cc pair.
FIG. 1: A comparison between the T cc and P cc in the doubly charmed molecular picture.
In fact, there are several predictions on the doubly charmed molecular pentaquarks [37][38][39]. For example, within the framework of chiral effective field theory [38], Chen et al performed a systematic study on the interactions of the Σ ( * ) c D ( * ) interactions. They found all the S −wave Σ ( * ) c D ( * ) systems with isospin I = 1/2 can be possible doubly charmed molecular pentaquarks, and their binding energies are larger than the corresponding Σ ( * ) cD ( * ) bound states. In Ref. [39], the authors obtained the similar conclusions in the resonance saturation model.
In this work, we adopt the one-boson-exchange (OBE) model to derive the effective potentials describing the Σ ( * ) c D ( * ) interactions, and consider the S − D wave mixing effects and the coupled channel effects. Our investigation will not only provide valuable information to experimental search for the doubly charmed molecular pentaquarks, but also give indirect test of the molecular state picture for the P c and T + cc states. This paper is organized as follows. After the introduction, we present the detailed deduction of the effective potentials for the Σ ( * ) c D ( * ) systems in Sec. II. In Sec. III, we present the corresponding numerical results by solving the coupled channel Shrödinger equation. The paper ends with the summary in Sec. IV.

II. INTERACTIONS
As a molecular state composed of two colorless hadrons, its wave function is constructed by three parts, i.e., the flavor wave function, the spin-orbit wave function, and the radial wave function. Here, the flavor wave functions |I, I 3 for the Σ ( * ) c D ( * ) systems are written as Here, G E stands for the G parity for the exchanged mesons, which include the scalar meson σ, pseudoscalar mesons π/η, and vector mesons ρ/ω. In our previous work, we have deduced the concrete OBE effective potentials for the Σ ( * ) cD ( * ) systems by employing the effective Lagrangian approach at the hadronic level. The general procedures can be divided into three steps. We first construct the effective Lagrangians relevant to the interactions between the S −wave charmed baryons/mesons and light mesons [40][41][42][43][44], and write down the corresponding scattering amplitudes M Σ ( * ) of the initial states (h 1 , h 2 ) and final states (h 3 , h 4 ), respectively. Finally, we perform the Fourier transformation to obtain the effective potential in the coordinate space V(r), i.e., In the above formula, we introduce a monopole type form factor F (q 2 , m 2 E ) = (Λ 2 −m 2 E )/(Λ 2 −q 2 ) at every interactive vertex to compensate the off-shell effects of the exchanged boson. Λ, m E , and q are the cutoff, the mass and four-momentum of the exchanged meson, respectively.
According to the relations in Eq. (1) and the expressions in Ref. [20], we finally obtain the OBE effective potentials for the Σ ( * ) c D ( * ) → Σ ( * ) c D ( * ) processes as summarized in Table I. And we define several useful functions, i.e., where D i j , E i j , and F i j are the spin-spin interactions and tensor force operators, their expressions are defined as follows, e.g.,  Once we start the numerical calculation through solving the coupled channel Shrödinger equation, these operators O i j should be replaced by a serial of matrix elements 2s +1 L J |O i j | 2s+1 L J as collected in Table I, where the notations 2s +1 L J | and | 2s+1 L J stand for the spin-orbit wave functions for the final and initial discussed channels, respectively.

III. NUMERICAL RESULTS
We perform a systematic investigation on the possible molecular pentaquarks composed of the S −wave Σ ( * ) c D ( * ) systems with all possible isospin I and spin J for the negative parity. To explore the roles of the S − D wave mixing effects, the coupled channel effects, the long-rang pion-exchange potential and the intermediate-and short-range from the ρ, ω, σ, and η exchanges in the formation of the loosely bound Σ ( * ) c D ( * ) states, we first perform the calculation for the single channel with both the one-pion-exchange (OPE) potentials and OBE potentials, the numerical results of which are given in Table II. We then include the coupled channel effects and perform the calculation again using the OPE potentials, and present the numerical results in Table III. Finally, we include the intermediate-and short-range ρ, ω, σ and η exchanges potentials in addition to the long-range interaction potentials, and the numerical results are presented in Table IV.
The isospin I and spin J for the S −wave Σ c D system with negative parity are (I, J) = (1/2, 1/2), (3/2, 1/2). For J = 1/2, the coupled channel wave function can be expanded as with isospin I = 1/2 and 3/2. For the single channel Σ c D| 2 S 1/2 with isospin I = 1/2, our results indicate that only the pion-exchange potential is not strong enough to bind the Σ c D system as the DDπ coupling is forbidden by the spin-parity conservation rule. After taking into account the heavier ρ, ω, and σ exchanges accounting for the intermediate-and short-range interactions, we obtain a loosely bound state with binding energy −0.23 MeV and rootmean-square (rms) radius 5.21 fm for a reasonable cutoff 1.14 MeV. As the cutoff increases to 1.34 GeV, the binding energy increases to −18.18 MeV while the rms radius decreases to 0.97 fm. Here, the intermediate-and short-range forces from the OBE model play an important role to form the single Σ c D molecular state with I(J P ) = 1/2(1/2 − ).
When we include the coupled channel effects from channels Σ c D * and Σ * c D * , we obtain a weakly bound state with binding energy −0.87 MeV and rms radius 3.41 fm for cutoff Λ = 1.21 GeV using only the long-range pion-exchange potential. The probability of the dominant channel Σ c D| 2 S 1/2 is 96.90%. Compared to the single channel case, the coupled c D ( * ) → Σ ( * ) c D ( * ) processes and the matrix elements O i j obtained from 2s +1 L J |O i j | 2s+1 L J for all the operators O i j . Here, G is the isospin factor, which is taken as −1 for the isospin-1/2 system, and 1/2 for the isospin-3/2 system. The values of the coupling constants are taken from [43][44][45][46], g S = 0.76, g = 0.59, β = 0.9, l S = 6.2, g 1 = 0.94, β S = −1.74, λ = 0.56 GeV −1 , λ S = −3.31 GeV −1 , and g V = 5.9. The variables in these functions are defined as     To summarize, we propose the Σ c D to be a good candidate of hadronic molecular state.
For the isospin I = 3/2 case, we could obtain bound state for the single channel Σ c D neither with the OPE potential nor with the OBE potential. After we include the coupled channel effects from channels Σ * c D as well as Σ c D * , a loosely bound state with binding energy −0.64 MeV and rms radius 3.65 fm appears when the cutoff is tuned to 1.62 GeV using the OPE potential. We further include the intermediate-and short-range interaction from the heavier ρ, ω, σ, and η exchanges, a loosely bound state is obtained with binding energy −0.53 MeV and rms radius 4.05 fm. The probability of Σ c D| 2 S 1/2 is 97.78% and that of Σ * c D * | 2 S 1/2 is 1.32%. The probabilities for other channels are very tiny, less than 1%. As the cutoff increases to 1.95 GeV, the binding energy increases to −13.52 MeV, and the rms radius decreases to 0.92 fm. Meanwhile, more coupled channel effects get involved. The probability of Σ c D| 2 S 1/2 is 82.43% while that for Σ * c D * | 2 S 1/2 is 13.62%. The probabilities for other channels are small, less than 3%. From the current numerical results, the system Σ c D[I(J P ) = 3/2(1/2 − )] may also be viewed as a candidate of doubly charmed hadronic molecule.
Due to the higher threshold of Σ * c D compared to Σ c D, we perform a calculation relative to the threshold of Σ * c D. The isospin I and spin J for Σ * c D with negative parity can be (I, J) = (1/2, 3/2) and (3/2, 3/2). The coupled channel wave function can be expanded as with isospin I = 1/2 and 3/2. In the heavy quark limit, the OBE effective potentials for the S −wave Σ * c D system are very similar to the S −wave Σ c D interactions. In the isospin I = 1/2 case, we could not obtain bound state solutions for the single Σ * c D with the long-range pion exchange potential only. After we take into account the intermediate-and short-range interaction from the heavier ρ, ω, σ, and η exchanges, a loosely bound state with binding energy −0.57 MeV and rms radius 4.05 fm emerges. If we include the coupled channel effects from Σ c D * and Σ * c D * channels, we obtain a loosely bound state with the binding energy −0.76 MeV and the rms radius 3.09 fm using the only longrange pion exchange when the cutoff is set to 1.48 GeV. When we further include the intermediate-and short-range interactions from the heavier ρ, ω, σ, and η exchanges, we obtain a loosely bound state with a smaller cutoff 1.05 GeV which is comparable to the value used for the study of deuteron [47,48]. Its binding energy is −1.30 MeV and rms radius is 2.84 fm. The probability for the dominant channel Σ * c D| 4 S 3/2 is 94.50% and that for Σ c D * | 4 S 3/2 is 5.14%. As the cutoff is tuned to 1.11 GeV, the binding energy becomes −12.32 MeV and rms radius decreases to 0.99 fm. The probability for the c D ( * ) systems using the OPE potential. The cutoff Λ, the root-mean-square radius r RMS , and binding energy E of the bound state are in units of GeV, fm, and MeV, respectively. P i (%) denotes the probability of the i−th channel. The results for the channel with the largest probability are marked with a bold typeface.

OPE
Channels (P i )   Σ * c D| 4 S 3/2 channel decreases to 73.36% whereas that for the Σ c D * | 4 S 3/2 channel increases to 25.60% due to the very close threshold for these two channels. In fact, there exists competition between the binding energy and the threshold difference for the channels involved. When the binding energy is small, the role of the threshold difference between the channels in-volved will be amplified. However, when the binding energy becomes bigger, this effect will disappear. For a conclusion, the Σ * c D is a good candidate of a hadronic molecule in the present OBE potential model.
For the isospin I = 3/2 case, we only obtain bound state solutions with the cutoff larger than 2.0 GeV which is larger c D ( * ) systems using the OBE potential. The cutoff Λ, the root-mean-square radius r RMS , and the binding energy E of the bound state are in units of GeV, fm, and MeV, respectively. P i (%) denotes the probability of the i−th channel. The results for the channel with the largest probability are marked with a bold typeface.

OBE
Channels (p i )  The isospin I and spin J for the system Σ c D * with negative parity can be (I, J) = (1/2, 1/2), (1/2, 3/2), (3/2, 1/2), and (3/2, 3/2). For the spin J = 1/2 case, the coupled channel wave function can be expanded as for isospin I = 1/2 and 3/2. For the (I, J) = (1/2, 1/2) case, we find bound states solutions for the single channel Σ c D * using the long-range pion exchange potential only when the cutoff is tuned to be larger than 2.40 GeV. After taking into account the intermediate-and short-range ρ, ω, σ, and η exchanges, a loosely bound state is obtained with smaller cutoff 1.30 GeV. Its binding energy is −0.51 MeV, and the rms radius is 4.41 fm. If we include the coupled channel effects instead of the intermediate-and short-range interactions, we obtain bound state solutions with cutoff tuned larger than 2.24 GeV. However, after we take into account the coupled channel effects as well the heavier ρ, ω, σ, and η exchanges, a loosely bound state emerges with a smaller cutoff 1.27 GeV, which is comparable to the value used for the study of deuteron and other hadronic molecules within the OBE model. Its binding energy is −0.41 MeV and the rms radius 4.70 MeV. The probability of Σ c D * | 2 S 1/2 is 97.56% while that of Σ c D * | 4 D 1/2 is 2.24%, and that for other channels is tiny. As the cutoff increases to 1.47 GeV, the binding energy increases to −11.22 MeV and rms radius decreases to 1.30 fm. Meanwhile, the probability of the S-wave channel Σ c D * | 2 S 1/2 decreases to 88.46% and that of the D-wave channel Σ c D * | 4 D 1/2 increases to 9.44%. From the current numerical results, the system Σ c D * with (I, J) = (1/2, 1/2) should be viewed as a good candidate of double charm molecular pentaquark, and the intermediate-and short-range interactions from the ρ, ω, σ, and η exchanges are very important, especially in the single channel case.
For the (I, J) = (3/2, 1/2) case, there exist bound state solutions with either OPE potential or OBE potential. When we take into account the coupled channel effects as well as the intermediate-and short-range interaction from the heavier ρ, ω, σ and η exchanges, a loosely bound state appears for the cutoff tuned to 1.41 GeV. Its binding energy is −0.48 MeV and the rms radius is 4.09 fm. It is almost a pure Σ c D * | 2 S 1/2 bound state with its probability of 99.53%. As the cutoff increases to 1.81 GeV, the binding energy increases to −17.34 MeV and the rms radius decreases to 0.84 fm. The probability of Σ c D| 2 S 1/2 decreases to 91.13% while that of the Σ * c D * | 2 S 1/2 channel increases to 8.04%. The reasonable cutoff parameter and the bound state properties support the Σ c D * state with 3/2(1/2 − ) as a candidate of the doubly charmed molecular pentaquark.
The system Σ c D * with (I, J) = (1/2, 3/2) is very interesting. We obtain a loosely bound state with only the longrange pion-exchange potential for the cutoff 1.20 GeV and with the OBE potential for the cutoff 0.91 GeV. When we take into account the coupled channel effects as well as the intermediate-and short-range interaction from the heavier ρ, ω, σ, and η exchanges, a loosely bound state is obtained with cutoff 0.91 GeV. The probability for the dominant channel Σ c D * | 4 S 3/2 is 97.47% while that of Σ c D * | 4 D 3/2 is 1.81%. As the cutoff increase to 1.01 GeV, the binding energy increases to −12.10 MeV while the rms radius decreases to 1.15 fm which is still comparable to the size of well-known deuteron. The probability of Σ c D * | 4 S 3/2 is 94.74% and that for Σ c D * | 4 D 3/2 is 3.32%. From the current numerical results, we propose the Σ c D * state with 1/2(3/2 − ) as a good candidate of doubly charmed molecular pentaquark.
For the isospin I = 3/2 case, we could not obtain bound state solutions for the single channel Σ c D * | 4 S 3/2 until tuning the cutoff to be as large as 4.0 GeV for the long-range pionexchange potential only and 3.90 GeV for the OBE potential. Although, after both the coupled channel effects and the intermediate-and short-range interactions are included, the cutoff with the bound state solutions is still too larger compared the reasonable value, which is around 1.0 GeV. Thus, the Σ c D * system with (I, J) = (3/2, 3/2) may not be a candidate of hadronic molecule.

IV. SUMMARY
We perform a systematic exploration of the possible doubly charmed molecular pentaquark of Σ ( * ) c D ( * ) with the one-bosonexchange potential model. To investigate the the S − D wave mixing effects, the coupled channel effects, the long-rang pion exchange interaction and the intermediate-and short-range interactions arising from the heavier ρ, ω, σ, and η exchanges, we performed four kinds of calculations. We first do the calculation for the single channel using the OPE potential and OBE potential individually. Then we include the coupled channel effects and do the calculations again within the OPE potential model. Finally, we take into account all the effects and do a full coupled channel calculation within the OBE potential model. We also find that for a loosely bound states in the coupled channel study, the binding energy and the threshold difference between different channels compete with each other. In other words, for a loosely bound state, the role of the threshold difference will be amplified by the small binding energy, which had already been emphasized in the study of the isospin breaking for the X(3872) [9] and T cc [35].
Our results reveal some general features for the coupled channel study of the Σ ( * ) c D ( * ) systems with the OBE model, which has already been discovered in previous coupled channel study of the hadronic molecules [49,50]. Overall, the coupled channel effects are helpful for the formation of the bound states. The long-range pion-exchange potential plays an important role in the formation of the loosely bound states while the intermediate-and short-range interaction from the heavier ρ, ω, σ, and η exchanges can also help strengthen the binding between Σ ( * ) c and D ( * ) . Very interestingly, we propose some good candidates of the doubly charmed molecular pentaquarks. From our results, the Σ c D state with I(J P ) = 1/2(1/2 − ), Σ * c D state with I(J P ) = 1/2(3/2 − ), and Σ c D * states with 1/2(1/2 − ), 1/2(3/2 − ) are good candidates of doubly charmed hadronic molecules. The Σ c D state with 3/2(1/2 − ) and Σ c D * with 3/2(1/2 − ) may also be viewed as the doubly charmed molecular candidates. We also find that the Σ * c D state with 1/2(3/2 − ) is more complicated due to the near threshold between the Σ c D * and Σ * c D systems. For a loosely bound state with the binding energy around 1 MeV, the dominant channel is the Σ * c D| 4 S 3/2 with the probability larger than 95% and small contributions from the Σ c D * | 4 S 3/2 channel. When the binding energy is around tens of MeV, the probability of the Σ * c D| 4 S 3/2 will be comparable with that of the Σ c D * | 4 S 3/2 channel.
The newly observed T cc with doubly charmed would definitely be a new hadronic state beyond the traditional baryons and mesons. Its observation opens a new window to search for the new hadronic state experimentally, and indicates there comes to a new era for the experimental research of the exotic hadronic states. The Σ c D and Σ * c D states can be searched for by analyzing the Λ c Dπ invariant mass spectrum of the bottom baryon and B meson decays. The Σ c D * states can be searched for in the invariant mass spectrum of Λ c D * π, Λ c Dππ and Λ c Dπγ. Since the width of Σ * c is much larger than that of D * , Σ * c D → Λ c Dπ would be the dominant decay mode. We sincerely hope these proposed doubly charmed molecular candidates will be searched for by the LHCb or BelleII Collaborations in the near future.