Evidence that the LHCb Pc states are hadronic molecules and the existence of a narrow Pc(4380)

narrow Pc(4380) Meng-Lin Du, ∗ Vadim Baru, 2, 3, † Feng-Kun Guo, 5, ‡ Christoph Hanhart, § Ulf-G. Meißner, 6, 7, ¶ José A. Oller, ∗∗ and Qian Wang 10, †† Helmholtz-Institut für Strahlenund Kernphysik and Bethe Center for Theoretical Physics, Universität Bonn, D-53115 Bonn, Germany Institute for Theoretical and Experimental Physics, B. Cheremushkinskaya 25, 117218 Moscow, Russia P.N. Lebedev Physical Institute of the Russian Academy of Sciences, 119991, Leninskiy Prospect 53, Moscow, Russia CAS Key Laboratory of Theoretical Physics, Institute of Theoretical Physics, Chinese Academy of Sciences, Zhong Guan Cun East Street 55, Beijing 100190, China School of Physical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China Institute for Advanced Simulation, Institut für Kernphysik and Jülich Center for Hadron Physics, Forschungszentrum Jülich, D-52425 Jülich, Germany Tbilisi State University, 0186 Tbilisi, Georgia Departamento de F́ısica, Universidad de Murcia, E-30071 Murcia, Spain Institute of Quantum Matter, South China Normal University, Guangzhou 510006, China Theoretical Physics Center for Science Facilities, Chinese Academy of Sciences, Beijing 100049, China

Introduction. The confinement property of quantum chromodynamics (QCD) in principle allows for the existence of a large variety of color neutral objects, however, it is not clear yet which configurations are realized in nature. As a result, searching for multiquark exotic hadrons beyond the conventional quark model has been one of the central issues in the study of the strong interactions. Tremendous developments have been made in the new era since 2003 when the B factories discovered the D * s0 (2317) [1] and X(3872) [2], whose properties are in notable contradiction with quark model predictions. The interest in studying such exotic hadrons was further boosted by the LHCb discovery of the hidden-charm pentaquarks P c (4450) and P c (4380) decaying into J/ψp in the Λ 0 b → K − J/ψp process in 2015 [3]. The experimental and theoretical efforts are summarized in a number of comprehensive reviews [4][5][6][7][8][9][10][11][12][13][14][15].
New surprises happened early this year when LHCb updated their measurements with a one-order-ofmagnitude larger data sample [16]: the narrow P c (4450) appears to be split into two narrower structures P c (4440) and P c (4457) and a third narrow peak P c (4312) shows up. At the same time, the broad P c (4380) loses its sig-nificance (whose existence needs to be verified in a complete amplitude analysis that is under way). A cornucopia of theoretical interpretations followed these new discoveries, including models of hadronic molecules , compact pentaquark states [39][40][41][42][43][44] and hadrocharmonia [45]. An amplitude analysis was performed in Ref. [46] focusing on the P c (4312) which was suggested to be a virtual state. Among the explanations, the hadronic molecular model stands out as it explains all of the three narrow P c states simultaneously as Σ cD (for P c (4312)) and Σ cD * (for P c (4440) and P c (4457)) bound states, see, e.g., Refs. [20,25,31], employing the approximate heavy quark spin symmetry (HQSS) of QCD. However, the model predicts in addition four more states, including one Σ * cD state at around 4.37 to 4.38 GeV and three states slightly below the Σ * cD * threshold. These seven P c states are in two heavy quark spin multiplets, labeled as j P , with j and P the total angular momentum of the light degrees of freedom and parity, respectively: three with j P = 1 2 − and four with j P = 3 2 − . States with the same spin in these two multiplets mix because the Σ c (D) is not degenerate with the Σ * c (D * ). While only three of them correspond to the ones reported by LHCb, it is cru-cial to check whether the existence of the whole two multiplets is consistent with the J/ψp distribution measured by LHCb. This is the question addressed in this letter: by constructing coupled-channel amplitudes analogous to those used in the analysis of the Z b states [47][48][49][50], we show that the observed J/ψp invariant mass distribution can be well described in the hadronic molecular scenario, which has seven Σ ( * ) cD ( * ) molecules, with the P c (4312) a Σ cD and P c (4440) and P c (4457) Σ cD * bound states, respectively, and for the first time we point at a clear signal from data for the existence of a narrow P c (4380) as a Σ * cD bound state. The remaining three predicted Σ * cD * states still await discovery. Framework. In order to describe the measured J/ψp distribution, we construct coupled-channel amplitudes considering all the Σ ( * ) cD ( * ) channels (to be called elastic channels following Refs. [47,48] since their thresholds are close to the P c masses) and the J/ψp channel (to be denoted as inelastic). HQSS is used to relate all the Σ withĤ I the effective Hamiltonian respecting HQSS. In the heavy quark limit, the contact interactions defined above are independent of s Q = 0 or 1. Since we work to leading order the above matrix elements are constants. In particular, the D-wave Σ ( * ) cD ( * ) contact terms, that turned out to be necessary in the study of the Z b states [50], will be neglected since data in the inelastic channels are insensitive to such operators. The contact terms in the particle basis are related to those in the spin basis via the rotation matrices introduced above, where j (i J ) denotes the light-quark spin of the i th channel for a given J-multiplet.
The OPE potential, V J OPE (p, q), can be obtained using the effective Lagrangian for the axial coupling of the pions to the charmed mesons and baryons [53,54] where . and Tr [.] denote traces in the spinor and isospin spaces, respectively, σ represents the Pauli matrices, S i andH are the heavy quark spin doublets for ground states (Σ c , Σ * c ) and (D,D * ) [55], , Φ = τ · π with τ and π the Pauli matrices in the isospin space and the pion fields, in order, and F π = 92.1 MeV is the pion decay constant. From the measured width of D * + → D 0 π + [56] one gets g = 0.57, and the coupling g 1 = 0.42 is taken from the lattice QCD calculation [57]. The OPE contributes to both S and D waves and can be important for describing the line shapes around thresholds [49,50,58]. Also the transitions between the elastic and inelastic channels can be related via HQSS. While the |1 ⊗ 1 2 component in Eqs. (1)-(3) couples to J/ψp in the S wave in the heavy quark limit, the |1⊗ 3 2 only couples to J/ψp in the D wave. We introduce two coupling strengths, where k is the magnitude of the J/ψ three-momentum in the c.m. frame of J/ψp. Then, the transition vertices V J αi between the α th elastic and i th inelastic channel, with i = 1, 2 denoting the S-wave and D-wave J/ψp, respectively, can be easily obtained by virtue of the decompositions in Eqs.
(1)-(3) as As for the direct J/ψp scattering, one notices that it is Okubo-Zweig-Iizuka suppressed. In fact, the recent lattice QCD results in Ref. [59] show that such an interaction is indeed very weak. Thus, the inelastic J/ψp channel is only included through its coupling to the elastic Σ ( * ) cD ( * ) channels, and it effectively modifies the contact terms for the elastic channels [47][48][49][50]60]. While the real part of its contribution can be absorbed by redefining the contact terms [50], C J αβ , the imaginary part cannot. We thus need to introduce into the effective elastic potential the term It is expected that, in addition to the J/ψp channels, there are more inelastic channels, most prominently Λ cD ( * ) and η c p [30,61,62]. While the latter is connected to the J/ψp channels via HQSS, the former is not and thus we are obliged to parametrize especially those via an additional imaginary part of the two contact terms. This introduces two more parameters. Thus the scattering problem contains in total 6 parameters and the full effective potential for the elastic channels can be written as Let us now come to the weak production amplitude . Since the energy region of interest is around the Σ ( * ) cD ( * ) thresholds, we only consider the elastic Σ ( * ) cD ( * ) channels produced in an S-wave. Parameterizing the weak production of the |s Q ⊗ j states with the total angular momentum J as where (s Q ⊗j ) J n refers to the n th state in the |s Q ⊗ j basis in Eqs. (1)-(3), the production contact term for the α th elastic channel for a given J may be parameterized as In total, there are additional seven parameters F J n . With the above ingredients, one can obtain the production amplitude, U J α , for the α th elastic channel by solving the following Lippmann-Schwinger equations (LSEs), and for the i th J/ψp inelastic channel, with E for the invariant mass of the system, p, q and k are the three-momenta in the c.m. frame for the α th , β th elastic channel and the J/ψp channel, respectively. The two-body propagator is with µ β and m β th the reduced mass and the threshold of the β th elastic channel. The Σ ( * ) c widths of 1.86 MeV (15 MeV) [56] are accounted for using complex mass m − iΓ/2 in m β th . The LSE is regularized using a hard cutoff, varied in the range from 1 to 1.5 GeV. Since the results barely depend on its value (effects of the cutoff variation can be largely absorbed into the refitted contact terms), the final results will be presented for the cutoff of 1 GeV. The equations given are unitary as long as the additional imaginary part of the contact terms is omitted. Unitarity can be restored once data on the Λ cD ( * ) channels are available, and we checked that introducing the Λ cD * channel in the way of Eq. (8) did not produce sizeable difference.
In order to fit the J/ψp invariant mass distribution, an incoherent smooth background is used to model possible contributions from misidentified non-Λ 0 b events, the Λ * resonances coupled to pK − , and possibly additional broad P + c structures. We here use the form which contains the parameters b 0 , b 1 , b 2 , g r , m and Γ. The backgrounds used in the experimental analysis [16] are also considered, and the results are similar which will be included in the uncertainties.
We perform fits of the measured J/ψp invariant mass spectrum considering two schemes including • scheme I: contact potential only, with the OPE potential, V J OPE , switched off; • scheme II: contact and OPE potentials as given in Eq. (9).
Results and Discussions. In this analysis, we do not consider isospin symmetry breaking effects which can be important to give rise to isospin-breaking decay modes [19,63] but not in describing the line shapes in the isospin-conserving J/ψp channel. Convolution with the experimental energy resolution is considered. For scheme I we find two solutions, denoted as A and B, describing the data almost equally well (with χ 2 /d.o.f. = 1.01 and 1.03, respectively). The corresponding best fits are shown in the left panel of Fig. 1. The two solutions produce different values of the parameters, in particular C 1 2 and C 3 2 (in fact, the values of C 1 2 and C 3 2 in solution A are very close to those of C 3 2 and C 1 2 in solution B, respectively), and thus give different pole locations. However, both solutions give seven poles, i.e. seven P c states, in the Σ ( * ) cD ( * ) scattering amplitudes: three with J = 1/2, three with J = 3/2 and one with J = 5/2. The masses of the generated P c states in solutions A and B are close to those of scenarios A and B for the corresponding quantum numbers in Refs. [20,31], respectively, though the TABLE I. The table shows the names of the states; their quantum numbers found from the fits within scheme II; the pole positions (on the sheets close to the physical one); the dominant channels (DCs) and their thresholds; the dimensionless couplings of the resonances in the DCs (from the T -matrix residues and defined as GDC ); the resonance couplings to the source derived from the residues, which are normalized by the event numbers and thus only the relative values are meaningful. The uncertainties given are from taking different backgrounds, the uncertainties from the fit for a given background are negligible.  widths are larger due to the inelastic channels and the Σ ( * ) c widths, and thus not shown here.
In both solutions, among the seven poles, the lowest one corresponds to the P c (4312) with 1 2 − , and is a Σ cD bound state: it is located in the second Riemann sheet (of the J/ψp channel), and would become a real bound state pole in the first Riemann sheet if the J/ψp channel were switched off. This is different from the virtual state scenario of Σ cD in Ref. [46] which only fits to data around the Σ cD threshold.
There are two Σ cD * bound states with quantum numbers is a narrow pole around 4.38 GeV whose dominant component is Σ * cD (see also [25]). This means that HQSS requires the existence of a P c (4380), which, however, is narrow and thus totally different from the broad one reported by LHCb in 2015 [3].
In scheme II the OPE has in particular a tensor force, whose importance is well-known for the nucleon-nucleon interaction. It leads to the mixing between S and D waves, and can have a sizeable impact on the line shape between thresholds [49,50,58]. Unlike in scheme I, once the full OPE is included, there is only one solution corresponding to the best fit with χ 2 /d.o.f. = 0.98, shown in the right panel of Fig. 1. It leads to poles presented in Table I which are similar to those in solution B of scheme I (see also Refs. [34,64]). The pole positions, the dominant channels (DCs) having the largest effective couplings (derived form residues of the T -matrix), and these effective couplings are listed in Table I. The results are insensitive to the form of the background, and the effects of using different backgrounds give the errors in Table I (the statistical errors propagating from the data are much smaller).
One sees that the P c (4312) couples dominantly to Σ cD with J P = 1 2 − , and both the P c (4440) and P c (4457) couple dominantly to the Σ cD * with quantum numbers 3 2 − and 1 2 − , respectively. They are all bound state poles and should be understood as hadronic molecules of the corresponding channels [11]. Comparing the fits from scheme I and scheme II, one sees that the OPE helps making the dip between the P c (4440) and P c (4457) more evident. In both schemes there is a narrow P c (4380) located at the right position where the data show a peak, though less prominent than those of the well-known three P c states. Its existence is a consequence of HQSS in the hadronic molecular picture. We have checked that it persists no matter whether or not the data around 4.38 GeV are included in the fit. Thus, this can be regarded as strong evidence for the observed P c states being hadronic molecules.
The three Σ * cD * molecules, which are expected to exist [20,31,36,51,52], do not have any unambiguous signal in the data. The production strengths related to the residues of the production amplitude U J α at the pole and shown in the last column of the table suggest that they are less strongly produced in the Λ b decays. One possible reason could be that the production of the three most pronounced P c structures gets enhanced by nearby triangle singularities discussed in Refs. [16,[65][66][67]. They need to be searched for in data with higher statistics.
Summary and Outlook. In summary, we investigated for the first time whether the appealing hadronic molecular model for the observed P c states is consistent with the LHCb data. A coupled-channel formalism is used to analyze the J/ψp invariant mass distribution, which contains much more information than the extracted pentaquark masses only. The relevant effective potential constructed based on HQSS involves all transitions between the elastic Σ ( * ) cD ( * ) channels, transitions from the elastic to the S-and D-wave J/ψp inelastic channels as well as the coupling to additional effective inelastic channels. We find that the data can be well described. In addition to the three established states, in our analysis a narrow P c (4380) state, identified as a 3 2 − Σ * cD molecule, with its mass basically fixed by HQSS, shows up a clear signal in the data. The three Σ * cD * bound states with masses from around 4.49 to 4.52 GeV are almost invisible because of their relatively low production rates in the Λ b decays. We expect that they can be resolved in the forthcoming data to be collected at the LHC Run-3 period or other production processes, such as the J/ψ photoproduction [62,[68][69][70][71][72][73][74][75][76]. It should be stressed that, if the P c states indeed are hadronic molecules, they have to show up as prominent structures also in the elastic channels, and data for those would therefore be extremely valuable. To further refine our approach, data are needed in the Λ cD ( * ) as well as in the η c p channels. The latter would provide additional information on the amount of spin symmetry violation in the system. All these studies will definitely shed important light on our understanding of how QCD forms hadrons.
We are grateful to Marek Karliner, Misha Mikhasenko, Sebastian Neubert, Juan Nieves and Qiang Zhao for useful comments and discussions. This work is supported in part by the National Natural Science