Pion-induced production of hidden-charm pentaquarks $P_{c}(4312),P_{c}(4440)$, and $P_{c}(4457)$

The production of the hidden-charm pentaquarks $P_{c}$ via pion-induced reaction on a proton target is investigated within an effective Lagrangian approach. Three experimentally observed states, $P_c(4312)$, $P_c(4440)$, and $P_c(4457)$, are considered in the calculation, and the Reggeized $t$-channel meson exchange is considered as main background for the reaction $\pi ^{-}p\rightarrow J/\psi n$. The numerical results show that the experimental data of the total cross section of the reaction $\pi^{-}p\rightarrow J/\psi n$ at $W\simeq 5$ GeV can be well explained by contribution of the Reggeized $t$ channel with reasonable cutoff. If the branching ratios $Br[P_{c}\rightarrow J/\psi N]\simeq 3\%$ and $Br[P_{c}\rightarrow \pi N]\simeq 0.05\%$ are taken, the average value of the cross section from the $P_{c}(4312)$ contribution is about 1.2 nb/100 MeV, which is consistent with existing rude data at near-threshold energies. The results indicate that the branching ratios of the $P_{c}$ states to the $J/\psi N$ and $\pi N$ should be small. The shape of differential cross sections shows that the Reggeized $t$-channel provides a sharp increase at extreme forward angles, while the differential cross sections from the $P_{c}$ states contributions are relatively flat. High-precision experimental measurements on the reaction $\pi ^{-}p\rightarrow J/\psi n$ at near-threshold energies are suggested to confirm the LHCb hidden-charm pentaquarks as genuine states, and such experiments are also helpful to understand the origin of these resonance structures.


I. INTRODUCTION
Study of the exotic hadrons beyond the constituent quark model is an important way to understand how quarks combine to form a hadron. As of now, many hidden-charm exotic states have been observed and listed in the Review of Particle Physics (PDG) [1]. Different from the charmonium-like states, only three possible candidates for the hidden-charm pentaquarks were reported in the literature [2]. The studies of the hidden-charm pentaquarks in both experiment and theory are very important to understand the exotic hadrons.
In fact, after the observation in 2015, many theoretical interpretations of these structures were proposed [24][25][26][27][28][29][30]. The P c (4450) and P c (4380) are close to the Σ cD * and Σ * cD thresholds. It is very natural to assign them as two molecular states of Σ cD * and Σ * cD , respectively [29,31]. However, the opposite parities of these two states make it difficult to assign both states as S-wave molecular states. The new LHCb results shows that the previous P c (4450) structure should be composed of two peaks of the P c (4440) and P c (4457). It is reasonable to expect that puzzling spin parities are from the low precision of previous observation and will be changed with more data accumulated. Combined with the observation of the P c (4312) which is close to the Σ cD threshold, these three states can be naturally interpreted as the S-wave molecular states. There are only two S-wave Σ cD * state with spin parities 1/2 − and 3/2 − , and only one S-wave Σ cD state with 1/2 − . In fact, before the LHCb observation of the P c states, there have been many predictions of the hidden-charm pentaquarks [32][33][34], the calculations in which support the existence of such bound states. It is interesting to see that two states P c (4440) and P c (4457) were observed near Σ cD * threshold and one state P c (4312) was observed near Σ cD threshold. It was further supported by the theoretical studies after the new LHCb results released [4, 6-8, 10, 13], that is, the P c (4312) can be assigned as a S-wave Σ cD bound state with spin parity 1/2 − , and the P c (4440) and P c (4457) as S-wave Σ cD * bound states with spin parities 1/2 − and 3/2 − , respectively.
Although the molecular-state interpretation is quite consistent with the current LHCb observation, there are still many other proposals to understand the origin of these P c states [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]. Moreover, up to now, these hidden-charm pentaquarks were still only observed in the Λ b decay at LHCb. If these states is really composed of a charm quark pair and three light quark, it should be easy to be produced by striking a nucleon by a particle, such as nucleon, photon or pion, to excite a charm quark pair in the nucleon. Confirmation of the existence of these resonance structure in the photon-or pion-induced production is very important to establish the P c states as genuine particles. Hence, it is urgent to search for the hidden-charm pentaquark in such production processes.
The productions of the hidden-charm pentaquarks with nucleon target are proposed even before the LHCb's first observation of the hidden-charm pentaquarks P c (4450) and P c (4380). In Ref. [35] the hidden-charm pentaquark states were predicted and suggested to be looked for in the reaction pp → ppJ/ψ. In Ref. [38], the photoproduction of the hidden-charm pentaquark was first suggested to be applied at Jefferson Laboratory. Then, the productions of these predicted pentaquark state via pion-or kaon-induced reaction were calculated [36,37]. After the states P c (4450) and P c (4380) were experimentally observed, many theoretical calculations about these two pentaquarks produced in different processes appeared [39][40][41][42][43][44]. After new LHCb results was released, we also updated the predictions about the photoproductions of three P c states [45], which is accessible at JLab.
Another important way to study the hidden-charm pentaquarks is pion-induced production [36,39]. At present, there are some experimental data for the π − p → J/ψn reaction [46,47]. Furthermore, we note that P c (4312), P c (4440) and P c (4457) are all observed on the J/ψp invariant mass spectrum. In terms of experiments, the secondary pion beam is accessible at J-PARC [48,49] and COMPASS [50] with high intensity. Therefore, combining these experimental data to examine the role of these three states in pion-induced reaction is very necessary. The reaction mechanisms of the reaction π − p → J/ψn are illustrated in Fig. 1. These include the production of pentaquark P c states via s-and u-channel (as shown in Fig. 1 (a)−(b)), and t-channel π and ρ exchanges as depicted in Fig. 1 (c). Considering the off-shell effect of the intermediate P c states, the contribution from u-channel will be omitted. In this work, within the frame of an effective Lagrangian approach, the productions of P c states via pion-induced reaction on a proton target will be investigated. In the calculation, three hidden-charm pentaquarks, P c (4457), P c (4440), and P c (4312), which are assumed to carry spin parities 3/2 − , 1/2 − , and 1/2 − , respectively, will be considered in the calculation. The Reggeized treatment will be applied to t channel to describe the main background of the productions. This paper is organized as follows. After the Introduction, we present the formalism including Lagrangians and amplitudes of the P c states productions in Section II. The numerical results of the cross section follow in Section III. Finally, the paper ends with a brief summary.

A. Lagrangians
To gauge the contributions of Fig. 1, one needs the following Lagrangians for the s-channel P c exchanges [36,37,[51][52][53], where N, π, P c and ψ are the nucleon, the pion, the P c state and the J/ψ meson fields, respectively, and τ is the Pauli matrix. The values of g 1/2 − πNPc and g 3/2 − πNPc can be determined by the corresponding decay widths, with where λ is the Källen function with a definition of λ(x, y, z) = (x − y − z) 2 − 4yz. For the values of g 1/2 − P c ψN and g 3/2 − P c ψN , we have conducted relevant research discussions in our previous studies about the photoproduction of the P c states [45]. Obviously, the coupling constants of the g P c ψN and g πNP c are proportional to the corresponding values of decay width of P c states. In Table. I, we present the values of coupling constants by assuming the J/ψN and πN channels account for 3% and 0.05% of total widths of the P c states, respectively. For the t-channel via π and ρ exchanges, the effective Lagrangians read as where N, π, ρ and ψ are the nucleon, the pion, the ρ and the J/ψ meson fields, respectively. Here, the g 2 πNN /4π = 12.96, g ρNN = 3.36, and κ ρNN = 6.1 are adopted [40,54,55]. Moreover, The g ψππ = 8.2 × 10 −4 and g ψρπ = 3.2 × 10 −2 will be used in the calculations, which was mentioned in Refs. [40,56].

B. Amplitudes
According to above Lagrangians, the scattering amplitude of the reaction π − p → J/ψn can be written as where ǫ µ J/ψ is the polarization vector of the J/ψ meson, and u is the Dirac spinor of the nucleon.
The reduced amplitudes A i,µ for the s channel with each J P assignment of P c state and the t-channel are written as with where s = (k 1 + k 2 ) 2 and t = (k 1 − k 2 ) 2 is the Mandelstam variables. For the s-channel P c -state exchange, a general form factor is adopted to describe the size of hadrons, i.e. [43,57], where q s and m P c are 4-momentum and mass of the exchanged P c state, respectively. Considering that it is a heavier hadron production, the typical value of cut off Λ s = 0.5 GeV will be taken as used in Refs. [43,51]. For the t-channel meson exchanges [37,40,[57][58][59][60][61][62], the general form factor F t (q 2 t ) consisting of are taken into account. Here, q V and m V are 4-momentum and mass of the exchanged meson, respectively. The value of the cutoff Λ t will be discussed in the next section.

C. Reggeized t-channel
The Reggeized treatment is often adopted to analyze hadron production at high energies [37,57,[59][60][61][62][63][64]. It can be introduced by replacing the t-channel Feynman propagator by the Regge propagator as, where the scale factor s scale is fixed at 1 GeV. Moreover, the Regge trajectories of α π (t) and α ρ (t) read as [39,58,61], One can observe that no additional parameter is introduced after the Reggeized treatment is introduced,.

III. NUMERICAL RESULTS
After above preparation, the cross section of the reaction π − p → J/ψn can be calculated and compared with experimental data [46,47]. The differential cross section in the center of mass (c.m.) frame is written as where s = (k 1 + p 1 ) 2 , and θ denotes the angle of the outgoing J/ψ meson relative to the π beam direction in the c.m. frame. k c.m. 1 and k c.m.
2 are the three-momenta of the initial π beam and final J/ψ, respectively.
In this work, total and differential cross sections of the reaction π − p → J/ψn are calculated as presented in Figs. 2-4. For the total cross section of the reaction π − p → J/ψn, there are currently two experimental data points located near the energy threshold and at center of mass (c.m.) energy W ≃ 5 GeV, respectively. We found that it is difficult to meet both experimental data points if considering only the t-channel background contribution. It can be seen from Fig. 2 that the data points at W ≃ 5 GeV are well matched by the cross section of t-channel by taking a cutoff Λ t = 2 GeV. However, at the same time, the data point near the threshold is more than an order of magnitude larger than the theoretical value of t-channel contribution. If we consider the contribution from the s-channel P c state, the data point near the threshold can be well explained. As shown in Fig. 2, one find that experimental data point near the threshold is consistent with the contribution from the P c (4312) state by assuming branching ratios Br[P c → J/ψN] ≃ 3% and Br[P c → πN] ≃ 0.05%. Due to adoption of the Regge propagator, we find that the cross section of the t channel reaches a maximum at W ≃ 5 GeV, and the total cross section decreases as the energy increases. If the Feynman propagator is adopted, the total cross section from t channel would become larger and larger with the increase of the c.m. energy. The difference between the Regge model and the Feynman model will help to clarify the role of Regge propagator in the future experiment.  . 2. (Color online) Total cross section for the reaction π − p → J/ψn. The black dashed, dark yellow dotted, green dot-dashed, blue dash-double-dotted, and red solid lines are for the background, the P c (4312), the P c (4440), the P c (4457) and total contributions, respectively. The bands stand for the error bar of the cutoff Λ t . The experimental data are from Refs. [46,47].
In order to distinguish the contributions from the three P c states more clearly, the Fig. 3 is presented, which is the same as the Fig. 2 except that the energy range is reduced. From  Fig. 3, one can see three distinct peaks, which are from the contributions of three P c states. As we discussed in our previous work about the photoproduction of the P c states [45], the P c (4312) can be observed within a bin of 0.1 GeV. But if one wants to distinguish two peaks from the P c (4440) and P c (4457), a bin at least at an order of 10 MeV is required. According to our calculation by assuming the branching ratios Br[P c → J/ψN] ≃ 3% and Br[P c → πN] ≃ 0.05%, if the width of a bin is 0.1GeV, the theoretical average value of the cross section from the P c (4312) contribution is about 1.2 nb in a bin interval, which is just in agreement with the experimental value near the threshold.
In Fig. 4, we present our prediction of the differential cross section of the reaction π − p → J/ψn at different c.m. energy. It  can be seen that the differential cross section has a large contribution at forward angles, which is caused by the Reggeized t-channel. In addition, we find that the shape of differential cross section tends to be flat at the c.m. energy W = 4.312, 4.44 and 4.457 GeV, which is due to the large contributions of the P c state at these energy points. Since the spin-parity quantum numbers of these P c states are selected to be 1/2 − or 3/2 − , these P c states can couple to both the initial πN and final J/ψN in S wave, and the couplings by higher partial waves can be ignored because the momentum between the final J/ψN is very small. Therefore, the shape of the differential cross section of these P c states is relatively flat, which reflects the characteristics of the S-wave coupling.

IV. SUMMARY AND DISCUSSION
We have studied the reaction π − p → J/ψn within the Regge model. The numerical results show that the experimental data near the threshold can be well explained if we consider the contribution from the pentaquark states. In addition, numerical results also indicate the experimental data at W ≃ 5 GeV is unlikely to come from the contribution of the P c state but should be caused by the t-channel background contribution.
At present, the branching ratios of P c decay to J/ψN and πN are still undetermined, but if the branching ratios Br[P c → J/ψN] ≃ 3% and Br[P c → πN] ≃ 0.05% are taken, then the average value of the cross section from the P c (4312) contribution is about 1.2 nb/100 MeV, which coincides with the experimental data point near the threshold. Combined with the results in our previous article about the photoproduction of the P c states [45], it is reasonable to think that the branching ratios of P c states to J/ψN and πN should be relatively small. Therefore, we suggest that experiments with high precision near the threshold can be performed, which is very important for determining the branch ratios and the internal structure of the P c states.
The differential cross sections for the reaction π − p → J/ψn are also calculated. One notices that the Reggeized t channel is very sensitive to the θ angle and gives considerable contributions at forward angles. On the contrary, the shape of cross section from the P c states contribution is relatively flat, and it is related to the spin-parity quantum numbers of these P c states, which can be checked by future experiment and may be an effective way to examine the validity of the Reggeized treatment and spin parities of these P c states.
J-PARC and COMPASS can generate pion beam covering the above energy regions, and provide high-precision experimental data. Our theoretical results will provide valuable reference information for the studies of the pentaquark states at these facilities.