Study of $P_c(4457)$, $P_c(4440)$, and $P_c(4312)$ in a quasipotential Bethe-Salpeter equation approach

Very recently, the LHCb Collaboration reported their new results about the pentaquarks at charmed energy region. Based on the new experimental results, we recalculate the molecular states composed of a $\Sigma_c^{(*)}$ baryon and a $\bar{D}^{(*)}$ meson. Two peak structures at about 4450 MeV can be interpreted as $\Sigma_c \bar{D}^*$ bound states with spin parities $1/2^-$ and $3/2^-$. The newly observed pentaquark $P_c(4312)$ can be assigned as a $\Sigma_c\bar{D}$ bound state with spin parity $1/2^-$. The experimental determination of spin parities will be very helpful to understand the internal structure of these pentaquarks.


Introduction
In 2015, the LHCb reported their first observation of the pentaquarks, P c (4450) and P c (4380), which carry puzzling opposite parities and large spin 5/2 for the P c (4450) [1].The masses of P c (4450) and P c (4380) are very close to the Σ c D * and Σ * c D thresholds, respectively.It is very alluring to assign those two pentaquark as corresponding molecular states.However, the puzzling parties force us to put one of the state in P wave if we accept the suggestion of the experimentists.Very recently, the LHCb Collaboration reported their new results about the pentaquarks at the same energy region [2].It is very interesting to see that the P c (4450) split into two peaks, P c (4457) and P c (4440) based on more accumulated data, and a new pentaquark near Σ c D threhsold was observed and named as P c (4312).
The new result is very helpful to deep our understanding of the hidden-charmed pentaquarks.Two higher states P c (4457) and P c (4440) correspond to original P c (4450) obviously.Though the partial wave analysis is on progress and their spin parities are not provided in the report, it is not surprising that the spin parity will be different from the a junhe@njnu.edu.cnsuggestied values in the previous experimental article [1].Besides, the two-peak structure suggests that the large spin 5/2 for the P c (4450) may be due to take two peaks as one.Hence, these two peaks may be related in to two Σ c D * molecular states with smaller spin, such as 1/2 and 3/2.
In fact, in early predictions about the hidden charmed pentaquarks in the molecular state picture, the Σ c D * bound states were predicted with spin parities 1/2 − and 3/2 − [3,4,5].For example, in our early work [4], two isovector Σ c D * bound state were predicted with 1/2 − and 3/2 − in the oneboson exchange model by solving the Schödinger equation.However, the best suggested spin parity of the P c (4450) as 5/2 + gave a blow to such assignment.To interpret such spin parity, in our previous work [6], we assign the S-wave 3/2 − Σ c D * bound state to the P c (4380) and the P c (4450) as a Pwave Σ c D * bound state [7].Even without try to give the opposite parities, the P c (4450) was assigned as a deeplybound Σ * D * state and the 3/2 − Σ c D * state is left to interpret the P c (4380) in Ref. [8].In their new work [9], the Σ ( * ) D ( * ) system were restudied in the one-boson-exchange model by solving the Schödinger equation.Three pentaquarks can be plausibly reproduced as the molecular states near their corresponding thresholds with reasonable cutoffs.
Based on the new results, as discussed above, we can expect that the P c (4457) and P c (4440) have smaller spins.Moreover, the observation of the P c (4312) near the Σ c D threshold in the new experiment confirms the idea that these pentaquarks should be the molecular states if we notice that all three states are very narrow, which is a character of the bound state.In this work, we will study the Σ ( * ) D( * ) system in a coupled-channel quasipotential Bethe-Salpeter approach.We expect that with smaller cutoff the Σ c D * abound state with 3/2 − in our previous work will move to its threshold and the states with spin 5/2 will vanish.And the existence of P c (4312) will be also studied in our approach.This work is organized as follows.After introduction, the detail of the dynamical study of coupled-channel Σ ( * ) D( * ) interactions will be presented, which includes relevant effective Lagrangian.The numerical results are given in Section 3. Finally, summary and discussion will be given.

D( * ) interactions and relevant Lagrangians
In this work, we will consider coupled-channel interactions between a charmed baryon and an anti-charmed meson as D * , which is described by the light meson exchanges.We need to construct effective Lagrangians depicting the couplings of light mesons and anti-charmed mesons or charmed baryons.
In terms of heavy quark limit and chiral symmetry, the couplings of light mesons to heavy-light charmed mesons P = ( D0 , D − , D − s ) are constructed as [10,11,12,13], L P Pσ = −2g s m P P † a Pa σ, L P * P * σ = 2g s m P * P * † a P * a σ.
(2) where constant f π = 132 MeV and P and V are the pseudoscalar and vector matrices The effective Lagrangians depicting the charmed baryons with the light mesons with chiral symmetry, heavy quark limit and hidden local symmetry read [11,14], where the partial ← → ∂ operates on the initial and final baryons and B matrix is To constrain the Lagrangians, The coupling constants should be determined.The values used in the calculation are listed in Table .1, which are from the literature [9,14,15,16].
Table 1 The parameters and coupling constants adopted in our calculation.The λ and λ S are in the unit of GeV −1 .Others are in the unit of 1.
0.9 0.59 5.8 0.56 0.76 -1.74 6.2 0.94 -3.31 With the above Lagrangians, the vertices for the heavy meson and the exchanged meson can be obtianed.With those vertices and the propagator of the exchanged meson, the interaction can be described.Here the propagators read, The vetices and the propagator will be input in to the code directly, hence, we do not give their explicit forms here.As our previous work [6], the Bethe-Saltpeter equation approach with a spectator quasipotential approximation, which was explained explicitly in the appendices of Ref. [17], to search the possible bound states.A bound state from the interaction corresponds to a pole of the scattering amplitude M which is described by potential kernel obtained in the above.The Bethe-Saltpeter equation for partial-wave amplitude with fixed spin-parity J P reads [17], ) Note that the sum extends only over nonnegative helicity λ ′′ .The partial wave potential is defined as where the initial and final relative momenta are chosen as p = (0, 0, p) and is the Wigner d-matrix.In this work we will adopt an exponential regularization by introducing a form factor in the propagator as with k 1 and m 1 being the momentum and mass of the charmed meson.The interested reader is referred to Ref. [17] for further information about the regularization.
A monopole form factor is introduced to compensate the off-shell effect of exchange meson as f (q 2 ) = e −q 2 with cutoff Λ e = m + α0.22 GeV.Here m and q are the mass and momentum of the exchanged light meson.In this work, we fix the α = 2.
After transforming the integral equation to a matrix equation, the pole of scattering amplitude M can be searched by variation of z to satisfy |1 − V(z)G(z)| = 0 with z = M + iΓ/2 equaling to the system energy M at the real axis [17].

Numerical results
In this work, the only free parameter is the cutoff in the exponential regularization.We will vary the cutoff Λ from 0.8 to 1.8 GeV to search for the poles which correspond to the bound states from the interactions.The coupled-channel D * interaction is considered in the current calculation.three poles can be found and listed in Tables 2 and 3.
Table 2 The position of the bound states with J P = 1/2 − in a unit of MeV.The Λ in a unit of GeV is the cutoff in the exponential regularization.The CC means coupled-channel calculation and Σ c D * and Σ c D mean the corresponding single-channel calculation.Form Table 2, one can find that two poles can be produced from the coupled-channel interaction.The upper state has a mass of about 4440 MeV, and the mass will decrease with the increase of the cutoff.The lower state has a mass of about 4320 MeV, which is considerably stable with the variation of the cutoff.For comparison, we also provide the results with single-channel calculation for Σ c D * and Σ c D interaction.Generally speaking, the coupled-channel effect is considerable small, which is consistent with the results for coupled-channel Σ * K − ΣK * interaction [18].The results suggest that the upper and lower states from the coupled-channel calculation can be taken as the Σ c D * and Σ c D molecular states.In the case of spin parity 3/2 − , only one state is found at about 4460 MeV as shown in Table 3.We also compare the coupled-channel results with the single-channel Σ c D * calculation.The results suggest this state with 3/2 − is mainly from the Σ c D * interaction.As the two 1/2 − states, this state is also not sensitive on the cutoff.

Summary and discussion
Inspired by new LHCb results about the hidden-charmed pentaquarks.We reanalyze the Σ ( * ) D( * ) interaction in the quasipotential Bethe-Salpeter equation approach.In our previous work [6,7], the P c (4450) is interpreted as a 5/2 + Σ c D * state.In the current work, we persist in the assumption that the hidden-charmed pentaquarks are from the molecular state with the nearest threshold, that is, two new pentaquarks about 4450 MeV are still expected to be the Σ c D * molecular states as we did for P c (4450) while we do not force them to have a large spin 5/2.The current results suggest that two states with spin parities 1/2 − and 3/2 − can be produced from the coupled-channel calculation and mainly from the Σ c D * interaction, which can be assigned as the P c (4440) and P c (4457), respectively.It is consistent with our original work in Ref. [4] and recent calculation in Ref. [9].
The newly observed pentaquark P c (4312) can be assigned as an 1/2 − Σ c D molecular state, which is the lower state in our coupled-channel calculation.In the current work, no bound state from Σ * D interaction can be found with the cutoff adopted.The P c (4380) is a broad state in the old experimental analysis.Hence, it is possible that the P c (4380) is not a molecular state near the corresponding threshold.
The report by LHCb collaboration does not provide the information of the spin parities of these three pentaquarks.So, other interpretations can not be excluded directly.For example, the two upper states can still be explained as 5/2 + and 5/2 − Σ c D * molecular state as suggested in Refs.[6,7], and the P c (4312) is a deeply bound Σ * D state.Hence, the partial wave analysis in experiment is essential to understand the origin and the internal structure of these pentaquarks.

Table 3
The position of the bound states with J P = 3/2 − in a unit of MeV.The Λ in the unit of GeV is the cutoff in the exponential regularization.The CC means coupled-channel calculation and Σ c