Molecular nature of P_{cs} (4459) and its heavy quark spin partners
aa r X i v : . [ h e p - ph ] F e b Molecular nature of P cs (4459) and its heavy quark spin partners C. W. Xiao, ∗ J. J. Wu, † and B. S. Zou
3, 2, 1, ‡ School of Physics and Electronics,Central South University, Changsha 410083, China School of Physical Sciences, University of ChineseAcademy of Sciences (UCAS), Beijing 100049, China Key Laboratory of Theoretical Physics, Institute of Theoretical Physics,Chinese Academy of Sciences, Beijing 100190, China (Dated: February 9, 2021)
Abstract
Inspired by the observation of the P cs (4459) state by LHCb recently, we reexamine the resultsof the interaction of the J/ψ
Λ channel with its coupled channels, exploiting the coupled channelunitary approach combined with heavy quark spin and local hidden gauge symmetries. By tuningthe only free parameter, we find a pole of (4459 .
07 + i .
89) MeV below the ¯ D ∗ Ξ c threshold, whichwas consistent well with the mass and width of the P cs (4459) state. Thus, we assume the P cs (4459)state to be a ¯ D ∗ Ξ c bound state with the uncertainties on its degeneracy with J P = − and J P = − . For the degeneracy, it would have two-poles structure, like P c (4450) before. Thereis another pole in the J P = − sector, (4310 .
53 + i .
23) MeV, corresponding to a deep boundstate of ¯ D Ξ c . Furthermore, the previously predicted loose bound states of ¯ D Ξ ′ c , ¯ D ∗ Ξ ′ c , ¯ D ∗ Ξ ∗ c with J = 1 / , I = 0 and ¯ D ∗ Ξ ′ c , ¯ D Ξ ∗ c , ¯ D ∗ Ξ ∗ c with J = 3 / , I = 0 may exist as either bound statesor unbound virtual states. We hope that future experiments can search for the ¯ D ( ∗ ) Ξ c molecularstates in their dominant decay channels of ¯ D ( ∗ ) s Λ c , also in the J/ψ
Λ and η c Λ channels to revealtheir different nature. ∗ [email protected] † [email protected] ‡ [email protected] . INTRODUCTION Early in 2010, with the coupled channel unitary approach (CCUA) [1–3] and the localhidden gauge (LHG) formalism [4–7] combined with SU(4) symmetry, several hidden charmand hidden charm strangeness resonances were predicted around the energy range of 4200MeV – 4600 MeV in Ref. [8], where the possible decay channel of η c N or J/ψN was suggestedfor searching for the hidden charm ones in the experiments and its further investigation wasgiven in detail in Ref. [9]. After that, the pentaquark states caught the theoretical interestonce again and had many predictions in the hidden charm sector in Refs. [10–17]. In 2015, theLHCb Collaboration reported two pentaquark-like resonances found in the
J/ψp invariantmass distributions of the Λ b → J/ψK − p decay [18, 19], denoted as P c (4380) + with a largewidth about 205 MeV and P c (4450) + with a small width of 39 MeV, which were confirmedby a model-independent re-analysis of the experimental data [20] and in the Λ b → J/ψpπ − decay [21]. Furthermore, in 2019, with the Run-2 data the LHCb Collaboration updatedthe new results for the P c states as three clear narrow structures [22], M P c = (4311 . ± . +6 . − . ) MeV , Γ P c = (9 . ± . +3 . − . ) MeV ,M P c = (4440 . ± . +4 . − . ) MeV , Γ P c = (20 . ± . +8 . − . ) MeV ,M P c = (4457 . ± . +4 . − . ) MeV , Γ P c = (6 . ± . +5 . − . ) MeV , where the P c (4450) was split into two states of P c (4440) and P c (4457), in addition to a newnarrow resonance P c (4312). Whereas, the broad old one P c (4380) could neither be confirmednor refuted in the updated results.In principle, due to the fact that the P c states were found in the experiments, the P cs states as their strangeness partners should also exist as predicted in Refs. [8, 9] from SU(4)flavour symmetry. Note that, the first predictions for the masses and the widths of themolecular resonances in Refs. [8, 9] were due to the lack of the experimental informations todetermine the only free parameter of a µ ( µ is not an independent one [3, 23, 24]) in the loopfunctions. Therefore, after the updated results of LHCb Collaboration [22] became available,this free parameter was fitted as a µ ( µ = 1 GeV) = − .
09 in Ref. [25] with the masses ofthree P c states based on the former work of [15], where seven hidden charm molecular stateswere predicted with the CCUA combined with the LHG symmetry and the heavy quark spinsymmetry (HQSS) [26–28]. Inspired by the experimental findings of the P c states, with thesame fitted parameter of a µ from the P c states, and also combined with the LHG symmetryand the HQSS, several bound states of ¯ D ( ∗ ) Ξ ( ∗ ) c and ¯ D ( ∗ ) Ξ ′ c were predicted in Ref. [29], wheresome of them were analogous to the ones in Refs. [8, 9] and seeking for them in the reactionof Ξ − b → J/ψ Λ K − was commented at the end as also suggested in the work of [30, 31]. Thepossibility to look for them in the Λ b decays was also discussed in the early work of [32, 33].Furthermore, the decay properties of the hidden charm strangeness resonances predicted inRefs. [8, 9] were investigated in detail in Ref. [34], where the partial decay widths of possibledecay channels were obtained by exploiting the effective Lagrangian framework throughthe triangle loops. On the other hand, with the one-boson-exchange model, the work ofRef. [35] also predicted that the possible ¯ D ∗ s Σ ( ∗ ) c and ¯ D ∗ Ξ ( ′ , ∗ ) c pentaquark states existed.Taking into account the spin-flavour symmetric states, the possible SU f (3) multiplets forthe charmonium compact pentaquark states were obtained in Ref. [31], where the possibledecay channels and the partial decay widths were discussed for these predictions. Using thechiral effective field theory up to the next-to-leading order, Ref. [36] found ten bound states2n the hidden charm strangeness systems of ¯ D ( ∗ ) Ξ ′ c and ¯ D ( ∗ ) Ξ ( ∗ ) c with different spin, where amass difference of about 6 MeV for two ¯ D ∗ Ξ c molecular states with spins J = and J = was obtained and looking for these P cs states in the decay channel of J/ψ
Λ was suggested.Just recently, the LHCb Collaboration had reported the results of Ξ − b → J/ψ Λ K − decayin Ref. [37], where a resonance structure of P cs (4459) state was found in the invariant massdistributions of J/ψ
Λ, given as M P cs = (4458 . ± . +4 . − . ) MeV , Γ P cs = (17 . ± . +8 . − . ) MeV , which is just about 19 MeV below the ¯ D ∗ Ξ c threshold. Note that, analogous to the de-generate P c (4450) state, the possible hypothetical structure as predicted in Refs. [29, 36]with different spins was unsure for the current data sample. The two-poles structure wasalso suggested in the results of the QCD sum rules in Ref. [38], where the P cs (4459) statewas assumed to be a ¯ D ∗ Ξ c molecular state with spin-parity J P = − or J P = − and a¯ D Ξ c bound state at 4 . +0 . − . GeV with J P = − was found too. Also with the QCD sumrules, Ref. [39] assigned the P cs (4459) state as a hidden-charm compact pentaquark statewith J P = − , which was confirmed in the later work of [40, 41]. On the other hand, with acombined effective field theory and phenomenological assumptions, the work of [42] proposedthat the P cs (4459) state could be a ¯ D ∗ Ξ c molecular pentaquark state with J P = − morelikely, or possibly J P = − with more uncertainties on its mass. Using a coupled channelanalysis based on the one-boson-exchange model, Ref. [43] concluded that the P cs (4459)state was not a pure ¯ D ∗ Ξ c molecular state and other possible resonances of ¯ D ( ∗ ) Ξ ( ′ , ∗ ) c couldexist, and its decay behavior was further discussed in Ref. [44]. In Ref. [45], the existenceof the ¯ D ( ∗ ) Ξ ( ′ , ∗ ) c molecular states were also found under the effective field theory by takinginto account the HQSS and SU(3) flavor symmetry, which were the SU(3)-flavour partnersof the ¯ D ( ∗ ) Σ ( ∗ ) c molecular states. With the quasipotential Bethe-Salpeter equation approach,Ref. [46] assigned the P cs (4459) state as the ¯ D ∗ Ξ c molecular state with J P = − and pre-dicted other ¯ D ( ∗ ) Ξ ( ′ , ∗ ) c states with different spins. Furthermore, the decay of Λ b → J/ψ Λ φ was suggested to search for the P cs (4459) state in Ref. [47].Motivated by the new findings of Ref. [37], we reexamine the results of Refs. [8, 9, 29]to understand the differences between our predictions and experimental data, then fix themodel parameters to make further predictions. II. FORMALISM
There are various approaches dealing with hadronic molecules as recently reviewed inRefs. [48, 49]. The LHG formalism seems working well to give a general consistent explana-tion for various observed hadronic molecular candidates with hidden charm [50, 51]. In Ref.[29], using the CCUA with LHG formalism, we considered the coupled channels of the
J/ψ
Λchannel, where there were nine channels of η c Λ, J/ψ
Λ, ¯ D Ξ c , ¯ D s Λ c , ¯ D Ξ ′ c , ¯ D ∗ Ξ c , ¯ D ∗ s Λ c , ¯ D ∗ Ξ ′ c ,¯ D ∗ Ξ ∗ c in the J P = − , I = 0 sector, and six channels of J/ψ
Λ, ¯ D ∗ Ξ c , ¯ D ∗ s Λ c , ¯ D ∗ Ξ ′ c , ¯ D Ξ ∗ c ,¯ D ∗ Ξ ∗ c in the J P = − , I = 0 sector. In addition, a single channel of ¯ D ∗ Ξ ∗ c with J P = − was also found in s wave within the HQSS, which was not taken into account in our worksince it could not couple to the J/ψ
Λ channel in s wave. Note that the interaction of the¯ D ∗ Ξ ∗ c , which can be specified with spin J = , , , was included in the coupled channelsin Ref. [29] under the constraint of the HQSS, and not considered in Refs. [8, 9].3 ABLE I. Potential matrix elements V ij of Eq. (1) for the J = 1 / , I = 0 sector. η c Λ J/ψ
Λ ¯ D Ξ c ¯ D s Λ c ¯ D Ξ ′ c ¯ D ∗ Ξ c ¯ D ∗ s Λ c ¯ D ∗ Ξ ′ c ¯ D ∗ Ξ ∗ c µ − µ − µ µ √ µ √ µ µ √ q µ µ √ µ √ µ µ √ µ µ µ − √ µ µ µ µ √ − q µ µ µ √ − q µ (2 λ + µ ) µ √ µ √ − λ − µ )3 √ q ( µ − λ ) µ µ
23 2 µ √ µ µ µ √ µ (2 λ + 7 µ ) √ λ − µ ) ( λ + 8 µ ) Within the CCUA, the scattering amplitudes ( T ) are evaluated by the coupled channelBethe-Salpeter equation with the on-shell prescription, T = [1 − V G ] − V, (1)where G is constructed by the loop functions with meson-baryon intermediate states and V are the potentials of the coupled channel interactions. Note that, G is a diagonal matrixwith elements of meson-baryon loop functions, where we take the ones with the dimensionalregularization, see more details in Refs. [8, 9]. Thus, the free parameter is the only one of a µ , see the discussions later. Respecting the HQSS, the elements of the potential V matrixare given in Tables I and II for the J = 1 / , I = 0 and J = 3 / , I = 0 sectors, respectively,where we only show V ij for j ≥ i for simplicity due to the fact that V ji = V ij in the CCUA.In Tables I and II, the coefficients µ i , µ ij ( i, j = 1 , , ,
4) and λ are the unknown low energyconstants with the HQSS constraint, see more details in Ref. [29].Using the LHG formalism, we obtain the values of these low energy constants [29], µ = µ = µ = µ = 0 (2) µ = µ / √ µ = λ = − F, F = 14 f ( p + p ′ ) (3) µ = − µ / √ µ / √ − r m V m D ∗ F, (4)with f π = 93 MeV and m V = 800 MeV, where p and p ′ are the energies of the incomingand outgoing mesons in a certain channel. Note that, we have explicitly taken the reduc-tion factor m V /m D ∗ in the matrix elements that involve the transition processes with theexchange of D ∗ meson. The null µ and µ values are due to the pion exchange neglect inour formalism, as done in Ref. [15], see more discussions in Ref. [24].4 ABLE II. Potential matrix elements V ij of Eq. (1) for the J = 3 / , I = 0 sector. J/ψ
Λ ¯ D ∗ Ξ c ¯ D ∗ s Λ c ¯ D ∗ Ξ ′ c ¯ D Ξ ∗ c ¯ D ∗ Ξ ∗ c µ µ µ − µ µ √ √ µ µ µ − µ µ √ √ µ µ − µ µ √ √ µ (8 λ + µ ) λ − µ √ √ ( λ − µ ) (2 λ + µ ) q ( µ − λ ) (4 λ + 5 µ ) III. RESULTS AND DISCUSSIONS
As discussed above, the subtraction constant a µ in the meson-baryon loop functions is afree parameter in our formalism, and thus, we cannot get a precise value for it theoretically.It is even worse that the prediction will significantly rely on its value for a loose boundsystem, since its value will influence the lowest strength of attractive potential to form abound state (we will discuss this issue in detail later). In practice, the only way to getits accurate value is using some experimental data to fix it. Therefore, using the newestexperimental results of [37], the value of a µ can be determined as a µ ( µ = 1 GeV) = − . P cs case. It is similar to what we have done in Ref. [25] for the P c case, where avalue of a µ ( µ = 1 GeV) = − .
09 was obtained. In the early prediction of Refs. [8, 9], thecentral value of a µ ( µ = 1 GeV) = − . . ρ and ω . Indeed, now the valuesof a µ ( µ = 1 GeV) = − .
94 and the one of a µ ( µ = 1 GeV) = − .
09 [25] are really aroundthe “natural values” of a µ = − a µ ( µ = 1 GeV) = − .
94, we obtain the results of the modulussquared of the amplitudes in Figs. 1 and 2 for J = 1 / , I = 0 and J = 3 / , I = 0,respectively, where the peak structures are analogous to the ones obtained in Ref. [29] butmore narrow and with higher energies. The corresponding poles and its coupling constantsto all the channels are given in Tables III and IV for J = 1 / , I = 0 and J = 3 / , I = 0,respectively, where the thresholds of each channel are shown accordingly. In Tables III andIV, with the couplings obtained, the partial decay widths and the branching ratios for eachchannel are evaluated. One thing should be mentioned, that the poles given in Tables IIIand IV are the ones located in the general “second Riemann sheet”, which means that allthe channels below the certain bound channel are extrapolated to the second Riemann sheet(these channels are always called the open channels), whereas, the other coupled channelsincluding the bound channel are in the first Riemann sheet. As found from our results of There is an extra pole around (4291 .
05 + i .
80) MeV in the ¯ D ∗ Ξ c channel, see the left panel of Fig. 1,which couples strongly to ¯ D ∗ s Λ c channel (threshold 4398 .
66 MeV) and looks like unnormal. It is due to thefact that the G functions for the channels of ¯ D ∗ Ξ c and ¯ D ∗ s Λ c (denoted as channel 6 and 7, respectively)become positive far below their thresholds, which also lead to the “effective” potential of V + V G changing to a positive one (see the discussion later). Thus, a repulsive potential lead to a bound stateunusually, see more discussions in Refs. [15, 52]. D ∗ Ξ c is tuned as 4459 MeV to make it consistent withthe mass of observed P cs state, which has increased about 30 MeV with respect to the oneobtained in Ref. [29]. Then the pole of the ¯ D ∗ Ξ c channel is (4459 .
07 + i .
89) MeV, where thewidth is quite consistent with the experimental results [37], and it just has 3 MeV difference.Note that, owing to the pion exchange neglect in our formalism as discussed above, this poleof the ¯ D ∗ Ξ c channel is degenerate with spins J = 1 / J = 3 /
2, as shown in Tables IIIand IV. Therefore, one can conclude from our formalism that the P cs (4459) state can be abound state of ¯ D ∗ Ξ c with spin uncertainty of J = 1 / J = 3 /
2. Besides, in Table IIIfor the J = 1 / , I = 0 sector, we have another very stable pole, (4310 .
53 + i .
23) MeV,which is bound by the ¯ D Ξ c channel and has 56 MeV binding energy. On the other hand,other three peak structures are all very close to the corresponding thresholds. The peakaround 4445 MeV is contributed from the pole at (4445 .
12 + i .
19) MeV bound by the ¯ D Ξ ′ c channel, of which the binding energy is just 0.23 MeV. Thus, in our model this bound statebecomes unstable if the parameter a µ changes a little to move it to the threshold, whichwill cause the threshold effect and then lead to no pole in the general second Riemannsheet, as shown in Tables III and IV for some channels. Furthermore, the other two peakstructures are actually due to the poles from the other Riemann sheets. Thus, these polescan not be recognized as normal bound states of the certain channels. It is proper to saythat these two peak structures are the threshold effects because of very weak attractiveinteraction potentials. Indeed, the three poles for the channels of ¯ D Ξ ′ c , ¯ D ∗ Ξ ′ c and ¯ D ∗ Ξ ∗ c werejust loosely bound as found in the results of Ref. [29], where these poles were only a fewMeV below the corresponding thresholds and had narrow widths, a few MeV. Similarly, inTable IV for J = 3 / , I = 0 sector, except for the one of ¯ D ∗ Ξ c stable, the other three polesof the channels ¯ D ∗ Ξ ′ c , ¯ D Ξ ∗ c and ¯ D ∗ Ξ ∗ c are not stable for the same reason. Three of themwere also loosely bound as given in the results of Ref. [29] with a few MeV for the bindingenergies and the widths.As shown in Tables III and IV, the degenerate molecular states of ¯ D ∗ Ξ c with J = 1 / J = 3 / .
07 + i .
89) MeV. Their main decaychannel is ¯ D ∗ s Λ c with the branch ratio larger than 80%, which can be understood as thischannel coming from the strong interaction by exchanging light vector meson in our model,and its large decay width was also found in Ref. [44]. But, the branch ratios of the J/ψ
Λchannel in the two cases are a little different because of the Clebsch-Gordan coefficientsin the HQSS model. Besides, the η c Λ channel is the only decay channel of the boundstate with J P = 1 / − , while it requires the d wave interaction for the J P = 3 / − state,which is neglected in our model. Therefore, if we assume the ¯ D ∗ Ξ c bound state to be the P cs (4459) state, it may have two-poles structure with different spins, like the one of P c (4450)before, and it can be helpful to reveal their nature with more experimental information inthe future for the partial decay widths and the branching ratios of the channels J/ψ
Λ and η c Λ. In Fig. 3, we show our results of Fig. 1 for J = 1 / , I = 0 compared with theexperimental data, where the full data are compared on the left and the parts with the cutof 1 . < m Λ K − < . J/ψ
Λ invariant mass distributions.For the results of the partial decay widths and the branching ratios shown in Table III, itseems strange that the bound state of ¯ D Ξ c with its pole at (4310 . i .
23) MeV decays morestrongly to the
J/ψ
Λ channel than to the η c Λ, whereas, the molecular state of ¯ D ∗ Ξ c withthe pole (4459 .
07 + i .
89) MeV is just the opposite, which is different from the cases of the6
ABLE III. Coupling constants to all channels for certain poles in J = 1 / , I = 0 sector.Chan. η c Λ J/ψ
Λ ¯ D Ξ c ¯ D s Λ c ¯ D Ξ ′ c ¯ D ∗ Ξ c ¯ D ∗ s Λ c ¯ D ∗ Ξ ′ c ¯ D ∗ Ξ ∗ c Thres. 4099.58 4212.58 4366.61 4254.80 4445.34 4477.92 4398.66 4586.66 4654.484310 .
53 + i . | g i | .
15 0 . . .
69 0 .
00 0 .
04 0 .
09 0 .
01 0 . i .
57 1 . − . − − − − − Br. 3.47% 7.16% − − − − − − .
12 + i . | g i | .
10 0 .
06 0 .
00 0 . . .
08 0 .
04 0 .
01 0 . i .
29 0 .
08 0 .
00 0 . − − . − − Br. 74.74% 21.22% 0.01% 0.01% − − − − .
07 + i . | g i | .
22 0 .
13 0 .
00 0 .
00 0 . . .
61 0 .
03 0 . i .
59 0 .
46 0 .
00 0 .
00 0 . − . − − Br. 11.57% 3.31% 0.00% 0.00% 0.70% − − − . | g i | − − − − − − − − − . | g i | − − − − − − − − − TABLE IV. Coupling constants to all channels for certain poles in J = 3 / , I = 0 sector..Chan. J/ψ
Λ ¯ D ∗ Ξ c ¯ D ∗ s Λ c ¯ D ∗ Ξ ′ c ¯ D Ξ ∗ c ¯ D ∗ Ξ ∗ c Thres. 4212.58 4477.92 4398.66 4586.66 4513.17 4654.484459 .
02 + i . | g i | . . .
61 0 .
02 0 .
04 0 . i . − . − − − Br. 14.68% − − − − . | g i | − − − − − − . | g i | − − − − − − . | g i | − − − − − − | T | [ M e V - ] √ s [MeV] |T D Ξ c | |T D * Ξ c | |T D * Ξ ’ c | |T D * Ξ c* | | T | [ M e V - ] √ s [MeV] |T D Ξ ’ c | FIG. 1. Results of the modulus squared of the amplitudes for J = 1 / , I = 0 sector. | T | [ M e V - ] √ s [MeV] |T D * Ξ c | |T D Ξ c* | |T D * Ξ c* | | T | [ M e V - ] √ s [MeV] |T D * Ξ ’ c | FIG. 2. Results of the modulus squared of the amplitudes for J = 3 / , I = 0 sector. ¯ D ( ∗ ) Σ c bound states studied in Ref. [24]. This is because of the constraint of the HQSS, seethe transition elements of µ and √ µ for the related channels in Table I. But, in Ref. [34]the bound state of ¯ D ∗ Ξ c predicted in Refs. [8, 9] was found to decay stronger to the J/ψ
Λchannel than the η c Λ. Indeed, the earlier work of [8, 9, 34] did not consider the HQSS aswell as the transition between Pseudoscalar-Baryon (PB) and Vector-Baryon(VB) channels.However, after taking into account the HQSS, the interactions between PB and VB arealmost the same except for the Clebsch-Gordan coefficients. On the other hand, the boundstates of ¯ D Ξ c and ¯ D ∗ Ξ c decay mostly to the channels of ¯ D s Λ c and ¯ D ∗ s Λ c , respectively, whichhave quite large partial decay widths and the branching ratios (see the results in Tables IIIand IV), whereas, the cases of the ¯ D ( ∗ ) Σ c bound states [24] cannot decay to the channels¯ D ( ∗ ) Λ c with the vector meson exchange. The reason is that the Λ c particle is in the samespin 1/2 antitriplet (the ¯3 multi-states) with the Ξ c particle in the SU(3) flavour symmetry,but not with the Σ c particle (belonging to the 6 multi-states).As discussed above, when we take the new value of a µ ( µ = 1 GeV) = − .
94 for theonly free parameter in the loop functions, the loose bound systems of ¯ D Ξ ′ c , ¯ D ∗ Ξ ′ c , ¯ D ∗ Ξ ∗ c in J = 1 / , I = 0 sector, and ¯ D ∗ Ξ ′ c , ¯ D Ξ ∗ c , ¯ D ∗ Ξ ∗ c in J = 3 / , I = 0 sector have becomeunstable with the poles moving to the thresholds, except for three strongly bound ones of¯ D Ξ c and ¯ D ∗ Ξ c . Note that, all of them were stable bound states in the results of Ref. [29]with a µ ( µ = 1 GeV) = − .
09, and even more bound with a µ ( µ = 1 GeV) = − . | T | [ M e V - ] √ s [MeV]LHCb|T D Ξ c | |T D * Ξ c | |T D * Ξ ’ c | |T D * Ξ c* | |T D Ξ ’ c | -5 0 5 10 15 20 25 30 4200 4300 4400 4500 4600 4700 | T | [ M e V - ] √ s [MeV]LHCb|T D Ξ c | |T D * Ξ c | |T D * Ξ ’ c | |T D * Ξ c* | |T D Ξ ’ c | FIG. 3. Results of the modulus squared of the amplitudes for J = 1 / , I = 0 sector comparedwith the experimental data, Left: with full data; Right: with 1 . < m Λ K − < . a u =- a (cid:0) =- a (cid:1) =- / V - - - - s ( MeV ) R e ( G ) a u =- / V / ( V + V G ) - - s ( MeV ) R e ( G ) FIG. 4. Results of the potential V versus the loop functions for the ¯ D ∗ Ξ c channel, where thevertical line locates at the threshold. Refs. [8, 9], taking the one of ¯ D ∗ Ξ c for example, the pole at 4370 MeV, which had beenbelow the threshold of the ¯ D ∗ s Λ c channel. Indeed, the reduced a µ leads to all of the polesless bound, and thus, some of the loosely bound states moved to the thresholds, see the leftpart of Fig. 4, where we plot the real parts of the loop functions G with different a µ andthe inverse potential V , taking the ¯ D ∗ Ξ c channel for example (denoted as the channel 6 inthe figure), see more discussions about the pole moving by the free parameter of the loopfunctions in Refs. [53, 54]. Furthermore, there is a strong coupling between the ¯ D ∗ s Ξ c (thechannel 6 in the figure) and ¯ D ∗ s Λ c (the channel 7 in the figure) channels, and then such off-diagonal transition potential will strengthen the attractive interaction from pure diagonalpotential V to the “effective” potential of V + V G owing to V = µ = 0 (see Table I) .In the right panel of Fig. 4, we compare the real parts of the loop functions G with the one of1 / ( V + V G ) (taking a µ = − . D ∗ Ξ c system will be more bound when a µ ( µ = 1 GeV) = − . D Ξ c system with thecontribution from the ¯ D s Λ c channel . 9 V. CONCLUSION
In summary, we revisit the interactions of the
J/ψ
Λ channel and its coupled channels in s wave, using the CCUA in combination with HQSS and LHG symmetry. With the newobservation of the P cs (4459) state by LHCb to fix the only free parameter, we obtain a poleof (4459 .
07 + i .
89) MeV located below the threshold of the ¯ D ∗ Ξ c channel, which representseffectively the two nearly degenerate states with spin parity J P = − and J P = − inour approach, which keeps in the potential only the constant leading term by dropping themomentum dependent terms. To remove the degeneracy, we will make further investigationsin the future by taking into account the momentum dependent terms, including the pionexchange mechanism. Thus, by assuming the LHCb observed P cs (4459) peak to be thedegenerated ¯ D ∗ Ξ c molecular states from our formalism, it would split to two peaks withhigher statistics in the future, like the previous P c (4450) peak observed by LHCb. From ourresults, there is also a ¯ D Ξ c bound state with J P = − and a pole at (4310 .
53 + i .
23) MeV,which is consistent with the predictions of Ref. [38] within the uncertainties. Owing to theuncertainties of the experimental data and hence their corresponding constraint to the freeparameter, the possible loosely bound states of ¯ D Ξ ′ c , ¯ D ∗ Ξ ′ c , ¯ D ∗ Ξ ∗ c with J = 1 / , I = 0and ¯ D ∗ Ξ ′ c , ¯ D Ξ ∗ c , ¯ D ∗ Ξ ∗ c with J = 3 / , I = 0, which were predicted previously, are in factsuffering a large model dependence, since there are no coupled channels to strengthen theattractive interaction. Thus, unfortunately even their existence is now put into question inour present work. Although their LHG potentials are attractive, whether they are boundstates or virtual states would sensitively depend on the model parameter and neglectedmomentum dependent terms. These results are consistent with recent general analysis inRef. [50]. To look for these molecular states, especially for the ¯ D Ξ c bound state and two¯ D ∗ Ξ c states (corresponding to the observed P cs (4459) peak), the decay channels of ¯ D s Λ c and ¯ D ∗ s Λ c , respectively, are strongly suggested due to their large decay branching ratios.Also searching for them both in the J/ψ
Λ and η c Λ channels can be helpful to distinguishtheir different nature. We hope that future experiments, the Run-3 in LHCb for example,can make further test on our predictions and suggestions to reveal the properties of the P cs states. ACKNOWLEDGMENTS
We thank useful discussions and valuable comments from Bo Fang about the experimentalinformation, and acknowledge Eulogio Oset for useful comments and careful reading thepaper. This work is partly supported by the Fundamental Research Funds for the CentralUniversities (J.J.Wu), and NSFC under Grant No. 12070131001 (CRC110 cofunded by DFGand NSFC), Grant No. 11835015, No. 12047503, and by the Chinese Academy of Sciences(CAS) under Grants No. XDB34030000 (B. S. Zou). [1] J. A. Oller and E. Oset, Nucl. Phys. A , 438-456 (1997) [erratum: Nucl. Phys. A ,407-409 (1999)] [arXiv:hep-ph/9702314 [hep-ph]].[2] E. Oset and A. Ramos, Nucl. Phys. A , 99-120 (1998) [arXiv:nucl-th/9711022 [nucl-th]].[3] J. A. Oller and U.-G. Meißner, Phys. Lett. B , 263 (2001),
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