Two states for the Ξ(1820) resonance

We recall that the chiral unitary approach for the interaction of pseudoscalar mesons with the baryons of the decuplet predicts two states for the Ξ(1820) resonance, one with a narrow width and the other one with a large width. We contrast this fact with the recent BESIII measurement of the K − Λ mass distribution in the ψ (3686) decay to K − Λ¯Ξ + , which demands a width much larger than the average of the PDG

Gradual progress in the description of the hadronic spectrum leads to the consequence that some resonances apparently well established actually correspond to two states.This is the case of the Λ(1405), for which two states around 1385 MeV and 1420 MeV were predicted in Refs.[1,2].After some time, these resonances found their place in the PDG [3] 1 .This is also the case of the K 1 (1270) axial vector resonance, where also two states were found in Ref. [4], for which experimental evidence was found in Ref. [5], and the saga continues with the two states also predicted for the D * (2400) [6] 2 .
Another case of two states, this time suggested from the experimental side, is the splitting of the Y (4260) resonance found by the BaBar collaboration [8,9] into two states Y (4230) and Y (4260) by the BESIII collaboration [10] 3 .A recent paper [14] shows that the Weinberg-Tomozawa interaction of the leading order chiral potentials produces in some cases a double pole structure.
In Refs.[20,21], the interaction of the octet of pseudoscalar mesons with the octet of baryons was studied and the two Λ(1405) states emerged.In Ref. [25], the study was extended to the interaction of the octet of pseudoscalar mesons with the decuplet of baryons and many resonances were generated that could be associated to well known 3 2 − existing states.Other resonances were predicted which were not found experimentally at the time the work was completed.One of them was a resonance, coming from the KΞ(1530) and ηΩ interaction, which was later identified with the recently found Ω(2012) state by the Belle collaboration [26].With ups and downs in the discussion of the nature of this resonance (see Ref. [27] for the latest update), the Belle collaboration concluded that the experimental information supported the molecular nature of this resonance [28]. 1 We should clarify that we call two states, indicating that we do not talk about different poles in different Riemann sheets, but two distinct poles in the same Riemann sheet. 2 Two states with these quantum numbers are found in Ref. [7] but with far less precision than in Ref. [6]. 3 We refer to splitting in states with the same quantum numbers.We do not consider in this block states close by with different quantum numbers, like the splitting of the P c (4450) of Ref. [11,12] into the P c (4440) and P c (4457) with time the work was done.Indeed, two resonances, one narrow and one with a large width, were predicted in the vicinity of Ξ(1820).The purpose of the present work is to show that support for this idea is now provided by the recent BESIII investigation of this resonance.
Actually, in Ref. [29] the ψ(3686) decay to K − Λ Ξ+ is investigated and in the K − Λ invariant mass two neat peaks, one for the Ξ(1690) and another one for the Ξ(1820), are observed.The surprising thing is that the width of the Ξ(1820) is reported as This result is much bigger, and incompatible with that of the PDG [3] of A solution to this problem is obtained with the acceptance of two states, as we show below.
In Ref. [25], four coupled channels were considered, The transition potential obtained from the chiral Lagrangians is given by where k 0 , k ′ 0 are the energies of the initial and final mesons, and the coefficients C ij are given in Table I.The above potential is the input of the Bethe-Salpeter (BS) equation to obtain the scattering amplitude, FIG. 1.The resonant mechanism for the production of Ξ+ K − Λ in the ψ(3686) decay.
In this way, two poles were obtained in Ref. [25], one narrow and the other one wide, in the vicinity of Ξ(1820) resonance.
We can see that the channel K − Λ, where the state is observed [29], is not any of the coupled channels of Table I.However, there is a way to make a transition to this state by means of the mechanism depicted in Fig. 1.This mechanism, considering the negative parity of Ξ+ , requires a P -wave in the ψ(3686) → Ξ+ P B * vertex and a D-wave in the 4 .The amplitude of Fig. 1 is then of the type where k is the momentum of the K − in the K − Λ rest frame, G j are the loop functions of the intermediate P B * states, regularized by means of a cutoff q max [21], and A j , C i , D ij are unknown coefficients that depend on the dynamics in Fig. 1.But the relevant thing here is that Eq. ( 5) involves a linear combination of the T ij amplitudes, accommodating the contribution of the two resonances.Clearly, the effect of both resonances should become visible in the experiment.
The invariant mass distribution can be written as, where pΞ is the momentum of the Ξ in the ψ(3686) rest frame, and k is the momentum of the kaon in the c.m. reference system of the with W an arbitrary weight.
We have redone here the calculations of Ref. [25] and corroborated the results obtained there.We have checked that the results are stable by varying the parameters (f and q max ), obtaining two poles, one with a small width and the other one broad.The best compromise with the experimental data is obtained by slightly changing the f parameter in Eq. ( 3) to 1.28f π , and q max = 830 MeV.The results are shown in Table II, together with the couplings of the states to the different channels, extracted from the behaviour at the pole, where the amplitude behaves like It is now clear why the two states have such a different width, since the only decay channel is πΞ * and the width goes as the square of the coupling to that channel, which is larger for the second state.
As already mentioned, the coefficients D ij are unknown.However, by looking at the strength of the different T ij matrices, we find that the ηΞ * channel has a large diagonal T 33 amplitude which shows evidence of the broad resonance (this is in agreement with Fig. 7 of Ref. [25]).We take then this amplitude characterizing the sum ij D ij T ji .Actually, we notice that the relevant T ij matrix elements have all a similar shape.Once this is done, we find a K − Λ mass distribution as shown in Fig. 2. We have added a background that follows the phase space, and adjusted C to the data.The strength is adjusted to the experimental data.7), in arbitrary units, with ij D ij T ji substituted by T 33 , with the experimental data taken from BESIII [29] and the background given by Eq. ( 8).
As we can see, the results obtained with the two resonances of Table II, together with the background, provide a fair description of the data.We perform a second test by performing a fit to the data, very similarly to what is usually done in experimental analyses.Thus, we take a coherent sum of amplitudes with R 1 , R 2 representing approximately the two resonances of Table II, with M R 1 = 1822 MeV, Γ 1 = 45 MeV, M R 2 = 1870 MeV, Γ 2 = 200 MeV.We adjust A and B and the background of Eq. ( 8).The coefficients A and B are found to have about the same strength.
We obtain a good description of the data, shown in Fig. 3, and most of the strength at higher invariant masses is provided by the contribution of the second resonance.We can see in Fig. 3 that the contribution of the wider resonance plays an important role filling up the strength in the higher part of the mass spectrum.Note that the background needed in Figs.
2 and 3 is practically the same.This means that in Fig. 2 the upper part of the spectrum comes from the T 33 amplitude, which contains information of the two resonances, with the wide one responsible for the strength in this region.Although this comment is unrelated to the discussion of this work, we would like to mention that around M K − Λ = 2100 MeV there seems to be a peak.We should note that in Ref. [25] and here, we do find a peak around 2100 MeV better seen in the KΩ diagonal T 44 matrix element.
In concluding remarks we stress the fact that the successful chiral unitary approach for meson-baryon interaction, applied to the interaction of pseudoscalar mesons with the baryondecuplet, gives rise to two states around the Ξ(1820), one of them narrow and the other one wide.This feature remains if reasonable changes are done in the strength of the interaction or the regulator of the loop functions, and is independent on whether one uses dimensional regularization [25] or the cutoff method as done here.We took advantage to show that this scenario provides a satisfactory description of the data in the ψ(3686) → K − Λ Ξ+ decay, solving the puzzle presented by the recent BESIII experiment [29], which provides a width for the Ξ(1820) much bigger than the one reported in the PDG [3].
2078] and ΩK[2165], where the threshold masses are written in brackets in units of MeV.As one can see, only the Ξ * π channel is open for decaying at 1820 MeV and the width of a state depends on the coupling to this channel.

FIG. 2 .
FIG.2.Results of Eq. (7), in arbitrary units, with ij D ij T ji substituted by T 33 , with the experimental data taken from BESIII[29] and the background given by Eq. (8).

FIG. 3 .
FIG.3.Results obtained adjusting Eq. (9) to the data together with a small background.In the figure we show the contribution of the background alone and the results obtained removing the contribution of the second pole.

ACKNOWLEDGEMENT
This work is partly supported by the National Natural Science Foundation of China under Grant No. 11975083 and No. 12365019, and by the Central Government Guidance Funds

TABLE II .
Pole positions and couplings for q max = 830 MeV.All quantities are given in units of