Theoretical study of the Xi(1620) and Xi(1690) resonances in Xi_c -> pi^+ MB decays
Kenta Miyahara, Tetsuo Hyodo, Makoto Oka, Juan Nieves, Eulogio Oset
KKUNS-2638, YITP-16-100
Theoretical study of the
Ξ(1620) and
Ξ(1690) resonances in Ξ c → π + M B decays
Kenta Miyahara ∗ Department of Physics, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan
Tetsuo Hyodo
Yukawa Institute for Theoretical Physics, Kyoto University, Kyoto 606-8502, Japan
Makoto Oka
Department of Physics, Tokyo Institute of Technology, Tokyo 152-8551, JapanAdvanced Science Research Center, Japan Atomic Energy Agency, Tokai, Ibaraki, 319-1195, Japan
Juan Nieves
IFIC, Centro Mixto Universidad de Valencia-CSIC Institutos de Investigaci´on de Paterna, Aptdo. 22085, 46071 Valencia, Spain
Eulogio Oset
Departamento de F´ısica Te´orica and IFIC, Centro Mixto Universidad de Valencia-CSIC,Institutos de Investigaci´on de Paterna, Aptdo. 22085, 46071 Valencia, Spain (Dated: March 16, 2017)Nonleptonic weak decays of Ξ c into π + and a meson ( M )-baryon ( B ) final state, MB , are analyzedfrom the viewpoint of probing S = − c is dominatedby a single quark-line diagram, preferred by the Cabibbo-Kobayashi-Maskawa coefficient, colorrecombination factor, the diquark correlation, and the kinematical condition. The decay processhas an advantage of being free from meson resonances in the π + M invariant mass distribution. Theinvariant mass distribution of the meson-baryon final state is calculated with three different chiralunitary approaches, assuming that the Ξ(1620) and Ξ(1690) resonances have J P = 1 / − . It is foundthat a clear peak for the Ξ(1690) is seen in the π Ξ and ¯ K Λ spectra. We also suggest that the ratiosof the π Ξ, ¯ K Λ and ¯ K Σ final states are useful to distinguish whether the peak is originated from theΞ(1690) resonance or it is a ¯ K Σ threshold effect.
PACS numbers: 13.75.Jz,14.20.-c,11.30.Rd
I. INTRODUCTION
The advent of the LHCb and its unexpectedly success-ful contribution to hadron physics has provided this areawith a plethora of new reactions that have stirred a re-vival of hadron studies. Thus in LHCb, Belle, BESIII,and other facilities, new reactions and decays of heavyhadrons have come under experimental study [1], whichhas triggered a large theoretical activity as well [2]. Oneof the interesting unexpected findings was the observa-tion of two structures in the
J/ψp invariant mass distri-bution in the Λ b → J/ψK − p decay in Refs. [3, 4] thatwere ascribed to two pentaquarks states. Prior to thisexperimental observation, the Λ b → J/ψK − p reactionwas studied theoretically in Ref. [5] and mass distribu-tions associated to the production of the Λ(1405) in the K − p and π Σ spectra were predicted. In particular, thecalculated K − p distribution was in good agreement withthe experimental findings. Furthermore, this informa-tion, together with predictions made for hidden charmstates of ¯ D ∗ Σ c − ¯ D ∗ Σ ∗ c molecular nature in Refs. [6–8]prompted a likely explanation in Ref. [9] for the narrow ∗ [email protected] state found in Refs. [3, 4] (see also related works alongthe same line in Refs. [10, 11]). Work has followed inRef. [12] with the study of the Ξ − b → J/ψK − Λ decay,suggesting that a strange hidden charm state also pre-dicted in Refs. [6, 7] could be seen in the
J/ψ
Λ massdistribution. Interestingly, in Ref. [13] the LHCb Collab-oration had also observed a peak at about the same massin the
J/ψp mass distribution of the Λ b → J/ψpπ − reac-tion, for which no comment was done in that paper nor inRefs. [3, 4] (see [14] for further comments). A work alongthe same lines as [9] was done for this latter reaction inRef. [15], showing consistency of the peak seen in theΛ b → J/ψpπ − reaction with the narrow one observedin the Λ b → J/ψK − p one. Very recently, a reanaly-sis of the experiment of Ref. [13] has been done by theLHCb Collaboration [16], concluding that the peak ob-served in Ref. [13] is indeed consistent with the claims oftwo states made in Refs. [3, 4]. The strange hidden charmstate of [6, 7] was also suggested to be searched for in the J/ψ
Λ mass distribution in the Λ b → J/ψη
Λ reaction inRef. [17], in the Λ b → J/ψK Λ reaction in Ref. [18] andin the Ξ − b → J/ψK − Λ in Ref. [12]. Discussions on theseand other reactions can be seen in Refs [19–22].The search for pentaquark states is not the only rele-vant information obtained from these reactions. Indeed, a r X i v : . [ nu c l - t h ] M a r one of the interesting findings in Ref. [5] was that the re-action, Λ b → J/ψ + π Σ( ¯ KN ), acted as a filter for I = 0baryon states, which was later confirmed by the analysisof [3, 4], where only the Λ ( I = 0) states were seen inthe K − p mass distribution. This was used [5] to makepredictions for the shape of the Λ(1405) in the π Σ massdistribution. Similarly, in Ref. [15] it was shown thatthe Λ b → J/ψpπ decay was a good filter for baryonswith I = 1 / b as spectators in thereaction. These filters make these decays particularlysuitable to study baryon resonances that in most reac-tions appear together with contributions of other isospinchannels. Taking advantage of this interesting propertythe Λ b → J/ψK
Ξ was suggested [23] as a tool to investi-gate the K Ξ interaction in the I = 0 sector. In a similarway, in Ref. [24] the Λ + c → π + π Σ( ¯
KN, η
Λ) was studiedand shown to be also a good filter for I = 0 baryon states,allowing one to see the Λ(1405) and Λ(1670) resonances.In the present work, we take advantage of these ideasand study the Ξ + c → π + ( ¯ K Σ , ¯ K Λ , π Ξ) reactions, show-ing that they provide a good filter for I = 1 / S = − K Σ threshold, and thus the influence of this thresholdon the nature of the Ξ(1690) deserves further study. Weshall also see that the Ξ + c → π + ( ¯ K Σ , ¯ K Λ , π Ξ) decay fil-ters the spin and parity of the final
M B pair, and hencethe observation of the Ξ(1620) and Ξ(1690) states in thisreaction might allow to determine the unknown spin andparity of these resonances.
II. Ξ RESONANCES
Although the number of Ξ states should be compara-ble with that of nucleon resonances from the viewpointof quark models, at present, the number of measured Ξstates is significantly smaller [25]. Therefore, the study ofΞ resonances is relevant in connection with the underly-ing baryon structure. The assignment of the spin-parity, J P , in most of the known Ξ resonances is also incom-plete, and thus these quantum numbers have been deter-mined only for few of them: the ground octet Ξ(1320)and decuplet Ξ(1530) states and the excited Ξ(1820) res-onance. The Ξ(1690) is a PDG three-star state, with( M, Γ) = (1690 ±
10 MeV , <
30 MeV), where M andΓ represent the mass and the width, respectively. It was first observed in the reaction, K − p → ( ¯ K Σ) Kπ , asa threshold enhancement in the neutral and negativelycharged ¯ K Σ mass spectra [26]. Subsequently, the reso-nance has been also observed in hyperon-nucleon inter-actions [27–29]. As explained in Sec. I, recently, heavyhadron decays have begun to emerge as a new analy-sis method for hadron spectroscopy. The Ξ(1690) hasbeen studied in some charmed hadron decays like thoseof the Λ c and ψ (3686) hadrons [30–33]. In one of suchrecent experiments, Λ + c → Ξ − π + K + , the BaBar Collab-oration [32] has found some evidence supporting spin-parity quantum numbers J P = 1 / − for this resonance.The spin J = 1 / + c → Λ ¯ K K + reaction [34]. Nevertheless to fully clarifythe Ξ(1690) quantum numbers, further experiments arecertainly required.In the theoretical side, the description of the Ξ(1690)has been somehow controversial [35–46]. In quark mod-els, the difficulty arises in assigning its spin-parity. Forexample, the nonrelativistic quark model in Ref. [35] pre-dicted the first radial excitation with J P = 1 / + around1690 MeV. On the other hand, Ref. [38] assigned theΞ(1690) to the first orbital excitation with J P = 1 / − ,and Ref. [40] supported this assignment analyzing its de-cay width. In addition, it is also difficult to reproduce itsmass, and several works predict masses for the Ξ(1690)significantly above the experimental value [36, 37, 39].There are other approaches based on large N c QCD [45]and the Skyrme model [46]. The former obtained a Ξ res-onance with J P = 1 / − which has a much larger massthan the Ξ(1690), and the latter predicted two Ξ res-onances with J P = 1 / − which have masses consistentwith the experimental values of the Ξ(1620) and Ξ(1690).In late years, the meson-baryon scattering in thestrangeness S = − J P = 1 / − and it would strongly couple to ¯ K Σ and η Ξ, having thuslarge molecular components [44]. However, this state didnot appear in the analysis of Ref. [41], where the au-thors suggested that the Ξ(1690) might not be a molecu-lar state. In all the chiral unitary approaches [41–44] theΞ(1620) is also generated, with a relatively large decaywidth. This state strongly couples to π Ξ and ¯ K Λ, and itis thought to be originated from the strong attraction inthe π Ξ channel [42–44]. The experimental evidence forthe Ξ(1620) is quite poor, and the PDG assigns to thisstate only one-star [25]. Considering such situation, theanalysis of these Ξ resonances is interesting, and impor-tant for the search of exotic states, which are not easilyaccommodated as three-body quark states.In this work, to study Ξ(1690) and Ξ(1620) we analyzethe Ξ c → π + ( M B ) i decay ( M and B represent the mesonand baryon, respectively, with the index i denoting themeson-baryon channel). To account for the final meson- FIG. 1. M K Λ and M K + ¯ K Dalitz plot for the Λ c → K + ( ¯ K Λ)reaction. The Ξ(1690) energy is shown by the vertical dottedline, while the horizontal band represents the mass and thewidth of the a (980).FIG. 2. M K Λ and M π + ¯ K Dalitz plot for the Ξ c → π + ( ¯ K Λ)reaction. As in Fig. 1, the Ξ(1690) energy is shown by thevertical dotted line. The horizontal band represents the massand the width of the K ∗ (892). baryon interaction, we examine the predictions deducedfrom the chiral unitary approaches of Refs. [41, 42, 44].As mentioned above, experimentally the Λ c decay hasbeen also examined to extract information about the Ξstates. We compare both, Λ c and Ξ c , decay reactions,from the viewpoint of the kinematics and we show Dalitzplots of the Ξ c → π + ( ¯ K Λ) and Λ c → K + ( ¯ K Λ) reac-tions in Figs. 1 and 2, respectively. In the Ξ(1690)energy region, the Λ c decay Dalitz plot overlaps greatlywith the a (980) meson resonance in the K ¯ K channel,which makes the Ξ(1690) analysis difficult [34]. On theother hand, in the Ξ c decay, the overlap with the corre-sponding meson resonance, the K ∗ (892) now in the π ¯ K channel, is much smaller. Furthermore, if we choose theΞ + c → π + ( ¯ K Λ) reaction instead of the Ξ c → π + ( K − Λ) FIG. 3. Dominant quark-line diagram for the Ξ + c → π + MB decay. The solid and the wiggly lines stand for the quarksand the W boson, respectively. decay, the π + ¯ K pair must be in an isospin I = 3 / /
2. This means that theanalysis of the Ξ + c decay should not be influenced by thepresence of meson resonances, since an isospin I = 3 / c decays, in particular that of the Ξ + c , isan ideal reaction for the study of strangeness Ξ baryons.There exist several excited baryon resonances in the ¯ K Λand π Λ channels around the Ξ(1690) energy region. How-ever, because such resonances are quite broad and theirlarge overlap, it is reasonable to suppose that their cor-responding bands would not be visible in the Dalitz plotin sharp contrast to the meson resonance cases discussedabove.
III. FORMULATION
Following our previous work [24], we show in Fig. 3 thedominant quark-line diagram for the Ξ + c → π + ( M B ) de-cay, when the final
M B pair is emitted close to threshold.We split the decay process in three parts. The first oneinvolves the c → s weak transition and the productionof a high momentum π + . Next we consider the ¯ qq cre-ation part, where the intermediate meson-baryon statesare constructed with certain weights. Finally, we havethe rescattering of the intermediate meson-baryon pairswhich will be taken into account in a coupled channelchiral unitary scheme.In what follows, we will focus on the Ξ + c decay. Theanalysis of the Ξ c decay runs in parallel, because thedominant quark-line diagram is similar to that shownin Fig. 3. There exist however some differences inducedby subdominant mechanisms, which will be discussed inSec. V C. A. Weak decay
The Cabibbo allowed reactions of interest for the Ξ c decay are c → su ¯ d and cd → su . When it is required theemission of high momentum π + , these reactions lead tothe two quark-line diagrams depicted in Figs. 3 and 4, FIG. 4. Subdominant mechanism for the Ξ + c → π + MB decay.Though its contribution is also Cabibbo allowed, it is howeversuppressed when compared to that depicted in Fig. 3 (see textfor details). respectively. However, the mechanism in Fig. 4 is sup-pressed in comparison with that shown in Fig. 3. Firstthere is a color enhancement factor in the latter one,which is not present in the diagram of Fig. 4. This is be-cause in the W boson- u ¯ d vertex, the color of the outgoingquarks is fixed by that of the u quark belonging to the Ξ c since a color singlet ( π + ) needs to be constructed. In con-trast, in the mechanism of Fig. 3 all the colors are allowedin the W vertex. On the other hand, the u and s quarksin the Ξ c form a strongly correlated antisymmetric di-quark configuration difficult to separate. Therefore, amechanism where the diquark state is destroyed like thatdepicted in Fig. 4 is expected to be suppressed. Finallykinematics also favor the diagram of Fig. 3 since we willbe interested in situations where the outgoing M B pairis produced at low invariant masses (see Fig. 2), whichin turn requires the emission of a high momentum π + .Because the u quark in the Ξ c is a spectator in Fig. 4, itis at rest in the Ξ c center of mass frame and thus, it isdifficult its association with a high momentum ¯ d , comingfrom the W decay, to construct the final high energy π + .For all the above arguments, we think the mechanismdepicted in Fig. 3 should be dominant in the Ξ c decay,and we will use it to study the influence of the Ξ reso-nances in the process. Attending to the structure of thequark degrees of freedom, the ground state of the Ξ c isalmost dominated by the flavor SU(3)-subgroup ¯3 con-figuration [47], | Ξ c (cid:105) = 1 √ | c ( su − us ) (cid:105) . (1)Therefore, the sus cluster formed after the charm quarkdecay will be 1 √ | s ( su − us ) (cid:105) . (2) B. ¯ qq creation The next step is the insertion of the vacuum-quantum-numbers ¯ qq -pair creation to construct the intermediatemeson-baryon state M B (cid:48) . To analyze decay modes wherefinal
M B state has J P = 1 / − spin-parity, the s quarkoriginated in the weak decay should carry one unit of an-gular momentum, L = 1. On the other hand, assuming ground states and a relative s wave for the final M B ,the ¯ qq creation should be attached precisely to this s quark. We further assume that the u and s quarks be-longing to the Ξ c baryon and spectators in the decay inthe mechanism of Fig. 3, keep the strong diquark corre-lation discussed in the previous section. Hence, after the¯ qq creation, these u and d quarks should be part of thebaryon, and the s quark originated in the weak decayshould form the meson, as shown in Fig. 3. The abovepicture leads to | M B (cid:48) (cid:105) = 1 √ | s (¯ uu + ¯ dd + ¯ ss )( su − us ) (cid:105) (3)As explained in the Appendix, we can connect two de-grees of freedom, the quarks and the hadrons. Using thequark representations of hadrons discussed in Appendix,we can rewrite the intermediate state as | M B (cid:48) (cid:105) = | K − Σ + (cid:105) − √ | ¯ K Σ (cid:105) + 1 √ | ¯ K Λ (cid:105) − √ | η Ξ (cid:105) . (4)In the isospin basis, this becomes | M B (cid:48) (cid:105) = 1 √ | ¯ K Λ (cid:105) − (cid:114) | ¯ K Σ (cid:105) − √ | η Ξ (cid:105) , (5)where the isospin quantum numbers of all states are I = 1 /
2. In Eqs. (4) and (5), we have neglected thecontribution from the η (cid:48) Ξ channel because its thresholdis located much higher in energy.Up to this point, we have considered mechanismswhere the high momentum π + is emitted right after theweak c → s transition, and its formation is independentof the vacuum quark-antiquark pair creation. It is natu-ral to consider these quark-line diagrams first, because weassume the emitted pion has relatively a large momentumso that the remaining system is close to the meson-baryonthresholds. Such diagram approach is known to be (quali-tatively) powerful in the hadronic weak decays. However,there are other quark-line diagrams where the π + is emit-ted after the ¯ qq insertion, as shown in Fig. 5. Although amomentum mismatch will suppress the emission of high-momentum pion for the soft ¯ qq pair creation, the contri-butions of such processes might be non-negligible. Thequark-line diagrams Figs. 5(c) and 5(d) are suppressedbecause of the color recombination factors, similarly asit was discussed above. Indeed, the color of the ¯ d or u We follow the convention of Ref. [48]: | π + (cid:105) = −| I = 1 , I z = 1 (cid:105) , | K − (cid:105) = −| / , − / (cid:105) , | Σ + (cid:105) = −| , (cid:105) , | Ξ − (cid:105) = −| / , − / (cid:105) . FIG. 5. Other possible mechanisms for the Ξ + c → π + MB decay. quarks, respectively, from the c weak decay is fixed sinceit should be coupled to the quarks in the ssu cluster.However, there are no robust reasons to exclude the con-tribution from the mechanisms depicted in Figs. 5(a) and5(b), except for the kinematical suppression produced byhaving a high energy quark emitted from a weak vertexpart of the low energy M B final pair. Using the same pro-cedure as above, we obtain that the intermediate
M B (cid:48)(cid:48) state for these additional diagrams would be | M B (cid:48)(cid:48) (cid:105) = 2 √ | η Ξ (cid:105) . (6)Because we do not specify the detailed mechanism for the¯ qq pair creation, the relative phase between the | M B (cid:48) (cid:105) and | M B (cid:48)(cid:48) (cid:105) intermediate states cannot be determined.We thus introduce a linear combination, | M B (cid:48) (cid:105) + x | M B (cid:48)(cid:48) (cid:105) (7)with an unknown weight factor x . As we will show inSec. V B, the qualitative features of the spectra are notsignificantly affected when values of x in the [ − ,
1] rangeare considered. For the sake of brevity, in what follows wewill mainly show results for x = 0, unless it is otherwisestated. C. Final-state interaction
The intermediate mesons and baryons [Eqs. (4) or (5)]re-scatter through strong interactions, and produce thedecay amplitude M j for the final meson-baryon M B pair.The schematic diagram is shown in Fig. 6, where the to-tal contribution is the sum of the tree diagram obtaineddirectly from the ¯ qq creation, and the rescattering termthat accounts for the final-state interaction (FSI) of the FIG. 6. Schematic diagram for the FSI of the meson-baryonpair (solid circle). The first and the second terms in the right-hand side stand for the tree and the rescattering contribu-tions, respectively. The latter diagram contains the meson-baryon loop function G and the scattering amplitude t . Be-sides, the factor h represents the intermediate meson-baryonweight introduced in Eqs. (4) or (5). intermediate meson-baryon pairs. The factor h repre-sents the coefficients in Eqs. (4) or (5). The meson-baryon loop function G and the meson-baryon scatteringamplitude t are calculated using a certain model. In thiswork, we use some chiral unitary approaches as explainedin Sec. IV.The decay amplitude M j to the final meson-baryonstate M B is expressed as M j = V P (cid:32) h j + (cid:88) i h i G i ( M inv ) t ij ( M inv ) (cid:33) , (8)where M inv represents the meson-baryon invariant mass.The dynamics before the FSI is all included in V P , whichis assumed to be constant in the relevant energy region.The actual value of V P may be determined from an ex-perimental measurement of the decay distribution in acertain decay channel. The coefficients h j for the physi-cal basis are h π Ξ = h π + Ξ − = 0 , h ¯ K Λ = 1 √ ,h K − Σ + = 1 , h ¯ K Σ = − √ , h η Ξ = − √ , (9)and for the isospin basis are h π Ξ = 0 , h ¯ K Λ = 1 √ , h ¯ K Σ = − (cid:114) , h η Ξ = − √ . (10)The reason for the vanishing π Ξ coefficients can be un-derstood from the decay mechanism as shown in Fig. 3,where the meson M consists of the s quark and cannot bethe π after the ¯ qq creation. For nonzero x in Eq. (7), h η Ξ is modified as h η Ξ = ( − x ) / √
3. To directly comparewith the experimental data, we rewrite the amplitude inthe isospin basis; M π Ξ = − √ M I =1 / π Ξ + (cid:114) M I =3 / π Ξ , M π + Ξ − = (cid:114) M I =1 / π Ξ + 1 √ M I =3 / π Ξ , M ¯ K Λ = M I =1 / K Λ , M K − Σ + = − (cid:114) M I =1 / K Σ + 1 √ M I =3 / K Σ , M ¯ K Σ = 1 √ M I =1 / K Σ + (cid:114) M I =3 / K Σ , M η Ξ = M I =1 / η Ξ . (11)Because the u quark in Fig. 3 is a spectator in the weakdecay, the final meson-baryon state retains the sameisospin as the Ξ c , and the I = 3 / M π Ξ = − √ M I =1 / π Ξ , M π + Ξ − = (cid:114) M I =1 / π Ξ , M ¯ K Λ = M I =1 / K Λ , M K − Σ + = − (cid:114) M I =1 / K Σ , M ¯ K Σ = 1 √ M I =1 / K Σ , M η Ξ = M I =1 / η Ξ . (12)With the above decay amplitudes, we can calculate thepartial decay width Γ j ,Γ j = (cid:90) d Π | M j | , (13)where d Π represents the three-body phase space. Theinvariant mass distribution is obtained by differentiatingthe width by M inv . IV. RESULTS WITH CHIRAL UNITARYAPPROACHES
In this section, we use the chiral unitary approachesas the final-state interaction, and show our predictionsfor different meson-baryon invariant mass distributionsin the Ξ c decay.To quantify systematic uncertainties, we will considerhere three chiral unitary approaches, that we will de-note by ROB, GLN, and Sekihara, and whose detailsand predictions can be found in Refs. [41], [42], and [44],respectively. The ROB and GLN approaches are formu-lated in the isospin symmetric limit, while the Sekihara In Refs. [41, 44], several parameter sets are introduced. Here, wechoose “Set 5” for the ROB model and that denoted by “Fit” inthe Sekihara one. model uses physical hadron masses, thus, including someisospin symmetry breaking corrections. In Tables I andII, we compile the pole positions and couplings g i to each M B channel of the resonances found in these references. The poles are found in the appropriate Riemann sheetsdefined by continuity with the real axis except for thecase of the Ξ(1690) in the Sekihara model, which is foundin a nonphysical Riemann sheet above, but quite close to,the K − Σ + threshold. The Ξ(1620) is dynamically gener-ated in the ROB and GLN models, with large couplingsto the π Ξ and ¯ K Λ channels. On the other hand, theΞ(1690) is found in the GLN and Sekihara approaches,with now large couplings to the ¯ K Σ and η Ξ channels,but not in the ROB model.In the above chiral unitary approaches, only the s -wavescattering for M B is considered. In this case, the decayamplitude M j depends only on M inv and the invariantmass distributions dΓ j / d M inv is reduced todΓ j d M inv = 1(2 π ) p π + ˜ p j M j M Ξ + c | M j | , (14)where M j is the baryon mass in the channel j , and p π + (˜ p j ) represents the three-momentum of the π + emittedin the weak decay part (meson in the final M B state) inthe Ξ c rest frame (in the M B rest frame), p π + = λ / ( M + c , m π + , M )2 M Ξ + c , ˜ p j = λ / ( M , M j , m j )2 M inv ,λ ( x, y, z ) = x + y + z − xy − yz − zx. (15)As the results of invariant mass distributions, first, wewill consider the π Ξ channel, which couples strongly tothe Ξ(1620) resonance in the ROB and GLN approaches.Later in this section, we will pay attention to the ¯ K Λ andthe ¯ K Σ invariant mass distributions. These two latterchannels are ideal to study the Ξ(1690) because theircouplings to this resonance are much larger than that of π Ξ channel, and in addition the Ξ(1690) lies near thesethresholds (see Tables I and II) .In Fig. 7, we show the π Ξ invariant mass distributionpredicted with the ROB and GLN models. Though bothmodels have the Ξ(1620) resonance pole in the meson-baryon scattering amplitudes, the peak structure can behardly seen around the M inv ∼ c decay distribution. The main reason of the absence ofthe peak is the large decay width. Especially in the GLNmodel, the width is larger than 250 MeV, and such a stateis difficult to see as a clear peak on the real energy axis.There is additional suppression of the signal related to In Table I, the values of the pole positions and the couplings areslightly different from the ones in the original papers [41, 42].This is because some small differences in the employed mesonand baryon masses. pole [MeV] g π Ξ g ¯ K Λ g ¯ K Σ g η Ξ ROB (Set 5) [41] Ξ(1620) 1606 − i . − . i . . i . − . i . . i GLN [42] Ξ(1620) 1568 − i . − . i . − . i . − . i . − . i Ξ(1690) 1667 − i . − . i . − . i . . i . . i TABLE I. Pole positions and couplings g i for the Ξ resonances found in the ROB [41] and the GLN [42] models.pole [MeV] g π Ξ g π + Ξ − g ¯ K Λ g K − Σ + g ¯ K Σ g η Ξ Sekihara (Fit) [44] Ξ(1690) 1684 − i − . . i . − . i . . i . . i − . − . i − . − . i TABLE II. Pole positions and couplings g i for the Ξ resonances found in the Sekihara model [44]. The Ξ(1690) pole appearsin a nonphysical Riemann sheet, above the K − Σ + threshold. Ξ c → π + ( π Ξ ) d Γ π Ξ / d M i n v [ a r b . un i t s ] M inv [MeV]ROB GLN FIG. 7. π Ξ invariant mass distribution obtained with theROB (red dashed curve) and the GLN (black solid curve)models. The vertical lines represent the ¯ K Λ and ¯ K Σ thresh-olds. the decay mechanism and the models for the final-stateinteraction. In the Ξ c decay process, the π Ξ channel doesnot appear in the intermediate state as in Eqs. (4) and(5). Hence, considering that the Ξ(1620) mainly couplesto the π Ξ and ¯ K Λ channels (see Table I), and neglectingfor simplicity the other channels, the decay amplitudecan be approximated in the energy region of interest forthe Ξ(1620) as M π Ξ ∼ V P √ G ¯ K Λ t ¯ K Λ ,π Ξ . (16)The Ξ(1620) appears below the ¯ K Λ threshold in both,ROB and GLN, chiral unitary approaches. Since thereis no π Ξ tree level contribution, the final π Ξ state isproduced only through the FSI, with its production ratejust determined by the ¯ K Λ loop function G ¯ K Λ . Gener-ally, a loop function becomes small below the threshold. d Γ j / d M i n v [ a r b . un i t s ] M inv [MeV] K- Λ π Ξ K - Σ + FIG. 8. ¯ K Λ (black solid curve), K − Σ + (red dash-dottedcurve), and π Ξ (blue dotted curve) invariant mass distribu-tions obtained with the GLN chiral unitary approach. Thevertical dashed line marks the ¯ K Σ threshold.
Especially, in the ROB model, G ¯ K Λ vanishes around theΞ(1620) energy region. As a consequence, it is difficultto see the Ξ(1620) signal in the ROB model, in which thewidth is relatively small ( ∼
130 MeV).In sharp contrast, the Ξ(1690) peak can be clearly seenin the π Ξ distribution of the Ξ c decay, as shown in theGLN result of Fig. 7. This is because in the GLN model,the decay width of the Ξ(1690) is quite small and the loopfunctions strongly related to the Ξ(1690) ( G ¯ K Σ and G η Ξ )do not vanish around the Ξ(1690) energy region. Thus,it is advisable to study the invariant mass distributionsaround the Ξ(1690) region for the rest of the channels.Predictions obtained with the GLN model are shown inFig. 8. We see the Ξ(1690) gives rise to a large peak inthe ¯ K Λ channel, which could be quite useful to extractdetails of this resonance.For a more detailed analysis, we consider also the Sek- d Γ j / d M i n v [ a r b . un i t s ] M inv [MeV] K- Λ π Ξ K - Σ + FIG. 9. ¯ K Λ (black solid curve), K − Σ + (red dash-dottedcurve) and π Ξ (blue dotted curve) invariant mass distri-butions obtained with the Sekihara model of Ref. [44]. Thevertical dashed lines mark the K − Σ + and ¯ K Σ thresholds. ihara model of Ref. [44] to the Ξ c decay analysis. Thedifferent M B invariant mass distributions are shown inFig. 9. Again in this case, the ¯ K Λ distribution presentsthe largest Ξ(1690) signal (peak). On the other hand,when comparing with the previous distributions obtainedwithin the GLN model, we see that the Ξ(1690) peaks inthe ¯ K Λ and π Ξ ( K − Σ + ) distributions predicted bythe Sekihara approach are smaller (larger) than thoseobtained with the GLN scheme. The reason is that, asmentioned above, in the Sekihara model the Ξ(1690) poledoes not show up in the proper “second Riemann sheet(SRS)”, i.e., the Riemann sheet obtained by continuityacross each of the two-body unitary cuts [49].Finally, we should note the existence of a cusp struc-ture around the Ξ(1690) region also in the ROB model(Fig. 7), despite the Ξ(1690) resonance is not being gen-erated in that approach. Indeed, the origin of this cusp isthe opening of the ¯ K Σ threshold and in the next section,we will discuss how to distinguish this situation from apeak produced by a dynamically generated resonance.
V. DISCUSSION
In the previous section, we have shown
M B distribu-tions from the Ξ c decay obtained with different chiralmodels. Here, first we propose a method to distinguishthe origin (cusp threshold effect, pole in the SRS or in anon-physical Riemann sheet) of the structures observedin the decay mass distributions. Next, we will estimatethe impact of the contributions from the mechanisms de-picted in the quark-line diagrams in Fig. 5, which arenot included in the dominant one of Fig. 3. Finally, wewill compare Ξ + c and Ξ c decays and show that the dif-ferences among them may be useful to better understandthe decay mechanisms of heavy hadrons. d Γ j / d M i n v [ a r b . un i t s ] M inv [MeV]K- Λ π Ξ K - Σ + FIG. 10. ¯ K Λ (black solid curve), K − Σ + (red dash-dottedcurve), and π Ξ (blue dotted curve) invariant mass distribu-tions obtained with the ROB model. The vertical dashed lineindicates the ¯ K Σ threshold.
A. Relation between the peak and decay ratios
As explained in Sec. IV, there are two possibilities forthe origin of the peak observed in the mass distributionsaround the Ξ(1690) energy region. It could be producedby a pole, either in the SRS or in a nonphysical Riemannsheet, or it might be a threshold effect. Here we proposethat the ratios of the decay fractions around the expectedposition of the Ξ(1690) resonance might be used to distin-guish one situation from the another one. In Fig. 10, weshow all the invariant mass distributions with the ROBmodel. Comparing these latter distributions with thethose presented earlier in Figs. 8, 9, we see that theheight of peak that appears in the ¯ K Λ distribution (solidcurve) is much larger in the schemes with a Ξ(1690) pole(GLN and Sekihara) than in the ROB approach, wherethe resonance is not dynamically generated. Indeed, inte-grating the invariant mass distributions over the Ξ(1690)region (1650 ≤ M inv ≤ ¯ K Λ Γ K − Σ + = 15 . , Γ ¯ K Λ Γ K − Σ + = 1 . , Γ ¯ K Λ Γ K − Σ + = 0 . . (17)The above ratios reveal a quite large difference due to theexistence or not, and in the former case to the exact na-ture (position) of the resonance pole. In the GLN chiralapproach, the Ξ(1690) is quite narrow and since the polelies below the K − Σ + threshold, the resonance does notaffect much the K − Σ + channel, while its influence forthe ¯ K Λ branching fraction becomes much larger. Onthe other hand, in the ROB model the Ξ(1690) is notgenerated, and the K − Σ + fraction largely exceeds the¯ K Λ one. In the Sekihara model, because the resonancepole does not directly affect the real axis, the predictedratio turns out to be between those obtained in the abovetwo cases, and G ¯ K Λ is comparable with G ¯ K Σ . B. Contribution from other diagrams
Up to now, we have set the parameter x in Eq. (7) to0, which amounts to consider only the decay mechanismof the quark-line diagram depicted in Fig. 3. Here, wetry to estimate the contribution from the other mech-anisms shown in Figs. 5(a) and 5(b). In these latterdiagrams, the only intermediate meson-baryon state isthe η Ξ, which weakly couples to the Ξ(1620) withinthe ROB and GLN chiral approaches (see Tables I andII). Thus, the addition of this decay mechanism shouldnot change much the situation described above for theΞ(1620). However, the Ξ(1690) has a substantial cou-pling to the η Ξ channel, and thus the Ξ(1690) produc-tion in the Ξ c decay might be affected. Predictions fromthe Sekihara approach for x = − ,
0, and 1 are shownin Fig. 11 (mass distributions) and in Eq. (18) (decayfraction ratios, as defined above),Γ ¯ K Λ Γ K − Σ + = . x = 0) , . x = +1) , . x = − . (18)Roughly speaking, the qualitative behavior and the rel-ative weight of the different mass distributions are notchanged by the addition of the mechanisms of Fig. 5, andthe major differences appear in the overall height of thedistributions. We thus conclude that the results with x = 0 presented in the previous section are reasonable,as long as the spectral shapes and the relative fractionsare concerned. C. Subleading diagrams for Ξ c decay Finally, in this subsection, we want to compare theΞ c and Ξ + c decays. We show in Fig. 12 all the Cabibbofavored quark-line diagrams, with a π + emitted beforethe q ¯ q -pair insertion, for both Ξ c and Ξ + c decays. Asdiscussed in Sec. III, diagram (b) of the Ξ + c decay is sup-pressed by color recombination factors, diquark corre-lations, and kinematics when a high momentum π + isrequired. Similarly, the diagram (b) of the Ξ c decay isalso suppressed, but the topology of the subdominantdiagrams is different from that of the Ξ + c decay. If the Note that with a fine tuning of the parameter x , the Ξ(1690) peakmay accidentally disappear in the π Ξ and ¯ K Λ spectra becauseof some destructive interferences. π Ξ d Γ π Ξ / d M i n v [ a r b . un i t s ] M inv [MeV] x=0 x=+1 x=-1 (a) Λ d Γ K - Λ / d M i n v [ a r b . un i t s ] M inv [MeV] x=0 x=+1 x=-1 (b) - Σ + d Γ K - Σ + / d M i n v [ a r b . un i t s ] M inv [MeV]x=0 x=+1 x=-1 (c) FIG. 11. (a) π Ξ , (b) ¯ K Λ, and (c) K − Σ + invariant massdistributions obtained in the Sekihara model using three dif-ferent values of x = 0 (black solid curves), +1 (red dash-dottedcurves) and − x is definedin Eq. (7) and controls the amount of the η Ξ component inthe intermediate MB state. resolution of the analysis is sufficient to extract the sub-dominant contributions, we may study the difference ofthe heavy hadron decay diagrams.0 FIG. 12. Cabibbo favored quark-line diagrams for the Ξ + c → π + ( ssu ) and Ξ c → π + ( ssd ) decays. VI. SUMMARY
We have studied the Ξ c → π + ( M B ) decay processas a tool to study Ξ resonances, such as the Ξ(1620)and the Ξ(1690). The M K Λ - M π + ¯ K Dalitz plot showsthe Ξ c → π + ( ¯ K Λ) decay is not affected by the presenceof resonances, in sharp contrast to the Λ c → K + ( ¯ K Λ)decay, where the a (980) in the K ¯ K channel considerablycomplicates the Ξ-resonance analysis.Taking into account Cabibbo-Kobayashi-Maskawa ma-trix and color suppressions, diquark correlations andkinematical restrictions, we have proposed a dominantmechanism (Fig. 3) to describe the Ξ c decays. This mech-anism determines the relative fractions of the intermedi-ate meson-baryon states, while the final M B state in-teraction has been incorporated and studied using threedifferent chiral unitary approaches. Thus, we have pre-dicted
M B invariant mass distributions in the three chi-ral schemes. We have seen that the Ξ c decay is not ade-quate to study the Ξ(1620) resonance because the openchannel π Ξ is not produced a tree level, and the otherpossible channels are closed. However, the Ξ(1690) peakcan be clearly seen in the π Ξ and ¯ K Λ mass distributions.We have further analyzed the peak around the Ξ(1690)energy region, because it could be produced by a pole,either in the second Riemann sheet or in a non-physicalRiemann sheet, or it might be just a threshold effect. Wehave shown that the ratios of the decay fractions aroundthe expected position of the Ξ(1690) resonance mightbe used to distinguish one situation from the anotherone. Comparing the several models for the final-stateinteraction, we have found that if the Ξ(1690) pole existsbelow the K − Σ threshold, the ¯ K Λ decay fraction willlargely exceed the K − Σ + one, as a consequence of thequite narrow decay width of the resonance. On the otherhand, if the pole is not placed in the SRS, defined bycontinuity with the physical sheet in the real axis, the¯ K Λ fraction becomes quite small and the K − Σ + modeturns out to be dominant.The above results are based on the mechanism shownin Fig. 3. However, there exist other quark-line diagrams where the high energy momentum π + is emitted after the¯ qq creation, which might provide also a sizable contribu-tion. We have estimated their contribution, and foundthat the neglected diagrams only alter the overall heightof the spectra. Thus, we have concluded that the resultsfrom the mechanism of Fig. 3 are reasonable, as longas only the spectral shape and the relative fractions areconcerned. VII. ACKNOWLEDGMENTS
This work is partly supported by Open PartnershipJoint Projects of JSPS Bilateral Joint Research Projects,JSPS KAKENHI Grant Nos. 16K17694 and 25247036,the Yukawa International Program for Quark-HadronSciences (YIPQS), the National Natural Science Founda-tion of China under Grant Nos. 1375024 and 11522539,the Spanish Ministerio de Economia y Competitivi-dad and European FEDER funds under contract nos.FIS2011-28853-C02-01 and FIS2011-28853-C02-02, andthe Generalitat Valenciana in the program Prometeo II-2014/068.
APPENDIX : QUARK REPRESENTATION OFHADRONS
In this appendix, we consider the quark representa-tion of hadrons, based on the SU(3) symmetry. For therepresentation of mesons, we use the SU(3) pseudoscalarmeson matrix as in the chiral perturbation theory [50].Respecting the SU(3) transformation, the meson degreesof freedom can be related to the quark degrees of freedomby the following equation, M = u ¯ u u ¯ d u ¯ sd ¯ u d ¯ d d ¯ ss ¯ u s ¯ d s ¯ s = π √ + η √ + η √ π + K + π − − π √ + η √ + η √ K K − ¯ K − η √ + η √ . (19)With regard to baryons, to obtain the similar relation, wereplace the antiquarks in the meson matrix by the flavorantitriplet diquark representation suited to the mixed an-tisymmetric representation of the baryons which appearsin Eq. (3),¯ u → √ ds − sd ) , ¯ d → √ su − us ) , ¯ s → √ ud − du ) , (20)1which leads to the following relation: B = 1 √ u ( ds − sd ) u ( su − us ) u ( ud − du ) d ( ds − sd ) d ( su − us ) d ( ud − du ) s ( ds − sd ) s ( su − us ) s ( ud − du ) (21)= Σ √ + Λ √ + Λ √ Σ + p Σ − − Σ √ + Λ √ + Λ √ n Ξ − Ξ − √ + Λ √ . (22)From these relations, we obtain the quark representationof the hadrons as summarized in Table III. This quarkrepresentation agrees with the one implicitly assumed inthe chiral Lagrangians, as shown in Ref. [52].Next, we consider the quark representation from a dif-ferent point of view, the assignment of the SU(3) mul-tiplets | N, Y, I, I (cid:105) , where N, Y, I, and I , respectively,represents the dimension of an irreducible representation,the hyper charge, the isospin, and the third componentof the isospin. This is because the M B fraction of heavyhadron decays can be understood as (the combinationof) the Clebsch-Gordan coefficients of SU(3) [53]. To la-bel the SU(3) multiplets, we use the SU(2) subgroupsof SU(3), T spin and V spin. The generators of thesesubgroups are T = I = λ ,T + = ( λ + iλ ) / , T − = ( λ − iλ ) / ,V = √ λ + 14 λ ,V + = ( λ + iλ ) / , V − = ( λ − iλ ) / , (24)where λ i is Gell-Mann matrix. T ± and V ± , respectively,correspond to the replacement, T ± : d ↔ u, − ¯ u ↔ ¯ d,V ± : s ↔ u, − ¯ u ↔ ¯ s. (25)This means that the the SU(3) multiplets | N, Y, I, I (cid:105) of the quarks can be labeled as [53] | u (cid:105) = | , + 13 , , + 12 (cid:105) , | d (cid:105) = | , + 13 , , − (cid:105) , | s (cid:105) = | , − , , (cid:105)| ¯ u (cid:105) = −| ∗ , − , , − (cid:105) , | ¯ d (cid:105) = | ∗ , − , , I z = + 12 (cid:105) , | ¯ s (cid:105) = | ∗ , + 23 , , (cid:105) . (26)Here, we consider the assignment of | N, Y, I, I (cid:105) to theoctet quark-antiquark state. The highest weight eigen-state of the octet meson for the T and V spin is | , , , +1 (cid:105) = | , +1 / , / , +1 / (cid:105) ⊗ | ∗ , − / , / , +1 / (cid:105) = | u ¯ d (cid:105) . (27)Using Eq. (25) and the Condon-Shortley phase conven-tion, we can determine other eigenstates. For example, | , , , +1 (cid:105) = | u ¯ d (cid:105) V − −−→ + | , − , / , +1 / (cid:105) = | s ¯ d (cid:105) , | , − , / , +1 / (cid:105) = s ¯ d T − −−→ + | , − , / , − / (cid:105) = −| s ¯ u (cid:105) , | , − , / , − / (cid:105) = −| s ¯ u (cid:105) V + −−→ √ | , , , (cid:105) + (cid:114) | , , , (cid:105) = | s ¯ s − u ¯ u (cid:105) . (28)In this way, we determine the quark representation of the | N, Y, I, I (cid:105) state. The SU(3) singlet state, | , , , (cid:105) , canbe constructed as | , , , (cid:105) = − √ | u ¯ u + d ¯ d + s ¯ s (cid:105) , (29)under two conditions, the orthogonality to | , , , (cid:105) and | , , , (cid:105) states and the Condon-Shortley phase conven-tion, (cid:104) j , j + M | ⊗ (cid:104) j , − j | J, M (cid:105) ≥
0, for both the T and V spins. Summarizing the assignment, we obtain u ¯ u u ¯ d u ¯ sd ¯ u d ¯ d d ¯ ss ¯ u s ¯ d s ¯ s = − | , , , (cid:105)√ − | , , , (cid:105)√ − | , , , (cid:105)√ | , , , +1 (cid:105) | , +1 , , + (cid:105)−| , , , − (cid:105) + | , , , (cid:105)√ − | , , , (cid:105)√ − | , , , (cid:105)√ | , +1 , , − (cid:105)−| , − , , − (cid:105) | , − , , + (cid:105) | , , , (cid:105)√ − | , , , (cid:105)√ . (30) In order to connect the η and η (cid:48) physical states with the η and η ones, we use the standard coupling of [51], η = 13 η + 2 √ η , η (cid:48) = 2 √ η − η . (23) The set of T spin and V spin, rather than the set of T spin and U spin, is chosen to be consistent with the phase convention inthe hadron matrices [53]. For V + , we use the phase of Eqs. (7.7) and (7.8) in Ref. [53]. Meson q ¯ q representation Baryon 3 q representation K + u ¯ s p √ u ( ud − du ) K d ¯ s n √ d ( ud − du ) π + u ¯ d Σ + 1 √ u ( su − us ) π √ ( u ¯ u − d ¯ d ) Σ [ u ( ds − sd ) − d ( su − us )] π − d ¯ u Σ − √ d ( ds − sd )¯ K s ¯ d Ξ √ s ( su − us ) K − s ¯ u Ξ − √ s ( ds − sd ) η √ ( u ¯ u + d ¯ d − s ¯ s ) Λ √ [ u ( ds − sd ) + d ( su − us ) − s ( ud − du )] η (cid:48) √ ( u ¯ u + d ¯ d + 2 s ¯ s )TABLE III. Quark representation of hadrons.Meson Baryon | N, Y, I, I z (cid:105) K + p | , +1 , , + (cid:105) K n | , +1 , , − (cid:105) π + Σ + | , , , +1 (cid:105) π Σ −| , , , (cid:105) π − Σ − −| , , , − (cid:105) ¯ K Ξ | , − , , + (cid:105) K − Ξ − −| , − , , − (cid:105) η Λ −| , , , (cid:105) η −| , , , (cid:105) TABLE IV. Assignment of the SU(3)-multiplet to hadron states.
Considering the chiral convention as in Eq. (19), we ob-tain the assignment of the SU(3) multiplet to hadronstates as in Table IV. Similarly, the assignment forbaryons is found from the assignment for the three-quark representation, starting from | , , , +1 (cid:105) = u ( su − us ) / √
2, and the baryon matrix in the chiral Lagrangianas in Eq. (22). The result is also shown in Table IV. Con-versely, when the assignment in the chiral Lagrangian istaken as in Table IV, the quark representation of hadronsis written as Table III.We note that at first sight, the assignment in Ta-ble IV and the phase convention in the chiral unitaryapproach [48], | π + (cid:105) = −| I = 1 , I z = +1 (cid:105) , | K − (cid:105) = −| I = 12 , I z = − (cid:105) , | Σ + (cid:105) = −| I = 1 , I z = +1 (cid:105) , | Ξ − (cid:105) = −| I = 12 , I z = − (cid:105) , (31) seem to be different for the following hadrons: π + , π , π − , η , η , Σ + , Σ , Σ − , Λ . (32)However, these assignments are physically equivalent.The difference for the hadrons in Eq. (32) means thatthe phase is different for both the s quark and ¯ s quark( ud diquark). Because the strangeness is the conservedquantum number and the sectors with the differentstrangeness do not mix under the strong interaction,these two assignments give the same results in physi-cal processes. Thus, from the usual phase conventionin Eq. (31) and the assignment of the SU(3) multipletto the quark representation as in Eq. (30), we can obtainthe physically equivalent quark representation of hadronsto the one in Table III.3 [1] S. Stone, PoS EPS-HEP2015 , 434 (2015).[2] E. Oset et al. , Int. J. Mod. Phys.
E25 , 1630001 (2016).[3] LHCb, R. Aaij et al. , Phys. Rev. Lett. , 072001(2015).[4] LHCb, R. Aaij et al. , Chin. Phys.
C40 , 011001 (2016).[5] L. Roca, M. Mai, E. Oset and U.-G. Meißner, Eur. Phys.J.
C75 , 218 (2015).[6] J.-J. Wu, R. Molina, E. Oset and B. S. Zou, Phys. Rev.Lett. , 232001 (2010).[7] J.-J. Wu, R. Molina, E. Oset and B. S. Zou, Phys. Rev.
C84 , 015202 (2011).[8] C. W. Xiao, J. Nieves and E. Oset, Phys. Rev.
D88 ,056012 (2013).[9] L. Roca, J. Nieves and E. Oset, Phys. Rev.
D92 , 094003(2015).[10] R. Chen, X. Liu, X.-Q. Li and S.-L. Zhu, Phys. Rev.Lett. , 132002 (2015).[11] J. He, Phys. Lett.
B753 , 547 (2016).[12] H.-X. Chen et al. , Phys. Rev.
C93 , 065203 (2016).[13] LHCb, R. Aaij et al. , JHEP , 103 (2014).[14] T. J. Burns, Eur. Phys. J. A51 , 152 (2015).[15] E. Wang, H.-X. Chen, L.-S. Geng, D.-M. Li and E. Oset,Phys. Rev.
D93 , 094001 (2016).[16] LHCb, R. Aaij et al. , Phys. Rev. Lett. , 082003(2016).[17] A. Feijoo, V. K. Magas, A. Ramos and E. Oset, Eur.Phys. J.
C76 , 446 (2016).[18] J.-X. Lu, E. Wang, J.-J. Xie, L.-S. Geng and E. Oset,Phys. Rev.
D93 , 094009 (2016).[19] H.-Y. Cheng and C.-K. Chua, Phys. Rev.
D92 , 096009(2015).[20] R. Chen, X. Liu and S.-L. Zhu, Nucl. Phys.
A954 , 406(2016).[21] H.-X. Chen, W. Chen, X. Liu and S.-L. Zhu, Phys. Rept. , 1 (2016).[22] E. Oset et al. , Nucl. Phys.
A954 , 371 (2016).[23] A. Feijoo, V. K. Magas, A. Ramos and E. Oset, Phys.Rev.
D92 , 076015 (2015).[24] K. Miyahara, T. Hyodo and E. Oset, Phys. Rev.
C92 ,055204 (2015).[25] Particle Data Group, K. A. Olive et al. , Chin. Phys.
C38 , 090001 (2014).[26] Amsterdam-CERN-Nijmegen-Oxford, C. Dionisi et al. ,Phys. Lett.
B80 , 145 (1978).[27] S. F. Biagi et al. , Z. Phys. C9 , 305 (1981).[28] S. F. Biagi et al. , Z. Phys. C34 , 15 (1987). [29] WA89, M. I. Adamovich et al. , Eur. Phys. J. C5 , 621(1998).[30] Belle, K. Abe et al. , Phys. Lett. B524 , 33 (2002).[31] FOCUS, J. M. Link et al. , Phys. Lett.
B624 , 22 (2005).[32] BaBar, B. Aubert et al. , Phys. Rev.
D78 , 034008 (2008).[33] BESIII, M. Ablikim et al. , Phys. Rev.
D92 , 092006(2015).[34] BaBar, B. Aubert et al. , Measurement of the Mass andWidth and Study of the Spin of the Ξ(1690) 0 Reso-nance from Λ + c → Λ ¯ K K + Decay at Babar, in
Proceed-ings of the 33rd International Conference on High EnergyPhysics (ICHEP ’06) , 2006, arXiv:hep-ex/0607043.[35] K.-T. Chao, N. Isgur and G. Karl, Phys. Rev.
D23 , 155(1981).[36] S. Capstick and N. Isgur, Phys. Rev.
D34 , 2809 (1986).[37] L. Ya. Glozman and D. O. Riska, Phys. Rept. , 263(1996).[38] M. Pervin and W. Roberts, Phys. Rev.
C77 , 025202(2008).[39] T. Melde, W. Plessas and B. Sengl, Phys. Rev.
D77 ,114002 (2008).[40] L.-Y. Xiao and X.-H. Zhong, Phys. Rev.
D87 , 094002(2013).[41] A. Ramos, E. Oset and C. Bennhold, Phys. Rev. Lett. , 252001 (2002).[42] C. Garcia-Recio, M. F. M. Lutz and J. Nieves, Phys.Lett. B582 , 49 (2004).[43] D. Gamermann, C. Garcia-Recio, J. Nieves and L. L.Salcedo, Phys. Rev.
D84 , 056017 (2011).[44] T. Sekihara, PTEP , 091D01 (2015).[45] C. L. Schat, J. L. Goity and N. N. Scoccola, Phys. Rev.Lett. , 102002 (2002).[46] Y. Oh, Phys. Rev. D75 , 074002 (2007).[47] W. Roberts and M. Pervin, Int. J. Mod. Phys.
A23 , 2817(2008).[48] E. Oset and A. Ramos, Nucl. Phys.
A635 , 99 (1998).[49] J. Nieves and E. Ruiz Arriola, Phys. Rev.
D64 , 116008(2001).[50] S. Scherer and M. R. Schindler, Lect. Notes Phys. ,pp.1 (2012).[51] A. Bramon, A. Grau and G. Pancheri, Phys. Lett.
B283 ,416 (1992).[52] R. P. Pavao, W. H. Liang, J. Nieves and E. Oset,arXiv:1701.06914.[53] J. J. de Swart, Rev. Mod. Phys.35