]>PLB34563S0370-2693(19)30243-610.1016/j.physletb.2019.04.011The AuthorsPhenomenologyFig. 1(a) Quark level diagram for Λc+→π+sud. (b) Hadronization through q¯q creation with vacuum quantum numbers.Fig. 1Fig. 2Triangle diagrams for the decay of Λc+→π+π0π−Σ+. (a) represents the final state interaction of K¯N→π−Σ+, while (b) represents the final state interaction of K¯0p→π0Σ+. The black ‘dot’ stands for the vertex of the final state interaction of K¯N→πΣ in S-wave.Fig. 2Fig. 3Transition amplitudes of K¯N→π−Σ+ in S-wave.Fig. 3Fig. 4Transition amplitudes of K¯0p→π0Σ+ in S-wave.Fig. 4Fig. 5Invariant π0π−Σ+ mass distribution of Λc+→π+π0π−Σ+ decay.Fig. 5Fig. 6Invariant π−Σ+ mass distribution of Λc+→π+π0π−Σ+ decay.Fig. 6Search for the Σ⁎ state in Λc+→π+π0π−Σ+ decay by triangle singularityJu-JunXiea⁎xiejujun@impcas.ac.cnEulogioOsetboset@ific.uv.esaInstitute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, ChinaInstitute of Modern PhysicsChinese Academy of SciencesLanzhou730000ChinabDepartamento de Física Teórica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Institutos de Investigacíon de Paterna, Aptdo. 22085, 46071 Valencia, SpainDepartamento de Física TeóricaIFICCentro Mixto Universidad de Valencia-CSICInstitutos de Investigacíon de PaternaAptdo. 22085Valencia46071Spain⁎Corresponding author.Editor: W. HaxtonAbstractA Σ⁎ resonance with spin-parity JP=1/2− and mass in the vicinity of the K¯N threshold has been predicted in the unitary chiral approach and inferred from the analysis of CLAS data on the γp→K+π0Σ0 reaction. In this work, based on the dominant Cabibbo favored weak decay mechanism, we perform a study of Λc+→π+π0Σ⁎ with the possible Σ⁎ state decaying into π−Σ+ through a triangle diagram. This process is initiated by Λc+→π+K¯⁎N, then the K¯⁎ decays into K¯π and K¯N produce the Σ⁎ through a triangle loop containing K¯⁎NK¯ which develops a triangle singularity. We show that the π−Σ+ state is generated from final state interaction of K¯N in S-wave and isospin I=1, and the Λc+→π+π0π−Σ+ decay can be used to study the possible Σ⁎ state around the K¯N threshold. The proposed decay mechanism can provide valuable information on the nature of the Σ⁎ resonance and can in principle be tested by facilities such as LHCb, BelleII and BESIII.1IntroductionInvestigating low-lying excited states of Σ(1193), Σ⁎, with isospin I=1 and strangeness S=−1 is one of the important issues in hadronic physics [1–3]. The Σ⁎ states were mostly produced and studied in antikaon-nucleon reactions, and the information on their properties is still rather limited [3]. Based on the chiral unitary approach, the low-lying S=−1 excited baryons were studied by means of coupled channels in Refs. [4–15]. In addition to the well reproduced properties of the Λ(1405), a possible resonance-like structure in I=1 around the K¯N threshold was found in Ref. [6] as a bound state and in Ref. [10] as a strong cusp effect. Then it was further investigated in Ref. [16] based on the analysis of the experimental data on the γp→K+π±Σ∓ reactions [17,18]. Such a state near the K¯N threshold is also discussed in Refs. [11,12], while in Ref. [15], a Σ⁎ state is found with mass around 1400 MeV, though it is not clear if it is related to one or two poles in the complex plane. On the other hand, the effect of this possible K¯N state with mass about 1430 MeV in the processes of χc0(1P)→πΣ¯Σ(Λ) decays was also studied in Refs. [19,20].The nonleptonic weak decays of charmed baryons, particularly Λc+ decays into two mesons and one baryon, have shown a great value to produce baryonic resonances with S=−1 and learn about their nature [21–24]. In some cases the production rate is enhanced by the presence of a triangle singularity (TS) in the reaction mechanism [25,26]. The TS appears from a loop diagram in the decay of a particle 1 into two particles 2 and 3 through the following process: at first the particle 1 decays into particles A and B, and the particle A subsequently decays into particles 2 and C, and finally the particles B and C merge and form the particle 3 in the final state.Following the work of Ref. [26], in this paper, we focus on the Λc+→π+π0π−Σ+ decay taking into account the π−Σ+ final state interaction from the triangle diagram. We consider the external W+ emission diagram for the transition of Λc+ into π+K¯⁎N, which gives the main contribution to the Λc+→π+π0π−Σ+ process [26]. Since we take into account the external W+ emission diagram, the W+ produces the π+ in one vertex and in the other one includes a c→s transition. Then we have a π+ and an sud cluster, with the ud diquark in I=0, because there these quarks are spectators. Hence the sud cluster hadronizes in K¯⁎N in I=0, and, since π has isospin 1, the πΣ system should be also in isospin 1 to keep the isospin conserved. We will show that the production of the K¯N state is enhanced by the TS in the π0π−Σ+ mass distribution, and that a narrow peak or cusp structure around the K¯N threshold in the πΣ mass distribution appears. The observation of the TS in this process would give further support to the existence of the K¯N resonance, 11In the following, we use Σ⁎(1430) to denote the K¯N resonance. and provide us better understanding on the triangle singularity.This article is organized as follows. In Sec. 2, we present the theoretical formalism for calculating the decay amplitude of Λc+→π+π0π−Σ+. Numerical results and discussions are presented in Sec. 3, followed by a summary in the last section.2FormalismWe consider the external emission mechanism of Fig. 1. The mechanism is Cabibbo favored. In Fig. 1 (a) we see that the original ud quarks of the Λc+ are in I=0 and furthermore they are spectators in the reaction, hence they continue to have I=0 in the final sud state. This state hadronizes to meson-baryon pairs after creating a q¯q pair with the quantum numbers of the vacuum. In our case we are interested in the K¯⁎N production. The insertion of q¯q can be done between any two quarks, but in our case it must involve the strange quark. The reason is that we want to have K¯⁎N in S-wave, hence negative parity. Since the ud quarks are spectators and have positive parity, it must be the strange quark that is produced in L=1. But we want it in its ground state in the K¯⁎ after the hadronization, hence it has to be involved in the hadronization process.The explicit flavor combination of meson and baryon pairs of the hadronization [see the process shown in Fig. 1 (b)], H, is shown in Ref. [26] and one finds,(1)H=K⁎−p+K¯⁎0n−63ϕΛ, and one ignores the ϕΛ component that has no role in the TS mechanism. We can see that H in Eq. (1) has I=0, since (K¯⁎0,−K⁎−) is our isospin doublet. This corresponds to the isospin of s(ud)I=0 just after the weak vertex, which is conserved after that.Once π+K¯⁎N is produced, the K¯⁎ decays to πK¯ and the K¯N interact to give πΣ. Since K¯⁎N is in I=0, so must be the ππΣ system, which forces the πΣ system, coming from the K¯N interaction, to have I=1 if there is isospin conservation, as we shall assume here. This is shown in Fig. 2.By filtering the I=1 πΣ system, the K¯N→πΣ amplitude incorporates the Σ⁎(1430) resonance and we should see the signal of the state clearly in the πΣ mass distribution. In addition, as we shall see, the mechanism develops a TS which enhances the production of the Σ⁎(1430) state. In Fig. 2 we have separated the two possible decay modes: Fig. 2 (a) shows the contribution from the final state interaction of K¯⁎N→π−Σ+, while Fig. 2 (b) stands for the transition of K¯0p→π0Σ+.We first consider the decay of Λc+→π+K⁎−p, proceeding via S-wave, and we take the decay amplitude as [26],(2)tΛc+→π+K⁎−p=Aσ→⋅ϵ→, with A constant. Then we can easily obtain the branching ratio Br(Λc+→π+K⁎−p), summing over the K¯⁎− polarizations, as(3)Br(Λc+→π+K⁎−p)=3mN|A|28π3MΛc+ΓΛc+×∫mK⁎−+mpMΛc+−mπ+pπ+p˜K⁎−dMK⁎−p, where pπ+ is the momentum of π+ in the Λc+ rest frame, and p˜K⁎− is the momentum of K¯⁎ in the K⁎−p rest frame with invariant mass MK⁎−p,(4)pπ+=λ1/2(MΛc+2,m2π,MK⁎−p2)2MΛc,(5)p˜K⁎−=λ1/2(MK⁎−p2,mK⁎−2,mp2)2MK⁎−p, with λ(x,y,z) the ordinary Källen function.By calculating the partial decay width of Λc+→π+K⁎−p, using the experimental branching ratio of Br(Λc+→π+K⁎−p)=(1.4±0.5)×10−2 and ΓΛc+=(3.3±0.1)×10−9 MeV [3], we can determine the value of the constant |A|2,(6)|A|2=(3.9±1.4)×10−16MeV−2, where the error is taken from the experimental error in the branching ratio of Br(Λc+→π+K⁎−p). In view of the weights in H in Eq. (1), the same value of |A|2 is used for the decay of Λc+→π+K¯⁎0n.Next, we write the total decay amplitude of Λc+→π+π0π−Σ+ for those diagrams shown in Fig. 2, 22More details can be found in Ref. [26].(7)ttotal=−Ag2(σ→⋅k→atTaMa+2σ→⋅k→btTbMb), where k→a and k→b are the momenta of the π0 and π− in the diagrams of Fig. 2 (a) and (b), respectively, calculated in the π0π−Σ+ rest frame, and the coupling g is given by g=mV/2fπ with mV=780 MeV and fπ=93 MeV. In Eq. (7), Ma and Mb are the two-body scattering amplitudes, which depend on the invariant masses of Mπ−Σ+ and Mπ0Σ+, respectively, and they have the explicitly forms as,(8)Ma=tK−p→π−Σ+−tK¯0n→π−Σ+,(9)Mb=tK¯0p→π0Σ+, where tK−p→π−Σ+ and tK¯0n→π−Σ+ depend on Mπ−Σ+, and tK¯0p→π0Σ+ depends on Mπ0Σ+. The factor 2 in Eq. (7) and the minus sign in Eq. (8) have their origin in the different K¯⁎→πK vertices. Note that the two combinations, K−p−K¯0n and K¯0p have I=1, as it should be.In addition, we give explicitly the amplitude tTa for the case of K⁎−, p and K− in the triangle loop, as an example,(10)tTa=∫d3q(2π)3mp2ωK⁎−ωpωK−1ka0−ωK−−ωK⁎−+iΓK⁎−2×1P0+ωp+ωK−−ka0(2+q→⋅k→a|k→a|2)×P0ωp+ka0ωK−−(ωp+ωK−)(ωp+ωK−+ωK⁎−)P0−ωK⁎−−ωp+iΓK⁎−2×1P0−ωp−ωK−−ka0+iϵ, with P0=Mπ−π0Σ+ the invariant mass of the final π0π−Σ+ system, ωp=|q→|2+mp2, ωK−=|q→+k→a|2+mK−2, and ωK⁎−=|q→|2+mK⁎−2. The energy ka0 and momentum |k→a| of π0 emitted from K⁎− are given by(11)ka0=Mπ−π0Σ+2+mπ02−Mπ−Σ+22Mπ−π0Σ+,(12)|k→a|=(k0)2−mπ02. While, kb0, |k→b|, and tTb can easily be obtained just applying the substitution to ka0, |k→a| and tTa with mπ0→mπ− and Mπ−Σ+→Mπ0Σ+.Then we obtain the final invariant masses distribution for four particles in the final state,(13)d3ΓdMπ0π−Σ+dMπ−Σ+dMπ0Σ+=g2|A|2128π5mΣ+MΛc+p˜π+×Mπ−Σ+Mπ0Σ+Mπ0π−Σ+(|k→a|2|tTaMa|2+2|k→b|2|tTbMb|2+22Re[tTaMa(tTbMb)⁎]k→a⋅k→b), with(14)p˜π+=λ1/2(MΛc+2,Mπ0π−Σ+2,mπ+2)2MΛc+. In Eq. (13), k→a⋅k→b is evaluated in terms of Mπ0π−Σ+, Mπ−Σ+, and Mπ0Σ+,(15)k→a⋅k→b=mπ02+mπ−2−Mπ0π−2+2ka0kb02, with(16)Mπ0π−2=Mπ0π−Σ+2+mπ02+mπ−2+mΣ+2−Mπ−Σ+2−Mπ0Σ+2.On the other hand, we have to regularize the integral in Eq. (10). In this work, we use the same cutoff of the meson loop that is used to calculate tK−p→π−Σ+ with θ(qmax−|q→⁎|), where q→⁎ is the q→ momentum in the R rest frame (see Ref. [27] for more details).3Numerical resultsIn Figs. 3 and 4 the two body transition amplitudes of K¯N→πΣ are shown. We plot the real and imaginary parts of those two body transition amplitudes. In Fig. 3, the blue curve and solid curve stand for the real parts of tK−p→π−Σ+ and tK¯0n→π−Σ+, respectively, while the blue-dashed curve and dashed curve stand for the imaginary parts of tK−p→π−Σ+ and tK¯0n→π−Σ+, respectively. In Fig. 4, the solid and dashed curves are the real and imaginary parts of tK¯0p→π0Σ+, respectively. It can be observed that the Re(tK¯N→π−Σ+) have peaks around 1430 MeV, and |Im(tK¯N→π−Σ+)| have peaks around 1420 MeV, and there are bump structures for Re(tK¯0p→π0Σ+) and Im(tK¯0p→π0Σ+) around 1430 and 1440 MeV, respectively. These results, shown in Figs. 3 and 4, are obtained using the Bethe-Salpeter equation, with the tree level potentials given in Ref. [5]. The loop functions for the intermediate states are regularized using the cutoff method with a cutoff of 630 MeV. This parameter is also used to evaluate the loop integral in the diagrams of Fig. 2.With all the ingredients obtained above, one can easily obtain dΓ/dMπ0π−Σ+ by integrating over Mπ−Σ+ and Mπ0Σ+ and using |A|2=3.9×10−16 MeV−2. We show the theoretical results of dΓ/dMπ0π−Σ+ in Fig. 5. We see a clear bump structure of the invariant π0π−Σ+ mass distribution around 1880 MeV for Λc+→π+π0π−Σ+ decay, which is due to the triangle singularity of the triangle diagrams as shown in Fig. 2.On the other hand, by integrating over Mπ0π−Σ+ and Mπ0Σ+, we obtain dΓ/dMπ−Σ+ which is shown in Fig. 6. We see a really narrow peak of the invariant π−Σ+ mass distribution around 1434 MeV for Λc+→π+π0π−Σ+ decay, which is the contribution from the K¯N resonance which is discussed above. Similar results can be also obtained for dΓ/dMπ0Σ+.From these results shown in Figs. 5 and 6, one can easily obtain the branching ratio of Br(Λc+→π+π0π−Σ+) is about (3±1)×10−4, with the error that is taken from the error in |A|2.One should stress the most remarkable feature in the distributions of Fig. 6: the width of the Σ⁎(1430) produced is about 10 MeV, remarkably smaller than the other Σ⁎ resonances of 30 MeV or even bigger [3]. Yet, as discussed in Ref. [16] the peak corresponds to a cusp at the K¯N mass threshold, however, very pronounced. The dynamics of these cusps corresponds to a state nearly bound. Theoretically, a small change in the parameters makes a pole appear. This situation is very similar to the one of the a0(980) resonance, which both theoretically [28] and experimentally [29] appears as a very pronounced cusp.4ConclusionsThe triangle singularities have recently shown to be very important in many hadronic decays. In this work we provide the first evaluation of the Σ⁎(1430) production in the decay of Λc+→π+π0π−Σ+. The decay mechanism for the production is given by a first decay of the Λc+ into π+K¯⁎N, then the K¯⁎ decays into K¯π and the K¯N merge to produce the Σ⁎(1430) through both the final state interaction of K¯N→πΣ transition and a triangle loop containing K¯⁎NK¯, which develops a singularity of the invariant mass of π0π−Σ+ system around 1880 MeV.It is found that a narrow peak, of the order of 10 MeV, tied to the Σ⁎(1430) state appears in the final π−Σ+ mass spectrum at the energy around the K¯N mass threshold of 1434 MeV. The line shape obtained here is intimately tied to the nature of the Σ⁎(1430) as a dynamically generated resonance from the meson baryon interaction, and shows up as a cusp structure. The theoretical calculations done here together with the experimental measurements would thus bring valuable information on the nature of this resonance. Corresponding experimental measurements could in principle be done by BESIII [30] and BelleII [31] Collaborations. In this sense the branching ratios obtained here for this Σ⁎(1430) signal are of the order of 10−4, which are well within the measurable range in these facilities. In view of that, the measurements of the Λc+→π+π0π−Σ+ decay is strongly encouraged. Besides, the mechanism studied in this work contributes also to the processes of Λc+→π+π0π0Λ and Λc+→π+π+π−Λ, and thus these processes are also very interesting and can be measured by future experiments.AcknowledgementsThis work is partly supported by the National Natural Science Foundation of China under Grant Nos. 11475227 and 11735003, and by the Youth Innovation Promotion Association CAS (No. 2016367). 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