$\Lambda(1405)$ production in the process $\chi_{c0}(1P)\to \bar{\Lambda}\Sigma\pi$

We have performed a theoretical study on the process $\chi_{c0}(1P)\to \bar{\Lambda}\Sigma\pi$, by taking into account the final state interactions of $\pi\Sigma$ and $\pi\bar{\Lambda}$ based on the chiral unitary approach. As the isospin filters of $I=0$ in the $\pi\Sigma$ channel and $I=1$ in the $\pi\bar{\Lambda}$ channel, this process can be used to study the molecular structure of the $\Lambda(1405)$ resonance, and to test the existence of the predicted states $\Sigma(1380)$ and $\Sigma(1430)$ with spin-parity $J^P=1/2^-$. Our results show that there is a peak around $1350 \sim 1400$~MeV, and a cusp around the $\bar{K}N$ threshold in the $\pi\Sigma$ invariant mass distribution, which should be the important feature of the molecular state $\Lambda(1405)$. We also find a peak around $1380$~MeV, and a cusp around $\bar{K}N$ threshold in the $\pi\bar{\Lambda}$ invariant mass distribution, which are associated to the $\Sigma(1380)$ and $\Sigma(1430)$ resonances.


I. INTRODUCTION
The nature of the baryon resonances is one of the important issues in hadron physics [1,2]. There are many facilities, such as BESIII, LHCb, Belle, et al., that have presented a lot of information about baryon resonances, which provide a good platform to extract the baryon properties. On the other hand, the theoretical work goes parallel, most of the existing states can be well described, and some predictions of the effective theories have been confirmed experimentally.
In order to further understand the molecular structure of the Λ(1405) state, we investigate the process χ c0 (1P ) →ΛΣπ, by considering the final state interaction of πΣ, which will dynamically generate the two poles of the Λ(1405) state in the chiral unitary approach. The χ c0 (1P ), with I G (J P C ) = 0 + (0 ++ ), is a cc state, * Electronic address: wangen@zzu.edu.cn † Electronic address: xiejujun@impcas.ac.cn and blind to SU(3), hence behaving like an SU(3) singlet. Since the outgoing particleΛ has isospin I = 0, the πΣ system must have isospin I = 0, to combine to the isospin I = 0 of the χ c0 (1P ). Thus, this process is a good filter of isospin that guarantees that the πΣ will be in I = 0. Besides, the lower Λ(1405) state couples strongly to the πΣ channel, so the πΣ final state in this process is an ideal channel to study the molecular structure of the Λ(1405) state.
The π andΛ can also undergo final state interaction, and the isospin of πΛ system is I = 1, which together with the Σ gives rise to the isospin of χ c0 . It should be noted that a baryon resonance around theKN threshold with J p = 1/2 − , strangeness S = −1 and isospin I = 1 was predicted in the chiral unitary approach [3,4] (we label this state as Σ(1430) in following), and can couple to the πΛ channel. It was also suggested to search for this state in the process χ c0 →ΣΣπ [40]. In addition, a Σ * state with J P = 1/2 − and M ∼ 1380 MeV and Γ ∼ 120 MeV (we label this state as Σ(1380) in following), has been predicted in the pentaquark picture [41], which were studied in the processes of J/ψ decay [42,43], K − p → Λπ + π − [44,45], Λp → Λpπ 0 [46], and Λ + c → ηπ + Λ [47], as so on. Thus, the χ c0 can also decay into a Σ and the intermediate resonanceΣ(1380) in S-wave, then theΣ(1380) go into the πΛ states in Swave. As a result, for the S-wave final state interaction of πΛ, we will consider the mechanism of the coupled channel in the chiral unitary approach, and the intermediate resonance Σ(1380) mechanism 1 . In this way, the shape of the πΛ mass distribution of this process can be helpful to test the existence of the Σ(1430) and Σ(1380) resonances, and to search for the unobserved state Σ *   21,43]. This paper is organized as follows. In Sec. II, we will give the formalism of mechanisms, in Sec. III, we will present our results and discussions, finally, the conclusion will be given in Sec. IV.

II. FORMALISM
In this section, we will describe the reaction mechanism for the process χ c0 (1P ) →ΛΣπ.

A. The model of χc0(1P ) →ΛΣπ
In the first step, χ c0 (1P ) can decay into the final states ΛΣπ directly in the tree level, as depicted in Fig. 1(a). Then the final particles πΣ can undergo final state interaction, which will give rise the dynamically generated resonance Λ(1405), as depicted in Fig. 1 For the mechanism of πΣ final state interaction, we must take into account that in the first step one can produce the other meson-baryon pairs that couple to the same πΣ quantum numbers, then reaching the final πΣ through re-scattering. Since χ c0 is SU(3) singlet, and Λ belongs to SU(3) anti-octet, hence, πΣ system should be in an octet state, which will give rise the same quantum numbers as Λ(1405). Since both π and Σ belong to SU(3) octets, we have the representation for 8(π) 8(Σ) going to the 8 s (symmetry) and 8 a (antisymmetry). With the SU(3) isoscalar factors in the PDG [21] of Table I, we can obtain the weights h i in the isospin basis of this states, which go into the primary production of each meson baryon channel, and stand for the weights of the tran- sition χ c0 →Λ P B (P B =KN, πΣ, ηΛ, ΞK), where the weightsD andF are unknown parameters.
The interaction of octet pseudoscalar mesons and the octet 1/2 + baryons, which can dynamically generate the Λ(1405), has be studied with the chiral unitary approach in Refs. [3,6,48]. In Ref. [22], a new strategy to extract the position of the two poles of Λ(1405) from πΣ photoproduction experimental data was done, based on small modifications of the unitary chiral perturbation theory amplitude, which will be adopted in this work. Because the thresholds of the channels ηΛ and KΞ lay far above the energies that we consider in this work, and the effect can be effectively reabsorbed in the subtraction constants, as discussed in Refs. [20,22], we do not consider these two channels in following calculations.
In addition to the πΣ final state interaction, the states π andΛ also can undergo the final state interaction in the process of χ c0 →ΛΣπ, as depicted in Fig. 2. In this case, we get the weights for different channels from Ref. [40], where the parametersD andF are are same as those of Eq. (1).
As predicted in Refs. [3,4], the πΛ system with isospin I = 1 can undergo the S-wave final state interaction, which will dynamically generate a cusp structure around theKN threshold, associated to the Σ(1430) resonance. Taking into the S-wave final state interaction, the total amplitude for the process χ c0 →ΛΣπ is, where the loop function G and the transition amplitude t of the πΣ final state interaction are taken from Refs. [22,40]. In this work, we work with R =F /D, and include the weightD of Eqs. (1) and (2) in the V p andṼ p factors, as done in Ref. [40]. In Eq.(3), the amplitude for the tree level of Fig. 1(a) is, If we consider the transition χ c0 → ΣPB (PB = KN , πΣ, πΛ, ηΣ,KΞ), the amplitude for the tree level can also be given as, Combing the Eqs. (4) and (5), we can obtain thatṼ p = V p ×(h πΣ /h πΛ ) = − √ 3V p . Now, the amplitude of Eq. (3) can be rewritten as, S-wave, as depicted in Fig. 3. The contribution of intermediate resonanceΣ(1380) should add coherently to the Eq. (6) 2 , as follows, where the normalization α stands for the amplitude strength, which will be chosen to provide a sizeable effect of the intermediate resonanceΣ (1380). M πΣ and M πΛ are the invariant masses of πΣ and πΛ, respectively. In this work, we take M Σ(1380) = 1380 MeV, Γ Σ(1380) = 120 MeV as fitted in Ref. [44].

III. RESULT AND DISCUSSIONS
Before presenting the results for the process χ c0 (1P ) → ΛΣπ, we show the module squared of the amplitudes |tK N,πΣ | 2 and |t πΣ,πΣ | 2 in I = 0 in Fig. 4, from where we can see that the peak of πΣ → πΣ amplitude mainly comes from the the lower pole, while the one ofKN → πΣ amplitude comes from the higher pole. In Fig. 5, we present the module squared of the transition amplitudes |t| 2 in I = 1. As we can see, a clear cusp structure around theKN threshold is found, same as the Refs. [22,40].
As we donot know the exact value of R =F /D, and the production weights ofKN and πΣ are expected to be same magnitude, we will vary the R from -2 to 2, as done in Ref. [40]. Firstly, we take R = 1, and α = 0.06 of the normalization in Eq. (8), which gives rise to a sizeable effect of intermediate Σ(1380). In Fig. 6, up to an  arbitrary normalization of V p , we show the πΣ invariant mass distribution, where the term of πΣ final state interaction (labeled as 'T πΣ ') gives rise to a peak around 1410 MeV, and the peak moves to low energy because of the interference with the tree level term (labeled as 'tree'). We also present the πΣ invariant mass distribution with different values of α in Fig. 7, which shows that the contribution from the intermediateΣ(1380) resonance does not significantly affect the peak position of the πΣ mass distribution. The πΣ invariant mass distribution with different values of R from -2 to 2 is shown in Fig. 8, where we can see that as the ratio R decreasing, the peak position of the Λ(1405) in the πΣ mass distribution moves to the region of low energies, and a cusp structure aroundKN threshold appears. Indeed, the value of R can be larger than 2 or less than -2, and the peak of the πΣ invariant mass distribution will vary from 1360 MeV, the position of the peak in t πΣ,πΣ , to 1400 MeV, the one in tK N,πΣ , as shown in Fig. 4.
As discussed in Ref. [49], one of the defining features associated to the molecular states that couple to several hadron-hadron channels is that one can finds a strong and unexpected cusp at the threshold of the channels corresponding to the main component of the molecular state, and one of the examples is the observations of the cusp, associated to the molecular state X(4160), in the B + → J/ψφK + decay [50]. The peak and the cusp, observed in Fig. 8, should be the important feature to confirm the existence of the Λ(1405) in the decay of χ c0 →ΛΣπ. Thus, we strongly encourage to measure the πΣ invariant mass distribution of the χ c0 →ΛΣπ decay.
In order to check the existence of the Σ(1430) and Σ(1380), we also present the πΛ invariant mass distribution in Fig. 9, where we can see that there is a clear bump structure around 1380 MeV, which is associated to the intermediate resonance Σ(1380), and a clear cusp structure around theKN threshold, which comes from the πΛ final state interaction, as shown in Eq. (6). By varying the value of the normalization α as depicted in Fig. 10, the bump structure ofΣ(1380) becomes smoother for a smaller α, and more clear for a larger one. It should be stressed that the bump structure ofΣ(1380), or the cusp structure around theKN threshold, if confirmed experimentally, should be related to the resonance Σ(1380) or Σ(1430).
In addition, we show the πΛ invariant mass distributions by varying the ratio R from -2 to 2 in Fig. 11. As we can see that the peak structure of theΣ(1380) and the cusp structure around theKN threshold are always clear for different values of the ratio R.
Finally, it should be noted that the SIDDHARTA measurement of kaonic hydrogen gives an accurate constraint on the K − p scattering length [51], and the interaction kernel with the next-to-leading order chiral perturbation theory is used with the systematic χ 2 analysis in Refs. [13,15,16]. However, as a motivation for measuring this process experimentally, we use the model of the final state interaction of πΣ of I = 0 developed in Ref. [22], which can also produce the main feature of the amplitudes of πΣ →KN and πΣ → πΣ given by the more accurate model.

IV. CONCLUSIONS
In this paper, we have studied the process χ c0 (1P ) → ΛΣπ by taking into account the final state interactions of πΣ and πΛ within the chiral unitary approach. As the isospin I = 0 filter in the πΣ system and the isospin I = 1 filter in the πΛ system, this process can be used to study the molecular structure of Λ(1405) state, and to search for the predicted states Σ(1380) or Σ(1430) with J P = 1/2 − . We have shown that, there is a peak of 1350 ∼ 1400 MeV with −2 ≤ R ≤ 2, and a cusp around theKN in the πΣ mass distribution, which should be the important feature the the molecular state Λ(1405).
In summary, the process χ c0 →ΛΣπ can be used to study the molecular structure of the Λ(1405) resonance, and also to test the existence of the predicted states Σ(1380) and Σ(1430).