Studying $\Lambda^*$ resonances in the $p \bar p \rightarrow \Lambda \bar\Lambda \eta$ reaction

In this work, we make a theoretical study on the $p \bar p \rightarrow \Lambda \bar\Lambda \eta$ reaction for antiproton beam energy from threshold to 4GeV within an effective Lagrangian approach and isobar model. By assuming this reaction is dominated by the excitation of $\Lambda$ and $\bar \Lambda$ resonances in intermediate states, we calculate the total cross sections and give the predictions of the angular distribution and invariant mass spectrum of final particles. In particular, we discuss the possibility to verify the existence of a narrow $\Lambda$ resonance found in the process of $K^- p\to \eta \Lambda$ in the present reaction. It shows that the $p \bar p \rightarrow \bar \Lambda \Lambda \eta$ reaction can provide us with valuable information about the $\Lambda$ resonances having significant couplings to $\bar K N$ and $\Lambda\eta$ channels. Thus the experimental data of this reaction will be a good supplement to the $\bar K N\to\eta \Lambda$ scattering data for studying $\Lambda$ resonances.


I. INTRODUCTION
The study on the properties of Λ resonances constitutes one important part of the research in the baryon spectroscopy, which offers us useful information about the strong interaction in the nonperturbative energy region and also tests of our knowledge in the strange particle channels. Up to now, most of the knowledge about Λ resonances is from the analysis of the data in theKN and πΣ channels. Studies on other channels, although very important, are still relatively lacking. Due to isospin conservation, the ηΛ channel is of special interest because it only couples to Λ resonance, which offers a relatively clean channel for studying the properties of the Λ resonances. But even with this advantage, the status of current knowledge on the coupling of Λ resonances to ηΛ channel is still not satisfying. In the Particle Data Group(PDG) book [1], there is only one Λ * state, i.e. Λ(1670), has well-established coupling with ηΛ channel.
The decay branch ratio of other Λ resonances to this channel is still not well identified. It is possible that other resonances indeed have weak coupling with this channel * Electronic address: liubc@xjtu.edu.cn and are therefore hard to study their couplings with ηΛ.
However, the relatively poor quality of experimental data in this channel is also a potentially important reason.
The Crystal Ball Collaboration data on the reaction K − p → ηΛ near threshold published in 2012 have much higher accuracy than before, which offers a good basis to investigate the reaction mechanism of this reaction and to extract the properties of Λ resonances in the ηΛ channel. Based on the new data an analysis within an effective Lagrangian approach and isobar model was performed in Refs. [2,3]. The main findings are, although the Λ(1670) gives the dominant contribution near threshold, the bowl structure appearing in the angular distribution may indicate a new narrow resonance. It was shown that the experimental data supported the existence of a D 03 resonance with M=1668.5 ± 0.5 MeV and Γ = 1.5 ± 0.5 MeV(denoted as Λ * D for convenience). Due to the very narrow width, this Λ resonance is obviously not any existing Λ resonance in the PDG book. The possible existence of a narrow Λ resonance in this channel was confirmed by another group based on a coupled-channel analysis [4,5].
However, in their analysis the proposed narrow resonance has the quantum numbers J P = 3 2 + (hereafter referred to as Λ * P ). Very interestingly, a narrow enhancement lying near the ηΛ threshold was also found in the mass spectrum of K − p in the decay of Λ c → pK − π + at Belle [6].
Until now, the origin of this enhancement is still not well identified. Very recently, it was argued that the enhancement might be caused by kinematical singularity [7].

II. THEORETICAL FORMALISM
In this work, we investigate the pp → ΛΛη reaction within an effective Lagrangian approach and isobar model. We assume that this reaction is dominated by the excitation of Λ andΛ resonances in the intermediate states with considering the contributions from the Λ(1670) and a very narrow Λ * D /Λ * P resonance suggested in Refs. [2][3][4][5]. The basic Feynman diagrams are depicted in Fig.1. The effective Lagrangians describing the KN Λ and K * N Λ interactions can be given as The value of g N KΛ can be determined by the SU (3) predictions, and we adopt g N KΛ = −13.24 in our calculations [17,18]. For the coupling constants g K * N Λ and κ K * , we take their values from the Nijmegen potential, i.e. g K * N Λ = 4.26 and κ K * = 2.66 [19,20].
The relevant interaction Lagrangians involving the Λ(1670) or the Λ * D /Λ * P resonances are used in the same forms as in Refs. [2,3], The coupling constant g Λ(1670)K * N = 0.753 is taken from Ref. [16], where its value is obtained based on a chiral quark model. For the Λ * DK N and Λ * D ηΛ couplings, we follow the results in Refs. [2,3], where the relevant coupling constants are fitted to the experimental data of the K − p → ηΛ reaction(Scenario I). The obtained parameters are shown in Table I. To get the parameters for the P 03 assignment, we fit them to the same data set as in the Scenario I but assuming the new Λ resonance is a P 03 state(Scenario II). The obtained mass and width of the narrow resonance are consistent with the results in Ref. [5] within uncertainties. Note that we also calculate the predictions of the Λ polarization for the K − p → ηΛ reaction and find that in Scenario II the predictions seem incompatible to the available data(see also Fig. 20 of Ref. [4]).
So more accurate Λ polarization data of the K − p → ηΛ reaction will be helpful to clarify the quantum numbers of this narrow resonance.
Because hadrons cannot be treated as elementary particles in the energy region under study, it is necessary to take into account the internal structures and off-shell effects. In phenomenological models, this is usually done by introducing form factors. In this work, we adopt the following form factor for various meson exchange vertices, where Λ M , m and q are the cutoff parameter, the mass of the exchanged particle and the exchanged momentum.
The cutoff parameters for the KN Λ and K * N Λ vertices are adopted as Λ K = 1.1 GeV and Λ K * = 0.9 GeV [17,18], respectively. While, the cutoff parameters for the Λ * K N vertices are not well determined in literatures. In present work, we use the same value as that for the ΛKN vertex and the uncertainties due to this parameter will be discussed in the next section.
The propagators for the Λ(1670), Λ * P/D , K and K * are adopted as the following forms: where the superscript + and -correspond to particle and antiparticle respectively.
With the ingredients given above, the amplitudes for various diagrams can be written by following the Feynman rules. Here we present the individual amplitudes explicitly, In the above formulas, the letters in the parentheses indicate the momentum of the exchanged particles and p η denotes the momentum of the η in the final state.
Based on the individual scattering amplitudes presented above, the general differential cross section of pp → ΛΛη reads where M f i represents the total amplitude, P i and P f represent the sum of all the momenta in the initial and final states, respectively. p a denotes the momenta of the three particles in the final state.
Before presenting the calculated results, we need to discuss the possible effects of the pp initial state interaction(ISI) and ΛΛ final state interaction(FSI) in the present reaction. It is known that the ISI may have important effects on the meson production in nucleonnucleon collisions [21,22], where the ISI reduces the cross section by an over factor with slight energy dependence.
The study on the pp → Λ cΛc reaction also shows that the ISI effect may reduce the cross section by a factor of 100 [11]. So it is natural to expect that the ISI effect may also be important for the reaction under study. When we consider the energy region near threshold, the interaction between final Λ andΛ may also become important [23].
A reliable description of the FSI between the Λ andΛ will rely on a good understanding of the ΛΛ interaction, for which our knowledge is still rather limited due to the absence of data. Therefore an accurate description of the FSI between Λ andΛ is still not possible. To take into account the ISI effect, we adopt a phenomenological approach as in Refs. [24,25]. Interestingly, in a recent work [12], the authors have adopted the same approach and applied it to study the pp →Λ − c Λ + c reaction. In their work, the parameters for the ISI were checked by reproducing the near threshold cross sections predicted by Juelich model, in which model ISI is taken into account more rigorously. Using the same parameters, they can also successfully reproduce the cross sections of the pp → ΛΛ reaction near threshold without considering FSI effect explicitly. For simplicity, in this work we choose to follow the approach in Ref. [12] and adopt their parameters. Thus we assume the effect of FSI has been effectively absorbed into the model parameters. Here we want to note even though we treat the ISI and FSI in a model dependent way, the main conclusions of the present work should not be changed significantly since our primary goal is to have an order of magnitude estimation of the total cross sections and to investigate the relative importance of various Λ resonances in this reaction.

III. RESULTS AND DISCUSSIONS
With the formulas and ingredients given in last section, the total and differential cross sections can be calculated in a straightforward way and we present the results in this section. To investigate the roles of the Λ(1670) and the possible narrow resonance in the pp → ΛΛη reaction, we will consider three scenarios. First, we include the contributions from both the Λ(1670) and the narrow Λ * D in the reaction(Scenario I). Second, we adopt the assumption that the narrow resonance is a P 03 state as in Refs. [4,5] and consider its contribution in this reaction(Scenario II).
Finally, we consider the case that the narrow resonance does not exist and thus there is no contribution from the narrow resonance(Scenario III). For all the three scenarios, the parameters of the models such as the coupling constants and relative phases are determined by fitting the total and differential cross sections of the K − p → ηΛ reaction, where these resonances play important roles. The adopted parameters have been listed in Table I. In Fig.2a, (Fig.2b) show that the production of the Λ * P dominates this reaction even at the near threshold region.
In fact, in our fitting of the K − p → ηΛ reaction data, we also find, even though the Λ(1670) gives the dominant contribution, the Λ * P contribution is significant as well. Compared to the s-wave state Λ(1670), the contribution of the Λ * P in the present reaction is also enhanced due to the large threshold momentum as mentioned above.
The band in the Fig.2b shows the uncertainties due to the cutoff parameter for the Λ * PK N vertex by varying it from 0.8 to 1.4 GeV.
In Fig.3, we show the differential cross sections obtained in Scenario I at P lab = 3.84 GeV, where the FSI of ΛΛ is expected to be small. As can be seen from the figure, there is a sharp peak appearing in the M Λη spec- The corresponding results for Scenario II are presented in Fig.4. In this case, there is no clear peak of the Λ * P in the M Λη spectrum. This is mainly because the Λ * P in our model lies very close to the Λη threshold and has a relatively large width(∼ 11 MeV). The enhancement in the M Λη spectrum compared to the phase space distribution is caused by a coherent sum of the contributions of the Λ(1670) and the Λ * P . On the other hand, the angular distribution of η shows distinct features compared to the results without the Λ * P contribution (Fig.5). After eliminating theΛ * contribution as done for Scenario I, the structure shown in the η angular distribution in the Λη rest frame also clearly indicates the higher partial wave contribution from the Λ * P . Compared to the results shown in Fig.3, we find that the η angular distributions have similar patterns in these two cases. Thus it is difficult to identify the quantum numbers of the narrow resonance by only analyzing the angular distributions, and the polarization data may be needed. A detailed study on the polarization observables will rely on a more rigorous treatment of both ISI and FSI and is out of the scope of present work. However, as can be seen from the figures an accurate measurement of the Dalitz plot or invariant mass spectrums can still offer valuable information about the narrow resonance, since the Scenarios I and II predict distinct features in these observables.  onance. It is also worth noting that even though the K * exchange contribution is included in the calculations, we find its contribution is very small and can be neglected.

IV. CONCLUSION
In this work, we investigate the production of Λ/Λ resonances in the pp → ΛΛη reaction within an effective Lagrangian approach and isobar model. Especially, we investigate the possibility to verify the existence of a new narrow Λ resonance found in the K − p → ηΛ reaction near threshold. Based on our model calculations, we find the narrow resonance, if exists, can give significant contribution in this reaction and the total cross sections of this reaction is found to be roughly at the order of 0.1 ∼ 10 nb at P lab = 3.1 − 4 GeV. Thus the measurements of this reaction will offer a good opportunity to verify the existence of this resonance. The predictions can be tested in the future by thePANDA experiment.