$\Xi(1690)^-$ resonance production via $K^-p \to K^+K^-\Lambda$

In this talk, we investigate $\Xi(1690)^-$ production from the $K^-p\to K^+K^-\Lambda$ reaction wit the effective Lagrangian method and consider the $s$- and $u$-channel $\Sigma/\Lambda$ ground states and resonances for the $\Xi$-pole contributions, in addition to the $s$-channel $\Lambda$, $u$-channel nucleon pole, and $t$-channel $K^-$-exchange for the $\phi$-pole contributions. The $\Xi$-pole includes $\Xi(1320)$, $\Xi(1535)$, $\Xi(1690)(J^p=1/2^-)$, and $\Xi(1820)(J^p=3/2^-)$. We compute the Dalitz plot density of $(d^2\sigma/dM_{K^+K^-}dM_{K^-\Lambda}$ at 4.2 GeV$/c$) and the total cross sections for the $K^-p\to K^+K^-\Lambda$. Employing the parameters from the fit, we present the cross sections for the two-body $K^-p\to K^+\Xi(1690)^-$ reaction near the threshold. We also demonstrate that the Dalitz plot analysis for $p_{K^-}=1.915 \sim2.065$ GeV/c makes us to explore direct information for $\Xi(1690)^-$ production, which can be done by future $K^-$ beam experiments.

Experimentally, Λ + c → ΛK 0 S K+ is particularly attractive, as high-statistics data are available from Belle/Belle-II and LHCb Collaborations. Nonetheless, the interference between Ξ(1690) − and a 0 (980) appears with a fixed crossing location in the phase space. The phase in the interference between the two resonances could change the spin analysis result. Hence, it is necessary to carry out a Ξ(1690) − production experiment using the (K − , K + ) reaction near the threshold. Ξ(1690) − is produced in the (K − , K + ) reaction and decays to ΛK − . In the K − p → K + K − Λ reaction, the φ(1020) → K + K − amplitude could interfere with the Ξ(1690) − production amplitude. However, the φ(1020) resonance is very narrow, so it can readily be isolated from the Ξ(1690) − resonance. Moreover, the relative location of the interference region can change with the K − beam momentum.
Here, we assume that the Ξ(1690) − has a spin-parity of J p = 1/2 − , as suggested by theoretical works [5,8] and reported by the BaBar Collaboration [13]. To compute the invariant amplitudes for the K − p → K + K − Λ reaction, we use the effective Lagrangian densities for the interaction vertices as follows: where B and B stand for baryons with spin-1/2 and spin-3/2, respectively. We should mention that, in the present calculation, we did not consider the KBB vertex for brevity, as there are no experimental data available for this reaction. The coupling constants for the ground-state hadron vertices, such as g KNΛ(1116) , are taken from the prediction of the Nijmegen soft-core potential model (NSC97a) [15]. The coupling constants for the s-wave resonances, Λ(1405) and Λ(1670), are obtained from the chiral unitary model [16], where the resonances are generated dynamically by the coupled-channel method with the Weinberg-Tomozawa (WT) chiral interaction. The couplings for Ξ(1690) and Ξ(1820) are estimated by ChUM [8] and the SU(6) relativistic quark model [5], respectively.
(Right) Differential cross section dσ/dM K − Λ as a function of the invariant mass squared M 2 K − Λ at p K − = 4.2 GeV. The green and blue areas indicate the results with and without the φ(1020) contribution, respectively. The experimental data [17] are overlaid as a histogram.
Regarding the coupling constants with two hyperon resonances, such as g KΛ * Ξ * and g KΣ * Ξ * , there is no experimental nor theoretical information available. Furthermore, it is also difficult and uncertain to simply employ the flavor SU(3)-symmetry relation, which is used to obtain g KΛ * Ξ and g KΣ * Ξ as in Ref. [18]. Hence, we set those coupling constants to be zero for simplicity, although in practice their unknown contributions can be absorbed into the cutoff parameters of the form factors.
We choose the phenomenological phase factors, e 3iπ/2 and e iπ/2 for the amplitudes with the spin-1/2 and spin-3/2 Ξ hyperons, respectively, as follows: Note that these phase factors are determined to reproduce the experimental data [17].

Numerical results
In this Section, we discuss the numerical results for the Ξ(1690) production. We first show the numerical results for the K − p → K + K − Λ reaction. The calculated Dalitz plot for the double differential cross section Fig. 2, where the Ξ * (1690) and Ξ(1820) resonances appear as vertical bands, while φ(1020) appears as a horizontal band in the bottom side. At this energy, there is no interference effect between Ξ * s and φ(1020). The Dalitz plot was projected on the K − Λ mass axis, as shown in the right panel of Fig. 2. The experimental data are taken from Ref. [17], which is the only data set available so far for the K − p → K + K − Λ reaction. The experiment was performed using the K − beam at 4.2 GeV/c to study Ξ(1820) and higher resonances. We then fit the data with the line shape of our calculation result in the low-mass region below M 2 K − Λ = 3.3 GeV 2 /c 4 . After fixing the model parameters by fitting with the three-body experimental data, the total cross sections for K − p → K + Ξ(1690) − are computed and represented as a function of K − beam momentum (p K − ) from threshold to 4 GeV/c in the left panel of Fig. 3. It increases rapidly from the threshold and peaks at p K − = 2.6 GeV/c (E cm = 2.47 GeV) with 1.5 µb, after which it decreases smoothly. As shown in the right panel of Fig. 3, the u-channel contribution is much larger than the s-channel contribution. In our present calculation, we set the coupling constant (g KY * Ξ * ) to zero to avoid further theoretical uncertainty. Shyam et al. [18] assumed that g KY * Ξ = g KY * N for the K − p → K + Ξ − reaction. However, there is no firmly established theoretical basis for the coupling constants (g KY * Ξ * ).

Summary
In this talk, we present our recent work on the Ξ(1690) − production in the K − p → K + Ξ(1690) − reaction within the effective Lagrangian approach. We consider the s-and u-channel Σ/Λ ground states and resonances for the Ξ-pole contributions, in addition to the s-channel Λ, u-channel nucleon pole, and t-channel K − -exchange for the φ-pole contributions. The Ξ-pole includes Ξ(1320), Ξ(1535), Ξ(1690)(J p = 1/2 − ), and Ξ(1820)(J p = 3/2 − ). We calculate the Dalitz plot density of (d 2 σ/dM K + K − dM K − Λ at 4.2 GeV/c) and the total cross sections for the K − p → K + K − Λ reaction near the threshold to determine the coupling constants and the form factors for the two-body K − p → K + Ξ(1690) − reaction. The calculated differential cross sections for the K − p → K + Ξ(1690) − reaction near the threshold show a strong enhancement at backward K + angles, caused by the dominant u-channel contribution.