$\Xi(1690)^-$ production in the $K^-p\to K^+K^-\Lambda$ reaction process near threshold

We investigate $\Xi(1690)^-$ production from the $K^-p\to K^+K^-\Lambda$ reaction within the effective Lagrangian approach at the tree-level Born approximation. We 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 calculate 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 $K^-p\to K^+K^-\Lambda$ reaction near the threshold. The calculation results are in good agreement with previously acquired experimental data. Using the parameters from the fit, we present the total and differential cross sections for the two-body $K^-p\to K^+\Xi(1690)^-$ reaction near the threshold. In our calculation, a strong enhancement at backward $K^+$ angles is predicted because of the dominant $u$-channel contribution. We also demonstrate that the Dalitz plot analysis for $p_{K^-}=1.915 -- 2.065$ GeV/c enables us to access direct information regarding $\Xi(1690)^-$ production, which can be tested by future $K^-$ beam experiments. The possible spin-parity states of $\Xi(1690)^-$ are briefly discussed as well.

Ξ(1820) is the only state for which spin-parity is determined (3/2 − ). The ordinary quark models predict that 3/2 − and 1/2 − should be almost degenerate, as in the case of N * . Recall that Λ(1405) lies near K N and it has a large mass difference (≈ 100 MeV) from the doublet Λ(1520). Therefore, either of Ξ(1620) or Ξ * (1690) − can be a spin partner of Ξ(1820), and the rest can be regarded as an S = −2 analogue state of Λ(1405), namely ΛK or ΣK molecular states [6].
Experimentally, Λ + c → ΛK 0 S K+ is particularly attractive, as high-statistics data are available from Belle/Belle-II and LHCb Collaborations. However, 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 reso-nances could change the spin analysis result.
In this respect, 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.
For the K − p → K + K − Λ reaction, there have been no experimental efforts since the era of the bubble chamber. Moreover, bubble chamber data are also very limited near the threshold. Schlein et al. [14] reported the first measurement of the K − p → K + K − Λ reaction using a 1.95 GeV/c K − beam with a 72-in hydrogen bubble chamber. They observed only 24 events for K − p → K + K − Λ and studied the φ resonance only. Badier et al. studied the K − p → K + K − Λ reaction using a 3 GeV/c K − beam with an 81-cm hydrogen bubble chamber [15]. Because of very limited statistics, no resonances were found in the ΛK − mass spectrum. Bellefon et al. [16] reported total cross sections for the K − p → K + K − Λ reaction from 1.934 to 2.516 GeV/c. A total of 271 events were recorded in a 2-m hydrogen bubble chamber. The highest statistics data are available from the K − p experiment at 4.2 GeV/c, involving 2935 events from a 2-m hydrogen bubble chamber [17]. The Dalitz plot for the K − p → K + K − Λ reaction is available. It is therefore crucial to perform a high-statistics experiment involving Ξ * production with a high-intensity K − beam and its decay distribution measurement to firmly determine their spin and parity; this type of experiment is possible at the J-PARC facility.
We calculate the total and differential cross sections for the K − p → Ξ(1690) − K + reaction in a beam momentum range from 2.1 GeV/c to 2.3 GeV/c. We also demonstrate that the Dalitz plot analysis of the K − p → K + K − Λ reaction enables us to access direct information concerning the Ξ(1690) − production. The double-polarization asymmetry turns out to be essential for determining the spin and parity quantum numbers of Ξ(1690) − via experiments.

II. THEORETICAL FRAMEWORK
In this Section, we introduce the theoretical formalism to calculate the Ξ * (1690) − production in the K − p → K + K − Λ reaction within the effective Lagrangian approach at the treelevel Born approximation. We consider five relevant Feynman diagrams for the K − p → K + K − Λ reaction with the Ξ-and φ-pole contributions, as shown in Fig. 1.
We define Γ, which depends on the parity P of the neighboring baryon in the above interaction Lagrangian densities, i.e., (ΓB) and (ΓB) for instance, as follows: The calculation of the invariant amplitudes is strightforward: where In the present calculation, the four-momenta of K − beam, target p, outgoing K + , outgoing K − , and Λ are denoted k 1 , k 2 , k 3 , k 4 , and k 5 , respectively, as shown in Fig. 1, while q i±j = k i ± k j are the relative four-momenta for two particles, where i and j range from 1 to 5.
The coupling constants for the ground-state hadron vertices, such as g K N Λ(1116) , are taken from the prediction of the Nijmegen soft-core potential model (NSC97a) [22]. The coupling constants for the s-wave resonances, Λ(1405) and Λ(1670), are obtained from the chiral unitary model [21], 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.
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. [20]. 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. The strong coupling constants used in the present calculation are listed in Table I. The full decay widths for Ξ * resonances are given as Γ Ξ(1532) = 9.1 MeV [1], Γ Ξ(1690) = 6 MeV [8], and Γ Ξ(1820) = 24 MeV [1,5]. In Eq. (6), we introduce the phenomenological form factors for the Ξ-and φ-pole contributions to take their spatial distributions into account: where s, t, and u are the Lorentz-invariant Mandelstam variables. In the present calculation, the cutoff parameters are determined to be Λ Ξ(1322) = 1.3 GeV, Λ Ξ(1532) = 1.3 GeV, Λ Ξ(1690) = 0.75 GeV, Λ Ξ(1820) = 1.1 GeV, and Λ φ = 0.44 GeV to reproduce the experimental data, which will be discussed in the next Section. We also 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:

III. NUMERICAL RESULTS FOR THE
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 d 2 σ/dM K + K − dM K − Λ at p K − = 4.2 GeV/c (E cm = 3.01 GeV) is represented in Fig. 2(a), 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 Fig. 2(b). 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 . The first bump structure near the threshold is due to the Ξ(1690) production, providing us with information on the cutoff parameters for the form factors, as given in the previous Section. The green and blue areas indicate the calculation results with and without the φ(1020) contribution, respectively. The mass range between 4.0 and 5.7 GeV 2 /c 4 for the large bump structure is consistent with the φ(1020) band crossing the limited phase space in the Dalitz plot. It should be noted that high-mass resonances decaying to K − K + cannot account for the bump structure in that mass range only. In the present calculation, high-mass K − K + resonance like f 2 (1525)(J p = 2 + ) is not included. Higher-mass Ξ * resonances could contribute to the bump structure.
Using the same cutoff parameters, we compute the total cross sections for the K − p → K + K − Λ reaction. The calculation results with (green area) and without (blue area) the φ(1020) contribution are compared with the world data taken from Ref. [24], as shown in Fig. 3(a). The yellow area at the bottom indicates interference between the φ and other contributions. It turns out that our theoretical model describes the experimental data for the K − p → K + K − Λ reaction qualitatively well. Enhanced K − Λ production between M 2 K − Λ = 4.0 GeV 2 /c 4 and 5.7 GeV 2 /c 4 could be associated with a contribution from higher-mass hyperon resonances. However, we did not include those high-mass resonances in our present calculation, as the mass region is far beyond the ΛK − threshold.
For the two-body K − p → K + Ξ(1690) − reaction, we computed the s-channel and u-channel diagrams in Fig. 4 with the same theoretical framework and the same parameters used for the K − p → K + K − Λ reaction. The total cross sections are represented as a function of K − beam momentum (p K − ) from threshold to 4 GeV/c in Fig. 3(b). 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 Fig. 3(b), 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. [20] 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 * ). The u-channel hyperon propagator and form factors also provide much larger strengths than that for the s-channel one, as previously shown in Ref. [23].
The differential cross sections dσ/dΩ for the K − p → K + Ξ(1690) − reaction are calculated as a function of cos θ in Fig. 3(c), where θ stands for the scattering angle of the outgoing K + in the center-of-mass (c.m.) frame. Because of the strong u-channel contributions, one observes backwardenhanced angular distributions for the various p K − values, as the energy increases. This backward-peaking behavior is a general feature for the double-charge and double-strangeness exchange (K − , K + ) process.
The threshold beam momentum for the K − p → K + K − Λ reaction is 1.687 GeV/c, while that for the K − p → K + Ξ(1690) − reaction is 1.878 GeV/c, which is accessible using the J-PARC Hadron-Hall Collaboration. The experiment for the K − p → K + K − Λ reaction near the threshold can be performed with the Hyperon Spectrometer [25] at the K1.8 beam line of J-PARC. One can measure all the charged particles not only from the Ξ(1690) − → K − Λ decay, but also Σ − K 0 , Σ 0 K − , Ξ − π 0 , and Ξ 0 π − decays. All the Ξ(1690) − decay modes contain three charged particles with one missing neutral particle in some channels, which enables us to reconstruct Ξ(1690) − without any kinematical ambiguity.
The sizable cross sections of a few µb for the K − p → K + Ξ(1690) − reaction also encourage future experiments using a high-intensity K − beam [26]. According to the calculated Dalitz plot density, simulated events for the K − p → K + K − Λ reaction are generated over the phase space available. We assume a uniform experimental acceptance for the K + K − Λ phase space. The Dalitz plots for the K − p → K + K − Λ reaction are plotted in Fig. 5 for four different K − beam momenta, from 1.915 GeV/c to 2.065 GeV/c. Because the φ(1020) production is predominant, it is difficult to identify Ξ(1690) − in the ΛK − mass distribution without the φ-band exclusion. The φ band is so narrow that we can remove the φ events by excluding the K + K − mass band for the φ. The crossing points between the Ξ(1690) − and the φ(1020) resonances change with the K − beam momentum. This enables us to study Ξ(1690) − in various kinematical regions, where the interference effects with the φ(1020) resonance are different. The calculated Dalitz plots show that we can neglect the interference effect between the Ξ(1690) − and the φ(1020) production channels in the K − p → K + Ξ(1690) − reaction. However, it is interesting to see that our theoretical model calculation predicts possible interference between the Ξ(1690) and tree-level Born-term amplitudes. Excluding the φ(1020) band, the projected ΛK − mass distributions for the beam momenta p K − = 2.015 and 2.065 GeV/c are displayed in Fig.  6(a) and (b), respectively. The lineshape of the Ξ(1690) − is clearly observed in the ΛK − mass spectrum. The tree-level Born-term contribution is subtracted from the projected ΛK − mass distribution, as shown with overlaid red distributions. The subtracted distributions are made of the Ξ(1690) − and the interference effect.
Finally, we want to discuss the spin and parity of Ξ(1690) − , which has not yet been fully determined experimentally, although the BaBar Collaboration reported that J P = 1/2 − assignment was favored [13]. Note that the theoretical predictions also support J P = 1/2 − [5,8], which we have employed for the numerical results shown above. For other possible spin-parity states, we use the following branching ratios suggested by the ChUM calculations [8], which are qualitatively consistent with experimental results [13]: for J P = (1/2 + , 3/2 + , 3/2 − ), respectively. Here, we have chosen the phase factor −1 between the two couplings for brevity [8]. The cutoff masses for the form factors are taken as Λ Ξ(1690) = (440, 2400, 650) MeV, which fairly reproduce the K − p → K + K − Λ data [24]. In Fig. 7(a), we plot the total cross sections for K − p → K + Ξ(1690) − for different spin-parity states. The total cross sections for J = 1/2 states increase rapidly near the threshold, whereas those for J = 3/2 states increase smoothly due to the p-wave nature near the threshold. The differential cross sections for p lab K − = 2.015 GeV/c in Fig. 7(b) show a strong enhancement at backward K + c.m. angles because of the dominant u-channel contributions for J = 1/2 + , 1/2 − and 3/2 − states. For J = 3/2 + state, it turns out that the s-and u-channel contributions compete strongly with each other.
Taking into account the lack of experimental and theoretical information on those quantum numbers, it is crucial to investigate theoretically physical observables that do not depend much on theoretical uncertainties, such as the form factors and coupling constants. One of the observables satisfying this criterion is double-polarization asymmetry for the present case K − p → K + Ξ(1690) − , which reads: Here, the subscripts (↑, ↓) denote the proton-target polarizations along a quantization axis, when the Ξ(1690) − possesses a fixed spin state (s R ). As understood from Eq. (17), the phenomenological form factors and coupling constants, being multiplied to the invariant amplitudes, are canceled approximately between the numerator and denominator, minimizing those theoretical uncertainties. In Fig. 8, we plot Σ(s R ) for different spin-parity states of Ξ(1690) − for p lab K − = 2.015 GeV/c. We chose J z = +1/2 for J P = 1/2 ± and summed over J z = +1/2 and J z = +3/2 contributions for J P = 3/2 ± . Because we do not have a meson exchange in the t channel here, the spin-conserving process dominates for J P = 1/2 ± , i.e., Σ ∼ 1, as shown in Fig. 8. Because the J P = 1/2 − state provides sizable spin-nonconserving contributions, it differs slightly from the J P = 1/2 + one. On the contrary, the J P = 3/2 ± states show both spin-nonconserving and spin-mixing contributions. Hence, from these observations, double-polarization asymmetry is a useful tool for determining the spin and parity quantum numbers of Ξ(1690) − .

IV. SUMMARY
In this study, we investigate 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 Ξpole 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. We also demonstrate that the Dalitz plot analysis for p K − = 1.915 − 2.065 GeV/c enables us to access direct information regarding the Ξ(1690) − production, which can be tested by future K − beam experiments. The doublepolarization asymmetry turns out to be essential to determine the spin and parity quantum numbers of Ξ(1690) − via experiments.