Double-strangeness production in $\Lambda p\to K^+X$ reaction

We investigate $S=-2$ production from the $\Lambda p\to K^+X$ reactions within the effective Lagrangian approach. The $\Lambda p\to K^+\Lambda\Lambda$ and $\Lambda p\to K^+\Xi^-p$ reactions are considered to find the lightest $S=-2$ system, which is $H$-dibaryon. We assume that the $H(2250)\to\Lambda\Lambda$, and $H(2270)\to\Xi^-p$ decays with the intrinsic decay width of 1 MeV. According to our calculations, the total cross-sections for $\Lambda p\to K^+\Lambda\Lambda$ and $\Lambda p\to K^+\Xi^-p$ reactions were found to be of the order of a few $\mu$b in the $\Lambda$ beam momentum range of up to 5 GeV$/c$. Furthermore, the direct access of information regarding the interference patterns between the $H$-dibaryon and non-resonant contributions was demonstrated.


I. INTRODUCTION
Double-strangeness baryon systems involve an -dibaryon, double hypernuclei, and possibly the inner core of neutron stars [1]. An observation of several double hypernuclei reveals that the ΛΛ interaction is weakly attractive. However, the Ξ − interaction was only studied in heavy-ion collisions, which indicates a strong, attractive interaction [2]. Recently, the Ξ -ΛΛ coupling was determined to be weak based on an initial observation of a Coulomb-assisted bound state for the Ξ − -14 N system [3], which was predicted to exist considering the evidence for a deeply-bound Ξ − -14 N state reported in a hybrid emulsion experiment at KEK-PS [4]. While strangeness = −2 baryon-baryon interactions provide critical information on exploring the smallest object ( -dibaryon), and the largest (the inner core of neutron stars), the experimental data is limited.
The lightest = −2 system is the -dibaryon, which can be decomposed into a compact 6-quark state, and two baryon states involving ΛΛ, Ξ , and ΣΣ components. The mass range of the -dibaryon is strongly connected with the existence of double Λ hypernuclei. Several double Λ hypernuclei have been reported: 6 ΛΛ He [5], 10 ΛΛ Be [6], and 13 ΛΛ B [7]. Because the ΛΛ → decay was not observed in the aforementioned studies, the must be heavier than > 2 Λ + ΛΛ ≈ 2.22 GeV/ 2 .
Recently, the HAL QCD collaboration has indicated that the ΛΛ( 1 S 0 ) interaction is not sufficiently attractive to generate a bound or resonant state close to the ΛΛ threshold, whereas the Ξ ( 1 S 0 ) phase shift increases sharply just above the Ξ threshold [8]. Experimental confirmation of the -dibaryon would be a significant accomplishment for a better understanding of hyperon interactions.
Enhanced ΛΛ production close to the ΛΛ threshold was reported in 12 C( − , + ) reactions at − = 1.65 GeV/ [9,10]. This threshold enhancement may provide insight for the possible existence of an -dibaryon near the ΛΛ or Ξ − thresholds. A high-statistics experimental reconfirmation should be awaited until the dedicated -dibaryon search experiment E42 [11] is performed using a high-intensity − beam at J-PARC.
The simplest method for producing the -dibaryon is to employ the double-strangeness and double-charge exchange ( − , + ) reaction on a light nuclear target to retain two units of strangeness in a 12 C nucleus, similar to the J-PARC E42 with a diamond target at − = 1.8 GeV/ . Furthermore, the -dibaryon is also available in other reactions, such as , , , and , most of which involve nuclear targets that contain at least two nucleons coupled to the -dibaryon production; therefore, the overlap of wavefunctions for hyperons and intranuclear nucleons should be considered. A crosssection measurement for the ΛΛ production was reported to be 6.7 ± 1.5 mb in a Ta reaction at 4 GeV/ [12]. Heavy-ion reactions can be used to produce Λ and Ξ − hyperons copiously so that the coalescence of two of these particles into thedibaryon may be observed. However, the H-dibaryon should be observed in a high-multiplicity environment for high-energy heavy-ion collisions.
Because the -dibaryon can be formed directly via Ξ and ΛΛ interactions, the production reaction involving the minimum number of vertices is the Ξ − → reaction with a proton target. However, in this case, the mass range of the -dibaryon is accessible only above the Ξ − mass threshold. Because a ΛΛ scattering experiment is unavailable, the second-best choice is a Λ → + reaction via a strangenessexchange process, with which the -dibaryon can be observed in the mass range below the ΛΛ threshold to a higher mass region.
A Λ beam is available via photoproduction and − -induced reactions by tagging + , 0 , or (892) * in the final state. For example, the − -induced reactions can either be a − → 0 Λ or − → (892) * Λ reaction. As the detection of a 0 → 2 decay triggers the production of both = +1 0 and = −1 0 with nearly equal probability, the production of Λ particles cannot be uniquely tagged. Therefore, the − → (892) * Λ reaction is selected as a primary reaction for the Λ elastic scattering measurement using an 8 GeV/ − beam at J-PARC [13]. In this case, the Λ beam is available in the momentum ranging from 0.2 to 2.0 GeV/ , and it is unavailable for doublestrangeness production above the threshold Λ momentum of 2.6 GeV/ .

arXiv:2101.10114v1 [hep-ph] 25 Jan 2021
However, the → + Λ reaction facilitates the production of a high-momentum Λ with Λ polarization in the photon beam energy region above 2.5 GeV. The measurement of Λ → + ΛΛ and Λ → + Ξ − is viable with the CLAS data [14] and the upcoming LEPS2 data [15]. This (Λ, + ) reaction measurement leads to a decisive conclusion regarding the existence of the H-dibaryon near the ΛΛ and Ξ − thresholds. Moreover, possible interference effects among the + ΛΛ and + Ξ − channels are noteworthy. In this study, numerical calculation results for the Λ → + ΛΛ and Λ → + Ξ − reactions within the effective Lagrangian approach have been reported. We calculate the Dalitz plot densities ( 2 / ΛΛ Λ + ) for the Λ → + ΛΛ reaction and ( 2 / Ξ − Ξ − + ) for the Λ → + Ξ − reaction. The -dibaryon states are assumed to appear at 2.25 GeV/ 2 and 2.27 GeV/ 2 in the ΛΛ and Ξ − channels, respectively. The intrinsic width of the -dibaryon was chosen to be 1 MeV. Based on calculations, the total cross-sections for the Λ → + ΛΛ and Λ → + Ξ − reactions were determined to be within the order of a few b in the Λ beam momentum of up to 5 GeV/ . Furthermore, we demonstrated that information regarding the interference patterns between the -dibaryon and non-resonant contributions can be directly accessed.
resulting in the following invariant amplitudes: where ± ≡ ± and ℎ indicate the dressed propagator for a hadron ℎ with spin . Its explicit form in the present work is as follows: 1/2 Here, the factors ℎ , Γ ℎ , and , denote the mass and full decay width of the hadron ℎ, and the transferred momentum, respectively. The values of Γ ℎ are listed in Table I. The phenomenological form factor ℎ presents the spatial extension of the hadron ℎ. In this study, we employ the following type of the form factors: The cutoff mass Λ ℎ is determined from other experimental data in the next section. Notably, the interchange of the Λ baryons in the final state (4 ↔ 5) in Eq. (2) gives a negative sign, owing to the Fermi-Dirac statistics. All the relevant meson-baryon couplings are obtained from the Nĳmegen soft-core potential model [18], as listed in Table II. The invariant amplitudes for the diagram ( ) can be evaluated as follows: for the ( , , ) meson exchange in the -channel. The superscripts in M ℎ 1 ℎ 2 ( ) denote the intermediate hadrons as shown in Fig. 1. Regarding the nucleon-resonance and Δ-baryon contributions, we only consider the couplings to the meson to avoid theoretical uncertainties.
The scalar meson represents 0 (500, 0 + ) [19]. For the production of -dibaryon near the threshold (  √  ≈ 2725 MeV), only the nucleon resonance * (1650, 1/2 − ) becomes relevant to the amplitude, M * ( ) . The strong coupling constants corresponding to * (1650, 1/2 − ) are also obtained from the chiral coupled-channel method [20], as listed in Table II. Similarly, the ( , , * ) meson-exchange contributions are as follows: where the strange scalar meson denotes − (800) [19]. The background contributions, which do not form resonant band structures in the Dalitz plot, are given by the following: and All the relevant coupling constants are provided in Table II.
For the Λ → + Ξ − reaction, we can compute the invariant amplitudes similarly without the particle-exchange (4 ↔ 5) terms in the final state. The relevant coupling constants for this reaction are provided in Table II. In this reaction, the 27-plet Θ ++ pentaquark contribution can be considered in diagram (c). However, the existence of this exotic baryon has never been confirmed experimentally. Hence, we ignore this contribution for brevity. In diagrams ( ) and ( ), there are two baryon-pole contributions, that is, Λ and Σ 0 , which differ from the ΛΛ channel.
Unlike the electromagnetic hadron production involving the Ward-Takahashi identity, determining the phase factors between strong-interaction amplitudes is a relatively difficult task owing to the lack of symmetry. In the present calculation, we employ a free parameter for the phase difference between the tree-level invariant amplitudes as follows: The phase factors among the invariant amplitudes of diagrams ( -), except for the nucleon-resonance contributions, are determined from the coupling constants in the Nĳmegen model [18]. Although we do not have a theoretical reasoning for fixing the phase factor for the nucleon-resonance contributions, we simply assume that * is positively real. Moreover, the nucleon-resonance contributions were numerically verified to be negligible, owing to its significant full decay width as provided in Table I. Thus, we introduce a single phase factor between the -dibaryon contributions and others, as shown in Eq. (22). The phase angle is considered a free parameter in the numerical calculations.
After attaining the aforementioned, the final-state interaction (FSI) contributions can be considered. The total amplitude including the FSI contributions is defined in the on-shell approximation (OnF) [21] as follows: whereM cc,OnF 1 2 → 3 4 stands for the flavor-singlet two-baryon coupling constant for the = 0 and = −2 channels in the coupled-channel (cc) method. Based on the isospin symmetry, its elementary amplitude can be expressed as follows [23]: The value of 1 to reproduce the binding energy of the HAL QCD data is given by −12.8/GeV 2 [23]. Note, in Eq. (23), only the baryon-baryon re-scattering for FSI is considered for simplicity, and the = 0 meson-baryon re-scattering is ignored because we are interested in the baryon-baryon invariant mass spectrum. The two-baryon propagator G OnF 1 2 applies the onshell factorization. The integration of G OnF 1 2 over the loop momentum can be regularized simply using the dimensionalregularization method [21] as follows: where we define Hence, in terms of the on-shell factorization, the coupledchannel amplitude for the 1 2 → 3 4 channel reads: Here, channels (1,2,3) are defined by the (ΛΛ, Ξ, ΣΣ) scattering states.
Regarding the ΛΛ → ΛΛ scattering for FSI, we consider only M cc 11 = M cc ΛΛ→ΛΛ in Fig. 3. Following extensive calculations, the analytical form is obtained as follows: Similarly, we can derive the Ξ − channel total amplitude as follows: . (32)

III. NUMERICAL RESULTS AND DISCUSSIONS
In this section, we provide the numerical calculation results with details regarding the Λ → + ΛΛ (ΛΛ channel) and Λ → + Ξ − (Ξ − channel) reaction processes. In this calculation, the -dibaryon is assumed to be unbound above the ΛΛ threshold. The mass range of the -dibaryon is strongly connected with the observation of the double Λ hypernuclei, which imposes that the -dibaryon mass should be larger than 2.22 GeV/ 2 . Recent Lattice QCD calculation results indicate that the mass ranges between ΛΛ and Ξ − thresholds [8,16,17,25,26]. Two -dibaryon states, below and above the Ξ − threshold, were chosen considering the (2250) → ΛΛ and (2270) → Ξ − decays.
As indicated in Section II, we employ the coupling constants for the dibaryon from the bare -dibaryon model, in which the values were determined to fit the flavor SU(3) symmetric HAL-QCD data [16]. Therefore, the values of may be different from reality, where the flavor symmetry is heavily broken. Nonetheless, as guidance for the present theoretical calculations, these symmetric values were adopted as a trial. In Ref. [17], the full decay width of the dibaryon was Γ = 2.7 MeV at the physical point. First, the cutoff mass was fixed in the form factors in Eq. (7). In Ref. [27], a few events of the 12 C(Ξ − , ΛΛ) reaction were reported. Using the eikonal approximation, the total crosssection was deduced to = 4.3 +6.3 −2.7 for the Ξ − → ΛΛ reaction at Ξ − = 0.5 GeV/ . We reproduced this value in the present theoretical framework. For simplicity, we only considered the , , and * exchanges in the -channel, and ignored a possible -dibaryon contribution in the -channel. Moreover, the cutoff masses for the three meson exchanges were chosen to be the same for brevity. The relevant invariant amplitude is then obtained as follows: where All relevant inputs are listed in Tables I and II. Thus, the cutoff mass is determined to be Λ = 435 GeV. The numerical results are shown in Fig. 4. In the left panel of Fig. 5, the numerical results for the total cross-sections of the ΛΛ (square) and Ξ − (circle) channels are presented for the total (thick) and -dibaryon-only (thin) contributions as a function of the Λ beam momentum lab . We determined that the total cross-sections for the two channels are of the order of approximately a few b, which is smaller than that for the → + Λ from the COSY experiment [28]. The total cross-sections from the ΛΛ channel are approximately twice as large as that of the Ξ − because there are more possible contributions, as shown in the relevant Feynman diagrams in Fig. 1, in addition to the larger Nĳmegen coupling constants. On the contrary, if we only consider the -dibaryon, the order of the cross-sections is reversed, owing to the value of ΛΛ being smaller than Ξ − by a factor of two considering the isospin factor. Note, the production cross-section for the dibaryon is a few tens of nanobarn. As shown in the right panel of Fig 5, to test the -dibaryon mass dependence of the total cross-sections, they are depicted with = (2.25 ∼ 2.29) GeV/ 2 for the two reaction channels. The effects from the mass changes are unapparent, while considerable difference can be observed for the Ξ − channel with = 2.25 GeV/ 2 .
In Fig 6, the numerical results for the differential crosssections of the ΛΛ (left) and Ξ − (right) channels are presented as the function of the scattering angle of + in the center-ofmass frame (cm) . We also analyzed the differential crosssections in the energy range of cm = 2.8-3.0 GeV. The thick and thin lines denote the cases with and without the -dibaryon contributions, respectively. The angular dependence for the two channels is relatively flat at a low energy and forwarding as the energy increases. Note, the angular dependence of thedibaryon production is nearly flat at high energies, indicating the -wave nature of the particle.
To investigate the production mechanisms more carefully, we present the numerical results for the differential crosssections for each contribution individually at cm = 2.8 GeV ( lab = 2.83 GeV/ ), in the same manner as presented in Fig 7. Regarding the ΛΛ channel, the and − exchanges in the proton-pole diagrams ( and ) are predominant owing to the combinations of the larger Nĳmegen coupling constants. Moreover, the -dibaryon production diagram with the Λ pole ( ) is significantly larger than that of the Ξ − -pole diagram ( ). Regarding the Ξ − channel, the -dibaryon production dia- grams are considerably larger than others, and the exchange in the channel ( ) provides a meaningful contribution. Generally, we determined that the -dibaryon production diagram ( ) with the Mandelstam variable = ( Λ − + ) 2 enhances forward scattering, and vice versa for the diagram ( ) with = ( − + ) 2 , as expected. The Dalitz plot for the Λ → + ΛΛ and Λ → + Ξ − reactions are plotted in Fig. 8 for a Λ beam momentum of 2.83 GeV/ . Because no background processes form structure in the Dalitz plots, the -dibaryon band appears predominant. The numerical results for the invariant-mass plots are provided in Fig. 9 with = 2.25 GeV/ 2 and 2.27 GeV/ 2 for the ΛΛ (left) and Ξ − (right) channels, respectively, at cm = 2.8 GeV. The width of the dibaryon is assumed to be 1 MeV, whereas the phase angle is tested for 0 (thick) and (thin). The shaded areas indicate the cases without the dibaryon. The light and heavy shared areas indicate the cases without and only with the dibaryon. We observed that the -dibaryon production rates are larger for the Ξ − channel by a factor of two, than that for the ΛΛ channel, and vice versa for the total background contributions, as shown in Fig. 9. The signal-to-background ratio is approximately 0.3 for the ΛΛ channel, whereas the larger value of 1.6 is for the Ξ − channel. Therefore, the Ξ − channel enables us to search for the dibaryon significantly easier than the ΛΛ channel. The production cross-sections for (2250) → ΛΛ and (2270) → Ξ − are approximately 40 nb and 38 nb, respectively. Significant changes are obtained by the different phase factors, clearly shown in the Ξ − channel, owing to the smaller interference with the background processes. Furthermore, we note that the channel opening effects from the final-state interactions were small, resulting in cusp-like structures being hardly observed.
Finally, considering the decay-angle distribution of the -dibaryon, the decay angle distribution of the -wave -dibaryon is isotropic at the rest frame of the dibaryon. We define the double differential cross-section as 2 / cos cm cos rest , where the angle cm denotes that of the outgoing + in the cm frame. The angle rest is defined by where ì rest and ì + cm indicate the three momenta of the one decaying baryon in the final state at the -dibaryon rest frame and the outgoing + in the cm frame, respectively. In Fig. 10, we depict the numerical results for the double differential crosssections as a function of cos cm and cos rest with thedibaryon contribution only. As expected, we clearly observe that the decay-angle distribution, i.e., the double differential cross-sections, are nearly flat for the various cos cm values.

IV. SUMMARY
In this study, we investigated the ( = 0, = 0)-dibaryon production via Λ → ΛΛ + theoretically. Thus, we employed the effective Lagrangian approach at the tree-level Born approximation. We considered the mass and decay width of the dibaryon as the theoretical input parameters, and they were chosen by considering presently available theoretical and experimental results, such as the lattice-QCD data analyses with the flavor SU(3) breaking effects: 2 Λ ≤ ≤ ( Ξ − + ) and Γ = (1 ∼ 10) MeV. The critical observations made in this study are as follows: • The total cross-sections for the ΛΛ and Ξ − channels are determined to be within the order of a few b in the Λ beam momentum of up to 5 GeV/ , while the production cross-section for the -dibaryon is approximately 100 nb. Here, we determined our model parameters such as the cutoff masses for the form factors, based on the experimental data for the Ξ − elastic and Ξ − → ΛΛ scattering cross-sections.
• The total cross-sections do not change siginificantly with the -dibaryon mass from 2.25 GeV/ 2 to 2.27 GeV/ 2 . Because the ΛΛ production channel involves more background processes than the Ξ − channel, by ignoring the channel via an exotic pentaquark-state, the -dibaryon contribution appears to be relatively large in the Ξ − channel.
• We observed that the differential cross-sections for the Λ → + ΛΛ and Λ → + Ξ − channels peak at the forward + angles in the cm frame, owing to the -channel meson and baryon exchange processes. The -dibaryon-pole contributions are significant near the threshold and depend minimally on the + angle.
• From the invariant mass distributions, the signal-tobackground ratios are approximately 0.3 and 1.6 for the ΛΛ and Ξ − channels, respectively, owing to the smaller background contributions in the Ξ − channel. Note, the -dibaryon peak areas yield 40 nb and 38 nb for the ΛΛ and Ξ − channels, respectively.
• We also explored the change in the interference patterns between the -dibaryon and background amplitudes with the relative phase angle for the Ξ − channel. The channel opening effects from the final-state interactions were small; therefore, cusp-like structures were hardly observed.
• Lastly, we calculated the decay angular distributions of the → ΛΛ and → Ξ − decays in the helicity frame in which the quantization axis is in the opposite direction of + in the -dibaryon rest frame. The angular distributions are flat over the -dibaryon mass region, as expected for the -wave resonance.
Considering the aforementioned factors, we conclude that the -dibaryon could be clearly identified in the Λ → + ΛΛ and Λ → + Ξ − reactions close to the production threshold, if it exists close to the Ξ − threshold. Further studies related to other -dibaryon production reactions are in progress and will appear elsewhere.