Production Optimization of Exotic Hypernuclei via Heavy-Ion Beams at GSI-FAIR

Abstract

Building on the successful demonstration of hypernuclear spectroscopy using heavy-ion beams, the HypHI Collaboration is shifting its focus to investigating proton- and neutron-rich hypernuclei. A crucial component of this research is the implementation of a fragment separator, which facilitates the production and separation of rare isotope beams and is vital for accessing hypernuclei far from the stability line. High-precision spectroscopy of these exotic hypernuclei is planned to be conducted at GSI first, which will be followed by experiments at the FAIR facility utilizing the FRS and Super-FRS fragment separators. A thorough systematic investigation paired with an optimization analysis was employed to establish the most favorable experimental setup for producing high-isospin hypernuclei. Theoretical models describing heavy-ion-induced reactions and hypernuclear synthesis guided this process, which was complemented by Monte Carlo simulations to obtain experimental efficiencies for the production and transmission of the exotic secondary beams. The outlined methodology offers insights into the anticipated yields of ${}_{\Lambda }{}^6\mathrm{He}$, ${}_{\Lambda }{}^9\mathrm{C}$, and a range of both proton- and neutron-rich hypernuclei.

1. Introduction

The study of strangeness production in nuclear and hadronic collisions has gained increasing interest over the past few decades. Standard nuclear matter consists of ordinary nucleons, which are formed by triplets of the lightest up and down quarks. However, a deeper understanding of dense matter properties necessitates the inclusion of the strange (s) quark [1,2]. Hypernuclei, which are bound systems containing both nucleons and hyperons (baryons with at least one s-quark), are invaluable for probing hyperon–nucleon and hyperon–hyperon interactions [3].

Since 2006, the HypHI Collaboration has conducted extensive research on hypernuclei at the GSI facility [4]. Moving forward, the forthcoming experiments of the HypHI Collaboration will take place not only at GSI but also at the future GSI-FAIR [5]. The GSI facility currently offers a wide array of ion beams, ranging from protons to uranium, enabled by the 18 Tm heavy-ion synchrotron (SIS18) [6,7], and produces exotic beams using the fragment separator (FRS) [8,9]. The SIS18 synchrotron is capable of delivering ion beams with kinetic energies up to 2 AGeV for nuclei with $A/Z=2$. The future FAIR facility represents a substantial upgrade, incorporating an additional synchrotron ring with a magnetic rigidity of 100 Tm (SIS100). Various experimental setups are under development at FAIR to support advanced research in nuclear and hadron physics. One of these initiatives, the NUSTAR program, includes the construction of a superconducting fragment separator (Super-FRS) with a 20 Tm magnetic rigidity, enabling powerful in-flight separation of exotic nuclei [10].

Research activities at HypHI Collaboration have employed a novel experimental method distinct from the conventional missing mass techniques that use meson or electron beams. These traditional approaches are utilized at renowned institutions such as KEK, JPARC, DA$\Phi$NE, JLab, and MAMI-C accelerator [11,12,13]. The production of hypernuclei in the projectile rapidity was first undertaken with ¹⁶O beams at 2.1 AGeV on a polyethylene target [14]. Subsequently, the hypernucleus production was aimed via the ⁴He and ⁷Li beams on a polyethylene target with the respective beam energies of 3.7 AGeV and 3.0 AGeV [15,16]. The obtained results were not conclusive because of the too small number of observed events.

Notwithstanding, the Phase 0 experiment of HypHI, conducted at GSI in 2009 and 2010, successfully demonstrated the feasibility of high-precision hypernuclear spectroscopy through heavy-ion induced reactions [17,18,19]. This experiment involved bombarding a stable ¹²C target with a ⁶Li beam at 2 AGeV, leading to the production and identification of hypernuclei like ${}_{\Lambda }{}^3\mathrm{H}$, ${}_{\Lambda }{}^4\mathrm{H}$, and ${}_{\Lambda }{}^5\mathrm{He}$. These achievements validated the experimental method, which diverges from traditional approaches by enabling the reconstruction and identification of decay vertices of $\Lambda$ particles in a more inclusive setup.

Due to the experimental setup’s open geometry, characteristic of inclusive experiments, several hypernuclear bound states were identified by measuring all possible decay fragments within the same data taking. Furthermore, an indication of a potential new bound state, a neutral hypernucleus formed by two neutrons and a ${\Lambda }^0$ hyperon, was observed when reconstructing the invariant mass of $d+{\pi }^{-}$ and $t+{\pi }^{-}$ [18]. On the theoretical side, recent calculations have challenged the existence of this ${}_{\Lambda }{}^3\mathrm{n}$ bound state [20,21,22,23,24], highlighting the need for precise exclusive mass measurements to either confirm or refute this hypothesis.

The Phase 1 experiment, the so-called WASA-FRS HypHI Experiment, aimed at the study of the hypernuclei ${}_{\Lambda }{}^3\mathrm{H}$, ${}_{\Lambda }{}^4\mathrm{H}$, and nn$\Lambda$ [25,26,27]. The primary objectives of this experiment are twofold: to shed light on the hypertriton puzzle and to investigate the existence of the previously proposed nn$\Lambda$ bound state. Firstly, the significantly shorter hypertriton lifetime reported by three independent state-of-the-art experiments, namely ALICE [28], STAR [29], and HypHI [17], compared to the predictions of theoretical models remains poorly understood. Therefore, obtaining new accurate results for the invariant mass and lifetime of ${}_{\Lambda }{}^3\mathrm{H}$ (and ${}_{\Lambda }{}^4\mathrm{H}$) is crucial to reach a definitive conclusion. Secondly, the observed enhancement in the invariant mass distributions of the $d+{\pi }^{-}$ and $t+{\pi }^{-}$ final states, as reported by the HypHI Collaboration [18], cannot be accounted for by existing theoretical calculations, which indicate the absence of a neutral nn$\Lambda$ bound state. Consequently, the WASA-FRS HypHI Experiment aims to produce more precise and statistically significant experimental results that can provide clarification on the potential existence of nn$\Lambda$. For this purpose, the distinctive combination of the high-resolution spectrometer FRS [9] and the high-acceptance detector system WASA [30] is used. The hypernuclear production was by means of heavy-ion induced reactions, specifically a beam of ⁶Li at 2 AGeV bombarding a stable ¹²C target. The experiment was successfully conducted at GSI in 2022 within the Super-FRS Experiment Collaboration. Currently, the data from the experiment are under analysis.

The next phases of the HypHI project are focused on the exploration of exotic hypernuclei approaching the proton and neutron drip lines. This ambitious endeavor will require rare-isotope beams to extend the hypernuclear chart toward proton-rich ${}_{\Lambda }{}^{22}\mathrm{Si}$ and neutron-rich ${}_{\Lambda }{}^{14}\mathrm{Li}$ hypernuclei. Actually, significant charge symmetry breaking effects are anticipated in these hypernuclei, potentially altering the drip-line positions due to differences between $\Lambda$-proton and $\Lambda$-neutron interactions.

The production of exotic hypernuclei can be influenced by the isospin of the beam projectile, as explored in this study. The objective of this investigation was to determine the experimental conditions required for the production of proton-rich or neutron-rich hypernuclei. This article presents a feasibility study demonstrating the potential for operating the Super-FRS at energies around 2 AGeV. To achieve the desired exotic hypernucleus, the appropriate primary beam and target isotopes must be selected. The exotic beam generated will then collide with a secondary production target to produce the hypernucleus of interest.

This analysis is an extension of a previous work [31] to new exotic hypernuclei that can be produced and studied with the combination WASA-FRS at GSI.

2. Materials and Methods

Achieving the production of a desired hypernucleus largely depends on the choice of the exotic secondary beam that will be generated and directed at a secondary target. The Super-FRS separator’s configuration is illustrated in Figure 1. Initially, a primary beam from the synchrotron impinges on a primary target, leading to the production of exotic fragments. The pre-separator then selects and purifies the desired exotic beam. This beam is transported to the focal plane FMF2 (pointed at with a pink arrow in Figure 1) of the main separator, where a secondary target is positioned for hypernuclei production. The subsequent segment of the main separator serves as a high-resolution spectrometer to analyze decay fragments from the mesonic-weak decay of the hypernucleus of interest. By configuring the fragment separator to focus on a particular momentum range or the magnetic rigidity of the decay fragment, the experiment can achieve a precise invariant mass measurement of the reconstructed hypernucleus. The FRS and Super-FRS spectrometers offer a momentum resolving power of $p/\Delta p=1500$, enhancing measurement accuracy. To maximize experimental efficiency, a dedicated data-processing method was developed to predefine the optimal experimental settings needed to produce a specific exotic hypernucleus.

The optimization procedure followed in this study is similar to the one already reported in Ref. [31]. Nonetheless, the current analysis differed from the previous in one important step: the adjustment of the theoretical models of the hypernuclear production cross-section to the existing experimental data, as it is detailed in the next paragraphs.

The initial systematic study employed the EPAX model [32,33], which provides an energy-independent model for the fragmentation of cross-sections at relativistic energies during heavy-ion collisions. This phenomenological model uses a universal analytical formula derived from fragmentation reaction data of medium and heavy-ion projectiles [34]. It effectively estimates the production cross-sections of both exotic and stable nuclei for a wide range of collision systems. A comprehensive set of beam–target combinations was analyzed, establishing normalized yields of exotic beams per centimeter of production target length. Simulations covered all possible exotic isotopes from hydrogen to scandium, which were generated from stable isotopes up to ⁴⁰Ca. However, the maximum acceptable magnetic rigidities of the FRS (18 Tm) and Super-FRS (20 Tm) limit the usable isotopes at 2 AGeV.

To simulate the ion-optic transmission, the study integrated these yield estimations into the MOCADI code [35,36]. The MOCADI framework allows rare-isotope separator facilities, including the existing FRS and the future Super-FRS, to evaluate the availability of specific exotic secondary beams. This approach enables precise estimation of the transmission and yield of each potential exotic secondary beam using the Super-FRS system. The secondary beam of interest, along with other exotic isotopes of similar magnetic rigidity, is transported from the production target to the focal plane FMF2 of the main separator of the Super-FRS, as depicted in Figure 1. At FMF2, a secondary target is set for hypernuclei production. The study also evaluates the secondary beam yield for each combination of beam and target species across varying target thicknesses. An optimization process within the MOCADI simulations was also developed to fine-tune the parameters of the ion-optical elements of the Super-FRS separator, aiming to maximize the intensity of the desired secondary beam at FMF2. This Monte Carlo study achieved systematic uncertainties of approximately 1%.

The theoretical model outlined in Ref. [37] was employed to analyze exotic hypernuclei production for each secondary beam of interest. This hybrid model combines the Dubna cascade model with a quark-gluon string model (DCM-QGSM) for simulating beam–target collisions [38] with a Fermi break-up model for describing spectator de-excitation [39]. Theoretical calculations included exotic beams from Li to Ne isotopes colliding with ¹²C or ⁹Be targets at 2 AGeV. Simulations were conducted with sufficient event counts to maintain a hypernuclei production cross-section uncertainty within 0.1 μb. To validate the model, theoretical predictions for ⁶Li + ¹²C collisions at 2 AGeV were compared against existing experimental data [19], showing good agreement. Because of this, the previously reported analysis [31] used the measured experimental cross-sections [19] and scaled the calculated ones with respect to the experimental values. However, in this study, only the theoretical cross-sections of the hypernuclear production are implemented without any modification.

Figure 2 shows the production cross-section of ${}_{\Lambda }{}^6\mathrm{He}$, ${}_{\Lambda }{}^9\mathrm{B}$, ${}_{\Lambda }{}^9\mathrm{C}$, and ${}_{\Lambda }{}^{10}\mathrm{C}$ hypernuclei at 2 AGeV based on neutron and proton numbers of the exotic beam interacting with a ¹²C target. This analysis helps identify the secondary beam isotope that maximizes each exotic hypernucleus production. However, the optimal scenario must also consider the feasibility of exotic beam production and adjustments for beam energy variations from 2 AGeV.

The study applied a parametrization method [40] to adjust the theoretical calculations according to the exotic beam’s kinetic energy, using the pp → pK${}^{+}\Lambda$ cross-section data [41] for scaling. For example, reducing the beam energy to 1.9 AGeV or 1.8 AGeV results in 73% or 46% reductions in hypernuclei production cross-sections, respectively.

A multivariate dataset was constructed to optimize the experimental parameters for hypernucleus production. This dataset consolidates the theoretical and simulation results, including the optimal primary beam, production target, its thickness, and the ideal exotic secondary beam for each specific hypernucleus. Nevertheless, it is not feasible to directly visualize the multivariate dataset and identify the best case by simply compiling all possible combinations of isotopes species. An optimization procedure was implemented to find the optimal solution.

During the analysis, all possible secondary beam combinations with beryllium or carbon targets were considered in the theoretical calculations of hypernuclei production. The procedure selected the appropriate secondary beam to determine the optimal conditions of the Super-FRS, allowing the calculation of the hypernuclear yield per second for a 4 centimeter secondary production target. This target thickness was chosen to match the experimental conditions used in previous HypHI experiments. The estimated hypernuclear yield incorporates experimental efficiency from ion-optical transmission and an exotic beam intensity derived from a primary beam of $5\times {10}^9$ ions/s. Additionally, a simpler case with a stable beam for hypernuclei production was included, using an intensity of ${10}^7$ ions/s to estimate the yield per second.

To solve the optimization problem, a cost function was defined considering several variables and parameters. The goal was to find the optimal set that maximizes or minimizes the cost function. The cost function is as follows:

$$F_{\alpha ,\beta }(C,E,T,I)=\alpha C+\beta E-\gamma T+\delta I$$

(1)

where C, E, T, and I represent the hypernuclear yield, secondary beam energy, production target thickness, and intensity of the secondary beam of interest, respectively. The coefficients $\alpha$, $\beta$, $\gamma$, and $\delta$ are weight factors that connect these variables. The coefficient $\delta$ for intensity was set to $1/2$ to ensure numerical stability during convergence. Additionally, the sum of the squared weights is constrained to 1, ${\alpha }^2+{\beta }^2+{\gamma }^2+{\delta }^2=1$, with $\gamma$ being determined as $\sqrt{3/4-{\alpha }^2-{\beta }^2}$. This ensures that $\alpha$ and $\beta$ lie within a circle of radius $\sqrt{3/4}$. The variables were normalized to the range [0, 1] to maintain the weight coefficients within a unit interval. $\alpha$ and $\beta$ are set based on the desired weight assigned to each variable. This cost function is specifically designed to maximize the production of the hypernucleus of interest by optimizing the energy and intensity of the secondary beam while minimizing the primary production target thickness. The optimization of these secondary beam characteristics is strongly related to the combination of primary beam and target species.

The search for the maximum of the cost function is performed with fixed values for $\alpha$ and $\beta$ to identify the optimal parameter set:

$$\underset{C,E,T,I}{argmax}\left\{F_{\alpha ,\beta }(C,E,T,I)\right\}$$

(2)

3. Results

The results can then be analyzed based on different values of $\alpha$ and $\beta$. Figure 3 presents the outcomes of the optimization for the ${}_{\Lambda }{}^6\mathrm{He}$ case with each point representing a possible optimal condition set. The best condition is determined using the maximax criterion, which is defined as shown below:

$$\underset{\alpha ,\beta }{argmax}\underset{C,E,T,I}{max}\left\{F_{\alpha ,\beta }(C,E,T,I)\right\}$$

(3)

The evolution of the variable set as a function of $\alpha$ and $\beta$ is illustrated in Figure 3. The images display the optimal primary beam and target species (Figure 3a,b), the target thickness (Figure 3c), the secondary beam isotope (Figure 3d) and its energy (Figure 3e), and the resulting yield of ${}_{\Lambda }{}^6\mathrm{He}$ (Figure 3f) as functions of $\alpha$ and $\beta$. The parameters $\alpha$ and $\beta$ are varied within the range [$-\sqrt{3/4}$, $\sqrt{3/4}$], where negative values of $\alpha$ aim to minimize hypernuclear yield, and negative values of $\beta$ aim to reduce the kinetic energy of the secondary beam. Calculating the optimal conditions within the weight limits is valuable, as it allows for the consideration of alternative decision criteria beyond the maximax approach presented in Equation (3). Notwithstanding, the global optimal conditions are obtained with $\alpha$ and $\beta$ values represented with a red star in Figure 3.

For the case of ${}_{\Lambda }{}^6\mathrm{He}$, the optimization procedure leads to the following experimental conditions: a primary beam of ²⁰Ne ions on a 2-centimeter ⁹Be target, with a selected exotic beam of ¹⁷F, which is transported to bombard a ¹²C target. The intensity of the ¹⁷F secondary beam is approximately $5.7\times {10}^6$ ions per second, assuming a primary beam intensity of $5\times {10}^9$ ions per second, which aligns with the expected intensity at the Super-FRS. Under these conditions, the ${}_{\Lambda }{}^6\mathrm{He}$ yield is approximately $1.35$ hypernuclei per second for a 4-centimeter secondary target.

Another example is shown in Figure 4, which presents the optimal conditions (marked by a red star) to produce ${}_{\Lambda }{}^9\mathrm{C}$. This experimental setup involves directing a ¹⁴N primary beam onto a ⁹Be target with a thickness of 5.5 cm. This interaction generates an exotic ¹²N beam, which is then transported to strike a 4-centimeter ¹²C target. Considering a primary beam intensity of $5\times {10}^9$ ions per second, the ¹²N secondary beam achieves an intensity of approximately $5.1\times {10}^6$ ions per second. Under these conditions, the hypernuclear yield is 0.25 ${}_{\Lambda }{}^9\mathrm{C}$ per second.

The same approach can be applied to other hypernuclei.

Table 1. Summary of the optimization procedure results, covering all $\Lambda$-hypernuclei up to carbon isotopes. For each hypernucleus, the optimal experimental conditions are detailed, including the reaction needed to generate the exotic beam, the target thickness, the chosen exotic beam for hypernuclei production on a 4-centimeter ¹²C target, as well as the exotic beam kinetic energy, the intensity, and the achieved hypernuclear yield.

	Reaction	Target (cm)	2nd Beam	${\mathrm{E}}_{\mathit{k}}$ (AGeV)	I (${10}^6$/s)	Yield (/s)
${}_{\Lambda }{}^2\mathrm{n}$	27Al + 10B	9	19O	1.82	5.2	0.06
${}_{\Lambda }{}^3\mathrm{n}$	20Ne + 9Be	2	17F	1.97	5.7	0.71
${}_{\Lambda }{}^4\mathrm{n}$	20Ne + 9Be	2	17F	1.97	5.7	0.36
${}_{\Lambda }{}^5\mathrm{n}$	20Ne + 9Be	2	17F	1.97	5.7	0.10
${}_{\Lambda }{}^6\mathrm{n}$	23Na + 11B	15.5	12B	1.79	11	0.06
${}_{\Lambda }{}^3\mathrm{H}$	20Ne + 9Be	2	17F	1.97	5.7	1.49
${}_{\Lambda }{}^4\mathrm{H}$	20Ne + 9Be	2	17F	1.97	5.7	1.29
${}_{\Lambda }{}^5\mathrm{H}$	20Ne + 9Be	2	17F	1.97	5.7	1.00
${}_{\Lambda }{}^6\mathrm{H}$	20Ne + 9Be	2	17F	1.97	5.7	0.50
${}_{\Lambda }{}^7\mathrm{H}$	20Ne + 9Be	2	17F	1.97	5.7	0.18
${}_{\Lambda }{}^8\mathrm{H}$	23Na + 11B	15.5	12B	1.79	11	0.08
${}_{\Lambda }{}^3\mathrm{He}$	14N + 9Be	5.5	12N	1.94	5.1	0.58
${}_{\Lambda }{}^4\mathrm{He}$	20Ne + 9Be	2	17F	1.97	5.7	1.24
${}_{\Lambda }{}^5\mathrm{He}$	20Ne + 9Be	2	17F	1.97	5.7	1.56
${}_{\Lambda }{}^6\mathrm{He}$	20Ne + 9Be	2	17F	1.97	5.7	1.35
${}_{\Lambda }{}^7\mathrm{He}$	20Ne + 9Be	2	17F	1.97	5.7	1.01
${}_{\Lambda }{}^8\mathrm{He}$	20Ne + 9Be	2	17F	1.97	5.7	0.50
${}_{\Lambda }{}^9\mathrm{He}$	23Na + 11B	15.5	12B	1.79	11	0.25
${}_{\Lambda }{}^4\mathrm{Li}$	20Ne + 9Be	2	17F	1.97	5.7	0.39
${}_{\Lambda }{}^5\mathrm{Li}$	20Ne + 9Be	2	17F	1.97	5.7	0.74
${}_{\Lambda }{}^6\mathrm{Li}$	20Ne + 9Be	2	17F	1.97	5.7	1.29
${}_{\Lambda }{}^7\mathrm{Li}$	20Ne + 9Be	2	17F	1.97	5.7	1.49
${}_{\Lambda }{}^8\mathrm{Li}$	20Ne + 9Be	2	17F	1.97	5.7	1.26
${}_{\Lambda }{}^9\mathrm{Li}$	16O + 9Be	5.5	14O	1.93	5.5	0.74
${}_{\Lambda }{}^{10}\mathrm{Li}$	23Na + 11B	15.5	12B	1.79	11	0.34
${}_{\Lambda }{}^5\mathrm{Be}$	14N + 9Be	5.5	12N	1.94	5.1	0.20
${}_{\Lambda }{}^6\mathrm{Be}$	14N + 9Be	5.5	12N	1.94	5.1	0.61
${}_{\Lambda }{}^7\mathrm{Be}$	14N + 9Be	5.5	12N	1.94	5.1	1.24
${}_{\Lambda }{}^8\mathrm{Be}$	20Ne + 9Be	2	17F	1.97	5.7	1.25
${}_{\Lambda }{}^9\mathrm{Be}$	20Ne + 9Be	2	17F	1.97	5.7	1.29
${}_{\Lambda }{}^{10}\mathrm{Be}$	20Ne + 9Be	2	17F	1.97	5.7	0.92
${}_{\Lambda }{}^{11}\mathrm{Be}$	20Ne + 9Be	5	18Ne	1.92	5.0	0.16
${}_{\Lambda }{}^7\mathrm{B}$	14N + 9Be	5.5	12N	1.94	5.1	0.22
${}_{\Lambda }{}^8\mathrm{B}$	14N + 9Be	5.5	12N	1.94	5.1	0.88
${}_{\Lambda }{}^9\mathrm{B}$	14N + 9Be	5.5	12N	1.94	5.1	1.12
${}_{\Lambda }{}^{10}\mathrm{B}$	14N + 9Be	5.5	12N	1.94	5.1	0.82
${}_{\Lambda }{}^{11}\mathrm{B}$	20Ne + 9Be	2	17F	1.97	5.7	0.44
${}_{\Lambda }{}^8\mathrm{C}$	14N + 9Be	5.5	12N	1.94	5.1	0.07
${}_{\Lambda }{}^9\mathrm{C}$	14N + 9Be	5.5	12N	1.94	5.1	0.25
${}_{\Lambda }{}^{10}\mathrm{C}$	14N + 9Be	5.5	12N	1.94	5.1	0.49
${}_{\Lambda }{}^{11}\mathrm{C}$	14N + 9Be	5.5	12N	1.94	5.1	0.29

summarizes the optimization results for $\Lambda$-hypernuclei up to carbon hypernuclei. It lists the necessary reaction for optimal exotic beam production, the target thickness, the selected exotic beam with its optimal kinetic energy and intensity, and the yield per second for each hypernucleus produced on a 4-centimeter ¹²C target. These optimal experimental conditions will be valuable for designing future hypernuclear experiments at the Super-FRS facility.

4. Discussion

A systematic approach was developed to identify the optimal experimental conditions for producing exotic hypernuclei within the Super-FRS fragment separator at the new FAIR facility, where future hypernuclear spectroscopy experiments will be conducted [31]. This optimization process integrates results from multiple theoretical models, including predictions of hypernuclei and exotic beam production through the fragmentation of the primary beam on the production target. It also incorporates Monte Carlo simulations of beam transport to estimate the experimental efficiency of the separator. These efficiencies correspond to the transmission from the secondary beam production site to the experimental area, where both hypernuclei production and spectroscopy will occur.

The optimization method combines these diverse inputs to determine the best conditions for any hypernucleus under consideration by maximizing the cost function. Specifically, for the ${}_{\Lambda }{}^6\mathrm{He}$ (${}_{\Lambda }{}^9\mathrm{C}$) hypernucleus, the optimal configuration involves using a ¹⁷F (¹²N) secondary beam derived from the fragmentation of a ²⁰Ne (¹⁴N) primary beam on a ⁹Be target. This setup could produce approximately 1.35 ${}_{\Lambda }{}^6\mathrm{He}$ (0.25 ${}_{\Lambda }{}^9\mathrm{C}$) nuclei per second on a 4-centimeter ¹²C target with an estimated production cross-section of 3.3 (0.7) μb. Given these conditions and the efficiency observed in previous experiments [17], around 11.700 (2.100) hypernuclei per day are expected.

The optimization process was extended to cover all hypernuclei up to carbon isotopes, providing valuable insights for future experiments on both proton-rich and neutron-rich hypernuclei. These results were achieved using a specific quasi-convex combination of the variables involved in the optimization. However, alternative approaches remain to be explored, particularly regarding the weight definitions in the cost function. Additionally, while the maximax criterion was applied in this study, other decision-making strategies could be considered to adjust the weights, offering new perspectives on the dataset.