The AMADEUS (Anti-kaonic Matter At DAΦNE: An Experiment with Unraveling Spectroscopy) Collaboration performed research of the low-energy K⁻-nucleon/nuclei interactions in light nuclear targets for over a decade [1–3]. The primary objective of these studies is an investigation of the poorly known Λ(1405) resonance and a deeper understanding of the K⁻ single- and multi-nucleon absorption processes, both at-rest and in-flight, including the possible formation of kaonic bound states.

The investigation of the in-medium modification of the K ̄ N interaction is of fundamental importance for the low-energy Quantum Chromodynamic (QCD) in the strangeness sector. Chiral Perturbation Theory (ChPT), an Effective Field Theory (EFT) that successfully describes interactions involving πN, ππ and NN in the low-energy regime [4, 5] is not applicable to the sector with s quarks due to the broad Λ(1405) and Σ(1385) resonances emerging just below the K ̄ N threshold. The resonances appearance causes an attractive K ̄ N interaction in the far subthreshold region, whereas it looks repulsive at threshold, as demonstrated by the SIDDHARTHA measurements of the K⁻p scattering length [6].

Two main theoretical approaches have been developed to overcome these difficulties, namely, phenomenological potential models based on the K ̄ N and NN interactions [7–13] and chiral unitary models involving the non-perturbative Chiral SU(3) dynamics [14–20]. The two models, constrained by the existing scattering data, describe the K ̄ dynamics above the threshold very well, however, a large difference appears in the subthreshold extrapolations. In particular, significantly weaker attraction is predicted by the chiral SU(3) models than by the phenomenological potential approach which leads to contrasting predictions for the Λ(1405) (I = 0) resonance and related kaonic nuclear bound states. Although the Particle Data Group (PDG) [21] lists the Λ(1405) as a four-star resonance (spin 1/2, isospin I = 0, strangeness S = −1), decaying into (Σπ)⁰ through the strong interaction, its nature remains still an open issue. According to the phenomenological potential models, the Λ(1405) is a pure strongly attractive K ̄ N bound state with a mass of about 1,405 MeV/c², binding energy of about 30 MeV and a width of 40 MeV/c² [7, 13]. Conversely, the chiral models [14–20] predict that Λ(1405) occurs as a superposition of two states, a high-mass state predominantly coupled to the K ̄ N production channel and a low-mass state mainly coupled to the Σπ channel which are located around 1,420 MeV/c² and at 1,380 MeV/c², respectively. The two different theoretical scenarios for Λ(1405) reflect the strength of the K ̄ N interaction and thus influence the possibility of K⁻ multi-nucleon bound states formation. Deeply bound nuclear states with narrow widths and large binding energies (up to 100 MeV/c²) are predicted by phenomenological models as a consequence of the strongly attractive K ̄ N interaction, while SU(3) models result in much less attractive K⁻N interaction, which leads to the prediction of slightly bound kaonic nuclear states. Till now, the bound kaonic nuclear states have been searched for in several experiments, using two main approaches: proton-proton and heavy ion collisions (DISTO [22]) as well as in-flight and at-rest K⁻ interactions in light nuclei (FINUDA [23] and KEK-PS E549 [24, 25] experiments). The first K⁻pp bound kaonic nuclear system signal has been currently observed and investigated at J-PARC in the ³ He(K⁻, Λp)n reaction [25].

Recently the ALICE Collaboration confirmed the couple-channel character of the K ̄ N interaction [26] indicating that the (Σπ)⁰ invariant mass spectral shape depends on the decay channel (since the isospin interference term contributes to Σ^± π ^∓ cross section with opposite sign and vanishes for Σ⁰ π ⁰) as well as on the production channel. In this case, K ̄ N absorption represents the golden channel for examining the predicted high mass pole of the Λ(1405). Moreover, experimental studies of spectral shape and yield of non-resonant contribution in hyperon–pion final states, allow to constraint the chiral predictions which are strongly model dependent. This has been performed by investigating the single-nucleon absorption K⁻n → Λπ ⁻ channel [27].

Investigation of K⁻ multi-nucleon absorption contributions plays a very important role in the determination of the K⁻-nucleus optical potential. Existing K⁻ single-nucleon optical potentials, combined with phenomenologically determined K⁻ multi-nucleon absorption term (based on global absorption bubble chamber data) do not reproduce the kaonic atoms data along the periodic table of the elements [28, 29]. Therefore, it is crucial to improve the theoretical model by providing complete characteristics of the absorption processes, by extracting the two-, three-, and four-nucleon absorptions (2NA, 3NA, and 4NA). The first comprehensive measurement of K⁻ multi-nucleon absorptions, including a contribution of the possible K⁻pp bound state, has been completed [30].

The purpose of the article is to provide an overview of the current status of the research performed by AMADEUS collaboration. It begins with an introduction of the experimental facility, namely, the DAΦNE accelerator and the KLOE detector. Thereupon, K⁻ single- and multi-nucleon absorption studies and their impact on the field are discussed, which is followed by Conclusions.

The AMADEUS studies are based on an experimental data sample, corresponding to 1.74 fb⁻¹ integrated luminosity, collected with the KLOE detection system [31, 32], installed at the double electron-positron ring of the DAΦNE collider [33] located at the National Laboratory in Frascati of INFN (Italy).

The DAΦNE facility (so-called ϕ meson factory) was designed to work at the center of mass energy of the ϕ meson. The acceleration complex deliver low-momentum (∼127 MeV/c) monochromatic charged kaon beam, characterized by a very small hadronic background, originating from the ϕ-meson decays (BR(K⁺K⁻) = (48.9 ± 0.5)%) which, in turn, is produced in e ⁺ e ⁻ collisions (beam energies of 0.51 GeV). The back-to-back topology of the kaons pair production allows to extrapolate non-identified charged kaon tracks.

The KLOE detector system has a 4π geometry and surrounds the DAΦNE interaction region (geometrical acceptance of 98%). The detection setup consists of two basic components: a large cylindrical Drift Chamber (DC) [31, 34] and an electromagnetic calorimeter (EMC) consisting of groved lead with scintillating fibers [31, 35]. The detection system was immersed in a 0.52 T magnetic field along the beam axis, provided by a superconducting solenoid.

The DC, designed for tracking and identification of charged particles, containing a total of about 52,000 wire, was filled with a mixture of helium (90%) and isobutane (10%) C₄H₁₀. Its inner radius, outer radius, and length were equal to 0.25, 2, and 3.3 m, respectively. The DC entrance wall was built of 750 μm layer of low-density carbon fiber and 150 μm layer of aluminum. The momenta of charged particles were determined with excellent relative accuracy of σ ( p ) p = 0.4 % . The spatial resolution of the particle tracks reconstruction was of σ _ρϕ ∼ 200 μm in the transverse and of σ _z ∼ 2 mm along the z-axis, while the accuracy of decay vertices reconstruction was about 1 mm.

The EMC composed of a cylindrical barrel with an inner radius of 2 m and two end-caps was dedicated to neutral particles detection. It also provided Time-of-Flight (TOF) information for the charged particles. The volume ratio of lead-scintillating fibers (lead/fibers/glue = 42:48:10) was optimized to achieve high light yield and high efficiency for photons in the 20–300 MeV/c energy range. The cluster position resolution along the fibers was σ ‖ = 1.4 c m ( E / 1 G e V ) , while in the orthogonal direction it was σ _⊥ = 1.4 cm. The energy and time resolutions for photon clusters are given by σ E E γ = 0.057 ( E γ / 1 G e V ) and σ _t = 57 p s ( E γ / 1 G e V ) ⊕ 100 ps, respectively.

Since the Λ(1405)’s resonance line shape is expected to depend on both, the production mechanism and the observed decay channel [26], experimental investigation of its properties is challenging. Additionally, extraction of the shape of the Λ(1405) invariant mass in reactions induced by negatively charged kaons (K⁻) is complicated by two biases. The first bias arises from the threshold of the Σπ invariant mass, which is limited by the last nucleon binding energy. This threshold is approximately 1,412 MeV/c² for K⁻ capture at rest on ⁴He and around 1,416 MeV/c² on ¹²C. Therefore, to verify the existence of the predicted high mass pole of the Λ(1405), which is expected to be located at approximately 1,420 MeV/c², it is necessary to explore the K⁻ absorption in flight. As shown in [36], the experimental Σ⁰ π ⁰ invariant mass threshold for K⁻ captures in ¹²C in flight (p _K ∼ 100 MeV/c) is shifted upwards by about 10 MeV with respect to the capture at-rest, thus opening the access to the energy range of interests. The Σ⁰ π ⁰ is the so-called “golden decay channel” since it provides a clear Λ(1405) signature in the I = 0 isospin.

Another crucial bias impacting the (Σπ)⁰ invariant mass spectrum is associated with the non-resonant contribution, which needs to be subtracted in order to extract the shape and investigate the characteristics of the Λ(1405) resonance. Chiral SU(3) meson-baryon coupled channels interaction models (Barcelona (BCN) [37], Prague (P) [38], Kyoto-Munich (KM) [39], Murcia (M1,M2) [40], Bonn (B2,B4) [41]) provide the K⁻n → Λπ/Σπ scattering amplitudes, which, however, strongly differ in the K ̄ N subthreshold region. To determine the appropriate model for explaining the observed spectra of Σ⁰ π ⁰, the K⁻n → Λπ ⁻ process was investigated by AMADEUS for the single-nucleon K⁻ absorption in ⁴He [27]. The non-resonant transition amplitude, below the K ̄ N threshold, was extracted for the first time for the K⁻n → Λπ ⁻ channel, based on the well known resonant part corresponding to the formation of Σ⁻(1385) (I = 1). The multidimensional fit of experimental distributions (Λπ ⁻ invariant mass, momentum, and angular spectrum) with dedicated Monte Carlo simulations for the contributing processes (non-resonant and resonant reactions, the primary production of a Σ followed by the ΣN → ΛN’ conversion process, the contamination of K⁻¹²C) was performed. The Monte Carlo simulations are based on the phenomenological K⁻-nucleus absorption model developed in Ref. [42]. The non-resonant transition amplitude modulus was found to be | A K − n → Λ π − | = ( 0.334 ± 0.018 s t a t . − 0.058 + 0.034 s y s t . ) fm at (33 ± 6) MeV/c² below the K ̄ N threshold. A measurement at s ∼ 33 MeV/c² below threshold was possible due to the binding energy of the absorbing nucleon as well as to the recoil energy of the K⁻n pair with respect to the residual ³He nucleus. The AMADEUS experimental result together with the theoretical predictions rescaled for the K⁻n → Σπ transition probabilities, is shown in Figure 1.

The obtained result enables to test the chiral predictions in the subthreshold region which allow constraining the corresponding non-resonant background for I = 0 channel (Σπ)⁰ and hence to determine the Λ(1405) properties.

Apart from single-nucleon capture studies, AMADEUS conducted research specifically focused on K⁻ absorptions on two or more nucleons. The investigation is highly significant for the determination of K⁻-nucleus/nucleon optical potential which has a strong impact on various sectors of physics, like nuclear and particle physics as well as astrophysics [44–46]. A detailed characterization of the K⁻ two-, three- and four-nucleon absorption processes (2NA, 3NA, and 4NA) in K⁻ capture on ¹²C nuclei was obtained by investigating Λ(Σ⁰)p decay channels [30, 47]. A comprehensive study was performed [30] based on the phenomenological model for the K⁻ captures at-rest and in-flight on light nuclei [42, 48]. The simultaneous fit of the experimental Λp invariant mass, Λp angular correlation, Λ and proton momentum spectra with the corresponding simulated distributions of the contributing processes (including the Σ⁰ productions followed by Σ⁰ → Λγ decay and for the 2NA: 1) the Quasi-Free (QF) processes, 2) elastic Final State Interaction (FSI) processes and 3) inelastic FSI processes due to conversion (ΣN →ΛN′)), allowed to extract the K⁻ 2NA, 3NA and 4NA branching ratios (BRs) and cross sections for low-momentum kaons in Λp and Σ⁰p final states. The obtained BRs and cross sections are summarized in Table 1.

The determined global BR (sum of 2NA, 3NA, and 4NA BRs) (21 ± 3 ( s t a t . ) − 6 + 5 (syst.)) % aligns with the K⁻ multi-nucleon absorption BRs measured in bubble chamber experiments [49, 50]. Combining the experimental BRs for processes leading to Λp pair production (16.1 ± 2.9 ( s t a t . ) − 0.9 + 1.0 (syst.))% [30] with a component corresponding to processes without Λp in the final state (5.5 ± 0.1 ( s t a t . ) − 0.9 + 1.0 (syst.))% [51] (determined based on theoretical and experimental information [49, 52]), the total BR for K⁻ 2NA in ¹²C was found to be (21.6 ± 2.9 ( s t a t . ) − 5.6 + 4.4 (syst.))% [51].

The performed studies show that the experimental BR of the Λp QF production in K⁻ 2NA interaction is lower than that of Σ⁰p QF production: R = B R ( K − ( p p ) → Λ p ) B R ( K − ( p p ) → Σ 0 p ) = 0.7 ± 0.2 ( s t a t . ) − 0.3 + 0.2 ( s y s t . ) indicating a dominance of Σ⁰p final state, which in contradiction to the ratio of corresponding phase spaces R′ = 1.22. This result was found to be consistent with theoretical calculations of Barcelona and Prague groups when considering the in-medium effect caused by the Pauli blocking [52].

The potential contribution of the K⁻pp bound system to the Λp spectra was explored revealing the entire overlap of the signal associated with the formation of the K⁻pp cluster in K⁻-induced reactions on carbon with the K⁻ 2NA-QF process [30]. Repeating the analysis in the FINUDA-like measurement [23] conditions (selection of back-to-back Λp events (cosθ _Λp < −0.8)) yielded the same results (BRs are in agreement with those obtained from entire data sample), indicating that if the bound system exists, it cannot be distinguished from the two-nucleon capture process within this type of analysis.

In this paper, the results obtained by the AMADEUS collaboration in studying the low-energy K⁻ interactions with light nuclei inducing single- and multi-nucleon absorptions, were reviewed. The investigation of K⁻-nucleons/nuclei interactions is fundamental for a better understanding of the non-perturbative quantum chromodynamics QCD in the strangeness sector.

By conducting studies on the K⁻n single nucleon absorption in ⁴He, it was possible to provide the first characterization of the non-resonant K⁻N → Yπ production below the K ̄ N threshold which is crucial for investigating the properties of the puzzling Λ(1405) resonance. Additionally, investigations of low-energy K⁻ capture on a solid carbon target led to a comprehensive understanding of the two-, three-, and four-nucleon absorptions in the Λp and Σ⁰p final states, including their branching ratios (BRs) and cross sections. Furthermore, it was discovered that the potential contribution from a K⁻pp bound state completely overlaps with the K⁻ two-nucleon quasi-free process. The presented results demonstrate that the DAΦNE collider is a unique research facility with outstanding capabilities for studying kaon physics at low energies.

The AMADEUS Collaboration is currently completing studies of K⁻ 4NA in the Λt golden channel and analyses related to K⁻p → Σ⁰ π ⁰(Λ π ⁰) cross section determination for kaon momentum below 100 MeV/c [53] which will provide additional new experimental constraints to the K ̄ N strong interaction.