Preamble

Reexamined mass of 22C via the constraint from the recently experimental extraction of its radius

Yi-Le Fan¹, Qing-Rui Sun¹, Cheng-Jun Feng¹, Yi-Bin Qian¹,† and Dong Bai²
¹Department of Applied Physics and MIIT Key Laboratory of Semiconductor Microstructure and Quantum Sensing, Nanjing University of Science and Technology, Nanjing 210094, China
²College of Mechanics and Engineering Science, Hohai University, Nanjing 211100, China

The neutron-rich nucleus 22C, located at the neutron dripline, exhibits intriguing structural properties, such as its Borromean nature and potential two-neutron halo configuration. Despite experimental advancements, uncertainties persist in the two-neutron separation energy S2n and the matter radius of this fascinating nucleus. In this work, we employ the three-body Faddeev approach to investigate the ground-state properties of 22C, constrained by the recently deduced matter radius. By optimizing the neutron-core and three-body interactions to reproduce the experimental radius, the two-neutron separation energy S2n is redetermined, revealing a weakly bound system dominated by s-wave configuration. Additionally, an excited state exhibiting an Efimov-like pattern is identified through analysis of the specific density distributions and relative distances in the three-body system, highlighting the geometric similarity between the ground and excited states.

Keywords: Two-neutron separation energy, Nuclear radius, Three-body approach, Neutron-rich nucleus

INTRODUCTION

With the advent of radioactive ion beam technology and facilities, the nuclear landscape has been extended to a large map encompassing Z = 118 and reaching toward both the proton and neutron driplines [1–3]. Contrary to the proton-rich side of the nuclear chart, experimental exploration of neutron-rich nuclei appears quite limited due to their short lifetimes and tiny cross sections [4–7]. Among these dripline nuclei, the heaviest carbon isotope experimentally determined to be 22C is of particular interest in both nuclear physics and Efimov physics, owing to its Borromean nature [8–11]. Specifically, about two decades ago, 22C was predicted to be an ideal s-wave two-neutron halo nucleus with a dominant ν(s1/2)² configuration [12, 13], which was subsequently confirmed in measurements of reaction cross sections despite an unexpectedly large matter radius [14].

Furthermore, this dripline nucleus 22C plays a crucial role in other exotic structural properties such as possible new magicity at N = 16 and shape decoupling [15–20]. As mentioned previously, 22C is a typical Borromean nucleus with a bound three-body (20C + n + n) system, while no binary subsystem—namely 20C + n or n + n—is bound. However, substantial uncertainties and puzzles remain regarding its nuclear quantities, such as the separation energy and matter radius [2, 21–23].

Since the remarkable experiment measuring the reaction of 22C on a liquid hydrogen target [14], extensive efforts have been made to clarify ambiguities about the bulk properties of this intriguing nucleus [9, 21, 22]. For example, the interaction cross sections (σI) of 22C on a carbon target were accurately measured at 235 MeV/nucleon, yielding σI = 1.280 ± 0.023 b [9]. Within a four-body Glauber reaction model, the root-mean-squared matter radius of 22C was then deduced to be 3.44 ± 0.08 fm, which is significantly smaller than previous extractions. Several other experiments have also been performed to understand the 22C nuclear structure, including neutron removal reactions from carbon isotopes and reconstruction of the 20C + n decay-energy spectrum [8, 21, 22].

As another key quantity for dripline nuclei, the binding energy or two-neutron separation energy of 22C remains uncertain. To date, there appears to be only one direct mass measurement of 22C, establishing an upper limit of S2n ≤ 320 keV [23]. According to the recent atomic mass evaluation (AME20), this value was determined to be 35 ± 20 keV [2]. From a theoretical perspective, different predictions for separation energies exist within both shell model and three-body approaches [12, 24–26]. Additionally, extensive efforts have been devoted to studying this exotic nucleus 22C and its isotopes via ab initio methods [27–29]. Nevertheless, debate continues regarding our understanding of 22C.

It is well established that a consistency exists between neutron separation energy and matter radius for neutron-rich nuclei [12, 24, 30, 31]. This correlation allows us to place constraints on the structural properties of 22C, even if absolute values cannot yet be determined precisely. Based on extracted matter radii from reaction measurements, the two-neutron separation energy of 22C can be obtained via empirical formulas relating energy and radius [23, 24, 32]. For instance, under such a procedure, the S2n of 22C would be about 10 keV from the large radius reported decades ago [24]. Very recently, reaction cross sections of neutron-rich carbon isotopes on 12C have been systematically measured over a wide range of incident energies from 30 to 950 MeV/nucleon [33]. The matter distributions and radii of carbon isotopes were then determined using the finite-range Glauber model with Coulomb correction, including for the exotic nucleus 22C. Impressively, the simultaneously obtained charge radii are in excellent agreement with those directly extracted from charge-change cross sections [33, 34].

Keeping these developments in mind, it is particularly significant to determine what can be learned from constraints imposed by these new high-accuracy data. Against this background, we employ the three-body Faddeev method to further understand the binding mechanism of 22C through its two-neutron separation energy, with the requirement of reproducing its matter radius. Details of the theoretical approach and parameterization choices will be presented in the next section, with specific results and discussions on nucleon density distributions given in Sect. III. A summary is provided in the final section.

II. THEORETICAL APPROACH

The Faddeev equations provide a rigorous mathematical framework for three-body systems, whose essence lies in decomposing the total wave function into three Faddeev components (Ψ₁, Ψ₂, Ψ₃), each corresponding to and describing the pairwise interaction between particles [35]. For instance, in the 22C (n + n + 20C) system, Ψ₁ may correspond to the interaction between the two neutrons, while Ψ₂ and Ψ₃ correspond to neutron-core interactions. Each component is associated with a different Jacobi coordinate system (as illustrated in Figure 1 [FIGURE:1]). The equation takes the form:

(Tᵢ + h + Vᵢ - E) ψᴶᴹᵢ = -Vᵢ (ψᴶᴹⱼ + ψᴶᴹₖ)

The left-hand side represents the independent dynamics of the subsystem, including the kinetic energy Tᵢ, the core Hamiltonian ĥ, and the two-body interaction potential Vᵢ (encompassing both nuclear and Coulomb forces). The right-hand side represents how the superposition of the other two components feeds back into the evolution of the current component through potential energy terms, reflecting the three-body coupling that constitutes the root cause of complexity in three-body problems.

Through this decomposition, the Faddeev equations transform the three-body problem into a coupled system of integro-differential equations, thereby avoiding the complexities of directly solving the high-dimensional Schrödinger equation. However, even with this simplified form, numerical solution still presents significant challenges due to multi-dimensional coordinate systems. To further reduce computational dimensionality and achieve a unified description of dynamical behavior across different Jacobi coordinates, the introduction of hyperspherical coordinates becomes essential.

By adopting the hyperspherical coordinate system, which includes the hyperradius ρ and hyperangles θᵢ, the two-dimensional system of partial differential equations is transformed into a set of coupled one-dimensional equations. The transformation of Jacobi coordinates into hyperspherical coordinates takes the following form [36]:

ρ² = xᵢ² + yᵢ², θᵢ = arctan(xᵢ/yᵢ)

The hyperradius ρ characterizes the global scale of the three-body system, being invariant under translational and rotational transformations as well as permutations of particle pairs (i, j), while correlating with the size of the nuclear core. In contrast, the hyperangle θᵢ describes the relative configuration between the particles, which exhibits radial dependence and is correlated with the relative magnitudes of the two Jacobi coordinates. The wave function is expanded within the hyperspherical coordinates as [36, 37]:

Ψ(xᵢ, yᵢ) = ρ⁻⁵ᐟ² ∑ₙ Rₙ(ρ)φₖᵢ(θᵢ)

φₖᵢ(θᵢ) = Nₗₓᵢₗᵧᵢ (sin θ)ˡˣⁱ (cos θ)ˡʸⁱ Pₗₓᵢ⁺¹ᐟ²,ˡʸᵢ⁺¹ᐟ²(cos 2θᵢ)

Rₙ(ρ) ∝ Lₙ⁵(z)e⁻ᶻᐟ²; z = 2ρ/β

Here φₖᵢ denotes the hyperangular component described by the Jacobi polynomial P; Nᵢ is the normalization coefficient and Kᵢ represents the hyperangular momentum directly related to the order of the corresponding Jacobi polynomial through Kᵢ = lₓᵢ + lᵧᵢ + 2nᵢ (nᵢ = 0, 1, 2, ···). χₖᵢ represents the hyperradial component, expanded in terms of Laguerre polynomial basis functions.

Upon introducing the hyperspherical expansion into the Faddeev coupled equations, one obtains a set of simultaneous linear equations:

Ha = Ea

Within the Faddeev equations, the matrix H requires several types of matrix elements:

V̂ᵢ = V̂ᵢᶜ(xᵢ) + V̂ˢᴼVᵢˢᴼ(xᵢ) + V̂ᴼVᵢᴼ(xᵢ) + V̂ˢˢVᵢˢˢ(xᵢ) + V̂ᵀVᵢᵀ(xᵢ)

V̂ᵢᶜ(xᵢ) denotes the central interaction. The spin-orbit (SO) coupling is described by the operator V̂ˢᴼ and radial form factor Vᵢˢᴼ(xᵢ). The tensor operator V̂ᴼ and radial form factor Vᵢᴼ(xᵢ) account for the multipole deformed potential. The standard tensor interaction is given by operator V̂ᵀ with radial dependence Vᵢᵀ(xᵢ), while the spin-spin interaction is represented by V̂ˢˢ and its form factor Vᵢˢˢ(xᵢ).

The central potential depends solely on the interparticle distance, with its operator form expressed as V̂ᵢᶜ(xᵢ) and is diagonal in the angular momentum and spin quantum numbers. Within a fixed Jacobi coordinate system, the matrix elements are given by:

⟨i : α′ᵢ|V̂ᵢᶜ|i : αᵢ⟩ = δₗ′ₓᵢₗₓᵢδₗ′ᵧᵢₗᵧᵢδₛ′ᵢₛᵢ ∫ φₗ′ₓᵢₗ′ᵧᵢ(θᵢ) Vᵢᶜ(xᵢ) φₗₓᵢₗᵧᵢ(θᵢ) dθᵢ

where φₗₓᵢₗᵧᵢ(θᵢ) represents the hyperspherical basis function.

The spin-orbit potential takes the form V̂ˢᴼVᵢˢᴼ(xᵢ) = Γᵢⱼlᵢ·sⱼ + Γᵢₖlᵢ·sₖ, where its matrix elements incorporate scalar products of the orbital angular momentum with particle spins sⱼ or sₖ.

III. NUMERICAL RESULTS AND DISCUSSIONS

In this section, we present detailed calculations of the ground-state properties for the 22C nucleus using a three-body model based on the Faddeev equations. As a typical neutron-dripline nucleus, the experimentally deduced root-mean-square (r.m.s.) matter radius of 22C, rₘᵉˣᵖ = 3.296 ± 0.123 fm [33], serves as a key constraint in this study. Based on this, the parameters of the Woods-Saxon potential for the neutron-core interaction are determined through an inverse optimization approach.

In developing the n + 20C interaction model, critical assumptions are implemented due to the absence of experimental data on core excitations: the 20C core is treated as a rigid structure with spin-parity Jπ = 0⁺, enforcing Pauli blocking of the core orbitals (1s₁ᐟ₂)², (1p₃ᐟ₂)⁴, (1p₁ᐟ₂)² and (1d₅ᐟ₂)⁶ [38]; valence neutrons predominantly occupy the (2s₁ᐟ₂) orbital with very weak binding [21].

The core-neutron interaction is described using a Woods-Saxon potential under these assumptions, as shown in Eq. (7) [12, 37, 39], which includes the first two terms from Eq. (5):

Vₙ₋꜀ₒᵣₑ(r) = V₀꜀ / [1 + exp((r - r₀)/a)] + Vₛₒ(l·s)

Determination of the interaction potential parameters is achieved by reproducing the experimentally deduced r.m.s. matter radii. Ref. [33] reports the experimental r.m.s. matter radius of 22C as rₘᵉˣᵖ = 3.296 ± 0.123 fm. Achieving convergence within experimental constraints necessitates reducing the radius parameter to r₀ = 1.13A¹ᐟ³ fm (versus the conventional 1.25A¹ᐟ³ fm used in Ref. [12, 37])—motivated by the monotonic increase of r.m.s. matter radii with r₀.

When setting r₀ = 1.13A¹ᐟ³ fm, the minimum achievable r.m.s. matter radius falls below the experimental lower limit, thereby encompassing the experimental range. The core-neutron potential strength V꜀ˡ⁼² is incrementally adjusted until rₘᶜᵃˡ precisely matches the experimental value of 3.173 fm. The maximum attainable r.m.s. matter radius through variation of V꜀² remains below the critical experimental values (3.296 fm and 3.419 fm). Consequently, a systematic adjustment of r₀ is implemented to translate the theoretical radius distribution, ensuring full coverage of the experimental value range. Specifically, distinct r₀ values are sequentially fixed to ensure the adjustable range of r.m.s. matter radii—achieved by tuning the neutron-core potential—encompasses each target experimental value.

From Table 1 [TABLE:1], it is observed that rₘᶜᵃˡ also depends on the three-body interaction potential. Entries 2 and 4 demonstrate that identical r.m.s. matter radius values can correspond to multiple distinct sets of neutron-core and three-body interaction potentials. Analysis of entries 3 and 5 reveals that the neutron-core interaction potential plays a predominant role, while the three-body interaction potential also contributes non-negligibly. Unique determination of this parameter set requires incorporation of an additional constraint, namely the two-neutron separation energy S2n. Comparison of entries 3 and 5 reveals that within a constrained range of neutron-core potentials, S2n exhibits stronger correlation with the three-body interaction potential. Consequently, S2n provides an approximate constraint for the three-body potential, enabling subsequent precise calibration of V꜀².

The two-neutron separation energy S2n, obtained as an outcome of the calculation constrained by the matter radius, is then compared with theoretical and experimental values for validation. Ref. [9] reports S2n = 0.56⁺⁰.²⁷₋₀.₂₀ MeV, but calculations considering only two-body potentials exhibit significant deviations from this value. This discrepancy is mitigated through the introduction of a Gaussian-form three-body interaction in Eq. (8), which emulates effects of core deformation or core excitation [35, 40, 41]:

V₃ₑ(ρ) = s₃ₑ exp[-(ρ/r₃ₑ)²]

When identical three-body potential parameters s₃ₑ = 3, r₃ₑ = 14 are applied to parameter sets 1, 4, and 5, an inverse correlation is observed: stronger neutron-core interaction potentials V꜀² yield larger two-neutron separation energies S2n. These results exhibit significant deviations from theoretical values. The deviation is reduced when the three-body potential strength is increased from the baseline of parameter set 5. This adjustment brings the calculated two-neutron separation energy S2n into closer agreement with the theoretical value of S2n = 0.56⁺⁰.²⁷₋₀.₂₀ MeV reported in Ref. [9]. The rationale for avoiding universally larger three-body potentials lies in Set 1's extremely shallow S2n binding. Further enhancement would induce a transition from bound states into the continuum. Consequently, each parameter set necessitates a customized three-body potential strength.

At this stage, the input parameters—including two-body and three-body potentials—have been calibrated through fixation of the r.m.s. matter radii and benchmarking against theoretical values of the two-neutron separation energy. These parameters are compiled in Table 1, with Sets 1, 2, and 3 selected as our primary configurations. Across these sets, only three parameters (V꜀², s₃ₑ, r₀) were varied while all others remained fixed.

The hyperangular momentum cutoff Kₘₐₓ = 20 was employed throughout calculations to guarantee numerical convergence. The choice of Kₘₐₓ is briefly discussed in Ref. [42]; larger values of Kₘₐₓ yield better convergence behavior and enable higher accuracy of the numerical solution. However, the influence of Kₘₐₓ on the results is significantly smaller than that of the two-body potential. Moreover, our study of the nucleus 22C does not involve long-range interactions. Therefore, choosing Kₘₐₓ = 20 is adequate for our calculations.

The neutron-neutron interaction is described using the Gogny-Pires-Tourreil (GPT) potential [43], with central, tensor, and spin-orbit terms included while omitting the spin-spin contribution. This potential provides good fits to the low-energy properties of nucleon-nucleon scattering.

Based on the parameter sets (Sets 1, 2, and 3) determined in Table 1, the orbital energy levels, spatial configurations, and density distributions were systematically calculated. Bound states prohibited by the Pauli principle, such as the 1s₁ᐟ₂ and 1p₃ᐟ₂ orbitals, are eliminated through supersymmetric transformations, restricting valence neutrons to the allowed 1d₅ᐟ₂ orbital. Furthermore, adjustment of the s-wave and p-wave potential strengths along with the spin-orbit coupling strength revealed that the (1s₁ᐟ₂), (1p₃ᐟ₂), and (1p₁ᐟ₂) states depend mainly on s-wave and p-wave contributions, which are not listed explicitly.

Table 1 lists the key single-particle orbital energies (units: MeV), where the orbital energy 1d₅ᐟ₂ ranges from −0.323 to −0.72 MeV, indicating that this orbital is in a weakly bound state. The deeply bound 1s₁ᐟ₂ orbital (−18.306 to −19.889 MeV) has a compact wave function and is Pauli-blocked for the valence neutrons.

It should be noted that our research results do not rely on these energy values but rather on their single-particle wave functions. The different potential energies selected in this paper produce almost identical single-particle wave functions for each occupied orbital, and all potential energy sets are configured so that the energy of the 2s₁ᐟ₂ single-particle state approaches zero, consistent with the prescription in Ref. [12].

Since there is no centrifugal potential barrier for neutrons in the 2s₁ᐟ₂ orbital in the average potential field, the extremely weak neutron binding energy leads to significant tunneling effects. This significant tunneling implies radial expansion of the s₁ᐟ₂ orbital wave function. This energy level structure is consistent with typical characteristics of neutron drip-line nuclei. Additionally, due to the expansion of the wave function, the dynamic coupling effects with other strongly bound orbitals are weakened beyond the static effects of the average potential field, indicating the presence of a pure (s₁ᐟ₂)² configuration and an unperturbed core.

If there are two neutrons in the 2s₁ᐟ₂ orbital, their interaction will be the only source of additional binding energy. Since the s-wave potential energy may be further weakened, the ground state energy of 22C is considered to be the minimum value in this analysis.

Table 2 [TABLE:2] presents the energies and r.m.s. radii of the ground state and excited state, calculated with three different three-body interaction parameter sets. Quantities with * correspond to the excited state. All energies are in units of MeV, while all lengths are in units of fm.

Using the two-body and three-body interaction models in Table 1, we solved the Faddeev equations to obtain the energy values and r.m.s. matter radii of the ground state and excited state (relative to the n + 20C threshold) of 22C. The calculation results are listed in Table 2. All energy values are given relative to the 20C + n + n threshold, and the values are close to each other.

Z = 6, N = 14 closed shell cores are called closed cores. The neutron part of the 20C ground state not only contains the 1d₅ᐟ₂ closed shell configuration but also other configurations such as 2s₁ᐟ₂. In the ground state, the two-neutron separation energy S2n = −E = 0.265–0.656 MeV. Since the parameters are set based on the experimental values in Ref. [33], S2n falls within the experimental range and is consistent with other theoretical values: S2n = 0.423 ± 1.140 MeV from Ref. [44] and S2n = −0.140 ± 0.460 MeV from Ref. [23], within the error range, confirming the reliability of the theoretical model.

In addition, the 22C ground state incorporates core correlation, featuring two valence neutrons occupying the halo s-orbital that must satisfy the orthogonality condition with the 20C core s-orbital. The excited state has a spin-parity of Jπ = 0⁺. It is noteworthy that the excited state energy E* is positive, indicating that the employed two-body and three-body potentials cannot bind the neutrons, which is consistent with the experimental observation that there is no bound state in the 21C nucleus [45].

To analyze the three-body configuration of 22C, the average distance parameters of valence neutrons were calculated (Table 3 [TABLE:3]). In the ground state, the average distance between two neutrons is rₙₙ ≈ 6.49 fm, and the distance from the core to the center of mass of the neutron pair is r꜀,ₙₙ ≈ 3.40 fm. In the excited state, these values extend to rₙₙ ≈ 8.90 fm and r꜀,ₙₙ ≈ 4.65 fm, corresponding to the size of a stable nucleus with mass number A ≈ 60. Therefore, the excited state of 22C can be described as a giant halo state. The key finding is that the ratio rₙₙ/r꜀,ₙₙ ≈ 1.91 remains constant. The identical proportions obtained in both states indicate that these two states have similar geometric structures, with the main difference being only a spatial scaling factor.

Table 3 presents the average distance rₙₙ between two valence neutrons in the ground and excited states of 22C, and the average distance r꜀,ₙₙ from the core to the center of mass of the valence neutron pair. The superscript ∗ denotes the excited state. All lengths are in units of fm.

Additionally, the correlation density distribution between the ground and excited states of 22C was analyzed. These values were calculated using the Jacobi coordinate system, with 20C as the spectator particle. The formula for calculating the spatial correlation density distribution is as follows [41]:

P(rₙₙ, r꜀,(ₙₙ)) ≡ x²y² ∫ |Ψᴶᴹ(x, y)|² dΩₓdΩᵧ

The spatial distribution of the two valence neutrons in the ground and excited states is shown in Figures 2 and 3, respectively. The spatial distribution function peaks at (rₙₙ, r꜀,ₙₙ) ≈ (6.5, 3.5) fm in the ground state (Figure 2 [FIGURE:2]), corresponding to the maximum probability density. This configuration is consistent with a compact three-body arrangement. In contrast, the main peak of maximum probability density for the excited state is at (9.5, 4.5) fm, with a secondary peak at (12, 5.5) fm, which originates from contributions of the (1d)² orbital (Figure 3 [FIGURE:3]). This double-peak structure reflects orbital rearrangement in the excited state: some neutrons transition from s-wave to d-wave, causing the spatial configuration to differentiate into compact and extended modes.

Orbital occupancy analysis reveals the s-wave dominant characteristics of 22C. In the ground state, (2s₁ᐟ₂)² accounts for 97.77%, while the d-wave component accounts for only 2.23%; in the excited state, (2s₁ᐟ₂)² decreases to 72.90%, while (1d₅ᐟ₂)² increases to 27.10%. This change coincides with the secondary peak position of the density distribution, indicating s-d orbital mixing in the excited state. The reason may be that the d-wave component acquires significant weight, and the extended halo structure enhances coupling between the core and valence neutrons, inducing orbital mixing. Notably, despite the increase in d-wave components, s-waves still dominate (>70%), so the excited state is essentially the same as the ground state—an s-wave halo nucleus [12]—but exhibits more complex multi-orbit coupling effects.

The Efimov effect is a universal quantum phenomenon in three-body systems, where an effective long-range attraction emerges from short-range two-body interactions near resonance, leading to a series of weakly bound states exhibiting discrete scale invariance [46]. Given the significantly larger r.m.s. radii and more diffuse spatial distribution of the excited state, it can be characterized as an Efimov state in the halo nucleus [47, 48]. Since 22C has been confirmed to be a double neutron halo nucleus, Ref. [49] points out that the Efimov effect is most likely to be observed in double neutron halo nuclei. Our study also reveals an interesting geometric similarity between the ground and excited state configurations—in Efimov physics, two consecutive Efimov states can be related through discrete spatial scaling factors [50]. The identical ratio relationship between rₙₙ and r꜀,ₙₙ in the ground and excited states of 22C indicates that the nuclear configurations of the two states have highly similar geometric shapes, meaning that discrete scaling symmetry exists. The features we discovered in 22C are highly consistent with the theoretical explanation of Efimov states [51].

IV. CONCLUSION

In this study, we systematically investigated the structural properties of the neutron-rich nucleus 22C using a three-body Faddeev approach, constrained by the experimentally determined matter radius. Our calculations employed a Woods-Saxon potential for the neutron-core interaction and introduced a Gaussian-form three-body potential to account for core deformation effects. By fine-tuning the potential parameters, the experimental matter radius (rₘ = 3.296 ± 0.123 fm) was successfully reproduced, while the derived two-neutron separation energy S2n showed agreement with both theoretical and experimental values, confirming the weakly bound nature of 22C.

The ground state of 22C exhibits a dominant (2s₁ᐟ₂)² configuration with a weakly bound 1d₅ᐟ₂ orbital, yielding S2n = 0.265–0.656 MeV, consistent with previous results. An excited state displays an extended s-d hybrid structure (s-wave > 70%) and a double-peak spatial distribution. The constant ratio rₙₙ/r꜀,ₙₙ ≈ 1.91 between states suggests discrete scaling symmetry, supporting 22C as an Efimov candidate among Borromean nuclei.

Our results highlight the interplay between neutron-core interaction and three-body forces in shaping the properties of 22C. The consistency between our predictions and experimental data validates the three-body approach as a powerful tool for studying neutron-rich nuclei. Future work could explore dynamical effects of core excitation and the role of higher-order interactions in refining the description of 22C and similar dripline nuclei. These insights contribute to a deeper understanding of exotic nuclear structures and their connections to universal few-body phenomena.

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