Probing nuclear interactions à la Rutherford: insights on ⁴He from α scattering

Abstract

Nuclear interactions play a key role for the stability of atomic nuclei and stellar environments. Successful parametrization and models of these interactions, developed in the last decades, accurately reproduce all the proton and neutron scattering data, besides the properties of few-body nuclear systems. However, recent electron scattering results focusing on the first excited resonant state of ⁴He nucleus, reveal a puzzling situation suggesting potential gaps in our understanding of the nuclear phenomenology. Here, we report a study of such ⁴He resonance by ⁴He + ⁴He scattering featuring data of unprecedented sensitivity and state-of-art analyses of the spectral line shape together with a phenomenological reaction modeling that incorporates the same nuclear densities employed in electron-scattering studies. Our analysis of the full set of experimental observables yields a reasonable description within the framework of current nuclear-interaction physics, thereby highlighting the need for further advancing the modeling of few-body open quantum systems.

The first excited state of the 4He nucleus represents a long-standing puzzle in nuclear physics, with theoretical and experimental inconsistencies on the estimation of several features. Here, the authors report refined experimental measurements of 4He+4He scattering, opening potential pathways towards resolution.

Introduction

Ernest Rutherford’s scattering experiments with α particles, aka ⁴He nuclei in their ground state, led to the groundbreaking discovery of the atomic nucleus over a century ago¹, revolutionized our understanding of atomic structure and laid the foundation for modern nuclear physics. Building on this rich legacy, in this paper, we delve into the contemporary exploration of α scattering employing new technologies and methods developed since Rutherford’s time with the goal of shedding light onto an intriguing quantum mechanical problem involving the α particle itself.

The ⁴He nucleus, is one of the simplest and most stable nuclei, constituted by two protons and two neutrons. Its ground state features a spin-parity J^π = 0⁺, and it is commonly indicated as the $0_1^{+}$ state. Since its discovery², ⁴He has captivated the scientific community as it represents the most stable nuclear system, which makes it the seed of clustering in light- to medium-mass nuclei³ and of α-decay radioactivity in heavy nuclei—the very same used by Rutherford. ⁴He also plays a crucial role in astrophysics as a fundamental component of stellar matter and a vital element in stellar nucleosynthesis⁴.

The modeling of ⁴He in terms of 4 nucleons requires the solution of a quantum mechanical 4-body problem, where nuclear interactions are a key element. The task of solving for the ground state could be achieved already in 2001⁵ by several so-called ab initio methods⁶. Progress accomplished in the past two decades has been in increasing the precision of nuclear interactions (referred to as nuclear Hamiltonians), either in a phenomenological way^7,8, or by introducing the more systematic effective field theory (EFT) approach, particularly in the variant based on chiral symmetry^9,10. Today, ground-state properties of ⁴He, such as binding energies and radii, are correctly described at about the percent level^11,12, and even nuclei as heavy as ²⁰⁸Pb¹³ can be investigated by ab initio nuclear theory. However, the description of the ⁴He first excited state, a resonant state also featuring a spin-parity J^π = 0⁺ and therefore commonly indicated as the $0_2^{+}$ state, still represents a challenge. This state is, in fact, situated between the p + ³H breakup channel starting at the proton-emission threshold S_p = 19.813 MeV, and the n + ³He breakup channel demarcated by the neutron emission threshold S_n at 20.578 MeV.

The complication of computing such a state is intrinsic to its unbound resonant nature, as elucidated in a recent benchmark study¹⁴. The position E_r–typically indicated as the centroid-energy above the S_p threshold–and width Γ_r were calculated with various theories^14–20, obtaining quite different results depending on the method used and on the implemented nuclear force.

From an experimental perspective, determining the properties of the $0_2^{+}$ state is also challenging. Inelastic scattering studies on ⁴He targets using different projectiles, such as electrons^21–24, protons²⁵ and α-particles^26,27, have in the past measured the position E_r and width Γ_r of this state. A large spread was observed, with a variation of about 0.6 and 0.7 MeV for E_r and Γ_r, respectively, and with some data being inconsistent with the other (see Fig. 1 and evaluation from ref. ²⁸). Recently, the form factor F_M of the transition $0_1^{+}\to 0_2^{+}$, connected by an electric monopole operator, has attracted much attention in the nuclear physics community, see refs. ^16,19,20,29 and Supplementary Methods (1.2). As a matter of fact, calculations present stark differences, and none of them reproduces all the three known properties (E_r, Γ_r, and F_M) of the $0_2^{+}$ state. This is an unsatisfactory situation for what should be one of the best known nuclei. The properties of the first excited state of the α particle and their direct link to basic nuclear Hamiltonians are thus calling for further investigations.

Fig. 1. Measured centroid-energy E_r and width Γ_r of the $0_1^{+}$ resonance^21–27.

The E_r is defined with respect to the proton-emission threshold S_p  = 19.813 MeV. The quoted error bars for the experimental datum of the present work (red) are discussed in section “Line shape analysis”. The error bars of the other data points are given in refs. ^21–27.

Among the different probes which allow to populate the $0_2^{+}$ state, inelastic scattering is the ideal toolbox as it requires minimal rearrangement of the ⁴He nuclei. Nonetheless, notable differences arise depending on the projectiles employed in these studies. While electron scattering has the advantage of being perturbative, thus allowing for a direct connection of the cross section to the target properties, α scattering offers larger cross-sections, better state-selectivity and sensitivity to the density of the resonant state (see Supplementary Methods (1.1)), making it very effective for identifying resonant features, such as those of the $0_2^{+}$ state in ⁴He. On the other hand, in the α − α scattering the many-body initial (ISI) and final state interactions (FSI) need to be accurately modeled to get detailed information of the involved states. Although this procedure can affect the overall uncertainty of the data analysis, modern nuclear reaction methods are able to describe the cross section with enough accuracy to disentangle different nuclear structure theories. In this framework, the resulting uncertainties can be significantly reduced by constraining the ISI and FSI with supplementary measurements of other observables such as elastic scattering and total reaction cross section.

The main challenge in the description of the $0_2^{+}$ state is its location in the continuum, where other excited states may overlap and/or interfere. Theoretical calculations¹⁷ show that below 26 MeV of excitation energy (i.e., about 6 MeV above the S_p threshold) the relative angular momentum of the fragments is limited to L = 0 (S-wave) and L = 1 (P-wave). Nonetheless, a number of different partial waves, namely the ¹S₀, ³P₀, ³P₁, ³P₂, may contribute. Here, the molecular notation ^2S+1L_J is adopted to label the ⁴He states, where S is the quantum number relative to the sum of the spins of the two fragments (namely n and ³He or p and ³H), and J(L) the total (orbital) angular momentum of the 4-body system. In addition, due to the t = 1/2 isospin of the two fragments, the ⁴He excited states can either have isospin T = 0 (isoscalar) or T = 1 (isovector) quantum number. While in electron scattering all of these partial waves are involved, in α scattering only isoscalar T = 0 states with S = 0, 1 and J = L can be populated. Therefore, besides the sought for ¹S₀ resonance, only the ³P₁ partial wave located at 24.25 MeV contributes, making α scattering advantageous for our purposes.

Herein we report on a new study of the elastic and inelastic α + α scattering at 53 MeV incident energy. The $0_2^{+}$ resonance and a small portion of the spectrum above the resonance energy are investigated while keeping low the number of open reaction channels. In particular, the α + α → α + p + ³H and the α + α → α + n + ³He channels are experimentally accessed for the first time by exclusive experiments, making it possible to reconstruct the ⁴He excitation energy with unprecedented low background, high resolution and selectivity. In this work, we extract the energy and width of the $0_2^{+}$ state from a modern analysis of a new α − α experiment (red point in Fig. 1), achieving the highly desired precision and providing a benchmark for future calculations.

While significant progress has recently been made in the ab initio treatment of nuclear reactions^30–35, no existing method can yet describe the inelastic α – α scattering process as an eight-body problem in the continuum. In this work, we model the process as a two coupled-channel system composed of the ground state $0_1^{+}$ and the excited state $0_2^{+}$ (treated as a bound state), with a $0_1^{+}\to 0_2^{+}$ coupling channel. This approach is rooted in the well-established double-folding potential reaction theory. However, instead of using phenomenological densities as done in the traditional theory, we employ here state-of-the-art ab initio densities that stem from different nuclear Hamiltonians, the same used in analyzing electron scattering data. This represents a significant advance, and constitutes a solid framework for a first analysis of the α − α scattering data presented here, that incorporates the microscopic physics presently available. Our analysis of the line shape of the $0_2^{+}$ state and of the elastic and inelastic differential cross sections, indicates that densities from chiral EFT lead to a better agreement with experiment. However, space is left for an improvement of nuclear reaction models towards a full ab initio description of the process.

Results

Experimental details

We performed an α-scattering experiment at the K800 Superconducting Cyclotron of the Istituto Nazionale di Fisica Nucleare—Laboratori Nazionali del Sud (INFN-LNS) in Catania (Italy), using a beam energy of 53 MeV. Thin metallic films with implanted ⁴He atoms were used as targets. The ⁴He ejectiles emitted at forward angles were detected by the MAGNEX magnetic spectrometer³⁶ while the ³H and ³He fragments from the decays of the populated ⁴He excited state (⁴He^*) were detected by the OSCAR array of solid state telescopes³⁷. A schematic view of the set-up is given in Supplementary Fig. 1. The ejectiles were identified event by event in atomic number (Z), mass (A) and charge (see Methods “Experimental set-up and particle identification” and Supplementary Fig. 2). The trajectories of the reaction ejectiles were then reconstructed and their momenta deduced at the target position (see Methods “Energy spectra”).

This set-up allows, on the one hand, to characterize the $0_2^{+}$ resonance with unprecedented precision, and on the other hand, together with theoretical modeling, to provide stringent quality checks for ab initio nuclear Hamiltonians.

The resonance characterization

In the inelastic process, the breakup channels p + ³H and n + ³He are experimentally accessed for the first time in hadronic experiments, making it possible to reconstruct the ⁴He excitation energy with very low background, high resolution and selectivity. Figure 2 shows the reconstructed ⁴He energy spectra from the p + ³H (Fig. 2a) and n + ³He (Fig. 2b) exit channels. In the p + ³H the resonant $0_2^{+}$ state is clearly populated over the non-resonant background, presenting an overall signal to background ratio of ≈12. We describe the p +  ³H channel including: (i) the ¹S₀ partial wave, featuring the resonant $0_2^{+}$ state and the non-resonant breakup; (ii) the ³P₁ component, through the 1⁻ resonant state at 24.25 MeV (E_r =  4.437 MeV)³⁸; (iii) the pick-up breakup processes α + α → ³H +  ⁵Li → α + p + ³H centered at 21.78 MeV (E_r = 1.967 MeV), as it emerges from the results of transfer reaction simulations. Although this structure could partially overlap with the $0_2^{+}$ state, we estimate an overall contribution of less than 0.2% in the region of the resonance between 19.72 and 20.84 MeV, which gives a negligible uncertainty in the $0_2^{+}$ cross section once we subtract its yield.

Fig. 2. Energy spectra for the p + ³H and n +  ³He channels.

a Experimental inelastic scattering spectrum (black dots) of the p + ³H channel compared with the sum of three components (black dotted): ¹S₀ (red solid), ³P₁ resonance at 24.25 MeV (E_r = 4.437 MeV) (red dashed) and α + α → ³H + ⁵Li  → α + p + ³H pick-up breakup transfer (gray hatched area). b The n + ³He channel, dominated by the simulated (black dotted) ³P₁ resonance at 24.25 MeV (E_r = 3.672 MeV). In both panels, error bars refer to the statistical and efficiency estimation uncertainties. The E_r (upper horizontal axis in top) refers to the centroid-energy above the S_p (a) and S_n (b) threshold, respectively.

For the n + ³He channel we account for: (i) the ¹S₀ non-resonant breakup (the $0_2^{+}$ resonance is below the threshold for this channel); (ii) the ³P₁ resonance at 24.25 MeV (E_r = 3.672 MeV above the S_n threshold); (iii) the α + α → ³He + ⁵He  → α + n +  ³He pick-up breakup reaction.

In the p + ³H channel, the interference of the $0_2^{+}$ resonance with the ¹S₀ non-resonant continuum and the presence of the tail of the ³P₁ partial wave clearly creates an asymmetry in the line shape. A powerful way to study such an interference was originally proposed by U. Fano³⁹ and progressively extended in the realm of modern physics^40–43. Recently, it was used to characterize low-lying resonances in the ¹⁰Li spectrum⁴⁴, inspiring us to use it also in the present study, leading to the very precise datum shown in Fig. 1. Upon implementing the Fano analysis (see details in Methods, section “Line shape analysis”), E_r is found at 0.50 ± 0.02 MeV (corresponding to an excitation energy on top of the ground state of E_x = 20.31 ± 0.02 MeV). Including the uncertainties, our result is found within the overlapping range of values from previous studies from electron scattering (0.29– 0.45) and hadronic scattering (0.47–0.48). Remarkably, the maximum of the observed peak is at E_r = 0.59 (E_x = 20.4 MeV), which is about 0.1  MeV higher than the actual resonance centroid-energy as a result of distortion in the line shape due to the interference. The extracted width Γ_r is 0.29 ±  0.04 MeV, in agreement with electron-scattering studies but not with the α scattering analysis of ref. ²⁷, reporting 0.89 MeV. Instead, assuming a symmetric Lorentzian shape E_r = 0.59 ± 0.04 and Γ_r = 0.33 ± 0.05 are found. Thus, a systematic shift of about 90 keV for the resonance centroid and about 40 keV for the width is observed when the Lorentzian shape is adopted.

Angular-dependent cross section

Here, we perform new calculations for the elastic and inelastic scattering in the framework of the coupled-channel quantum scattering theory and compare them to our experimental data. The ISI and FSI are built in a semi-microscopic approach by folding the nucleon–nucleon interaction with the $0_1^{+}$ and $0_2^{+}$ densities as well as the $0_1^{+}\to 0_2^{+}$ transition density obtained from four different ab initio calculations, all treating the $0_2^{+}$ resonance as a bound state. The obtained ISI and FSI were validated through comparisons with elastic scattering and reaction cross-section data (see Methods, section “Assessment of the reaction modeling uncertainties” for details). More details on the reaction modeling uncertainties can be found in the section “Assessment of the reaction modeling uncertainties” and Supplementary Figs. 5 and 6. First, we consider densities from refs. ^16,45 and ref. ²⁹, which uses a chiral EFT Hamiltonian based on a two-body force at next-to-next-to-next-to leading order⁴⁶, and a different parametrization of the three-body force at next-to-next-to-leading order^47,48. We will display cross sections obtained from these two sets of densities in one band encompassing the differences and name it “the chiral EFT” family. Second, we consider densities obtained from calculations with two simplified Hamiltonians: one from ref. ¹⁵, where the three-body force is purely central, and one from ref. ²⁰, where the Hamiltonian is based on the SU(4) isospin symmetry. We will display cross sections obtained from these two sets of densities as a band encompassing the differences and denote it as the “simplified Hamiltonian” family. These two bands serve as an uncertainty estimate quantifying the dependence of the cross section on the input densities.

Figure 3 shows the experimental angular distributions of the differential cross section in comparison to theoretical calculations. On panel (a), we have our experimental data obtained at a beam energy of 53 MeV (available also in Supplementary Data 1), while on panel (b), we also show previously published data at a beam energy of 64 MeV²⁶. In both cases, we consider only the contribution of the $0_2^{+}$ resonance extracted, angle by angle, from the peaked spectral shape centered at 20.4 MeV, after a polynomial modeling of the background underneath. This accounts, on average, for the ³P₁ as well as the ¹S₀ non-resonant and interference contributions.

Fig. 3. Inelastic scattering angular distributions.

Inelastic scattering differential cross section for the $0_2^{+}$ resonance as a function of the scattering angle in the center-of-mass frame (lower horizontal axis) or the momentum transfer (upper horizontal axis) at 53 MeV (a) and 64 MeV²⁶ (b) incident beam energy. The experimental data (red stars) are compared with the theoretical results obtained using the (blue band) chiral effective field theory densities (solid¹⁶ and dashed²⁹) and the (gray band) simplified Hamiltonian densities (solid²⁰ and dashed¹⁵). The error bars correspond to the combination of uncertainties coming from the solid angle and detection efficiency estimation, the uncertainties in the beam current and the target thickness determination, the statistical error and the background subtraction. See text for details.

A fair agreement is found between the data and the calculations adopting the chiral EFT densities, both at 53 and 64 MeV. The pronounced minimum present in the data at 64 MeV at θ_c.m. ≈ 18^∘ (corresponding to q ≈ 0.85 fm⁻¹) is well reproduced. However, no clear sign of the minimum observed at 53 MeV at θ_c.m. ≈ 23^∘ is present in the calculations. Different from chiral EFT calculations, results from the simplified Hamiltonian family deviate from the data at 53 MeV as well as at small angles for the 64 MeV energy. At large angles for 64 MeV, all theories predict a minimum at θ_c.m. ≈ 65^∘, but the data were significantly less accurate there.

Confronting the experimental inelastic cross sections at 53 and 64 MeV incident energy, a marked difference appears, especially in the region at low momentum transfer. Indeed, due to the high excitation energy, the population of the $0_2^{+}$ state also determines a sudden change of the kinetic energy and wavelength for the colliding systems. This unveils an interesting situation where a small change in the projectile energy generates a substantial change in the diffraction pattern, paving the way for a strong sensitivity of this process to fine details of the scattering dynamics, namely the spatial distributions of the ⁴He $0_1^{+}$ and $0_2^{+}$ states. The pronounced first minimum observed at ≈0.85 fm⁻¹ for the cross section at 64 MeV energy is damped at 53 MeV and shifted at ≈0.95 fm⁻¹, while it almost disappears in the calculations.

An expansion of the calculated inelastic scattering cross sections in terms of the angular momentum l of the relative motion of the outgoing nuclei allows to get insight on this situation. At 64 MeV, the cross section is mainly fed by the l = 4 partial wave, with minor roles for the l = 0 and l = 2 and negligible contributions for l ≥ 6. The l = 4 partial wave features a pronounced minimum at about 20^∘, common to all calculations as well as to the data. Instead, the l = 2 contribution is the leading one at 53 MeV, also due to the less available momentum for the scattering waves, with a minimum at about 55^∘ and a growing trend down to zero degrees. In the case of the simplified Hamiltonian, the same analysis still indicates a similar behavior at both energies, although the l = 4 contribution at 53 MeV is more relevant, and a minimum starts to appear at forward angles (≈15^∘). As a consequence, the data at 53 MeV suggest the need for a mild redistribution of the cross-section partial waves toward an enhanced l = 4 contribution. In terms of radial distribution of the cross section, this shift toward larger l reflects the need for more density overlap of the two colliding nuclei at larger distances (R ≳ 4 fm). This is significantly larger than the size of ⁴He ground state radius (r(⁴He_g.s.) = 1.7 fm⁴⁹), typically explored in electron-scattering experiments, indicating that with α scattering we are probing more peripheral regions of the density. In particular, while the latter is sensitive only to the transition density, α–α scattering is particularly sensitive to the density of the $0_2^{+}$ excited state, for which the prediction of the four different ab initio calculations differ substantially.

We find that the best result is given by the chiral EFT density by ref. ²⁹ and prove that inelastic α–α scattering poses stringent tests to nuclear structure models, emphasized by the strong sensitivity of the diffractive patterns in the cross-section angular distributions to the details of the resonance density profile.

Discussion

Our study allows us to give insight into the nature of the $0_2^{+}$ resonance of ⁴He and provides strong constraints for present and future models of the nuclear Hamiltonian. The adopted experimental technique allowed for an unprecedented clean and background-free spectral response. Our results within the Fano theory of resonances give a centroid-energy of E_r = 0.50 ± 0.02 MeV and a width Γ_r = 0.29 ± 0.04 MeV, providing the most precise ever measurements of the $0_2^{+}$ resonance, in disagreement with previous α-scattering data^26,27 but in agreement, as for the width, with the latest electron-scattering experiment by ref. ²⁴. Since the Fano analysis indicates a significant interference between the $0_2^{+}$ resonance and the underlying non-resonant continuum, employing a similar analysis on other scattering data would be interesting.

The comparisons of the experimental cross sections with calculations based on coupled channels quantum scattering theory, with ab initio densities used in input, has allowed to set a robust test of ⁴He wave functions of the ground state $0_1^{+}$, the $0_2^{+}$ resonance, and the $0_1^{+}\to 0_2^{+}$ transition density. Interestingly, we find that densities from chiral EFT better describe the α scattering data. Due to the importance of the $0_2^{+}$ density in the α − α scattering calculation, we conclude that while the densities from refs. ^15,20 well reproduce the transition form factor probed by electron scattering, they do not well describe the density of the excited state probed by the α scattering. The chiral densities of ref. ²⁹ provides the best overall agreement with data from both electron and α scattering, despite a slight deviation from the diffraction pattern observed in α scattering at 53 MeV.

At the same time, the fact that the interaction used in ref. ²⁹ predicts a resonance position and width that deviates from the experimentally extracted values (see Fig. 1) highlights a deeper challenge. Indeed, establishing a transparent link between the measured cross section in an open quantum system, such as α−α scattering and the underlying resonance in one of its sub-systems, remains a challenging problem in quantum mechanics. Despite notable advances in reaction theory, a fully quantitative framework applicable to inelastic α−α scattering is still lacking. Our analysis—based on a robust yet necessarily approximate and not fully consistent reaction model—demonstrates that combining α-scattering with electron-scattering substantially tightens the sensitivity to the underlying microscopic physics. These results not only expose the current theoretical limitations but also clearly define the direction for future progress, including the long-term goal of constructing an ab initio optical potential for light nuclei. Ultimately, the high-precision data reported here, together with the insights gained from this study, provide a compelling impetus for advancing theoretical approaches to inelastic α−α scattering.

Methods

Experimental set-up and particle identification

Two experiments were performed at the MAGNEX facility³⁶ of the Istituto Nazionale di Fisica Nucleare—Laboratori Nazionali del Sud (INFN-LNS) in Catania.

The first experiment was dedicated to the elastic scattering measurement in the angular range 6.5^∘≤θ_Lab≤ 45^∘, corresponding to 13^∘≤θ_c.m.≤ 90^∘ in the center of mass reference frame. The ⁴He beam was accelerated by the K800 Superconducting Cyclotron at 53 MeV (13.2 MeV/u) incident energy and impinged on a solid target, built by the implantation of ⁴He atoms on a ¹⁸¹Ta sheet. The effective thickness of the ⁴He component of the target, measured by the Rutherford back-scattering (RBS) technique, was 2.5 ± 0.13 μg cm⁻² (3.7 × 10¹⁷ atoms cm⁻²) while the thickness of ¹⁸¹Ta was 950 ±  10 μg cm⁻² (3.2 × 10¹⁸ atoms cm⁻²). Residual impurities of oxygen (14.6 ± 0.4 μg cm⁻² (5.5 × 10¹⁷ atoms cm⁻²)) were observed in the RBS analyses. The beam current was measured by a Faraday cup (FC) mounted 15 cm downstream of the target. The solid angle of MAGNEX was 50 msr, defined by four 1-mm-thick tantalum slits located 25 cm downstream of the target. The α-particle ejectiles were detected by the MAGNEX focal plane detector (FPD)⁵⁰, which consists of a gas-filled tracker followed by a wall of 60 (5 × 7 cm² area, 1-mm-thick) silicon pad detectors.

In the second experiment, conducted at the same facility, the inelastic scattering was studied by performing a coincidence measurement between the ⁴He ejectiles and one of the breakup fragments emitted by the residual ⁴He (³H or ³He). Our study was designed to explore a significant portion of the phase space for both the ³H + p and ³He + n decay modes. The experimental setup is illustrated in Supplementary Fig. 1. Additional elastic scattering data were collected under experimental conditions similar to the first experiment to cross-check. The ⁴He beam was accelerated at 53 MeV incident energy and impinged on a solid target built by the implantation of ⁴He atoms on a ²⁷Al sheet. The replacement of the tantalum backing with an aluminium one was intended to further reduce the background observed in the first experiment, attributed to beam scattering on tantalum. The effective thickness of ⁴He was 1.3 ± 0.1 μg cm⁻² (1.9 × 10¹⁷ atoms cm⁻²) while the thickness of ²⁷Al was 210 ± 2 μg cm⁻² (4.6 × 10¹⁸ atoms cm⁻²). Residual impurities of oxygen (2.7 ± 0.1 μg cm⁻² (1.0 × 10¹⁷ atoms cm⁻²)) and carbon (0.2 ± 0.02 μg cm⁻² (1.2 × 10¹⁶ atoms cm⁻²)) were observed in the RBS analyses. The beam current was measured by an FC mounted 15 cm downstream of the target. The measurement was repeated with an ancillary target composed of aluminium, oxygen and carbon in the appropriate proportions for background estimation. The ⁴He ejectiles were momentum analysed by the MAGNEX spectrometer, whose optical axis was set at θ_opt = 6.6^∘, spanning the range of detection angles between 2^∘≤θ_Lab≤ 12^∘ in the laboratory reference frame. The momentum acceptance of the spectrometer (≈24%)³⁶ allowed the measurement of the ⁴He excitation energy (E_x) up to 4.0 MeV above the S_p threshold, covering a significant part of the available energy spectrum, which extends up to 6.7 MeV above S_p at the beam energy of the experiment. The α-particle ejectiles were detected by the MAGNEX FPD, while the ³H and ³He fragments were detected by the OSCAR solid state telescope (5 × 5 cm² area)³⁷. This device consists of two detection stages: a Single Sided Silicon Strip Detector (SSSSD), which measures the energy loss ΔE, followed by 16 Silicon pads (arranged in a 4 × 4 mode) for measuring the ions' residual energy E_resid. The thicknesses of 20 μm for the ΔE and 300 μm for the E_resid allow the unambiguous identification of light reaction fragments with energies of a few MeV. In particular, the energy range of ³H and ³He particles detected by OSCAR was 2.5–9.0 MeV and 5.0–11.2 MeV, respectively. For both of them, a large part of their phase space was explored. The OSCAR system was mounted inside the MAGNEX scattering chamber, 15 cm far from the target, subtending an angular range between 19^∘ and 38^∘ with an angular resolution of ~1^∘. The unambiguous particle identification is illustrated in Supplementary Fig. 2. Details on the particle identification technique can be found at refs. ^37,51.

Details of data reduction

Energy spectra

The trajectories of the ⁴He ejectiles detected at the MAGNEX FPD were reconstructed, and the momentum vector was deduced at the target position, adopting the methodology described in ref. ⁵². The missing mass technique, with the use of relativistic kinematics relations, was adopted to get ⁴He energy spectra. The inelastic scattering events were further characterized by the coincident measurement of ³H or ³He fragments by the OSCAR telescope (see section “Cross section determination”). The obtained energy resolution, in the region of the ⁴He($0_2^{+}$) state, ranges from δ_E ≈ 80 keV at small scattering angles to δ_E ≈ 100 keV for the largest explored angles. This was mainly determined by the spectrometer response, the beam energy spreading and the kinematic energy dependence on the scattering angle. A weighted energy resolution of 86 keV was estimated when summing the spectra of all explored angles.

Line shape analysis

An important aspect in analyzing the ⁴He($0_2^{+}$) resonance is to include the contribution of the L > 0 resonant and non-resonant strengths and the possible interference between the $0_2^{+}$ resonance with the underlying non-resonant continuum featuring the same 0⁺ quantum numbers. A powerful way to study this topic was proposed by the ref. ³⁹, stemming from the asymmetric line shape (modeled by the Fano function) generated by such interference.

An additional feature is the distortion of the resonance line shape due to the proximity of the particle emission threshold⁵³. Since the strength is zero below the threshold, an asymmetric line shape is always found, although the actual shape depends also on the multipolarity. Different from the Fano asymmetry, the threshold effect does not depend on the dynamic mechanism populating the resonance, thus providing a stable distortion of its line shape. The stronger the distortion, the larger is Δ_r = Γ_r /(E_r - E_t), the ratio between the resonance full width at half maximum (Γ_r) and the distance of the resonance center (E_r) from the threshold (E_t), especially when Δ_r exceeds unity. In the p + ³H channel, the threshold at S_p = 19.813 MeV can affect the line shape for both the ¹S₀ (Δ_r ≈ 0.7) and the isoscalar ³P₁ (Δ_r ≈ 1.4) resonances, with a smaller impact on the former thanks to its smaller width. Ideally, both the interference with the non-resonant continuum and the threshold effect should be included in the resonance line shape analysis. However, to our knowledge, such a theoretical framework is not available to date and approximate schemes are needed.

Our assumption for the ¹S₀ resonance is to account only for the Fano asymmetry, since Δ_r < 1. In reverse, we assume only the threshold effect for the isoscalar ³P₁, considering the small L = 1 non-resonant strength in the explored excitation energy region¹⁷. Figure 2a shows the best fit obtained adopting a Fano function, convoluted with experimental energy resolution, as a model of the observed structure in the region of the $0_2^{+}$ state. In this framework the cross section in the region of the resonance is given as

$$\sigma ={\sigma }_{cont}\frac{{∣q_F+ϵ∣}^2}{1+{ϵ}^2},$$ 1

where σ_cont is the inelastic cross section of the non-resonant continuum and ϵ = 2(E − E_r)/Γ_r, where E_r is the resonance center and Γ_r its width. The line shape is controlled by the Fano parameter q_F which, apart from a phase factor, is determined by the relative weight of the resonant and non-resonant amplitudes. An interesting aspect of the Fano theory is that the q_F parameter depends on the dynamics of the resonance population, while the resonance center and width do not change. When q_F ≫ 1, the line shape becomes progressively more symmetric, approaching the Lorentzian function when q_F goes to infinite. For this analysis, we find q_F = 2.54 ± 0.07 when we consider the spectrum in the fully explored angular region, which calls for a significant effect from the interference discussed here. Further analyses of the line shape in smaller angular slices of the measured cross section do reveal that the resonance center and width are stable, within the quoted uncertainties.

The quoted uncertainties for the resonance energy (±0.02 MeV) and width (±0.04 MeV) include the uncertainty in the energy threshold effect on the line shape model, the deconvolution of the spectrum in the three components (as shown in Fig. 2), the contribution of the finite bin size, the accuracies in energy or/and angle of the delivered beam, the spectrometer and the ray-reconstruction procedure as well as the variation of the experimental resolution as a function of angle (see Methods “Energy spectra“).

Cross-section determination

The elastic scattering yields were determined with angular steps of 0.5^∘ in MAGNEX over the explored angular range. The differential cross section was extracted through a careful determination of the experimental yield, the beam flux, the target thickness, the explored solid angle and the detection efficiency. For the inelastic scattering process, exclusive yields were determined with angular steps of 2^∘ in MAGNEX, combined with the whole OSCAR system, to reach a compromise with the reduced yield of the coincidence measurement. The background due to aluminium, oxygen and carbon impurities in the target, as well as that generated by spurious signals in the detectors, was negligible in the coincidence spectra, leading to an overall signal to instrumental background ratio ranging between 15 and 40 at the different detection angles. Nonetheless, this background was experimentally determined using data collected with the ancillary target and subtracted.

The efficiency of our detection system in the coincidence measurement was calculated by the MULTIP Monte Carlo simulation algorithm⁵⁴, taking into account the MAGNEX intrinsic efficiency⁵⁵. The exclusive yields, corrected for the detection efficiency, were transformed to laboratory double differential cross sections, taking into account the areal density of the ⁴He scattering centers, the beam current, the MAGNEX and OSCAR solid angles, the last one computed by a Monte Carlo simulation.

Calculation methodology

The theoretical cross-section angular distributions were calculated in the coupled- channel quantum scattering theory framework. The nucleus–nucleus potential was obtained by double-folding the nucleon–nucleon interaction with densities from four different ab initio theories. A first group consisted of two theories of the chiral EFT family by refs. ^16,45 and ref. ²⁹ while, a second one, by theories on which the densities were obtained from calculations with two simplified Hamiltonians by ref. ¹⁵ and by ref. ²⁰. The ${\rho }_{0_1^{+}}$ and ${\rho }_{0_2^{+}}$ densities for the $0_1^{+}$ and $0_2^{+}$ state respectively, as well the transition density ${\rho }_{0_1^{+}\to 0_2^{+}}$ are presented in Supplementary Fig. 3a–c. One can find more details on the reaction theory, the double-folding model and the densities in Supplementary Methods (1.1), on the adopted potentials in the reaction theory in Supplementary Methods (1.1.1) and on the used ab initio theory in Supplementary Methods (1.2). Finally, in Supplementary Methods (1.2.1), one may see how we successfully handle the $0_2^{+}$ lying in the continuum as an effective bound state. The reliability of the process is illustrated in Supplementary Fig. 3d. The calculated centroid-energy, assuming the $0_2^{+}$ as an effective bound state, is E_r = 0.5 ± 0.1 MeV.

Assessment of the reaction modeling uncertainties

As for the uncertainties in the reaction modeling, the main source comes from limitations in the adopted model space and uncertainties in the determination of the ISI and FSI. Indeed, our coupled-channel reaction scheme includes explicitly only the elastic scattering and the inelastic excitation of the $0_2^{+}$ resonance. Other inelastic channels, although kinematically accessible, are accounted for on average through a complex polarization nucleus–nucleus potential. A phase-shift analysis with the adopted potentials was performed as described in Supplementary Methods (1.1.1), whose results are illustrated in Supplementary Fig. 4. The availability of a rich and accurate dataset at ⁴He + ⁴He interaction at different energies and for different partial waves^56–65 gave us the chance to fine-tune the potential and evaluate a realistic uncertainty band. In particular, the range of the band was set in order to allow variations of the ∣S∣ matrix within an interval Δ∣S∣ = ±0.1 around the best value. The goodness of this approach is first evaluated through comparisons of the calculated elastic scattering angular distribution cross sections at E_Lab = 53 MeV with our data, as shown in Supplementary Fig. 5a, presenting an overall good agreement. The existing elastic scattering data at 53.4 MeV from ref. ⁶⁵ are also in good agreement with the new dataset. Our theoretical framework has been adopted to analyze also the existing elastic scattering data at E_Lab = 64 MeV from ref. ⁶⁵. Remarkably, the theoretical results are in very good agreement with these data, as visible in Supplementary Fig. 5b. Overall, the results point to a strong reliability of our phase-shift analysis and consequently of our ISI. Further comparisons were performed between the total reaction cross section (${\sigma }_{\mathrm{tot}}^{\mathrm{theo}}$), from coupled-channel calculations with that extracted from suitable experimental data and reliable fusion reaction calculations (${\sigma }_{\mathrm{tot}}^{\exp }$)^66–68. The σ_tot observable is notoriously related to the imaginary part of the ISI. The result of this comparison is presented in Supplementary Table 1, presenting a very good agreement between experimental (${\sigma }_{\mathrm{tot}}^{\exp }$) and theoretical (${\sigma }_{\mathrm{tot}}^{\mathrm{theo}}$) reaction cross sections, although the error bars are sizable in both cases. The effect of the nucleon–nucleon interaction was found to be negligible by changing the adopted Brazilian Nuclear Potential (BNP) with the Hamiltonians used in the ab initio densities.

Finally, the effect of the uncertainty band in the ⁴He + ⁴He interaction in the inelastic angular distribution cross sections is presented in Supplementary Fig. 6.

Author contributions

F.C., S.B., D.C., M.Cav., C.A., G.O., and L.C. proposed the experiment. F.C., V.S., D.C., M.Cav., and I.L. performed the experimental setup with a contribution from H.-W.B., S.C., C.C., M.Cin., O.S., A.S., and M.Vig. F.C., V.S., D.C., M.Cav., I.L., C.A., G.A.B., S.C., M.Cic., I.C., C.F., A.H., M.H., Y.K., O.S., A.S., D.T., and A.Y. performed the data taking. V.S. performed the data reduction with a contribution from F.C., D.C., M.Cav., M.F., M.Cic., M.Cin., D.D., A.H., T.M., and O.S. V.S., and F.C. performed the line shape analysis. L.C. performed coupled-channel cross- section calculations with a contribution from F.C., V.S., and Y.K. S.B., G.O., M.Viv., and A.K. collected or/and calculated densities from ab initio calculations. F.C., V.S., S.B., L.C., G.O., and M.Viv. wrote the manuscript. All the authors have revised the manuscript.

Peer review

Peer review information

Nature Communications thanks the anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

The project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 714625). S.B. acknowledges support by the Deutsche Forschungsgemeinschaft (DFG) through the Cluster of Excellence “Precision Physics, Fundamental Interactions, and Structure of Matter” PRISMA⁺ EXC 2118/1 (Project ID 39083149) and through CRC1660: Hadrons and Nuclei as discovery tools (Project No. 514321794). L.C. acknowledges financial support by Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) Proc. No 2019/07767-1, Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) Proc. No 302072/2022-5, and project INCT-FNA Proc. No 464898/2014-5.

Data availability

Relevant data supporting the key findings of this study are provided in a Supplementary file with this paper (Supplementary Data 1). Any additional data are available from the corresponding author upon request.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-74761-y.