The measurement of 12C+28Si fusion down to about 42 nb (Stefanini et al., Phys. Lett. B 872 (2026) 140084) gives the cleanest evidence so far of low-energy fusion hindrance in a light heavy-ion system — the class from which one must extrapolate to the astrophysically relevant carbon- and oxygen-burning reactions. It leaves a precise, untouched open problem with three linked components: (i) the small oscillations of the logarithmic slope systematically observed above the hindrance threshold in 12C+28Si and the neighboring systems 12C+24,26Mg and 12C+30Si have no accepted explanation, and the candidate mechanisms (elastic alpha-transfer, quasi-molecular resonances, low compound-nucleus level density, other reaction dynamics) remain undiscriminated; (ii) it is unknown whether the cross section at still lower energies continues to follow the one-dimensional no-coupling tunnelling limit or drops below it once ion-ion distances shrink inside the nominal touching point, where the two-body potential picture itself fails; (iii) the reaction-rate and nucleosynthesis consequences of a confirmed hindrance in light systems remain unquantified with the updated threshold systematics.

Background

Heavy-ion fusion below the Coulomb barrier is measured through the cross section for forming a compound nucleus — a hot, equilibrated system that decays by evaporating protons, neutrons and alpha particles, leaving detectable evaporation residues. Near the barrier (the energy hill of the ion-ion potential set by the balance of nuclear attraction and Coulomb repulsion) the cross section is strongly enhanced over simple one-dimensional tunnelling by channel couplings: surface vibrations and rotations of the two nuclei mix barriers of different heights, an effect described by coupled-channels (CC) calculations. In many medium-mass and heavy systems, however, progressively deeper below the barrier the measured cross sections fall below the expectations of standard CC calculations and even approach the no-coupling one-dimensional tunnelling limit of the same potential. This is called low-energy fusion hindrance; it is diagnosed experimentally by the logarithmic derivative L(E)=d[ln⁡(σ(E)E)]/dEL(E)=d[\ln(\sigma(E)E)]/dE increasing as the energy decreases, equivalently by the astrophysical S-factor S(E)=σ(E)Ee2πηS(E)=\sigma(E)Ee^{2\pi\eta} (with η\eta the Sommerfeld parameter, which strips off the dominant Coulomb-tunnelling energy dependence) developing a maximum.

Hindrance was discovered in the system 60Ni+89Y by Jiang et al. (2002) and systematized across medium-mass systems; the review by Jiang et al. (Eur. Phys. J. A 57 (2021) 235) documents it as a still-open theoretical challenge. Proposed microscopic explanations include the adiabatic model of Misicu and Esbensen, in which the coupling to intrinsic motion adjusts adiabatically at low collision velocity and effectively shallows the ion-ion potential (developed systematically by Ichikawa, 2015), and the time-dependent Hartree-Fock demonstration by Simenel et al. (2017) that the Pauli exclusion principle (antisymmetrization of nucleon wave functions) resists the overlap of the two nuclear densities, making the barrier thicker and higher. For heavy systems a Langevin dynamical mechanism based on early neck formation has also been proposed (Amano et al., 2023).

In light systems — those of astrophysical interest such as 12C+12C, 12C+16O, 16O+16O — the situation is unsettled. Their fusion Q-values are positive, so an S-factor maximum is not algebraically required (Jiang et al. 2006 systematics); the maxima, where seen, are broad and obscured by pronounced resonant oscillations. Those oscillations have competing attributions: elastic alpha-transfer for 12C+16O, quasi-molecular resonances (long-lived cluster-type configurations in the entrance channel), or, for 12C+12C, the low level density of the compound nucleus 24Mg (Jiang et al., PRL 110 (2013) 072701). The controversy is live: Jiang et al. (PRC 75 (2007) 015803) concluded from logarithmic-slope fits that the 12C- and 16O-induced cross sections should drop rapidly at astrophysical energies, while the re-examination by Uzawa and Hagino (2025), releasing a fixed exponent in the same fit, finds no hindrance within error bars for 12C+12C and 12C+13C; a fully microscopic resonating-group calculation by Descouvemont (2026) finds a low-energy S-factor decrease supporting hindrance for 12C+12C. Because direct measurements at stellar energies remain out of reach, the field strategy (Jiang et al., PRC 79 (2009) 044601) is to measure slightly heavier systems where hindrance is observable and extrapolate downward.

In this strategy the family 12C+24,26Mg and 12C+30Si was measured (Montagnoli et al. 2018; Montagnoli et al. 2022 for 12C+24Mg; Stefanini et al. 2023, whose title already pairs 'Hindrance and oscillations'), and the threshold systematics for light systems was recently updated using 16O+48Ca (Stefanini et al., PRC 111 (2025) 064620), predicting higher hindrance thresholds in the astrophysical region. A recent measurement (Stefanini et al., Phys. Lett. B 872 (2026) 140084) adds 12C+28Si: using an electrostatic beam separator and particle-gamma coincidences (AGATA gamma-ray array plus segmented silicon detectors), fusion cross sections were measured from 150 mb down to about 42 nb. CC calculations with a Woods-Saxon potential reproduce the data above roughly 0.1 mb; below that, hindrance sets in (around Ecm≃10.1E_{cm}\simeq10.1 MeV, σ≃15 μ\sigma\simeq15~\mub), the CC results overestimate, and the cross sections approach the one-dimensional no-coupling tunnelling limit — suggesting, as the adiabatic model predicts, that the coupling strengths gradually vanish. The paper leaves open: the origin of the small L(E)L(E) oscillations above the hindrance threshold, which are systematic across the whole family but 'still lack a realistic explanation'; the behavior below the measured energies, where the significance of a two-body potential becomes questionable once the ion-ion distance is smaller than the touching point of the two densities; the still-large error bars of the lowest points; and the stellar consequences, since Gasques et al. (2007) showed with the older systematics that hindrance could significantly modify carbon and oxygen burning in late stellar stages, but no quantification exists for the updated light-system thresholds.

Problem Statement

The problem, as left open by the 12C+28Si deep sub-barrier measurement (Stefanini et al., Phys. Lett. B 872 (2026) 140084), is to establish and explain low-energy fusion hindrance in light heavy-ion systems, using 12C+28Si as the benchmark where hindrance is cleanly observed. It comprises three linked components serving this single objective.

(1) Origin of the slope oscillations. Identify the microscopic mechanism that generates the small oscillations of the logarithmic derivative L(E)=d[ln⁡(σE)]/dEL(E)=d[\ln(\sigma E)]/dE systematically observed above the hindrance threshold in the 12C+28Si excitation function and in the neighbouring systems 12C+24,26Mg and 12C+30Si, and discriminate quantitatively among the candidate mechanisms advanced for light-system oscillations — elastic α\alpha-transfer, quasi-molecular resonances, low compound-nucleus level density, or other reaction dynamics — against the observed pattern (amplitude, energy spacing, energy dependence, and system dependence across the family).

(2) Behaviour at still lower energies. Determine whether the 12C+28Si fusion cross section — measured down to about 42 nb (lowest 28Si beam energy Elab=29.5E_{lab}=29.5 MeV, nominal Ecm≈8.85E_{cm}\approx8.85 MeV) and currently consistent with the one-dimensional no-coupling tunnelling limit — continues to follow that limit at lower energies or drops below it, in a regime where the classical ion-ion distance is smaller than the nominal touching point of the two nuclear densities and the two-body potential description itself becomes questionable. Relevant information may come from measurements at still lower energies with reduced statistical and systematic uncertainties, or from theoretical treatments that remain valid below the touching point, incorporating the adiabatic and Pauli effects implied by the observed gradual vanishing of the coupling strengths.

(3) Astrophysical consequences. Quantify how a confirmed hindrance in light systems modifies the astrophysical S-factors and thermonuclear reaction rates of the carbon- and oxygen-burning reactions (12C+12C, 12C+16O, 16O+16O) in the stellar environments identified as sensitive (late evolution of massive stars, type-Ia supernovae), and the consequent nucleosynthesis changes, using the updated hindrance-threshold systematics.

Components (1) and (2) jointly settle whether and how hindrance operates in light systems; component (3) translates that settlement into astrophysical input. A complete answer must resolve all three components; an answer that resolves only one or two is partial: the common objective is the established, mechanistically understood, and astrophysically quantified hindrance of light-system fusion.

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