Applicability of the causality-based dispersion relation to exotic, continuum-coupled nuclear systems: the abnormal threshold anomaly in ${}^{6}$He+${}^{209}$Bi and weakly-bound projectiles

Near the Coulomb barrier, the real and imaginary parts of the phenomenological nucleus-nucleus optical potential are linked by a causality-based (Kramers-Kronig-type) dispersion relation, the classical 'threshold anomaly' framework established for tightly bound systems. For weakly bound and halo nuclei, whose coupling to the breakup continuum persists below the barrier, Yang et al. (Phys. Rev. Lett. 119, 042503 (2017)) extracted the exit-channel optical potential of ${}^{6}$He+${}^{209}$Bi from transfer reactions down to deep sub-barrier energies and reported an 'abnormal threshold anomaly': the imaginary strength first grows as the energy decreases below the barrier, then falls to zero at a finite threshold, while the dispersion integral of the measured imaginary part fails to reproduce the measured real part. The open problem is threefold and jointly aims at one objective — a causal, predictive account of the real-imaginary connection for continuum-coupled systems: (i) identify the physical mechanism (transfer channels, breakup continuum coupling, soft dipole resonance, extended matter distribution) and its quantitative combination; (ii) establish mathematically and physically whether the standard dispersion relation, or a modified relation, holds when the breakup continuum replaces isolated singularities, including how much of the observed energy dependence is spurious, arising from fitting local effective potentials to an intrinsically nonlocal interaction; and (iii) determine empirically whether the abnormal threshold anomaly and the breakdown (or modification) of the classical dispersion relation are universal features of exotic systems. The question remains contested: later rigorous uncertainty-quantification analyses of neighbouring weakly-bound systems recover the standard threshold anomaly and dispersion-relation consistency, but with conclusions that depend explicitly on the imposed physical priors, while newer deep sub-barrier measurements on ${}^{6}$Li+${}^{208}$Pb again find the anomaly and state the dispersion relation is not applicable.

Background

In nuclear reaction physics, the interaction between projectile and target is described by a complex optical-model potential (OMP) $U(r;E)=V(r;E)+iW(r;E)$: the real part $V$ refracts (elastic scattering), while the imaginary part $W$ absorbs flux into nonelastic channels (fusion, transfer, breakup). For tightly bound projectiles at energies around the Coulomb barrier (the energy hill created by the competition of nuclear attraction and electrostatic repulsion, of order 20 MeV in total for heavy systems such as ${}^{6}$He+${}^{209}$Bi), a celebrated regularity appears: as the energy drops below the barrier, the nonelastic channels close, $W$ falls rapidly to zero, and $V$ develops a localized bell-shaped enhancement around the barrier. Mahaux, Ngô and Satchler showed that causality — the scattering amplitude vanishing for negative times — implies a Kramers-Kronig-type dispersion relation connecting the two parts, $\Delta V(r;E)=\frac{P}{\pi}\int_{0}^{\infty}\frac{W(r;E')}{E'-E}\,dE'$, where $P$ denotes the Cauchy principal value and $\Delta V$ is the energy-dependent (polarization) component of the real potential. This 'threshold anomaly' (TA) and its dispersive interpretation were consolidated in the classic literature (Mahaux, Ngô, Satchler 1986; Satchler 1991) and are a standard diagnostic in near-barrier heavy-ion physics. For weakly bound nuclei (${ }^{6,7}$Li, ${ }^{9}$Be) and halo nuclei, the situation changes qualitatively. A halo nucleus such as ${ }^{6}$He has two neutrons bound by only ~1 MeV and a greatly extended matter distribution, with low-lying 'soft dipole' strength; its coupling to the breakup continuum (a dense spectrum of unbound states) remains effective below the Coulomb barrier. Hussein, Gomes, Lubian and Chamon (2006) proposed the 'breakup threshold anomaly' (BTA): for weakly bound projectiles $W$ persists or even increases as the energy decreases below the barrier, a behavior still formulated within the dispersion-relation framework. Elastic-scattering studies of weakly bound systems, e.g. Santra et al. on ${ }^{6}$Li+${ }^{209}$Bi (2011), showed a variety of near-threshold behaviors. The critical step came from Yang et al. (2017), who bypassed the insensitivity of sub-barrier elastic scattering of radioactive beams by extracting the exit-channel OMP of ${}^{6}$He+${}^{209}$Bi from the transfer reaction ${ }^{208}$Pb(${ }^{7}$Li,${ }^{6}$He)${ }^{209}$Bi, analyzed in the coupled-reaction-channels (CRC) framework (an approach that couples the elastic channel to transfer channels explicitly), at energies reaching deep below the barrier, and validated against independent elastic-scattering data (Aguilera et al. 2001; Guimarães et al. 2016 found a consistent critical interaction distance from backscattering). At the sensitivity radius $r_s=13.5$ fm (the radial region where the data actually constrain the potential), they found: $W$ first increases with decreasing sub-barrier energy and then falls quickly to zero at a threshold $E_{th}=13.73\pm1.63$ MeV, about $0.68$ of the barrier height — the first absorption threshold ever determined in an exotic system; the real potential shows a bell-like shape around the barrier; but the dispersion integral of the measured $W$, evaluated with the standard schematic fit, differs qualitatively from the measured $V$. They concluded the dispersion relation is not applicable to this system, possibly common for exotic nuclei, citing three theoretical reasons: the Kramers-Kronig derivation assumes a finite set of isolated singularities whereas the breakup continuum is dense; phenomenological local OMPs carry spurious energy dependence because the fundamental interaction is nonlocal in space; and the fitted potential is constrained only near the sensitivity radius. The paper itself asks what mechanism (long-range transfer, breakup continuum, soft dipole resonance, extended matter distribution) dominates, what relation should replace the classical one, and whether the phenomenon is universal — and this is precisely what remains unresolved, with the additional complication that later statistical re-analyses (Navarro Pérez and Lei 2019; Yang et al. 2020) show that near-threshold OMP behavior extracted from near-barrier data is strongly prior-dependent, so even the existence of the anomaly is entangled with the analysis methodology.

Problem Statement

Determine whether, and in what form, the causality-based dispersion relation connects the real and imaginary parts of the nucleus-nucleus optical potential for weakly bound and exotic nuclear systems strongly coupled to a breakup continuum, and explain the observed abnormal threshold anomaly. Concretely, the source-faithful question decomposes into jointly linked parts serving this single objective: (1) Mechanism — identify which physical mechanism (long-range transfer channels, coupling to the breakup continuum, soft dipole resonance, extended halo matter distribution, or a combination) produces the anomalous increase of absorption with decreasing energy in the sub-barrier region, and quantify how the contributions combine, for the paradigmatic ${}^{6}$He+${}^{209}$Bi case where the imaginary strength at the sensitivity radius (the radial region where the data actually constrain the potential), $W(r_s;E)$, rises below the barrier and closes at $E_{th}\approx13.7$ MeV while $V(r_s;E)$ shows a bell-like variation that the standard dispersion integral of the measured $W$ fails to reproduce; (2) Theory — establish mathematically and physically whether a dispersion-like relation (the standard Kramers-Kronig/Mahaux-Ngô-Satchler relation, or a modified one appropriate to a dense continuum of breakup states) correctly links $V$ and $W$ for continuum-coupled systems, including quantifying how much of the apparent energy dependence is spurious, arising from the use of phenomenological local effective potentials in place of fundamentally nonlocal interactions and from the limited radial sensitivity of the fitted potentials; (3) Universality — determine empirically, for a wider range of unstable and weakly bound nuclei, whether the abnormal threshold anomaly and the failure (or modification) of the classical dispersion relation are universal features of exotic systems or specific to particular cases. An answer must address the causal $V$-$W$ connection itself: either demonstrating that the standard relation is satisfied once uncertainties and nonlocality-induced artifacts are properly handled, or deriving and validating the relation that replaces it, together with the mechanism that generates the observed $W(E)$ and $V(E)$ behavior.

References

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