For tightly-bound projectiles the optical-model potential for elastic scattering displays the threshold anomaly near the Coulomb barrier: the imaginary part W(E)W(E) falls rapidly as the energy drops below the barrier and the real part follows through a causality-linked dispersion relation. For weakly-bound projectiles such as 6^{6}Li, 7^{7}Li, 9^{9}Be and the halo nucleus 6^{6}He, conflicting analyses have reported instead a breakup threshold anomaly (BTA) — sub-barrier absorption persisting or growing while the dispersion relation still holds in modified form — or an abnormal threshold anomaly, in which the dispersion relation itself appears violated. A 2019 uncertainty-quantification study argued that these anomalies may be artifacts of underconstrained and overfitted analyses, but its conclusion depended on imposing an undemonstrated physical constraint (a decreasing sub-barrier W(E)W(E)), and a subsequent Bayesian reanalysis of the same data reaffirmed the abnormal behavior. The open problem is to determine, with statistically rigorous and microscopically grounded analysis across weakly-bound projectile-target combinations, the true near-threshold behavior of the potential — whether the conventional threshold anomaly is universal for weakly bound systems or BTA-type behavior is genuine — and to derive that behavior from the underlying reaction mechanisms (breakup versus fusion and other non-elastic channels).

Background

In the optical model, the elastic scattering of two nuclei is described by an effective complex potential U(E)=V(E)+iW(E)U(E)=V(E)+iW(E), whose real part VV refracts and whose imaginary part WW absorbs flux lost to non-elastic channels. For tightly-bound systems a well-documented behavior appears at energies around the Coulomb barrier (the energy at which the repulsive Coulomb and attractive nuclear forces balance): as the energy decreases below the barrier and non-elastic channels close, W(E)W(E) drops rapidly, and causality — expressed as a dispersion relation — forces the real part to vary through ΔV(E)=1πP∫0∞W(E′)E′−E dE′\Delta V(E)=\frac{1}{\pi}\mathcal{P}\int_0^{\infty}\frac{W(E')}{E'-E}\,dE', where P\mathcal{P} denotes the Cauchy principal value. This coupled behavior is the threshold anomaly (TA). Weakly bound nuclei complicate this picture: 6^{6}Li (breakup threshold Sα≈1.47S_\alpha\approx1.47 MeV) and 9^{9}Be are stable but fragile, and 6^{6}He is a two-neutron halo nucleus — two weakly bound neutrons orbiting an alpha-particle core, with two-neutron separation energy S2n≈0.98S_{2n}\approx0.98 MeV — so coupling to the breakup continuum remains strong even below the barrier. Hussein, Gomes, Lubian and Chamon (Phys. Rev. C 73, 044610 (2006)) argued that in this case the imaginary part need not fall below the barrier and coined the term breakup threshold anomaly (BTA) for the resulting modified dispersive behavior, with the real part often becoming more attractive with increasing energy. Numerous conventional (single-energy, covariance-matrix) fits reported BTA-type behavior for 6^{6}Li on targets from 27^{27}Al to 209^{209}Bi (e.g., Santra et al., Phys. Rev. C 83, 034616 (2011), for 6^{6}Li+209^{209}Bi). For the halo system 6^{6}He+209^{209}Bi, Yang et al. (Phys. Rev. Lett. 119, 042503 (2017)), using a transfer-reaction surrogate method (extracting the 6^{6}He+209^{209}Bi potential from the 208^{208}Pb(7^{7}Li,6^{6}He)209^{209}Bi reaction) to reach sub-barrier energies, reported an even stronger deviation — an abnormal threshold anomaly — and concluded that the dispersion relation itself is not applicable to such exotic systems; these fits typically had χ2/d.o.f.≪1\chi^2/\mathrm{d.o.f.}\ll1, suggesting overfitting. Navarro Pérez and Lei (Phys. Lett. B 795, 200 (2019)) then performed the first rigorous uncertainty quantification of near-barrier elastic scattering, fitting all energies simultaneously with Monte-Carlo error propagation for 6^{6}Li+209^{209}Bi and 6^{6}He+208^{208}Pb. They found that TA-consistent potential ensembles are obtained only after imposing a physical constraint that the imaginary strength decrease below the barrier; without it, the data do not discriminate, and previously claimed BTA/abnormal-TA behavior could reflect underconstrained fits. They acknowledged that their study covered only two projectiles, that extrapolation to 7^{7}Li, 9^{9}Be and other exotic nuclei remained untested, that the decreasing-WW constraint was imposed as physical input rather than derived microscopically, and that the interplay of large breakup coupling, the observed suppression of complete fusion — full capture of the projectile into a compound nucleus (Lei and Moro, Phys. Rev. Lett. 122, 042503 (2019); Canto et al., Phys. Rep. 596, 1 (2015)) — and the net sub-barrier absorption required clarification. The question stands unresolved: subsequent work is split between analyses reporting BTA for 6^{6}Li, 7^{7}Li, 9^{9}Be and deuteron projectiles, a Bayesian reanalysis (Yang et al., Phys. Lett. B 807, 135540 (2020)) concluding that the abnormal threshold anomaly is confirmed and the dispersion relation's applicability remains in doubt, and at least one system (6^{6}Li+144^{144}Sm, Kurmanaliyev et al., Eur. Phys. J. A 59, 282 (2023)) displaying the ordinary TA.

Problem Statement

Determine the genuine near-threshold energy dependence of the effective optical-model potential U(E)=V(E)+iW(E)U(E)=V(E)+iW(E) for reactions induced by weakly-bound projectiles at energies around the Coulomb barrier, and establish its microscopic origin. This comprises three jointly linked questions that a complete answer must address: (1) Universality — establish whether the conventional threshold anomaly, in which W(E)W(E) decreases rapidly as the energy falls below the Coulomb barrier and V(E)V(E) follows through the dispersion relation ΔV(E)=1πP∫0∞W(E′)E′−E dE′\Delta V(E)=\frac{1}{\pi}\mathcal{P}\int_0^{\infty}\frac{W(E')}{E'-E}\,dE', holds for weakly-bound and halo projectiles (6,7^{6,7}Li, 9^{9}Be, deuteron, 6^{6}He and other exotic systems) across a broad range of targets — at minimum those already analyzed in the literature (6^{6}Li+209^{209}Bi, 6^{6}He+208^{208}Pb and 6^{6}He+209^{209}Bi, 6^{6}Li+144^{144}Sm, 7^{7}Li+92,100^{92,100}Mo and 9^{9}Be+197^{197}Au) — or whether breakup threshold anomaly behavior (persistent or increasing sub-barrier absorption with the dispersion relation still satisfied in modified form) or abnormal threshold anomaly behavior (persistent or increasing sub-barrier absorption with the dispersion relation itself violated) is genuine for some systems. (2) Microscopic justification — derive from a microscopic reaction theory (for example a Feshbach projection, which formally eliminates the coupling to non-elastic channels to yield an energy-dependent effective potential, with explicit treatment of the breakup continuum) the sub-barrier behavior of W(E)W(E), including the relative contributions of breakup, complete and incomplete fusion, transfer and other non-elastic channels, and state under what physical conditions the imaginary potential should or should not decrease below the barrier. (3) Discrimination criterion — establish the data precision, angular coverage, energy density, and fitting constraints or priors under which TA-type, BTA-type and abnormal-TA-type potential behavior can be statistically distinguished, so that anomalous (BTA- or abnormal-TA-type) behavior can be certified as physics rather than an artifact of underconstrained fits. The statistical strand (1)+(3) and the microscopic strand (2), covering at least the systems named in (1), define what a complete answer must address; they jointly serve one overarching objective — a validated, predictive near-barrier optical potential for weakly-bound nuclei whose energy dependence is understood from the underlying reaction mechanisms — which is the goal they pursue rather than an additional required deliverable.

References

  1. R. Navarro Pérez, J. Lei, 'Is the unusual near-threshold potential behavior in elastic scattering of weakly-bound nuclei a precision error?', Physics Letters B 795 (2019) 200–205. DOI: 10.1016/j.physletb.2019.06.005; arXiv:1812.05641. https://arxiv.org/abs/1812.05641
  2. M. S. Hussein, P. R. S. Gomes, J. Lubian, L. C. Chamon, 'New manifestation of the dispersion relation: Breakup threshold anomaly', Physical Review C 73 (2006) 044610. DOI: 10.1103/PhysRevC.73.044610. https://doi.org/10.1103/PhysRevC.73.044610
  3. L. Yang, C. J. Lin, H. M. Jia, D. X. Wang, N. R. Ma, L. J. Sun, F. Yang, X. X. Xu, Z. D. Wu, H. Q. Zhang, Z. H. Liu, 'Is the Dispersion Relation Applicable for Exotic Nuclear Systems? The Abnormal Threshold Anomaly in the 6^{6}He+209^{209}Bi System', Physical Review Letters 119 (2017) 042503. DOI: 10.1103/PhysRevLett.119.042503. https://doi.org/10.1103/PhysRevLett.119.042503
  4. L. Yang, C. J. Lin, H. M. Jia, D. X. Wang, N. R. Ma, L. J. Sun, F. Yang, X. X. Xu, Z. D. Wu, H. Q. Zhang, Z. H. Liu, 'Abnormal behavior of the optical potential for the halo nuclear system 6^{6}He+209^{209}Bi', Physical Review C 96 (2017) 044615. DOI: 10.1103/PhysRevC.96.044615. https://doi.org/10.1103/PhysRevC.96.044615
  5. S. Santra, S. Kailas, K. Ramachandran, V. V. Parkar, V. Jha, B. J. Roy, P. Shukla, 'Reaction mechanisms involving weakly bound 6^{6}Li and 209^{209}Bi at energies near the Coulomb barrier', Physical Review C 83 (2011) 034616. DOI: 10.1103/PhysRevC.83.034616. https://doi.org/10.1103/PhysRevC.83.034616
  6. L. Yang, C. J. Lin, Y. X. Zhang, P. W. Wen, H. M. Jia, D. X. Wang, N. R. Ma, F. Yang, F. P. Zhong, S. H. Zhong, T. P. Luo, 'Bayesian analysis on interactions of exotic nuclear systems', Physics Letters B 807 (2020) 135540. DOI: 10.1016/j.physletb.2020.135540. https://doi.org/10.1016/j.physletb.2020.135540
  7. H. Kumawat, C. Joshi, V. V. Parkar, V. Jha, B. J. Roy, Y. S. Sawant, P. C. Rout, E. T. Mirgule, R. K. Singh, N. L. Singh, B. K. Nayak, S. Kailas, 'Elastic scattering for 6^{6}Li+51^{51}V and systematic study of breakup threshold anomaly', Nuclear Physics A 1002 (2020) 121973. DOI: 10.1016/j.nuclphysa.2020.121973. https://doi.org/10.1016/j.nuclphysa.2020.121973
  8. F. Torabi, E. F. Aguilera, O. N. Ghodsi, A. Gómez-Camacho, 'Systematic study of elastic scattering and fusion induced by weakly bound 6^{6}Li on medium mass targets. Threshold anomalies', Nuclear Physics A 994 (2020) 121661. DOI: 10.1016/j.nuclphysa.2019.121661. https://doi.org/10.1016/j.nuclphysa.2019.121661
  9. F. Gollan, D. Abriola, A. Arazi, M. A. Cardona, E. de Barbará, D. Hojman, R. M. Id Betan, G. V. Martí, A. J. Pacheco, D. Rodrigues, M. Togneri, 'Energy dependence of the optical potential of the weakly bound 9^{9}Be projectile on the 197^{197}Au target', Nuclear Physics A 1000 (2020) 121789. DOI: 10.1016/j.nuclphysa.2020.121789. https://doi.org/10.1016/j.nuclphysa.2020.121789
  10. C. Joshi, H. Kumawat, R. K. Singh, N. L. Singh, D. Patel, B. K. Nayak, J. Acharya, A. Parihari, K. Rani, S. D. Sharma, G. Kaur, I. Ahmed, K. S. Golda, N. Saneesh, M. Kumar, A. Jhingan, P. Sugathan, 'Exploring breakup coupling effect in 7^{7}Li+92,100^{92,100}Mo elastic scattering around Coulomb barrier energies', The European Physical Journal A 58 (2022) 40. DOI: 10.1140/epja/s10050-022-00690-1. https://doi.org/10.1140/epja/s10050-022-00690-1
  11. Zh. Kurmanaliyev, B. Mauyey, A. A. Ibraheem, K. W. Kemper, Sh. Hamada, 'Behavior of 6^{6}Li+144^{144}Sm elastic and inelastic angular distributions in the region of the Coulomb barrier', The European Physical Journal A 59 (2023) 282. DOI: 10.1140/epja/s10050-023-01191-5. https://doi.org/10.1140/epja/s10050-023-01191-5
  12. T. Giudice, D. Abriola, A. Arazi, E. de Barbará, M. A. Cardona, J. Gómez, D. Hojman, R. M. Id Betan, M. S. Kohen, N. Llaneza, G. V. Martí, B. Paes, D. Schneider, H. Soler, J. Lubian, 'Study of the threshold anomaly in the elastic scattering of d+197^{197}Au', Journal of Physics G: Nuclear and Particle Physics 50 (2023) 045103. DOI: 10.1088/1361-6471/acb452. https://doi.org/10.1088/1361-6471/acb452
  13. H. M. Maridi, N. Keeley, K. Rusek, 'Simultaneous calculation of elastic scattering, fusion, and direct cross sections for reactions of weakly-bound projectiles', Physical Review C 109 (2024) 034601. DOI: 10.1103/PhysRevC.109.034601; arXiv:2307.06875. https://arxiv.org/abs/2307.06875
  14. L. Yang, C. J. Lin, H. M. Jia, N. R. Ma, P. W. Wen, 'Experimental investigations on near-barrier reactions with weakly-bound nuclei at the China Institute of Atomic Energy', The European Physical Journal A 61 (2025) 124. DOI: 10.1140/epja/s10050-025-01598-2. https://doi.org/10.1140/epja/s10050-025-01598-2
  15. J. Lei, A. M. Moro, 'Puzzle of Complete Fusion Suppression in Weakly Bound Nuclei: A Trojan Horse Effect?', Physical Review Letters 122 (2019) 042503. DOI: 10.1103/PhysRevLett.122.042503. https://doi.org/10.1103/PhysRevLett.122.042503
  16. L. F. Canto, P. R. S. Gomes, R. Donangelo, J. Lubian, M. S. Hussein, 'Recent developments in fusion and direct reactions with weakly bound nuclei', Physics Reports 596 (2015) 1–86. DOI: 10.1016/j.physrep.2015.08.001. https://doi.org/10.1016/j.physrep.2015.08.001
  17. H. Liu, J. Lei, Z. Ren, 'Exact Treatment of Continuum Couplings in Nuclear Optical Potentials via Feshbach Theory', arXiv:2508.07584 (2025). https://arxiv.org/abs/2508.07584
  18. H. Liu, J. Lei, Z. Ren, 'Coulomb bridge mechanism for peripheral polarization of weakly bound projectiles', arXiv:2605.23541 (2026). https://arxiv.org/abs/2605.23541