In triaxial atomic nuclei, a valence proton, a valence neutron, and the rotating core can form two mirror-image (left- and right-handed) aplanar configurations of angular momentum, related by time reversal; spontaneous breaking of this chiral symmetry should manifest as a pair of near-degenerate bands with the same spin sequence (chiral doublet bands). A 2017 review by Starosta and Koike, which introduced an explicit chiral basis for the particle-rotor model, closed with a set of open questions: reconciling the electromagnetic selection rules of Koike et al. (2004) with the different transformation properties of the chiral basis states under the combined R2(π)T\mathcal{R}_2(\pi)\mathcal{T} symmetry; including pairing and residual proton-neutron interactions; extending the chiral basis beyond the idealized γ=90∘\gamma=90^\circ limit to realistic and γ\gamma-soft shapes; determining the nature of multiple pairs of ΔI=1\Delta I=1 bands (multiple chiral doublets, Mχ\chiD, e.g. in 103^{103}Rh and 133^{133}Ce); and the fact that a textbook example of a chiral doublet — energy degeneracy extending over several ℏ\hbar of angular momentum together with a consistent set of electromagnetic transition rates — had not been observed. The problem is open.

Background

Deformed atomic nuclei at high angular momentum display rotational bands — sequences of states of increasing spin II connected by fast electromagnetic transitions. In an odd-odd nucleus (both proton and neutron numbers odd), the two unpaired nucleons align their angular momenta with the deformed body-fixed axes, generating bands whose structure reflects the underlying shape. When the nuclear shape is triaxial — ellipsoidal with three unequal principal axes, conventionally parametrized by the angle γ\gamma (γ=0∘\gamma=0^\circ and γ=60∘\gamma=60^\circ are the axially symmetric prolate and oblate limits) — the three principal axes are physically distinct, conventionally called the long, short, and intermediate axes.

Frauendorf and Meng predicted in 1997 that in a maximally triaxial nucleus with a valence proton angular momentum along the short axis, a valence neutron (or neutron hole) angular momentum along the long axis, and collective core rotation along the intermediate axis, the three angular momenta form an aplanar, handed (chiral) geometry: its mirror image is obtained by time reversal, and spontaneous breaking of the resulting left-right degeneracy should produce two almost identical rotational bands of the same spin and parity, the chiral doublet bands. The first experimental candidates, near-degenerate ΔI=1\Delta I=1 bands (bands whose spins differ by 1ℏ1\hbar at each step) built on πh11/2⊗νh11/2\pi h_{11/2}\otimes\nu h_{11/2} proton-particle/neutron-hole configurations, were reported by Starosta et al. in 2001 in odd-odd N=75N=75 isotopes such as 132^{132}La and 134^{134}Pr; since then several dozen candidate nuclei have been reported in the A≈80A\approx80, 100100, 130130, and 190190 mass regions.

The standard diagnostics of chirality are indirect: near-degeneracy of the partner-band energies over a range of spin, similar ratios of magnetic-dipole to electric-quadrupole transition strengths, B(M1)/B(E2)B(M1)/B(E2), in the two bands, and similar staggering patterns. Koike, Starosta, and Hamamoto (2004) derived, in a particle-rotor model for the chiral geometry, selection rules for the electromagnetic transitions between and within chiral bands, expressed through a quantum number tied to an exact symmetry of the model Hamiltonian — a combination of a 180∘180^\circ rotation about the 2nd principal axis with time reversal, denoted R2(π)T\mathcal{R}_2(\pi)\mathcal{T}. These selection rules are among the sharpest experimentally testable predictions of the chirality picture.

The particle-rotor model (PRM) couples one or two valence nucleons to a rotating triaxial core; it is normally solved in a basis that does not distinguish handedness, so chirality is extracted a posteriori from expectation values of orientation operators. The 2017 review by Starosta and Koike instead constructed an explicit chiral basis for a proton particle coupled to a neutron hole: the proton angular momentum is quantized along the short axis, the neutron-hole angular momentum along the long axis, and the total angular momentum along the intermediate axis. In this basis the states separate exactly into right-handed, left-handed, and planar subspaces, the Hamiltonian matrix block-decouples (matrix elements between right- and left-handed states vanish), the symmetry operator R2(π)T\mathcal{R}_2(\pi)\mathcal{T} relates the two chiral sectors so that only one needs to be diagonalized, and the expectation value of the chirality operator σ=(j⃗π×j⃗ν)⋅I⃗\sigma=(\vec{j}_\pi\times\vec{j}_\nu)\cdot\vec{I} is the simple product of the basis quantum numbers. The framework, however, was derived at the idealized triaxial point the authors label γ=90∘\gamma=90^\circ (where the deformation potential within a single high-jj orbital, the single-jj approximation, takes the symmetric form proportional to j12−j22j_1^2-j_2^2), with irrotational-flow moments of inertia Jk=J0sin⁡2(γ−k⋅120∘)\mathcal{J}_k=\mathcal{J}_0\sin^2(\gamma-k\cdot120^\circ), and without pairing or residual proton-neutron interactions — idealizations that do not hold in real nuclei. The label γ=90∘\gamma=90^\circ reflects the convention of the chiral-basis construction rather than the γ\gamma convention introduced above: with the irrotational-flow formula it yields J3=4J1=4J2=J0\mathcal{J}_3=4\mathcal{J}_1=4\mathcal{J}_2=\mathcal{J}_0, the maximum difference between principal axes allowed by the quadrupole deformation of the mean field, i.e. the maximally triaxial point of that construction.

Two further notions are needed for the current status. A candidate showing the doublet pattern only on average, with handedness fluctuating between the two bands, realizes chiral vibration rather than static chiral symmetry breaking; a direct experimental discriminator between the two is the magnetic moment (g-factor) of a band member, since it measures the orientation of the constituent angular momenta. Multiple chiral doublets (Mχ\chiD) denote the coexistence of more than one pair of chiral doublet bands in a single nucleus, e.g. built on the same configuration at different excitation energy or on different configurations. Wobbling motion is an oscillation of the rotational axis about the principal axis with the largest moment of inertia; its coupling to chirality (the chiral wobbler) was proposed as an extension of the chiral-basis framework. The 2017 review closed by listing as open questions: the reconciliation of the chiral basis with the 2004 selection rules (the chiral basis states transform under R2(π)T\mathcal{R}_2(\pi)\mathcal{T} differently than assumed in the 2004 derivation, so detailed studies are needed to reconcile the two and derive testable predictions); the inclusion of pairing and residual proton-neutron interactions; the extension to realistic γ\gamma values and γ\gamma-soft nuclei; the fact that a textbook example of chiral doublet bands — energy degeneracy over several ℏ\hbar of angular momentum with a consistent set of electromagnetic transition rates — had yet to be observed; the interpretation of multiple pairs of ΔI=1\Delta I=1 bands such as in 103^{103}Rh and 133^{133}Ce as chiral doublets, Mχ\chiD, or other coupling schemes; and the extension of the chiral basis to wobbling motion and to more than one particle-hole pair.

Problem Statement

The problem, as posed by the 2017 review of Starosta and Koike, is to establish — or definitively refute — static chiral symmetry breaking in atomic nuclei as the mechanism of the near-degenerate ΔI=1\Delta I=1 doublet bands. The open questions with which the review closed fall into four linked components:

(1) Theoretical predictions from the chiral basis: derive the electromagnetic transition rates and selection rules within the chiral basis of the particle-rotor model, reconciling the fact that the chiral basis states transform under the R2(π)T\mathcal{R}_2(\pi)\mathcal{T} operator differently than assumed in the 2004 selection-rule derivation of Koike, Starosta, and Hamamoto, and obtain from this reconciliation experimentally testable predictions that discriminate chiral doublets from other interpretations.

(2) The observational core: observe a textbook example of chiral doublet bands — a pair of same-spin-parity bands whose energy degeneracy extends over several ℏ\hbar of angular momentum and whose electromagnetic transition rates (absolute B(M1)B(M1) and B(E2)B(E2) values, their intraband and interband pattern) are mutually consistent with the chiral geometry — or establish by direct observables (g-factors, static quadrupole moments, absolute transition probabilities) that the observed candidates realize chiral vibration or another mechanism instead.

(3) Multiple doublets: determine the nature of the multiple pairs of ΔI=1\Delta I=1 bands observed in single nuclei such as 103^{103}Rh and 133^{133}Ce — chiral doublets, multiple chiral doublets (Mχ\chiD), or other coupling schemes — through detailed comparison between theory and experiment.

(4) Realistic theory: extend the chiral-basis framework beyond the idealized γ=90∘\gamma=90^\circ single-jj construction with irrotational-flow moments of inertia to realistic γ\gamma values and γ\gamma-soft nuclei, and to Hamiltonians including pairing and residual proton-neutron interactions (and, as the review also envisaged, to wobbling motion and more than one particle-hole pair), so that the framework can decide components (1)–(3) for real nuclei rather than for the idealized model point.

The four components are linked but of unequal weight: component (2) is the observational core, and its resolution — in either direction — decides the posed question of whether static chiral symmetry breaking is the mechanism of the observed ΔI=1\Delta I=1 doublet bands; components (1), (3), and (4) are supporting components that need not all be completed for that question to be decided.

References

  1. K. Starosta and T. Koike, 'Nuclear chirality, a model and the data', Physica Scripta 92, 093002 (2017). DOI: 10.1088/1402-4896/aa800e, https://doi.org/10.1088/1402-4896/aa800e
  2. S. Frauendorf and Jie Meng, 'Tilted rotation of triaxial nuclei', Nuclear Physics A 617, 131 (1997). DOI: 10.1016/S0375-9474(97)00004-3, https://doi.org/10.1016/S0375-9474(97)00004-3
  3. K. Starosta, T. Koike, C. J. Chiara, D. B. Fossan, D. R. LaFosse, A. A. Hecht, C. W. Beausang, M. A. Caprio, J. R. Cooper, R. Krücken, J. R. Novak, N. V. Zamfir, K. E. Zyromski, D. J. Hartley, D. L. Balabanski, Jing-ye Zhang, S. Frauendorf, and V. I. Dimitrov, 'Chiral Doublet Structures in Odd-Odd N=75 Isotones: Chiral Vibrations', Physical Review Letters 86, 971 (2001). DOI: 10.1103/PhysRevLett.86.971, https://doi.org/10.1103/PhysRevLett.86.971
  4. T. Koike, K. Starosta, and I. Hamamoto, 'Chiral Bands, Dynamical Spontaneous Symmetry Breaking, and the Selection Rule for Electromagnetic Transitions in the Chiral Geometry', Physical Review Letters 93, 172502 (2004). DOI: 10.1103/PhysRevLett.93.172502, https://doi.org/10.1103/PhysRevLett.93.172502
  5. I. Kuti, Q. B. Chen, J. Timár, D. Sohler, S. Q. Zhang, Z. H. Zhang, P. W. Zhao, J. Meng, K. Starosta, T. Koike, E. S. Paul, D. B. Fossan, and C. Vaman, 'Multiple Chiral Doublet Bands of Identical Configuration in 103Rh', Physical Review Letters 113, 032501 (2014). DOI: 10.1103/PhysRevLett.113.032501, https://doi.org/10.1103/PhysRevLett.113.032501
  6. Q. B. Chen, K. Starosta, and T. Koike, 'Three-level mixing model for nuclear chiral rotation: Role of the planar component', Physical Review C 97, 041303(R) (2018). DOI: 10.1103/PhysRevC.97.041303, https://doi.org/10.1103/PhysRevC.97.041303
  7. B. W. Xiong and Y. Y. Wang, 'Nuclear chiral doublet bands data tables', Atomic Data and Nuclear Data Tables 125, 193 (2019). DOI: 10.1016/j.adt.2018.05.002, https://doi.org/10.1016/j.adt.2018.05.002
  8. E. Grodner, J. Srebrny, Ch. Droste, L. Próchniak, S. G. Rohoziński, M. Kowalczyk, M. Ionescu-Bujor, C. A. Ur, K. Starosta, T. Ahn, M. Kisieliński, T. Marchlewski, S. Aydin, F. Recchia, G. Georgiev, R. Lozeva, E. Fiori, M. Zielińska, Q. B. Chen, S. Q. Zhang, L. F. Yu, P. W. Zhao, and J. Meng, 'First Measurement of the g Factor in the Chiral Band: The Case of the 128Cs Isomeric State', Physical Review Letters 120, 022502 (2018). DOI: 10.1103/PhysRevLett.120.022502, https://doi.org/10.1103/PhysRevLett.120.022502
  9. E. Grodner, M. Kowalczyk, M. Kisieliński, J. Srebrny, L. Próchniak, Ch. Droste, S. G. Rohoziński, Q. B. Chen, M. Ionescu-Bujor, C. A. Ur, F. Recchia, J. Meng, S. Q. Zhang, P. W. Zhao, G. Georgiev, R. Lozeva, E. Fiori, S. Aydin, and A. Nałęcz-Jawecki, 'Examination of nuclear chirality with a magnetic moment measurement of the I=9 isomeric state in 128Cs', Physical Review C 106, 014318 (2022). DOI: 10.1103/PhysRevC.106.014318, https://doi.org/10.1103/PhysRevC.106.014318
  10. E. Grodner, Michał Kowalczyk, Julian Srebrny, Leszek Próchniak, Chrystian Droste, Jan Kownacki, Maciej Kisieliński, Krzysztof Starosta, and Takeshi Koike, 'The g-factor measurement as an ultimate test for nuclear chirality', Frontiers of Physics 19, 34202 (2023). DOI: 10.1007/s11467-023-1359-2, https://doi.org/10.1007/s11467-023-1359-2
  11. Shouyu Wang, Chen Liu, Bin Qi, Wenzheng Xu, and Hui Zhang, 'Experimental studies for nuclear chirality in China', Frontiers of Physics 18, 64601 (2023). DOI: 10.1007/s11467-023-1303-5, https://doi.org/10.1007/s11467-023-1303-5
  12. Shou-Yu Wang, 'Recent progress in multiple chiral doublet bands', Chinese Physics C 44, 112001 (2020). DOI: 10.1088/1674-1137/abaed2, https://doi.org/10.1088/1674-1137/abaed2
  13. R. J. Guo, S. Y. Wang, R. Schwengner, W. Z. Xu, B. Qi, C. Liu, A. Rohilla, F. Dönau, T. Servene, H. Schnare, J. Reif, G. Winter, L. Käubler, H. Prade, S. Skoda, J. Eberth, H. G. Thomas, F. Becker, B. Fiedler, S. Freund, S. Kasemann, T. Steinhardt, O. Thelen, T. Härtlein, C. Ender, F. Köck, P. Reiter, and D. Schwalm, 'Lifetime measurements in 80Br and a new region for observation of chiral electromagnetic selection rules', Physics Letters B 833, 137344 (2022). DOI: 10.1016/j.physletb.2022.137344, https://doi.org/10.1016/j.physletb.2022.137344
  14. R. J. Guo, S. Y. Wang, C. Liu, R. A. Bark, J. Meng, S. Q. Zhang, B. Qi, A. Rohilla, Z. H. Li, H. Hua, Q. B. Chen, H. Jia, X. Lu, S. Wang, D. P. Sun, X. C. Han, W. Z. Xu, E. H. Wang, H. F. Bai, M. Li, P. Jones, J. F. Sharpey-Schafer, M. Wiedeking, O. Shirinda, C. P. Brits, K. L. Malatji, T. Dinoko, J. Ndayishimye, S. Mthembu, S. Jongile, K. Sowazi, S. Kutlwano, T. D. Bucher, D. G. Roux, A. A. Netshiya, L. Mdletshe, S. Noncolela, and W. Mtshali, 'Evidence for Chiral Wobbler in Nuclei', Physical Review Letters 132, 092501 (2024). DOI: 10.1103/PhysRevLett.132.092501, https://doi.org/10.1103/PhysRevLett.132.092501
  15. Y. Wu and Q. B. Chen, 'Pairing effect on critical spin in chiral rotation', Physics Letters B 849, 138445 (2024). DOI: 10.1016/j.physletb.2024.138445, https://doi.org/10.1016/j.physletb.2024.138445
  16. Y. Wu and Q. B. Chen, 'Pairing effect on critical spin in chiral rotation. II. Triaxial deformation dependence', Physical Review C 111, 024302 (2025). DOI: 10.1103/PhysRevC.111.024302, https://doi.org/10.1103/PhysRevC.111.024302
  17. A. Krakó, D. Sohler, J. Timár, I. Kuti, Q. B. Chen, S. Q. Zhang, J. Meng, K. Starosta, T. Koike, E. S. Paul, D. B. Fossan, and C. Vaman, 'Multiple chiral doublet bands in 104Rh', Physics Letters B 855, 138850 (2024). DOI: 10.1016/j.physletb.2024.138850, https://doi.org/10.1016/j.physletb.2024.138850
  18. T. J. Gao, Jing-Bin Lu, Yingjun Ma, Yuhu Zhang, S. Q. Zhang, H. D. Wang, Jia-Qiang Liu, Pei-Yao Yang, Zhen Ren, Cheng-Qian Li, Q. B. Chen, Z. C. Gao, Jian Li, K. Y. Ma, and Guoxiang Dong, 'Evidence for possible multiple chiral doublet bands with identical configuration in the odd-odd nucleus 126Cs', Physical Review C 109, 024307 (2024). DOI: 10.1103/PhysRevC.109.024307, https://doi.org/10.1103/PhysRevC.109.024307
  19. Q. B. Chen, N. Kaiser, Ulf-G. Meißner, and J. Meng, 'Static quadrupole moments of nuclear chiral doublet bands', Physics Letters B 807, 135568 (2020). DOI: 10.1016/j.physletb.2020.135568, https://doi.org/10.1016/j.physletb.2020.135568
  20. B. Hu and Q. B. Chen, 'Static quadrupole moment as a criterion to distinguish chiral modes', Physical Review C 109, L021302 (2024). DOI: 10.1103/PhysRevC.109.L021302, https://doi.org/10.1103/PhysRevC.109.L021302
  21. B. Li, P. W. Zhao, and J. Meng, 'Chiral dynamics in soft triaxial nuclei', Physics Letters B 856, 138877 (2024). DOI: 10.1016/j.physletb.2024.138877, https://doi.org/10.1016/j.physletb.2024.138877
  22. E. O. Lieder, R. M. Lieder, R. A. Bark, Q. B. Chen, S. Q. Zhang, J. Meng, E. A. Lawrie, J. J. Lawrie, S. P. Bvumbi, N. Y. Kheswa, S. S. Ntshangase, T. E. Madiba, P. L. Masiteng, S. M. Mullins, S. Murray, P. Papka, D. G. Roux, O. Shirinda, Z. H. Zhang, P. W. Zhao, Z. P. Li, J. Peng, B. Qi, S. Y. Wang, Z. G. Xiao, and C. Xu, 'Resolution of Chiral Conundrum in 106Ag: Doppler-Shift Lifetime Investigation', Physical Review Letters 112, 202502 (2014). DOI: 10.1103/PhysRevLett.112.202502, https://doi.org/10.1103/PhysRevLett.112.202502
  23. P. W. Zhao, Y. K. Wang, and Q. B. Chen, 'Microscopic resolution of the nuclear chiral conundrum with crossing twin bands in 106Ag', Physical Review C 99, 054319 (2019). DOI: 10.1103/PhysRevC.99.054319, https://doi.org/10.1103/PhysRevC.99.054319
  24. Jie Meng and Yi-Ping Wang, 'The 25th anniversary for nuclear chirality', in 'Chirality and Wobbling in Atomic Nuclei' (CRC Press, 2024), chapter 3, pp. 62–74. DOI: 10.1201/9781032691633-3, https://doi.org/10.1201/9781032691633-3