In the relativistic Green's-function description of pseudospin symmetry (PSS) in nuclear single-particle bound and resonant states (T.-T. Sun and Z.P. Li, Phys. Lett. B 854 (2024) 138721), the breaking of PSS is quantified by the pseudo-spin-orbit (PSO) potential, whose global splitting integral VPSO=−∫dr F2(r) M−−2(dM−/dr) κ/rV_{\mathrm{PSO}}=-\int dr\,F^{2}(r)\,M_{-}^{-2}(dM_{-}/dr)\,\kappa/r with M−(ε,r)=ε−Σ(r)M_{-}(\varepsilon,r)=\varepsilon-\Sigma(r) could not be evaluated directly because M−M_{-} has zeros; the authors circumvented this by an algebraic ratio η\eta of PSO potentials between pseudospin partners, obtained under the approximation Σ(r)≈C\Sigma(r)\approx C (constant potential depth), justified only by the very small diffusivity of the Woods–Saxon potentials used. The open problem is twofold and unified by one objective: construct a general, regularized method to evaluate the PSO splitting integral that remains well defined when M−M_{-} has zeros or singular behavior — as it generically does for bound states — and establish the validity and quantitative accuracy of the constant-Σ formulas for ηE\eta_{E} and ηΓ\eta_{\Gamma} beyond the small-diffusivity regime, including other potential shapes and parameter sets.

Background

In relativistic mean-field descriptions of nuclear structure, nucleons move in a Dirac equation with a scalar potential S(r)S(r) and a vector potential V(r)V(r). Their sum Σ(r)=V(r)+S(r)\Sigma(r)=V(r)+S(r) (the Fermi-sea potential) governs the observed single-particle levels, while their difference Δ(r)=V(r)−S(r)\Delta(r)=V(r)-S(r) (the Dirac-sea potential) governs the negative-energy states. Pseudospin symmetry (PSS) concerns the near-degeneracy of pseudospin doublets — pairs of nuclear levels (n,l,j=l+1/2)(n, l, j=l+1/2) and (n−1,l+2,j=l+3/2)(n-1, l+2, j=l+3/2) that can be relabeled with a pseudo-orbital angular momentum l~=l+1\tilde{l}=l+1 and a pseudospin 1/21/2. Ginocchio showed that PSS is an exact symmetry of the Dirac Hamiltonian when dΣ(r)/dr=0d\Sigma(r)/dr=0, i.e. when Σ(r)\Sigma(r) is constant; this condition is approximately met in exotic nuclei with highly diffuse potentials, and PSS breaking is understood as a competition between the pseudo-centrifugal barrier and the pseudo-spin-orbit (PSO) potential (see the review by Liang, Meng, and Zhou for the history and status). The Dirac equation can be reduced to a Schrödinger-like equation for the lower component F(r)F(r) of the Dirac spinor, whose effective potential is energy-dependent through M−(ε,r)=ε−Σ(r)M_{-}(\varepsilon,r)=\varepsilon-\Sigma(r), where ε\varepsilon is the single-particle energy, complex (ε=E−iΓ/2\varepsilon=E-i\Gamma/2, with EE the resonance energy and Γ\Gamma its width) for resonant states. This potential separates into a pseudo-centrifugal barrier (PCB) VPCB(r)=(1/M−) l~(l~+1)/r2V_{\mathrm{PCB}}(r)=(1/M_{-})\,\tilde{l}(\tilde{l}+1)/r^{2} and a pseudo-spin-orbit part VPSO(r)=−(1/M−2)(dM−/dr) κ/rV_{\mathrm{PSO}}(r)=-(1/M_{-}^{2})(dM_{-}/dr)\,\kappa/r, where κ\kappa is the Dirac angular-momentum quantum number that distinguishes the two members of a spin or pseudospin doublet. Sun and Li (2024) studied PSS uniformly for bound and resonant states by locating the poles of the single-particle Green's function (the energy-domain propagator whose poles give exact single-particle energies and resonance widths) for spherical Woods–Saxon potentials Σ(r)=C/(1+e(r−R)/a)\Sigma(r)=C/(1+e^{(r-R)/a}) with depth CC varied from 00 (the PSS limit) to −80-80 MeV, radius R=7R=7 fm, diffuseness a=0.3a=0.3 fm, and Dirac-sea depth D=650D=650 MeV, tracing pseudospin doublets as they evolve from resonant into bound states with increasing depth. They found that the energy and width splittings of the doublets track the ratio η\eta of PSO potentials between partners, and derived practical algebraic formulas for ηE\eta_{E} and ηΓ\eta_{\Gamma} (their Eqs. (13)–(15)) under the replacement Σ(r)≈C\Sigma(r)\approx C. Two acknowledged limitations of this analysis remain unresolved. First, the 'global' PSO splitting energy of their Eq. (9), VPSO=−∫0Rboxdr F2(r) M−−2(dM−/dr) κ/rV_{\mathrm{PSO}}=-\int_{0}^{R_{\mathrm{box}}}dr\,F^{2}(r)\,M_{-}^{-2}(dM_{-}/dr)\,\kappa/r with RboxR_{\mathrm{box}} the box size of the coordinate-space calculation, 'failed' because of singular points of M−M_{-}: for a bound state (ε<0\varepsilon<0), Σ(r)\Sigma(r) rises monotonically from CC to 00, so M−M_{-} crosses zero at a turning radius where the integrand acquires a nonintegrable (r−r0)−2(r-r_{0})^{-2} pole — the quantity is not merely numerically delicate but ill-defined as an ordinary integral. Second, the constant-Σ approximation behind the η\eta formulas is motivated only by the very small diffusivity a=0.3a=0.3 fm of the specific Woods–Saxon shapes studied, and its accuracy for realistic diffuseness, other potential shapes, or other parameter sets was not established.

Problem Statement

Within the relativistic Green's-function framework of Sun and Li (Phys. Lett. B 854 (2024) 138721), provide a general, regularized method to evaluate the global pseudo-spin-orbit (PSO) splitting integral VPSO=−∫0Rboxdr F2(r) 1M−2(ε,r)dM−(ε,r)dr κr,M−(ε,r)=ε−Σ(r),V_{\mathrm{PSO}}=-\int_{0}^{R_{\mathrm{box}}}dr\,F^{2}(r)\,\frac{1}{M_{-}^{2}(\varepsilon,r)}\frac{dM_{-}(\varepsilon,r)}{dr}\,\frac{\kappa}{r},\qquad M_{-}(\varepsilon,r)=\varepsilon-\Sigma(r), in the cases — generic for bound states, where ε<0\varepsilon<0 and Σ(r)\Sigma(r) rises from its depth CC to 00, so that M−M_{-} crosses zero — in which M−M_{-} has zeros or singular behavior, so that the energy and width splittings of pseudospin doublets can be computed directly from the PSO integral rather than only through the algebraic ratio η\eta of PSO potentials between partners; and establish the validity and quantitative accuracy of the approximate replacement Σ(r)≈C\Sigma(r)\approx C underlying the practical splitting formulas for ηE\eta_{E} and ηΓ\eta_{\Gamma} (Eqs. (14)–(15) of the source) beyond the small-diffusivity (a=0.3a=0.3 fm) spherical Woods–Saxon regime in which they were derived, including other potential shapes and parameter sets. The two parts serve one scientific objective: making the PSO-based, uniform quantitative description of pseudospin-symmetry breaking — energy, width, and density splittings from resonant to bound states — well defined and controlled, rather than circumvented.

References

  1. T.-T. Sun and Z.P. Li, Uniform descriptions of pseudospin symmetries in bound and resonant states, Phys. Lett. B 854 (2024) 138721. DOI: 10.1016/j.physletb.2024.138721; arXiv:2311.14765, https://arxiv.org/abs/2311.14765
  2. T.-T. Sun, Z.P. Li, and P. Ring, Conservation and breaking of pseudospin symmetry, Phys. Lett. B 847 (2023) 138320. DOI: 10.1016/j.physletb.2023.138320; arXiv:2309.09266, https://arxiv.org/abs/2309.09266
  3. H. Liang, J. Meng, and S.-G. Zhou, Hidden pseudospin and spin symmetries and their origins in atomic nuclei, Phys. Rep. 570 (2015) 1. DOI: 10.1016/j.physrep.2014.12.005; arXiv:1411.6774, https://arxiv.org/abs/1411.6774
  4. J.N. Ginocchio, Pseudospin as a Relativistic Symmetry, Phys. Rev. Lett. 78 (1997) 436. DOI: 10.1103/PhysRevLett.78.436, https://doi.org/10.1103/PhysRevLett.78.436
  5. T.-T. Sun, W.-L. Lu, L. Qian, and Y.-X. Li, Green's function method for the spin and pseudospin symmetries in the single-particle resonant states, Phys. Rev. C 99 (2019) 034310. DOI: 10.1103/PhysRevC.99.034310; arXiv:1902.07442, https://arxiv.org/abs/1902.07442
  6. T.-T. Sun, B.-X. Li, and K. Liu, Probing spin and pseudospin symmetries in deformed nuclei by the Green's function method, Phys. Rev. C 109 (2024) 014323. DOI: 10.1103/PhysRevC.109.014323; arXiv:2310.17203, https://arxiv.org/abs/2310.17203
  7. X.-X. Shi, B. Huang, and X.-N. Cao, Exploring relativistic symmetry in shell evolution in the N=32 and N=34 isotones within a relativistic point-coupling model and the similarity renormalization group, Phys. Rev. C 111 (2025) 034304. DOI: 10.1103/PhysRevC.111.034304, https://doi.org/10.1103/PhysRevC.111.034304
  8. W. Gao, T.-T. Sun, and W.-H. Long, Single-particle resonant states in relativistic Hartree-Fock theory: A Green's function approach, Chin. Phys. C 50 (2026) 084109. DOI: 10.1088/1674-1137/ae6a85; arXiv:2603.04769, https://arxiv.org/abs/2603.04769
  9. Z.-Y. Zheng, Q. Liu, and J.-Y. Guo, Spin symmetry in antinucleon resonances within nuclei, Phys. Rev. C 113 (2026) 014314. DOI: 10.1103/j924-nypg, https://doi.org/10.1103/j924-nypg
  10. C.C. Soares, L.B. Castro, and A.S. de Castro, Scattering, bound states, and resonances in the one-dimensional Dirac equation via supersymmetric quantum mechanics, arXiv:2607.00306 (2026), https://arxiv.org/abs/2607.00306
  11. W. Zhang, S.-J. Zhang, and P.-H. Chen, Mechanism to synthesize superheavy element 120: A dinuclear system model approach with microscopic inputs, Phys. Rev. C 113 (2026) 054604. DOI: 10.1103/l4gb-dl4h; arXiv:2604.09287, https://arxiv.org/abs/2604.09287
  12. W. Sun, C.-J. Xia, and T.-T. Sun, Quarkyonic matter with strangeness in an extended relativistic mean-field model, Phys. Rev. D 113 (2026) 094021. DOI: 10.1103/bnc1-452p; arXiv:2601.17300, https://arxiv.org/abs/2601.17300