Conservation and breaking of pseudospin symmetry

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It is well known that the pseudospin symmetry (PSS) is a relativistic dynamical symmetry connected with the small component of the Dirac spinor.In this paper, the conservation and breaking of PSS in nuclear single-particle states are investigated in spherical Woods-Saxon potentials with the Green's function method which provides a novel way to precisely determine the resonance parameters and properly describe the spacial density distributions for all resonances of any widths, with which the process of symmetry restoration and breaking are perfectly shown.In the PSS limit, i.e., when the attractive scalar and repulsive vector potentials have the same magnitude but opposite sign, PSS is exactly conserved with strictly the same energy and width between the PS partners as well as identical density distributions of the lower components.In the cases of finite-depth potentials, the PSS is broken with obvious energy and width splittings and a phase shift between the density distributions.
Symmetries in the single-particle spectrum of atomic nuclei are of great importance on nuclear structures and have been extensively studied in the literature (see Refs. [1,2] and references therein).More than 50 years ago pseudospin symmetry (PSS) was found in atomic nuclei, i.e., the two single-particle states with quantum numbers (n, l, j = l + 1/2) and (n − 1, l + 2, j = l + 3/2) are quasi-degeneracy and can be redefined as the pseudospin (PS) doublets (ñ = n, l = l + 1, j = l ± 1/2) [3,4].The pseudospin symmetry (PSS) is a relativistic dynamical symmetry, which has been used to explain a number of phenomena in nuclear structures, such as deformation [5], superdeformation [6,7], identical rotational bands [8,9], magnetic moment [10], quantized alignment [11] and so on.In addition, PSS is also of great concern in atomic and molecular physics and has been discussed in some special atomic and molecular potentials [12][13][14].
Since the recognition of PSS in the nuclear spectrum, comprehensive efforts have been made to explore its origin until Ginocchio pointed out that PSS is a relativistic symmetry in the Dirac Hamiltonian which is exactly conserved when the scalar and vector potentials satisfying Σ(r) ≡ S(r) + V (r) = 0 [15].He also revealed that the pseudo-orbital angular momentum l is nothing but the orbital angular momentum of the lower component of the Dirac wave function [15] and there are certain similarities in the relativistic single-nucleon wave functions of the corresponding pseudospin doublets [16].However, there is no bound state in the PSS limit.Later, Meng et al. pointed out a more general condition of dΣ(r)/dr = 0, which can be approximately satisfied in exotic nuclei with highly diffuse potentials [17,18] and the onset of the pseudospin symmetry to a competition between the pseudo-centrifugal barrier (PCB) and the pseudospin-orbit (PSO) potential.Afterwards, PSS in nuclear spectra have been studied extensively such as PSS in deformed nuclei [19][20][21][22][23][24][25], SS in anti-nucleon spectra [26][27][28][29][30], PSS and SS in hypernuclei [31][32][33][34], perturbative interpretation of SS and PSS [30,[35][36][37][38], and PSS in supersymmetric quantum mechanics [39][40][41][42].
In recent years, the study of single-particle resonant states has attracted increasing attentions due to the essential roles in the exotic nuclei with unusual N/Z ratios.In exotic nuclei, the neutron or the proton Fermi surface is very close to the continuum threshold and the valence nucleons can be easily scattered to the single-particle resonant states in the continuum due to the pairing correlations and the couplings between the bound states and the continuum become very important [43][44][45][46].Therefore, the study of the PSS in the single-particle resonant states should also be of great interests and importance.Until now, there are already some investigations of the PSS in the single-particle resonant states.The PSS and SS in nucleon-nucleus and nucleon-nucleon scattering have been investigated [47][48][49][50].In 2004, Zhang et al. confirmed that the lower components of the Dirac wave functions for the resonant PS doublets also have similarity properties [51].Guo et al. investigated the dependence of the pseudospin breaking for the resonant states on the shape of the mean-field potential in a Woods-Saxon form [52][53][54] as well as on the ratio of neutron and proton numbers [55].In 2012, great progress has been achieved by Lu et al. in Ref. [56], where they gave a rigorous justification of PSS in single-particle resonant states and shown that the PSS in single-particle resonant states is also exactly conserved under the same condition for the PSS in bound states, i.e., Σ(r) = 0 or dΣ(r)/dr = 0 [56].However, the wave functions of the PS partners in the PSS limit are still absent.And also their research is mainly based on a radial square-well potential [57].Furthermore, a uniform description for the conservation and breaking of PSS, i.e., from the PSS limit to cases with finite-depth potentials, is highly expected.
In this letter, we will illustrate the exact conservation and breaking of PSS in the nuclear single-particle states in spherical Woods-Saxon potentials.Green's function (GF) method [58][59][60][61][62] is employed, which has been confirmed to be one of the most efficient tools for studying the single-particle resonant states with the following advantages: treating the single-particle bound and resonant states on the same footing, precisely determining the energies and widths for all resonances regardless of their widths, and describing properly the spatial density distributions [63][64][65][66].Besides, this method can describe the resonant states in any potentials without any requirement on the potential shape.
In a relativistic description, nucleons are Dirac spinors moving in a mean-field potential with an attractive scalar potential S(r) and a repulsive vector potential V (r) [67].The Dirac equation for a nucleon reads where α and β are the Dirac matrices and M is the nucleon mass.Based on the Dirac Hamiltonian ĥ(r), a relativistic single-particle Green's function G(r, r ′ ; ε) can be constructed, which obeys With a complete set of eigenstates ψ n (r) and eigenvalues ε n , the Green's function can be simply represented as which is a 2 × 2 matrix because of the upper and lower components of the Dirac spinor ψ n (r).Equation ( 3) is fully equivalent to Eq. ( 2).For a spherical nucleus, the Green's function can be expanded as where Y l jm (θ, φ) is the spin spherical harmonic, G κ (r, r ′ ; ε) is the radial Green's function, and the quantum number κ = (−1) j+l+1/2 (j + 1/2).The Eq. ( 2) can be reduced as where Σ(r) ≡ V (r)+S(r), ∆(r) = V (r)−S(r), and I is a two-dimensional unit matrix.A radial Green's function G κ (r, r ′ ; ε) could be constructed with exact asymptotic behaviors of the Dirac wave functions for bound states and continuum.For these details, please see Refs.[62,63] To study the conservation and breaking of PSS in resonant states, radial Woods-Saxon potentials are consid-ered both for Σ(r) and ∆(r), Here, the potential depths C = −66 MeV and D = 650 MeV, the width R = 7 fm, and the diffusivity parameter a = 0.3 fm are adopted.
On the single-particle complex energy plane, bound and resonant states respectively distribute on the negative real energy axis and in the fourth quadrant.The energy ε n is real for bound states while complex for the resonant states and in latter case ε n = E − iΓ/2 with E and Γ respectively being the resonant energy and width.Meanwhile, these eigenvalues are the poles of Green's function as shown in Eq. ( 3).Thus, a direct approach that by searching for the poles of Green's function to determine the single-particle energies ε n has been proposed naturally [64][65][66].In practice, one can do this by calculating the integral function of the Green's function G κ (ε) for each partial wave κ at different energies ε [66] G κ (ε) = dr |G (11)   κ (r, r; ε)| + |G (22)   κ (r, r; ε)| , (7) where |G  (r, r; ε)| are the moduli of the Green's functions respectively for the "11" and "22" matrix elements.To search for the bound and resonant states, Green's functions in a wide energy range are calculated by scanning the single-particle energy ε.For the bound states, the energies ε are taken along the negative real energy axis.For the resonant states, the energies ε are complex ε = ε r + iε i which are scanned in the fourth quadrant of the complex energy plane ε, both along the real and imaginary energy axes.In Fig. 1, the resonant parameters of the state 3d 5/2 are exactly determined to be E = 2.2728 MeV and Γ/2 = 1.9949MeV by searching for the poles of the Green's functions in the fourth quadrant of the complex energy plane ε, where a sharp peak is observed at ε r = 2.2728 MeV and ε i = −1.9949MeV.Calculations are done with an energy step of 0.1 keV for the integral functions G κ (ε) in a coordinate space with size R max = 20 fm and a step of dr = 0.05 fm.This approach has been certified to be highly effective for all resonant states regardless of whether they are wide or narrow [65,66].
Besides, with the Green's function method, the density distributions in the coordinate space can also be examined by exploring ρ κ (r, ε) defined at the energy ε = E, Im G (11)   κ (r, r; E) + G (22)   κ (r, r; E) , where the terms G (r, r; E) are respectively related with the upper and lower components of the Dirac wave functions (c.f.Eq. ( 3)).In Fig. 2   the density distributions for the resonant state 3d 5/2 are also plotted, with the red and blue lines the contributions from the upper and lower components.In order to better display the density distribution, here and hereafter, we adjust the highest peak of ρ κ (r, ε) to be 1.0 fm −1 •MeV −1 and ensure relative sizes of the different components remaining unchanged.Since 3d 5/2 is a low lying narrow resonant state, the density distribution of the lower component is reduced to be zero very soon while very slight oscillation can be observed in the upper component.
In the following, the PSS in resonant states will be studied with the Green's function method.In Fig. 3, we show the solutions in different potential depths on the complex energy plane for the PSS doublets with pesudospin angular momentum l = 3, i.e., d 5/2 with κ = −3 and g 7/2 with κ = 4.In the PSS limit, i.e., the potential depth C = 0, all the roots locate in the lower half-plane and there is no bound states.Three pairs of resonant PS doublets with exact the same energy and width are obtained, indicating the exact conservation of PSS in resonant states.Besides, one single intruder state 1d 5/2 appears near the continuum threshold.With finite potential depths, one finds the breaking of the PSS with obvious energy and width splitting between the PS partners.More in details, for most PS partners, g 7/2 with pseudospin s = +1/2 owns lower energy and smaller width compared with the PS partner d 5/2 with s = −1/2 due to the higher PCB potential of g 7/2 .One exception is the PS partner (3d 5/2 , 2g 7/2 ) obtained when C = −66 MeV, the energies of which are ε(3d 5/2 ) = 2.2728 − i1.9949 MeV and ε(2g 7/2 ) = 2.5422 − i0.1019 MeV, respectively.Meanwhile, PS partners move down and some resonant PS partners evolve to be bound states.For PS doublets with other values of l, similar behaviors concerning the exact conservation and the breaking of the PSS could be observed.
To study the conservation and breaking of PSS, the similarities of the lower component of the Dirac wave functions for the PS doublets are also examined.In Fig. 4, the spacial density distributions ρ κ (r, ε) of the PS partners 3d 5/2 and 2g 7/2 in different potential depths C are plotted.The left and right columns are respectively the contributions from the upper and lower components of Dirac wave functions.In the PSS limit, i.e., C = 0, the density distributions for the PS partner are exactly the same for the lower component while differ one node for the upper component, which provides a direct evidence for the exact conservation of the PSS in resonant states and also certifies that PSS is a symmetry of the Dirac Hamiltonian related with the lower component of the Dirac spinor.In the case of the finite-depth potentials, the lower components of the density distributions for the PS partners are no longer exactly the same, but it still remains strong similarity.Their difference is manifested as an obvious phase shift, which increases with the deepening of the potential depths.For example, when the potential depth C = −45 MeV, there is almost one π phase shift between the PS partners for the density distributions outside the potential in the area of coordinate r > 7.5 fm.
In summary, the conservation and breaking of the PSS in nuclear single-particle states are investigated within the relativistic framework by exploring the poles of Green's function in spherical Woods-Saxon potentials.Great advantages of the Green's function method that precisely determining the energies and widths for all resonances regardless of their widths and describing properly the spatial density distributions provide a good platform to study restoration and breaking of PSS.In the PSS limit, i.e., Σ(r) ≡ V (r) + S(r) = 0, the PSS in resonant states is confirmed to be strictly conserved with the exact same energy and width between the PS partners.Besides, we also find identical density distributions of the small components for the first time, which provides a direct evidence that the PSS is a relativistic dynamical symmetry connected with the small component of the Dirac spinor.In the cases with finite-depth potentials, the PSS is broken with obvious energy and width splitting for the PS partners as well as a phase shift between the spatial density distributions of the small components.

FIG. 2 :
FIG.2:(Color online) Density distributions ρκ(r, ε) of the resonant state 3d 5/2 with the contributions from the upper and lower components of the Dirac wave functions.

FIG. 4 :
FIG. 4: (Color online).Evolutions of the density distributions ρκ(r, ε) for the PS doublets 3d 5/2 and 2g 7/2 from the PSS limit with C = 0 MeV to the cases with finite-depth potentials C = −15, −33, −45 MeV.Densities plotted in the left and right columns are contributed by the upper and lower components of the Dirac wave functions, respectively.