Higher-order similarity-renormalization-group and effective-mass corrections to pseudospin symmetry breaking, and the origin of its better conservation than spin symmetry in nuclei

Pseudospin symmetry (PSS) — the quasi-degeneracy of single-nucleon doublets $(n, l, j=l+1/2)$ and $(n-1, l+2, j=l+3/2)$ — was traced by Shen, Liang, Zhao, Zhang, and Meng [Phys. Rev. C 88, 024311 (2013)] to a supersymmetric partner Hamiltonian obtained after a similarity renormalization group (SRG) transformation of the Dirac equation, and the pseudospin splittings of a Woods-Saxon potential were shown to be perturbative even when the leading spin-orbit term of order $1/M^{2}$ is kept. That analysis, however, deliberately omitted the effective-mass-related (Darwin) term and all higher-order SRG corrections, and worked with a fixed potential rather than a self-consistent mean field. The open problem is to complete this program: quantify the impact of the omitted higher-order SRG terms and of an explicit, self-consistently coupled effective mass on the pseudospin-orbit potential and on the perturbative character of PSS, and establish the intrinsic relations between the spin-orbit potential and the central potential or the nucleon effective mass that explain why PSS is typically better conserved than spin symmetry in realistic nuclei.

Background

In the nuclear shell model, single-particle states come in two types of near-degenerate pairs. Spin doublets $(n, l, j = l\pm 1/2)$ are split by the strong spin-orbit interaction that generates the traditional magic numbers, and the approximate degeneracy of the two $j$ values for fixed $n$ and $l$ is called spin symmetry (SS). Pseudospin symmetry (PSS) is the empirical quasi-degeneracy of the pairs $(n, l, j=l+1/2)$ and $(n-1, l+2, j=l+3/2)$, relabelled as pseudospin doublets $(\tilde{n}=n-1, \tilde{l}=l+1, j=\tilde{l}\pm 1/2)$; it underlies pseudo-SU(3) models of deformed and superdeformed nuclei and is relevant to shell evolution in exotic nuclei. In relativistic descriptions a nucleon moves in a Dirac equation with a large scalar potential $S(r)$ (about $-350$ to $-400$ MeV deep) and vector potential $V(r)$ (about $+300$ MeV). Ginocchio showed in 1997 that PSS is exactly conserved when $\Sigma(r) \equiv S(r)+V(r) = 0$, and the condition $d\Sigma/dr = 0$ was later identified as the general exact limit; dually, SS is controlled by $\Delta(r) \equiv V(r)-S(r)$ through $d\Delta/dr = 0$. Since $\Sigma$ is only about $70$ MeV deep while $\Delta$ is about $700$ MeV deep, their surface gradients have very different sizes, which is the empirical backdrop to the question of why PSS near the Fermi surface is often better respected than SS.

Two technical tools define the framework of this problem. First, the similarity renormalization group (SRG) — a continuous unitary transformation that drives the Dirac Hamiltonian into block-diagonal form, decoupling the large and small components of the Dirac spinor — rewrites the Dirac equation as a Hermitian Schrödinger-like equation expanded in powers of $1/M$ (the nucleon mass $M$), whose relativistic correction terms include a spin-orbit (SO) potential (first appearing at order $1/M^{2}$) and a Darwin term (a derivative operator tied to the nucleon effective mass $M^{*}(r) = M + \Sigma(r)$, the 'Dirac mass'). Hermiticity is what makes quantitative perturbation theory possible. Second, supersymmetric (SUSY) quantum mechanics factorizes a Hamiltonian $H_{1} = B^{+}B^{-}$ through first-order operators built from a superpotential, producing a partner Hamiltonian $H_{2} = B^{-}B^{+}$ with an isospectral (one-to-one) set of eigenstates. For the radial Schrödinger equation with the reduced superpotentials $q_{\kappa}(r)$ ($\kappa$ is the Dirac quantum number labelling $(l,j)$), the SUSY partner Hamiltonian $\tilde{H}$ exhibits an explicit $\kappa(\kappa-1)/r^{2}$ centrifugal barrier — exactly the structure shared by pseudospin doublets — so the PSS-conserving dynamics sits naturally in $\tilde{H}$, and the PSS-breaking part $\tilde{W}^{\rm PSS}$ can be treated as a perturbation about the PSS-conserving limit $\tilde{H}_{0}^{\rm PSS}$.

In part I of the series [Phys. Rev. C 87, 014334 (2013)] this program was carried out at lowest SRG order (no SO term). In part II [Phys. Rev. C 88, 024311 (2013)], using a spherical Woods-Saxon potential fitted to the neutron potential of ${}^{132}$Sn with bare mass $M = 939$ MeV, the authors included the leading SO term arising at order $1/M^{2}$: the SO potential itself (a direct effect) substantially reduces the pseudospin-orbit (PSO) splittings, while its indirect effect through the modified superpotentials $q_{\kappa}(r)$ is small, and the perturbative nature of PSS survives. However, relative to self-consistent relativistic mean-field calculations, the higher-order SO terms and all effective-mass terms (including the Darwin term) were neglected, and the potential was externally imposed rather than self-consistent. The authors' own summary therefore poses the completion of the program as an open task, noting in particular that the intrinsic relations between the SO potential and the central potential or the effective mass should be taken into account to explain why PSS is conserved better than SS in realistic nuclei.

Problem Statement

Complete the quantitative SRG + supersymmetric-quantum-mechanics + perturbation-theory account of pseudospin symmetry initiated in Phys. Rev. C 87, 014334 and 88, 024311 (2013). Concretely: (1) include the terms omitted there — the effective-mass-related Darwin term and the higher-order spin-orbit terms of the $1/M$ SRG expansion, with an explicit and self-consistently determined effective mass $M^{*}(r)$ — and determine their quantitative impact on the pseudospin-orbit (PSO) potential $\tilde{V}_{\rm PSO}(r)$, on the pseudospin splittings $\Delta E_{\rm PSO}$, and on whether PSS remains perturbative (i.e., whether the PSS-breaking part stays a small perturbation about the PSS-conserving partner Hamiltonian $\tilde{H}_{0}^{\rm PSS}$), extending the analysis from the fitted Woods-Saxon potential to self-consistent Dirac mean-field potentials; and (2) establish the intrinsic relations between the spin-orbit potential and the central potential or the nucleon effective mass in such a self-consistent framework that explain why PSS is typically better conserved than spin symmetry in realistic nuclei. All the parts of (1) and (2) above are jointly required and none is independently sufficient: the inclusion of the Darwin and higher-order spin-orbit terms with a self-consistently determined $M^{*}(r)$; the quantitative impact on $\tilde{V}_{\rm PSO}(r)$ and on $\Delta E_{\rm PSO}$; the determination of whether PSS remains perturbative; the extension from the fitted Woods-Saxon potential to self-consistent Dirac mean-field potentials; and the intrinsic relations behind the better conservation of PSS than SS. 'Quantitative impact' is to be reported as explicit results for the change produced by each omitted term (its magnitude and radial behaviour), not merely its sign; 'intrinsic relations' are to be stated as explicit relations — analytic or numerical — between the spin-orbit potential and the central potential or the nucleon effective mass. Together these deliverables serve one overarching objective: a complete, quantitative, and self-consistent theory of pseudospin-symmetry breaking in realistic nuclei within this framework, against which the perturbative character of PSS and the SS/PSS conservation asymmetry can be decided.

References

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