Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 47 (2014)

RIKEN Accel. Prog. Rep. 00 (0)Spin-orbit eﬀects on pseudospin symmetry†
H. Z. Liang,∗1 S. H. Shen,∗2 P. W. Zhao,∗2 S. Q. Zhang,∗2 and J. Meng∗1,∗2Pseudospin symmetry (PSS)1,2) was introduced in1969 to explain the near degeneracy between pairs ofnuclear single-particle states with the quantum numbers (n−1, l+2, j = l+3/2) and (n, l, j = l+1/2). They
are regarded as the pseudospin doublets with modiﬁed
quantum numbers (ñ = n − 1, ˜l = l + 1, j = ˜l ± 1/2).
Although this concept was introduced for more than 40years ago, the origin of PSS and its breaking mechanism in realistic nuclei have not been fully understood.
Speciﬁcally, determining whether its nature is perturbative remains an unsolved problem.
Recently, we used the perturbation theory to investigate the symmetries of the Dirac Hamiltonian andtheir breaking in realistic nuclei3) , which provides aclear and quantitative way for investigating the perturbative nature of PSS. On the other hand, supersymmetric (SUSY) quantum mechanics can provide aPSS-breaking potential without singularity, and naturally interpret the unique feature that all states with
˜l > 0 have their own pseudospin partners except for theintruder states4) . Then, the similarity renormalizationgroup (SRG) technique ﬁlls the gap between the perturbation calculations and the SUSY descriptions bytransforming the Dirac Hamiltonian into a diagonal
form and keeping every operator Hermitian5,6) .
Therefore, understanding the PSS and its breakingmechanism in a quantitative manner by combining theSRG technique, SUSY quantum mechanics, and perturbation theory is considered promising.
Here, we highlight the PSS-breaking potentials
ṼPSO (r), which are derived from the Dirac equation
with the SRG and SUSY transformations.
In the upper panel of Fig. 1, the ṼPSO (r) obtained
without and with the spin-orbit (SO) term are shownfor the f˜ orbitals. These potentials show several
special features, which are crucial for understanding
the PSS: (i) They are regular functions of r. (ii)
Their amplitudes directly determine the sizes of reduced pseudospin-orbit (PSO) splittings ∆EPSO ≡(Ej< − Ej> )/(2˜l + 1) according to the perturbation
theory. (iii) Their shape, being negative at small radius but positive at large radius with a node at thesurface region, can explain the general tendency of
the PSO splittings becoming smaller with increasing
single-particle energies.
To identify the SO eﬀects, the ṼPSO (r) obtained with
the SO term is further decomposed into the contributions of the indirect and direct SO eﬀects, because theformer one represents the SO eﬀects on ṼPSO (r) via the†∗1
∗2

superpotentials, while the latter is the SO potential itself. Comparison with the result obtained without theSO term shows that the indirect eﬀect is small andeventually results in less inﬂuence due to the cancellation between the inner and outer regions. On theother hand, the SO potential is always positive with apeak at surface. It substantially raises the ṼPSO (r), inparticular for the surface region.
All of these properties are shown in the lower panel of Fig. 1, in which ∆EPSO are shown as a functionof Eav = (Ej< + Ej> )/2. ∆EPSO match the amplitudes of ṼPSO (r). The decreasing PSO splittings withincreasing single-particle energies is due to the special
shape of ṼPSO (r). The SO term reduces ∆EPSO systematically, and this eﬀect can be understood now ina quantitative manner.

Fig. 1. Upper panel: PSS-breaking potentials ṼPSO (r) obtained with and without SO term. The former one isdecomposed into the indirect and direct the SO eﬀects.
Lower panel: ∆EPSO vs Eav with and without the SOl ∓ 1/2 states.
term, where j< , j> stand for the ˜References
1) A. Arima, M. Harvey, and K. Shimizu: Phys. Lett. B
30, 517 (1969).
2) K. Hecht and A. Adler: Nucl. Phys. A 137, 129 (1969).
3) H. Z. Liang et al.: Phys. Rev. C 83, 041301(R) (2011).
4) S. Typel: Nucl. Phys. A 806, 156 (2008).
5) J.-Y. Guo: Phys. Rev. C 85, 021302(R) (2012).
6) J.-Y. Guo et al.: Phys. Rev. Lett. 112, 062502 (2014).

Condensed from the articles in Phys. Rev. C 87, 014334(2013) and ibid. 88, 024311 (2013)
RIKEN Nishina Center
School of Physics, Peking University- 50 -1