Ⅱ-2. Nuclear Physics (Theory)

RIKEN Accel. Prog. Rep. 53

Non-relativistic expansion of Dirac equation with spherical vector andscalar potentials†
Y. X. Guo∗1,∗2,∗3 and H. Z. Liang∗2,∗1Since the 1970s, the density function theory (DFT)in both the non-relativistic and relativistic frameworks has succeeded in microscopically describing theground-state and excited-state properties of thousandsof nuclei in a self-consistent manner. However, theconnection between these two frameworks remains unclear. The non-relativistic expansion of the Dirac equation is considered to be a potential bridge connecting
these two frameworks.1,2)
In 2012, Guo3) ﬁrst applied the similarity renormalization method (SRG)4,5) to perform the nonrelativistic expansion of the Dirac equation. By using
SRG, the Dirac Hamiltonian can be transformed intoa diagonal form after inﬁnite steps of unitary transformations, i.e., the eigenequations for the upper andlower components of the Dirac spinors can be decoupled. Furthermore, the Hamiltonian H thus obtained
was expanded in powers of 1/M (with M being thebare mass of the nucleon).
The results up to the 1/M 3 order were calculated.3)
However, the diﬀerences between the corresponding
results and the exact ones were still approximately
1 MeV. Therefore, it is necessary to include higherorder corrections. In this work, the results up to the1/M 4 order are calculated, and the diﬀerences between
the calculated and exact values are less than 0.2 MeV
for all states.
From a simple observation, some geometric progressions are determined in the results of the conventional SRG method. This observation is reminiscentof the idea of resummation. In the reconstituted SRG
method, replacing M with the Dirac mass M ∗ = M +S(with S being the scalar potential) not only yields thenon-relativistic expansion up to a certain order, butalso sums up the terms that belong to their families
up to inﬁnite order. Consequently, the convergence ofthe reconstituted SRG is much faster than that of theconventional SRG. The spectrum of the HamiltonianH̃ thus obtained is shown in Fig. 1.
The Foldy-Wouthuysen (FW) transformation6–9) isanother non-relativistic expansion method that hasbeen widely used. Both the FW transformation andSRG method provide a systematic way to perform thenon-relativistic expansion of the Dirac equation up toan arbitrary order. Thus, we also apply the FW transformation to a general case of the covariant DFT. In†∗1
∗2
∗3-10
3p3/2
2f 5/2E (MeV)-20-302f 7/2

-40

1h9/2

2p3/2

2d5/2

1f 5/2-50
IIIIII

Exact

1g7/2

I

IIIII

Exact

Fig. 1. Energy spectrum of H̃ for the four pseudospin partners calculated using the reconstituted SRG method.
The first, second, and third columns show the singleparticle energies up to the first, second, and third orders, respectively. The last column, labeled “Exact,”shows the eigenenergies of the Dirac equation.

this work, the results with the FW transformation havealso been obtained up to the 1/M 4 order and comparedwith the corresponding results obtained using conventional SRG method. It seems that the FW transformation and the conventional SRG method produce thesame results up to the second order, but they show differences from the third order. By introducing a blockdiagonal transformation Ξ, we determined that the results obtained using the conventional SRG method areequal to those obtained using the FW transformation
with an additional unitary transformation Ξ. Consequently, the spectrum of the single-particle energy obtained using the FW transformation is identical to thatobtained using the SRG method.
With the applications of the SRG method and FW
transformation, the bridge connecting the DFT in therelativistic and non-relativistic frameworks is now conceivable.
References

1) P. G. Reinhard, Rep. Prog. Phys. 52, 439–514 (1989).
2) M. Bender, P. H. Heenen, P. G. Reinhard, Rev. Mod.
Phys. 75, 121–180 (2003).
3) J. Y. Guo, Phys. Rev. C 85, 021302 (2012).
4) F. Wegner, Ann. Phys. Berlin 506, 77–91 (1994).
5) A. B. Bylev, H. J. Pirner, Phys. Lett. B 428, 329–333(1998).
6) M. H. L. Pryce, Proc. Royal Soc. A 195, 62–81 (1948).
7) L. L. Foldy, S. A. Wouthuysen, Phys. Rev. 78, 29–36(1950).
8) S. Tani, Prog. Theor. Phys. 16, 267–285 (1951).
9) L. L. Foldy, Phys. Rev. 87, 688–693 (1952).

Condensed from the articles in Phys. Rev. C 99, 054324(2019) and Chin. Phys. C 43, 114105 (2019)
Department of Physics, The University of Tokyo
RIKEN Nishina Center
Department of Modern Physics, University of Science andTechnology of China- 67 -