Nonrelativistic expansion of Dirac equation with spherical scalar and vector potentials by similarity renormalization group

Abstract

By following the conventional similarity renormalization group (SRG) expansion of the Dirac equation developed in [J.-Y. Guo, Phys. Rev. C 85, 021302 (2012)], we work out the analytic expression of the 1/M 4 order and verify the convergence of this method. As a step further, the reconstituted SRG method is proposed by using the re-summation technique. The speed of convergence of the reconstituted SRG becomes much faster than the conventional one, and the single-particle densities with the reconstituted SRG are also almost identical to the exact values.

Nonrelativistic expansion of the Dirac equation with spherical scalar and vector potentials by a reconstituted Foldy-Wouthuysen transformation

“…In order to make the comparisons and discussions with the corresponding results obtained by the SRG methods in Ref. [20], we use the Woods-Saxon forms for the scalar and vector potentials, Σ(r) = Σ 0 f (a 0 , r 0 , r) and ∆(r) = ∆ 0 f (a 0 , r 0 , r), with f (a 0 , r 0 , r) = 1…”

“…Thus the detailed 1/M 4 -order results of the conventional SRG method derived in Ref. [20] can help us investigate the results in Fig. 1 Table. VIII in Ref.…”

“…In contrast, the orbital that shows the biggest positive difference is the 1h 9/2 state at −14.027 MeV, whose deviation from the corresponding exact value is 89 keV. In order to understand these energy differences for the states with large angular momenta l, we recall that, although a discrepancy between the results of the conventional SRG method [20] and FW transformation [32] appears starting from the 1/M 3 order, their spectrum of the single-particle energies are identical to each other [32]. Thus the detailed 1/M 4 -order results of the conventional SRG method derived in Ref.…”

“…Inspired by the reconstituted SRG method [20], here we develop the reconstituted FW transformation. Replacing the bare mass M with the Dirac mass M * = M + S and considering the Hermitian property of the Hamiltonian, the corresponding operators are defined as…”

“…In addition to the improvement of the convergence, the single-particle densities ρ v (r) = ψ † (r)ψ(r), which are also called the signle-particle vector or baryon densities, calculated by the novel reconstituted SRG are also more identical to the exact values than the conventional ones [20]. In contrast, in order to calculate the singleparticle scalar densities ρ s (r) = ψ † (r)γ 0 ψ(r), the γ 0 matrix should be transformed in the same way as the Dirac Hamiltonian.…”