Pseudo-spin symmetry in density-dependent relativistic Hartree-Fock theory

The pseudo-spin symmetry (PSS) is investigated in the density-dependent relativistic Hartree-Fock theory by taking {the} doubly magic nucleus $^{132}$Sn as a representative. It is found that the Fock terms bring significant contributions to the pseudo-spin orbital potentials (PSOP) and make it comparable to the pseudo-centrifugal barrier (PCB). However, these Fock terms in the PSOP are counteracted by other exchange terms due to the non-locality of the exchange potentials. The pseudo-spin orbital splitting indicates that the PSS is preserved well for the partner states $\lrb{\nu 3s_{1/2}, \nu2d_{3/2}}$ of $^{132}$Sn in the relativistic Hartree-Fock theory.

approach with a no-sea approximation, the nucleons interact via the exchange of mesons and photons. For the description of nuclear structure, the relevance of relativity is not in the need of relativistic kinematics but it lies in a covariant formulation which maintains the distinction between scalar and vector fields (more precisely, the zeroth component of the Lorentz four-vector field). The representations with large scalar and vector fields in nuclei (of the order of a few hundred MeV) provide more efficient descriptions of nuclear systems than non-relativistic approaches, for example the origin of the nuclear spin-orbit potential [21] and that of the PSS [5,9].
Although there exist some attempts to include the exchange terms in the relativistic description of nuclear matter and finite nuclei [22,23,24], the relativistic Hartree-Fock (RHF) method was still not comparable with the RMF theory in the quantitative description of nuclear systems. Recently, it was shown that the density-dependent relativistic Hartree-Fock (DDRHF) theory [25] gives a successful quantitative description of nuclear matter and finite nuclei at the same level as the RMF [26]. Compared with RMF, the relevance of the relativity is still kept well in DDRHF although the covariant formulation becomes much more complicated due to the exchange terms. Since the DDRHF describes quite well nuclear systems, it is worthwhile to investigate the role of exchange terms in the PSS, especially the influence of the non-locality on the conservation of the PSS. In this work, we study the role of the Fock terms on the conservation of PSS in the covariant relativistic Hartree-Fock theory. The numerical study is done with the effective Lagrangian PKO1 [25] in the case of DDRHF, and the results are compared with those obtained with the RMF model PKDD [20].
For spherical nuclei, the Dirac spinor can be written as, where χ 1 2 (τ a ) is the isospinor, G a and F a correspond to the radial parts of upper and lower components, respectively. Y la jama is the spinor spherical harmonics and Y l ′ a jama (r) = −σ rY la jama (r) with l ′ a = 2j a − l a . In the spherical nuclei, the total angular momentum j a , its projection on the z axis m a andκ = −β(σ ·L + 1) are conserved. The eigenvalues ofκ are κ a = ±(j a + 1 2 ) (− for j a = l a + 1 2 and + for j a = l a − 1 2 ). In the following, the sub-index a will be omitted in the notations. Within the DDRHF, the radial Dirac equations, i.e., the relativistic Hartree-Fock equations for spherical nuclei, are expressed as the coupled differential-integral equations [22,26] where the scalar potential Σ S and the time component of the vector potential Σ 0 contain the contributions from the direct terms and the rearrangement term due to the density-dependence of the meson-nucleon couplings. The X and Y functions represent the results of the non-local Fock potentials acting on F and G, respectively [26]. By introducing the functions X G , X F , Y G and Y F as in Ref. [22], the coupled differential-integral equations (2) can be transformed into the equivalent local ones, The coupled equations (4) can be solved by using the same numerical method as in RMF [27]. In Eqs. (4), the functions X G , X F , Y G and Y F might be taken as the effective potentials from the exchange (Fock) terms in the DDRHF. Eqs. (4) must be solved iteratively since the potentials depend on the solution (G, F ).
¿From the radial Dirac equations (4), the Schrödinger-type equation for the lower component F is obtained as, with where V PCB and V PSO correspond to the pseudo-centrifugal barrier (PCB) and pseudo-spin orbital potential (PSOP), respectively. In the limit of V PSO = 0, the pseudo-spin becomes a good symmetry and the PSS can be labeled by the pseudo radial numberñ = n − 1, pseudo-orbitl = l ′ , and pseudo-spins = s = 1 2 , with the total angular momentum j =l ±s for the two partner states. For instance, the partners are ns 1/2 , (n − 1)d 3/2 forl = 1, np 3/2 , (n − 1)f 5/2 forl = 2, etc. Notice that X G and Y F in the V PSO are new contributions from the Fock terms compared with RMF. The potential V 2 entirely originates from the Fock contributions. The direct (Hartree) and exchange (Fock) contributions of the PSOP and V 1 can be separated as, while ∆ comes from both the Hartree and Fock contributions.
For both DDRHF (filled symbols) and RMF (open symbols) results, the pseudo-spin splitting between ν3s 1/2 and ν2d 3/2 is more than ten times smaller than that between ν2s 1/2 and ν1d 3/2 . As shown in Fig. 1 and Fig. 2, there exist some differences in the single-particle energies (especially the deeply bound states) between the DDRHF and RMF, whereas the monotonous decreasing behavior of ∆E PSO with the decreasing binding energies is observed in the both models. As a reference, the spin-orbit splitting ∆E SO = (E lj=l−1/2 − E lj=l+1/2 )/(2l + 1) versus the average binding energyĒ SO = (E lj+l−1/2 + E lj=l+1/2 )/2 is also shown in Fig. 2 for the spin-orbit partners 1p, 1d, 1f , and 1g, and 2p, 2d. A clear difference between the pseudo-spin orbital and spin-orbit splittings can be seen in their energy-dependence, i.e., a strong energy-dependence is found for the pseudo-spin orbital splitting, while the spin-orbit splitting shows a weak energy-dependence.
This difference can be understood from the comparison between the expressions for the PSOP (7c) and the corresponding spin-orbit potential. The energy E and the potential ∆ in the denominator ∆ − E of Eq. (7c) are comparable so that it brings a strong energy-dependence for the PSOP. On the other hand, the corresponding denominator of the spin-orbit potential in RHF is V − E [9], which gives much weaker energy-dependence because the single-particle energy E is much smaller than the potential V : the value of E is a few tens MeV while V is several hundred MeV.   In RMF it has been proved that the PSS is well obeyed if the PCB is much stronger than the PSOP [9]. Fig. 5 shows the PCB and PSOP multiplied by the factor F 2 / V D − E for the pseudospin partners ν2s 1/2 , ν1d 3/2 and ν3s 1/2 , ν2d 3/2 in 132 Sn. Due to the denominator ∆ − E in Eq.
(7c), there exists singular points at r ≃ 6 fm for the partner ν2s 1/2 , ν1d 3/2 , and at r ≃ 7.5 fm for ν3s 1/2 , ν2d 3/2 . The other local peaks in the PSOP are due to the nodes of upper component G. For the s states (l = 0), the PCB is much stronger than the PSOP since the contributions for the PSOP around the nodes or the singular points are more or less mutually cancelling. On the other hand, for the d states, the PSOP are comparable to the PCB even after taking account of the cancellation around the nodes or singular points. Comparing the shapes of the PSOP in Fig.   5 and X G , Y F in Fig. 4, one can find that the Fock terms present significant contributions to the PSOP, especially for the d states in the inner part of the nucleus. It is also seen that the Fock terms in Fig. 6 have substantial contributions to the PSOP in Fig. 5. ¿From Eq. (9), one can estimate the contributions of the potentials V PCB ,V D andV E to the single-particle energy E. For example, the PCB contribution can be expressed as, The results calculated by the DDRHF with PKO1 and the RMF with PKDD are shown in Table   I for the pseudo-spin partners 1p and 2p. For both the DDRHF and RMF, the terms F ′′ , ∆ D andV D in Eq. (9) show substantial differences between the partner states ν2s 1/2 , ν1d 3/2 and ν3s 1/2 , ν2d 3/2 whereas the differences in the PCB and the exchange termsV F are negligible. The differences in F ′′ and ∆ D reflect those of the lower components in the two pseudo-spin partners as shown in the left panel of Fig. 3.
The large differences between the partner states can be seen in the F ′′ , ∆ D andV D contributions.
However, these three terms cancel largely one another and the PSS is preserved to a good degree, especially in the partners ν3s 1/2 , ν2d 3/2 in both DDRHF and DDRMF. The Fock termsV E become small although each term on the right hand side of Eq. (10b) shows appreciable differences  between the partner states: for example, V E PSO is large for d-states. The contributions from the exchange potentials V E PSO and V E 1 in Eq. (9) are shown in Fig. 6. For the s states, the exchange terms V E PSO and V E 1 give small contributions because of their changing signs. On the other hand, for the d states there are significant cancellations between V E PSO and V E 1 , especially in the inner part of the nucleus. Although the Fock terms bring substantial contributions to the PSOP, these contributions are cancelled by the other exchange term V E 1 , which stems mainly from the nonlocality (the state-dependence) of the exchange potentials. Thus, the PSS still remains preserved even after the inclusion of Fock terms due to these large cancellations as can be seen in Table I.
Let us now discuss the reason why the cancellations among the exchange terms occur. From the similar radial dependence between the non-local terms X (Y ) and Dirac wave functions F (G) in Fig. 3, we might be able to validate the following relations, where X 0 and Y 0 are supposed to be state-independent potentials due to the Fock terms. Using Eq.
(12), the non-local RHF equations (2) can be reduced to local ones similar to the RMF equations in which the terms X G and Y F do not appear in the PSOP. Thus, the realization of the PSS will be similar to the RMF case. Therefore, the cancellation of the Fock contributions in Table I is not occasional but it is because of the similar radial dependence between the Fock-related terms (X, Y ) and the wave functions (F, G).
In summary, the PSS in the DDRHF theory was investigated in the doubly magic nucleus 132 Sn.