Pseudospin symmetry as a relativistic dynamical symmetry in the nucleus

Abstract

Pseudospin symmetry in nuclei is investigated by solving the Dirac equation with Woods-Saxon scalar and vector radial potentials, and studying the correlation of the energy splittings of pseudospin partners with the nuclear potential parameters. The pseudospin interaction is related to a pseudospin-orbit term that arises in a Schroedinger-like equation for the lower component of the Dirac spinor. We show that the contribution from this term to the energy splittings of pseudospin partners is large. The near pseudospin degeneracy results from a significant cancelation among the different terms in that equation, manifesting the dynamical character of this symmetry in the nucleus. We analyze the isospin dependence of the pseudospin symmetry and find that its dynamical character is behind the different pseudospin splittings observed in neutron and proton spectra of nuclei.Comment: 13 pages, 9 figures, uses REVTeX4 macro

Spin and pseudospin symmetries in the Dirac equation with central Coulomb potentials

“…In this case, one cannot have exact spin symmetry (i.e., V = S) since now − = −(V − S) acts as binding potential. This behavior has been related by several authors to the perturbative nature of spin and pseudospin symmetry, namely that in the case of fermions (antifermions) the pseudospin (spin) symmetry is nonperturbative for nuclear mean-field potentials [6,[8][9][10]. Note, however, that for harmonic oscillator systems one can have exact pseudospin and spin symmetries for fermions and antifermions respectively [11].…”

“…In previous works in which the perturbative nature of pseudospin (or spin in the case of antifermions) was related to the onset of the respective symmetry, one would either examine the effect the pseudospin-orbit or spin-orbit term on the energy splittings of the respective doublets [6,8,9] or analyze the convergence of a perturbation expansion in the respective potential strength [10]. In Ref.…”

“…These terms can be obtained from the second-order radial equations for g κ (r) and f κ (r), respectively (see, for instance, Ref. [9]) (hc) 2 …”

“…(45) and (47) are of first order in α and α respectively. Since those terms are responsible, when potentials go to zero, for breaking spin and pseudospin symmetries, respectively, one could associate them with the spin-orbit and pseudospinorbit coupling terms as was done with Woods-Saxon-type potentials [6,9]. However, contrary to what happens with those potentials, in the present case, because of the functional form of Coulomb potentials, the spin-orbit term (51) has a double pole at r =hc α /(E + mc 2 ), so it cannot be calculated separately from the so-called Darwin term, coming from the derivative term in Eq.…”

Further investigation of relativistic symmetry with the similarity renormalization group

“…In Refs. [22][23][24], it was pointed out that the observed pseudospin splitting arises from a cancelation of the several energy components, and the PSS in nuclei has a dynamical character. A similar conclusion was reached in Refs.…”