We develop a complex scaling method for describing the resonances of deformed nuclei and present a theoretical formalism for the bound and resonant states on the same footing. With

Self-consistent proton-neutron quasiparticle random phase approximation based on the spherical nonlinear point-coupling relativistic Hartree-Bogoliubov theory is established and used to investigate the β + /electroncapture (EC)-decay half-lives of neutron-deficient Ar, Ca, Ti, Fe, Ni, Zn, Cd, and Sn isotopes. The isoscalar proton-neutron pairing is found to play an important role in reducing the decay half-lives, which is consistent with the same mechanism in the β decays of neutron-rich nuclei. The experimental β + /EC-decay half-lives can be well reproduced by a universal isoscalar proton-neutron pairing strength.Nuclear β decays play important roles in many subjects of nuclear physics. Specifically, the investigation of β decay provides information on the spin and isospin dependence of the effective nuclear interaction, as well as on nuclear properties such as masses [1], shapes [2], and energy levels [3]. Moreover, nuclear β decays are also important in nuclear astrophysics, because they set the time scale of the rapid neutron-capture process (r-process) [4][5][6][7][8], which is a major mechanism for producing the elements heavier than iron. In addition, nuclear β decays can provide tests for the electroweak standard model [9][10][11]. With the development of radioactive ion beam facilities, the measurement of nuclear β-decay half-lives has achieved great progress in recent years [12][13][14][15].On the theoretical side, apart from the macroscopic gross theory [16], two different microscopic approaches have been widely used to describe and predict the nuclear β-decay rates. They are the shell model [5] and the proton-neutron quasiparticle random phase approximation (QRPA) [17][18][19]. While the shell model takes into account the detailed structure of the β-strength function, the proton-neutron QRPA approach provides a systematic description of β-decay properties of arbitrarily heavy nuclei. In order to reliably predict properties of thousands of unknown nuclei relevant to the r-process, the self-consistent QRPA approach has become a current trend in nuclear structure study, including those based on the Skyrme-Hartree-Fock-Bogoliubov (SHFB) theory [20] and the covariant density functional theory (CDFT) [21][22][23].In the CDFT framework, the self-consistent proton-neutron RPA was first developed based on the meson-exchange relativistic Hartree (RH) approach [24]. To describe the spinisospin excitations in open shell nuclei, it has been extended to the QRPA based on the relativistic Hartree-Bogoliubov (RHB) approach [25] and employed to calculate the β-decay half-lives of neutron-rich nuclei in the N ≈ 50 and N ≈ 82 regions [21,22]. In addition, based on the meson-exchange * haozhao.liang@riken.jp relativistic Hartree-Fock (RHF) approach [26,27], the selfconsistent proton-neutron RPA has been formulated [28] and well reproduces the spin-isospin excitations in doubly magic nuclei, without any readjustment of the parameters of the covariant energy density functional [28,29]. Recently, the self-consistent QRPA b...

Symmetry is an important and basic topic in physics. The similarity renormalization group theory provides a novel view to study the symmetries hidden in the Dirac Hamiltonian, especially for the deformed system. Based on the similarity renormalization group theory, the contributions from the nonrelativistic term, the spin-orbit term, the dynamical term, the relativistic modification of kinetic energy, and the Darwin term are self-consistently extracted from a general Dirac Hamiltonian and, hence, we get an accurate description for their dependence on the deformation. Taking an axially deformed nucleus as an example, we find that the self-consistent description of the nonrelativistic term, spin-orbit term, and dynamical term is crucial for understanding the relativistic symmetries and their breaking in a deformed nuclear system.

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