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The beyond mean-field approach for low-lying hypernuclear states is extended by mixing the configurations associated with both single-particle and quadrupole-octupole collective excitations within the generator coordinate method based on a covariant density functional theory. The method is demonstrated in the application to the low-lying states of $^{21}_\Lambda$Ne, where the configurations with the $\Lambda$ hyperon occupying the first ($\Lambda_s$) and second ($\Lambda_p$) lowest-energy states are considered. The results indicate that the positive-parity states are dominated by the $\alpha+^{12}$C+$\alpha+\Lambda_s$ structure. In contrast, the low-lying negative-parity states are dominated by a strong admixture of $\alpha+^{16}$O+$\Lambda_s$ structure and $\alpha+^{12}$C+$\alpha+\Lambda_p$ structure due to the inclusion of octupole correlations. As a result, the low-lying negative-parity states become much lower than what is expected from the previous studies without the mixing, and the electric multipole transition strengths are significantly quenched., Comment: 9 pages with 5 figures and 2 tables

The uncertainty in the nuclear matrix elements (NMEs) of $0νββ$ decay for $^{76}$Ge, $^{82}$Se, $^{128}$Te, $^{130}$Te, and $^{136}$Xe in the self-consistent quasiparticle random phase approximation (QRPA) method is investigated by using eighteen Skyrme interactions supplemented with either a volume- or surface-type of pairing interactions. The NMEs for the isotopes concerned (except $^{136}$Xe) are less sensitive to the particle-hole ($ph$) interactions, while strongly dependent on the employed isovector particle-particle ($pp$) pairing interactions even though the pairing strengths are optimized to the same pairing gap. The results indicate that a precise determination of the isovector $pp$ pairing interaction in the Skyrme energy density functional is of importance to reduce the uncertainty in the NMEs within the QRPA framework., 7 pages, 5 figures

Symmetry techniques based on group theory play a prominent role in the analysis of nuclear phenomena, and in particular in the understanding of observed regular patterns in nuclear spectra and selection rules for electromagnetic transitions. A variety of symmetry-based nuclear models have been developed in nuclear physics, providing efficient tools of choice to interpret nuclear spectroscopic data. This chapter provides a pedagogical introduction to the basic idea of symmetry-breaking mechanism and symmetry-restoration methods in modeling atomic nuclei., Comment: 35 pages with 8 figures. Contribution to the "Handbook of Nuclear Physics", Springer, 2022, Eds. I. Tanihata, H. Toki, and T. Kajino

The beyond mean-field approach for low-lying hypernuclear states is extended by mixing the configurations associated with both single-particle and quadrupole-octupole collective excitations within the generator coordinate method based on a covariant density functional theory. The method is demonstrated in the application to the low-lying states of $^{21}_Λ$Ne, where the configurations with the $Λ$ hyperon occupying the first ($Λ_s$) and second ($Λ_p$) lowest-energy states are considered. The results indicate that the positive-parity states are dominated by the $α+^{12}$C+$α+Λ_s$ structure. In contrast, the low-lying negative-parity states are dominated by a strong admixture of $α+^{16}$O+$Λ_s$ structure and $α+^{12}$C+$α+Λ_p$ structure due to the inclusion of octupole correlations. As a result, the low-lying negative-parity states become much lower than what is expected from the previous studies without the mixing, and the electric multipole transition strengths are significantly quenched., 9 pages with 5 figures and 2 tables

The generator coordinate method (GCM) is an important tool of choice for modeling large-amplitude collective motion in atomic nuclei. The computational complexity of the GCM increases rapidly with the number of collective coordinates. It imposes a strong restriction on the applicability of the method. In this work, we propose a subspace-reduction algorithm that employs optimal statistical ML models as surrogates for exact quantum-number projection calculations for norm and Hamiltonian kernels. The model space of the original GCM is reduced to a subspace relevant for nuclear low energy spectra and the NME of ground state to ground state $0\nu\beta\beta$ decay based on the orthogonality condition (OC) and the energy-transition-orthogonality procedure (ENTROP), respectively. For simplicity, the polynomial ridge regression (RR) algorithm is used to learn the norm and Hamiltonian kernels of axially deformed configurations. The efficiency and accuracy of this algorithm are illustrated for 76Ge and 76Se by comparing results obtained using the optimal RR models to direct GCM calculations. The low-lying energy spectra of $^{76}$Ge and $^{76}$Se, as well as the $0\nu\beta\beta$-decay NME between their ground states, are computed. The results show that the performance of the GCM+OC/ENTROP+RR is more robust than that of the GCM+RR alone, and the former can reproduce the results of the original GCM calculation accurately with a significantly reduced computational cost., Comment: 15 pages with 19 figures

The generator coordinate method begins with the variational construction of a set of non-orthogonal mean-field states that span a subspace of the full many-body Hilbert space. These states are then often projected onto states with good quantum numbers to restore symmetries, leading to a set with members that can be similar to one another, and it is sometimes possible to reduce this set without greatly affecting results. Here we propose a greedy algorithm that we call the energy-transition-orthogonality procedure (ENTROP) to select subsets of important states. As applied here, the approach selects on the basis of diagonal energy, orthogonality, and contribution to the matrix element that governs neutrinoless double-$��$ decay. We present both shell-model and preliminary ab initio calculations of this matrix element for the decay of $^{76}$Ge, with quadrupole deformation parameters and the isoscalar pairing strength as generator coordinates. ENTROP converges quickly, reducing significantly the number of basis states needed for an accurate calculation., 7 pages, 9 figures, authors added, some details clarified

Working with Hamiltonians from chiral effective field theory, we develop a novel framework for describing arbitrary deformed medium-mass nuclei by combining the in-medium similarity renormalization group with the generator coordinate method. The approach leverages the ability of the first method to capture dynamic correlations and the second to include collective correlations without violating symmetries. We use our scheme to compute the matrix element that governs the neutrinoless double beta decay of $^{48}$Ca to $^{48}$Ti, and find it to have the value $0.61$, near or below the predictions of most phenomenological methods. The result opens the door to ab initio calculations of the matrix elements for the decay of heavier nuclei such as $^{76}$Ge, $^{130}$Te, and $^{136}$Xe., 6 pages, 4 figures and 1 table. supplementary material included. version to be published

We use a reference state based on symmetry-restored states from deformed mean-field or generator-coordinate-method (GCM) calculations in conjunction with the in-medium similarity-renormalization group (IMSRG) to compute spectra and matrix elements for neutrinoless double-beta ($0������$) decay. Because the decay involves ground states from two nuclei, we use evolved operators from the IMSRG in one nucleus in a subsequent GCM calculation in the other. We benchmark the resulting IMSRG+GCM method against complete shell-model diagonalization for both the energies of low-lying states in $^{48}$Ca and $^{48}$Ti and the $0������$ matrix element for the decay of $^{48}$Ca, all in a single valence shell. Our approach produces better spectra than either the IMSRG with a spherical-mean-field reference or GCM calculations with unevolved operators. For the $0������$ matrix element the improvement is slight, but we expect more significant effects in full ab-initio calculations., 12 pages, 5 figures, 1 tables

Background: The discovery of neutrinoless double-beta ($0������$) decay would demonstrate the nature of neutrinos, have profound implications for our understanding of matter-antimatter mystery, and solve the mass hierarchy problem of neutrinos. The calculations for the nuclear matrix elements $M^{0��}$ of $0������$ decay are crucial for the interpretation of this process. Purpose: We study the effects of relativity and nucleon-nucleon short-range correlations on the nuclear matrix elements $M^{0��}$ by assuming the mechanism of exchanging light or heavy neutrinos for the $0������$ decay. Methods: The nuclear matrix elements $M^{0��}$ are calculated within the framework of covariant density functional theory, where the beyond-mean-field correlations are included in the nuclear wave functions by configuration mixing of both angular-momentum and particle-number projected quadrupole deformed mean-field states. Results: The nuclear matrix elements $M^{0��}$ are obtained for ten $0������$-decay candidate nuclei. The impact of relativity is illustrated by adopting relativistic or nonrelativistic decay operators. The effects of short-range correlations are evaluated. Conclusions: The effects of relativity and short-range correlations play an important role in the mechanism of exchanging heavy neutrinos though the influences are marginal for light neutrinos. Combining the nuclear matrix elements $M^{0��}$ with the observed lower limits on the $0������$-decay half-lives, the predicted strongest limits on the effective masses are $|\langle m_��\rangle|3.065\times 10^8~\mathrm{GeV}$ for heavy neutrinos., 24 pages, 4 figures

We discuss anharmonicity of the multi-octupole-phonon states in $^{208}$Pb based on a covariant density functional theory, by fully taking into account the interplay between the quadrupole and the octupole degrees of freedom. Our results indicate the existence of a large anharmonicity in the transition strengths, even though the excitation energies are similar to those in the harmonic limit. We also show that the quadrupole-shape fluctuation significantly enhances the fragmentation of the two-octupole-phonon states in $^{208}$Pb. Using those transition strengths as inputs to coupled channels calculations, we then discuss the fusion reaction of $^{16}$O+$^{208}$Pb at energies around the Coulomb barrier. We show that the anharmonicity of the octupole vibrational excitation considerably improves previous coupled-channels calculations in the harmonic oscillator limit, significantly reducing the height of the main peak in the fusion barrier distribution., Comment: 5 pages with 4 figures, to be published in Phys. Rev. C as a Rapid Communication

We describe low-lying collective excitations of atomic nuclei with the multi-reference covariant density functional theory, and combine them with coupled-channels calculations for heavy-ion fusion reactions at energies around the Coulomb barrier. To this end, we use the calculated transition strengths among several collective states as inputs to the coupled-channels calculations. This approach provides a natural way to describe anharmonic multi-phonon excitations as well as a deviation of rotational excitations from a simple rigid rotor. We apply this method to subbarrier fusion reactions of $^{58}$Ni+$^{58}$Ni, $^{58}$Ni+$^{60}$Ni and $^{40}$Ca+$^{58}$Ni systems. We find that the effect of anharmonicity tends to smear the fusion barrier distributions, better reproducing the experimental data compared to the calculations in the harmonic oscillator limit., Comment: 11 pages, 9 eps figures

We present a detailed formalism of the microscopic particle-rotor model for hypernuclear low-lying states based on a covariant density functional theory. In this method, the hypernuclear states are constructed by coupling a hyperon to low-lying states of the core nucleus, which are described by the generator coordinate method (GCM) with the particle number and angular momentum projections. We apply this method to study in detail the low-lying spectrum of $^{13}_{~��}$C and $^{21}_{~��}$Ne hypernuclei. We also briefly discuss the structure of $^{155}_{~~��}$Sm as an example of heavy deformed hypernuclei. It is shown that the low-lying excitation spectrum with positive parity states of the hypernuclei, which are dominated by $��$ hyperon in $s$-orbital coupled to the core states, are similar to that for the corresponding core states, while the electric quadrupole transition strength, $B(E2)$, from the 2$^+_1$ state to the ground state is reduced according to the mass number of the hypernuclei. Our study indicates that the energy splitting between the first 1/2$^-$ and 3/2$^-$ hypernuclear states is generally small for all the hypernuclei which we study. However, their configurations depend much on the properties of a core nucleus, in particular on the sign of deformation parameter. That is, the first $1/2^-$ and $3/2^-$ states in $^{13}_{~��}$C are dominated by a single configuration with $��$ particle in the $p$-wave orbits and thus providing good candidates for a study of the $��$ spin-orbit splitting. On the other hand, those states in the other hypernuclei exhibit a large configuration mixing and thus their energy difference cannot be interpreted as the spin-orbit splitting for the $p$-orbits., 18 pages, 20 eps figures

Taking the hypernucleus $^{13}_{~��}$C as an example, we illustrate the miscroscopic particle-rotor model for low-lying spectra of hypernuclei. This approach is based on the beyond-mean-field method, with the particle number and angular momentum projections. The quantum fluctuation of the mean-field is also taken into account for the core nucleus using the generator coordinate method. We show that the impurity effect of $��$ hyperon, such as a change in $B(E2)$, is well described with this model. Our calculation indicates that the most important impurity effect in $sd$-shell hypernuclei is a change in a deformation parameter rather than in a nuclear size., 4 pages, 2 eps figures. A talk given at the 12th International Conference on Hypernuclear and Strange Particle Physics (HYP2015), September 7-12, 2015, Sendai, Japan

A study of 4$��$ linear-chain structure in high-lying collective excitation states of $^{16}$O with a covariant density functional theory is presented. The low-spin states are obtained by configuration mixing of particle-number and angular-momentum projected quadrupole deformed mean-field states with generator coordinate method. The high-spin states are determined by cranking calculations. These two calculations are based on the same energy density functional PC-PK1. We have found a rotational band at low-spin with the dominated intrinsic configuration considered to be the one that 4$��$ clusters stay along a common axis. The strongly deformed rod shape also appears in the high-spin region with the angular momentum $13-18\hbar$; however whether the state is pure $4��$ linear chain or not is less obvious than that in the low-spin states., 10 pages with 11 figures and 3 tables

Neutrinoless double-$��$ ($0������$) decay is related to many fundamental concepts in nuclear and particle physics beyond the standard model. Currently there are many experiments searching for this weak process. An accurate knowledge of the nuclear matrix element for the $0������$ decay is essential for determining the effective neutrino mass once this process is eventually measured. We report the first full relativistic description of the $0������$ decay matrix element based on a state-of-the-art nuclear structure model. We adopt the full relativistic transition operators which are derived with the charge-changing nucleonic currents composed of the vector coupling, axial-vector coupling, pseudoscalar coupling, and weak-magnetism coupling terms. The wave functions for the initial and final nuclei are determined by the multireference covariant density functional theory (MR-CDFT) based on the point-coupling functional PC-PK1. The low-energy spectra and electric quadrupole transitions in ${}^{150}$Nd and its daughter nucleus ${}^{150}$Sm are well reproduced by the MR-CDFT calculations. The $0������$ decay matrix elements for both the $0_1^+\rightarrow 0_1^+$ and $0_1^+\rightarrow 0_2^+$ decays of ${}^{150}$Nd are evaluated. The effects of particle number projection, static and dynamic deformations, and the full relativistic structure of the transition operators on the matrix elements are studied in detail. The resulting $0������$ decay matrix element for the $0_1^+\rightarrow 0_1^+$ transition is $5.60$, which gives the most optimistic prediction for the next generation of experiments searching for the $0������$ decay in ${}^{150}$Nd., 17 pages, 9 figures; table added

The impurity effect of hyperon on atomic nuclei has received a renewed interest in nuclear physics since the first experimental observation of appreciable reduction of $E2$ transition strength in low-lying states of hypernucleus $^{7}_��$Li. Many more data on low-lying states of $��$ hypernuclei will be measured soon for $sd$-shell nuclei, providing good opportunities to study the $��$ impurity effect on nuclear low-energy excitations. We carry out a quantitative analysis of $��$ hyperon impurity effect on the low-lying states of $sd$-shell nuclei at the beyond-mean-field level based on a relativistic point-coupling energy density functional (EDF), considering that the $��$ hyperon is injected into the lowest positive-parity ($��_s$) and negative-parity ($��_p$) states. We adopt a triaxially deformed relativistic mean-field (RMF) approach for hypernuclei and calculate the $��$ binding energies of hypernuclei as well as the potential energy surfaces (PESs) in $(��, ��)$ deformation plane. We also calculate the PESs for the $��$ hypernuclei with good quantum numbers using a microscopic particle rotor model (PRM) with the same relativistic EDF. The triaxially deformed RMF approach is further applied in order to determine the parameters of a five-dimensional collective Hamiltonian (5DCH) for the collective excitations of triaxially deformed core nuclei. Taking $^{25,27}_��$Mg and $^{31}_��$Si as examples, we analyse the impurity effects of $��_s$ and $��_p$ on the low-lying states of the core nuclei..., 15 pages with 18 figures and 1 table (version to be published in Physical Review C)

We discuss low-lying collective excitations of $��$ hypernuclei using the self-consistent mean-field approaches. We first discuss the deformation properties of $��$ hypernuclei in the $sd$-shell region. Based on the relativistic mean-field (RMF) approach, we show that the oblate deformation for $^{28}$Si nucleus may disappear when a $��$ particle is added to this nucleus. We then discuss the rotational excitations of $^{25}_��$Mg nucleus using the three-dimensional potential energy surface in the deformation plane obtained with the Skyrme-Hartree-Fock method. The deformation of $^{25}_��$Mg nucleus is predicted to be slightly reduced due to an addition of $��$ particle. We demonstrate that this leads to a reduction of electromagnetic transition probability, $B(E2)$, in the ground state rotational band. We also present an application of random phase approximation (RPA) to hypernuclei, and show that a new dipole mode, which we call a soft dipole $��$ mode, appears in hypernuclei, which can be interpreted as an oscillation of $��$ particle against the core nucleus., 8 pages, 5 figures. A talk given at XI International Conference on Hypernuclear and Strange Particle Physics (HYP2012), Oct. 1-5, 2012, Barcelona, Spain

The non-relativistic Skyrme energy density functional (EDF) based collective Hamiltonian, that takes into account dynamical correlations related to the restoration of broken symmetries and fluctuations of quadrupole collective variables, is applied to quantitatively study the impurity effect of $��$ hyperon on the collectivity of $^{44}$S. Several Skyrme forces for both the nucleon-nucleon ($NN$) and $��$-nucleon ($��N$) interactions are used. The influence of pairing strengths on the polarization effect of $��$ hyperon is also examined. It is found that although these Skyrme forces with different pairing strengths give somewhat different low-lying spectra for $^{44}$S, all of them give similar and generally small size of $��$ reduction effect (within 5%) on the collective properties., 8 pages, 5 figures, based on the talk presented at 18th Nuclear Physics Workshop "Maria and Pierre Curie", 2011, Kazimierz, Poland

The configuration-fixed constrained triaxial relativistic mean-field approach is extended by including time-odd fields and applied to study the candidate multiple chiral doublets (MKiD) nucleus 106Rh. The energy contribution from time-odd fields and microscopical evaluation of center-of-mass correction as well as the modification of triaxial deformation parameters beta, gamma due to the time-odd fields are investigated. The contributions of the time-odd fields to the total energy are 0.1-0.3 MeV and they modify slightly the gamma values. However, the previously predicted multiple chiral doublets still exist., Comment: 9 pages, 3 figures, accepted for publication as a Brief Report in Physical Review C