Preamble

Mechanisms of Mirror Energy Difference for States Exhibiting Thomas-Ehrman Shift: Gamow Shell Model Case Studies of $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O

K. H. Li,$^{1,2}$ P. Y. Wang,$^{2,3}$ J. G. Li,$^{2,3,4,\dagger}$ N. Michel,$^{2,3,\ddagger}$ M. R. Xie,$^{2,3}$ C. W. Ma,$^{1,5,\S}$ and W. Zuo$^{2,3,4}$

$^{1}$College of Physics, Center for Theoretical Physics, Henan Normal University, Xinxiang 453007, China
$^{2}$CAS Key Laboratory of High Precision Nuclear Spectroscopy, Institute of Modern Physics, Chinese Academy of Sciences, Lanzhou 730000, China
$^{3}$School of Nuclear Science and Technology, University of Chinese Academy of Sciences, Beijing 100049, China
$^{4}$Southern Center for Nuclear-Science Theory (SCNT), Institute of Modern Physics, Chinese Academy of Sciences, Huizhou 516000, Guangdong Province, China
$^{5}$Institute of Nuclear Science and Technology, Henan Academy of Sciences, Zhengzhou 450015, China

The mirror energy difference (MED) of mirror states, particularly for states exhibiting the Thomas-Ehrman shift, serves as a sensitive probe of isospin symmetry breaking. We employ the Gamow shell model, which includes inter-nucleon correlations and continuum coupling, to investigate the MED for sd-shell nuclei using $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O as examples. Our GSM calculations provide good descriptions of the excitation energies and MEDs for both mirror pairs. Moreover, our calculations reveal that large MEDs arise from significant occupation of weakly bound or unbound $s_{1/2}$ waves, which makes the radial density distribution of states in the proton-rich nucleus more extended than that of their mirror states in deeply bound neutron-rich nuclei. Additionally, our GSM calculations show that the Coulomb contribution differs for low-lying states in proton-rich nuclei, which significantly contributes to the MEDs of mirror states, as is well recognized. Furthermore, our GSM calculations demonstrate that nucleon-nucleon interaction contributions differ for mirror states, particularly for proton-rich states bearing the Thomas-Ehrman shift, which also contributes to significant isospin symmetry breaking with large MED values.

Keywords: isospin symmetry breaking, Thomas-Ehrman shift, mirror energy difference, continuum coupling, Gamow shell model

Introduction

Exotic nuclear structures in drip-line nuclei have become a subject of great interest in recent years, as they exhibit unique properties that differ significantly from those of stable nuclei \cite{1-7}. One of the most significant phenomena observed in these systems is the Thomas-Ehrman shift (TES) \cite{8,9}. This effect is most pronounced in nuclei near the proton drip line, where the balance between the strong force and Coulomb force is particularly delicate. States exhibiting TES effects are often weakly bound or unbound, characteristic of open quantum systems, while their neutron-rich mirror counterparts remain deeply bound, resulting in large mirror energy differences (MED) in their isobaric states \cite{8-13}. These large MEDs are attributed to proximity to threshold effects, where continuum effects must be properly treated. Consequently, a thorough understanding of the Thomas-Ehrman shift is pivotal for elucidating the dynamics of weakly bound and unbound nuclear systems and for understanding the mechanisms underlying isospin symmetry breaking in mirror nuclei.

Two possible explanations exist for states with large MEDs: external or internal in character. If extended single-particle wave functions of weakly bound or unbound $s$- or $p$-waves are significantly occupied in the considered states, the large MED is of external nature, as in TES states \cite{11,14,15}. The second possibility, related to configuration mixing (see Refs. \cite{16,17}), is of internal nature. In this case, the extended wave function arises from strong configuration mixing in which a few nodal states of $s$ or $p$ waves are included in the calculations. These external and internal effects are distinct but can be intertwined in complex ways. For instance, the inversion of ground states in the $^{16}$F and $^{16}$N mirror nuclei is primarily due to the unbound proton $1s_{1/2}$ orbital, which can also be well described in GSM calculations within the configuration mixing framework \cite{15,18}.

The sd-shell nuclei, situated at the boundary between light and heavy nuclei, exhibit a wide range of nuclear structure phenomena that remain somewhat mysterious \cite{19}. In recent years, these nuclei have been extensively studied using various experimental techniques \cite{20-23}. A wealth of information on the Thomas-Ehrman shift has been gleaned from sd-shell proton drip-line nuclei, where numerous states exhibiting significant TES effects have been identified \cite{10-12,24}. For instance, the mirror pairs $^{18}$Ne/$^{18}$O \cite{10,25} and $^{19}$Na/$^{19}$O \cite{24} serve as notable examples. For sd-shell nuclei, TES is mainly driven by $s$-waves. Indeed, the proton $1s_{1/2}$ orbital is weakly bound or unbound in proton drip-line nuclei, whereas the neutron $1s_{1/2}$ is well bound in their mirror neutron-rich nuclei.

Several theoretical models have been developed to probe isospin asymmetry in mirror nuclei, such as the standard shell model (SM) \cite{26-30}, mean-field calculations \cite{31,32}, and ab initio approaches \cite{25,33-37}. Within standard SM calculations, weakly bound and unbound wave functions are indirectly considered via phenomenological adjustment of matrix elements related to the $1s_{1/2}$ orbital \cite{11,14}. Mean-field calculations, such as Skyrme-Hartree-Fock, have also been extensively employed in MED studies \cite{31,32}. However, these models involve parameters constrained by data \cite{26,31,38}.

In recent years, ab initio approaches such as the ab initio valence-space in-medium similarity renormalization group have also been applied to study MEDs of sd-shell nuclei \cite{12,13,25,33-36,39}, in which extended many-body wave functions are partially described using a large number of harmonic oscillator (HO) spaces. Moreover, current theoretical calculations have pointed out that TES is caused by the repulsive Coulomb interaction and the occupation of weakly bound or unbound $s$- or $p$-waves for valence protons. However, detailed studies on the mechanism of TES are lacking. In recent shell model calculations \cite{30}, TES has been investigated using calculated spectroscopic factors. Furthermore, in our previous work \cite{39}, we compared MED results calculated using the shell model with spectroscopic factors and the ab initio valence-space in-medium similarity renormalization group approach, finding that both models give similar results.

One of the major challenges in studying drip-line nuclei is accounting for the interplay between configuration mixing and continuum effects. The Gamow shell model (GSM) \cite{16,17,25,33,40-45} has emerged as a powerful tool in this regard, as it provides a unified framework to describe the structure of nuclei close to the particle emission threshold and enables accurate understanding of exotic properties in drip-line nuclei. Based on this situation, we employ GSM to investigate significant isospin symmetry breaking with large MED values and the underlying mechanisms for sd-shell nuclei, using the $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O mirror pairs as examples.

II. Method

GSM is a multiconfiguration shell model framework that works in the picture of a core plus valence nucleons \cite{42,46-49}. At the heart of GSM lies the utilization of the one-body Berggren basis \cite{50}, which includes bound, resonance, and scattering states generated by a finite-range potential, typically of Woods-Saxon (WS) type (see details in Refs. \cite{42,46,50}). The GSM Hamiltonian matrix is characterized as complex symmetric \cite{42,46}. The overlap method along with the Jacobi-Davidson method extended to complex-symmetric matrices are adopted to diagonalize and identify many-body resonance eigenstates \cite{42,46,51}. Consequently, GSM calculations include both inter-nucleon correlations and continuum coupling \cite{42,46,47}.

The many-body Schrödinger equation of the GSM Hamiltonian can be solved within the so-called cluster orbital shell model (COSM) formalism \cite{52} (see Refs. \cite{46,53,54}). The GSM Hamiltonian in COSM coordinates reads \cite{46,53,54}:

$$
\hat{H}{\text{GSM}} = \sum}^{A_{\text{val}}} \left( \frac{p_i^2}{2\mu_i} + \hat{Ui^{(c)} \right) + \sum \right)}^{A_{\text{val}}} \left( \hat{V}_{ij} + \frac{\mathbf{p}_i \cdot \mathbf{p}_j}{M_c
$$

where $A_{\text{val}}$ is the number of valence nucleons, $\mu_i$ is the effective mass of the nucleon, $\hat{U}^{(c)}$ is represented by a one-body WS potential mimicking the inert core, $\hat{V}_{ij}$ is the residual inter-nucleon interaction modeled by a pionless effective field theory (EFT) interaction \cite{55,56} in which only two-body contact terms up to next-to-next-to-leading order are considered. The regularization approach adopted in Refs. \cite{48,57-60} is used. The last term embodies the recoil effects induced by the finite mass of the core $M_c$. The EFT interaction is optimized to reproduce the low-lying states of selected nuclei.

In the present work, the $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O mirror partners are taken as examples. The doubly magic nucleus $^{16}$O is chosen as the inert core, and the $s_{1/2}$, $p_{1/2,3/2}$, and $d_{3/2,5/2}$ partial waves are represented by the Berggren basis, in which 40 discretization points are used in total for continuum states in each partial wave, while the $f_{5/2,7/2}$ partial waves are treated using the HO basis with 6 HO states adopted for each partial wave. Only the Coulomb force is considered for the isospin non-conserving part of the GSM Hamiltonian. The contribution of the isospin-dependent part of the nuclear interaction to TES is small and is neglected in the present GSM calculations. The Hamiltonian used in Ref. \cite{61} is adopted in the present work. The calculated excitation energies of the $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O mirror partners are presented in Tables 1 and 2, which show good agreement with experimental data \cite{62}. In the following section, the mechanisms of the mirror energy differences for the $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O mirror partners are investigated in detail.

III. Results

Our GSM calculations accurately describe the excitation energies of low-lying states for the mirror partners $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O. To delve deeper into the significant isospin symmetry breaking observed in these mirror nuclei, we define the MED for a given state $J^\pi$ as $\text{MED}(J^\pi) = E_x(T_z^<, J^\pi) - E_x(T_z^>, J^\pi)$, where $T_z^<$ and $T_z^>$ denote the negative and positive isospin projection $T_z = (N - Z)/2$, respectively, for a mirror pair. We have calculated MED values for mirror states in $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O, as presented in Tables 1 and 2, along with experimental data. The calculated MED values for low-lying states in the mirror partners $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O align well with experimental data, with the exception of the $0^+_2$ state in the $^{18}$Ne/$^{18}$O mirror nuclei, where our GSM calculations yield larger values than the experimental data. Both our GSM calculations and experimental data highlight significant isospin symmetry breaking in the $3^+_1$ state of the $^{18}$Ne/$^{18}$O mirror nuclei and the $1/2^+_1$ state of the $^{19}$Na/$^{19}$O mirror nuclei, evidenced by their large MED values.

To investigate the significant isospin symmetry breaking and associated large MEDs, we calculate the average occupations of low-lying states through GSM, focusing particularly on the $s_{1/2}$ and $d_{5/2}$ partial waves above the $^{16}$O core for the $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O mirror nuclei, as illustrated in Figs. 1 and 2. Other partial waves such as $d_{3/2}$ and $f_{5/2,7/2}$ exhibit negligible occupations and are therefore excluded from these figures. The calculated average occupations reveal almost identical patterns for mirror states within the $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O pairs. Our GSM calculations further indicate that states exhibiting significant isospin symmetry breaking with large MED values also show significant occupancy in the $s_{1/2}$ partial waves—markedly higher than in their respective ground states. For instance, the occupations of the $s_{1/2}$ partial wave for the $3^+1$ and $1/2^+_1$ states in the $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O mirror pairs, respectively, are substantially greater than those of the ground states. Additionally, our calculations show that the $0^+_2$ states of $^{18}$Ne/$^{18}$O demonstrate notable $s$O plus valence particles picture.}$ partial wave occupations compared to the ground states, resulting in a large MED. In contrast, experimental data give a smaller MED value, hinting at a complex structure of the $0^+_2$ states in $^{18}$Ne/$^{18}$O that might not be fully captured by the $^{16

To elucidate the underlying mechanism of large MEDs, we conduct a detailed analysis of the radial density distributions of mirror states in the $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O pairs using GSM. The results allow us to systematically compare the radial distributions of valence protons in proton-rich nuclei and valence neutrons in their neutron-rich mirror counterparts. The results are presented in Figs. 3 and 4 for the $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O mirror partners, respectively. Our GSM results reveal that states characterized by minor isospin symmetry breaking with small MEDs exhibit almost identical radial density distributions, which decline sharply in the asymptotic regions, such as the ground states of both $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O. This phenomenon is largely attributed to the dominance of $d_{5/2}$ partial waves, which are constrained within the nuclear region by high centrifugal and Coulomb barriers, despite the state being unbound. Conversely, GSM calculations depict the radial density distributions of the $3^+1$ state of $^{18}$Ne and the $1/2^+_1$ state of $^{19}$Na as more extended in the asymptotic region than their neutron-rich counterparts, $^{18}$O and $^{19}$O, respectively. This disparity stems from the non-existent centrifugal barrier for the $s. Consequently, our GSM calculations provide direct calculations for radial density distribution and unveil that mirror states demonstrating significant isospin symmetry breaking with large MEDs possess many-body wave functions in proton-rich nuclei that are more extended than those in their neutron-rich mirror states, adding a new dimension to our understanding of the role of isospin symmetry breaking in shaping their properties.}$ partial wave, leading to a more pronounced distribution in the proton-rich nucleus due to the weakly bound or unbound nature of the $s_{1/2}$ partial wave. A similar mechanism underlies the formation of halo nuclei, where valence nucleons occupy weakly bound $s$- or $p$-partial waves, resulting in an extended density distribution due to the minimal or absent centrifugal barrier \cite{16,53,65

The GSM Hamiltonian, as shown in Eq. (1), can be divided into nuclear interaction (encompassing core-nucleon and nucleon-nucleon interactions) and Coulomb interaction (including one-body Coulomb (1BC) interaction between the inner core and valence protons, and two-body Coulomb (2BC) interaction between valence protons). We perform further calculations to dissect the contributions from different parts of the Hamiltonian, aiming to shed light on the underlying mechanisms in mirror states exhibiting significant isospin symmetry breaking with large MEDs. The computed energies for low-lying mirror states in the pairs $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O, along with experimental data \cite{62}, are showcased in Figs. 5 and 6, respectively. To gain deeper insights, we also present the energy minus 2BC contribution (GSM-2BC) and energy minus 1BC and 2BC contributions (GSM-1BC-2BC) in the proton-rich nuclei $^{18}$Ne and $^{19}$Na.

Indeed, GSM-1BC-2BC corresponds to the contribution of nuclear interaction. Within the isospin symmetry picture, the difference in mirror state energies should solely stem from Coulomb interactions, implying that the GSM-1BC-2BC values for a state in a proton-rich nucleus would be the same as its mirror state in the neutron-rich nucleus. Our GSM calculations show that the GSM-1BC-2BC values for the ground states of $^{18}$Ne and $^{19}$Na closely align with the computed ground-state energies of their neutron-rich counterparts, $^{18}$O and $^{19}$O, respectively, indicating the preservation of isospin symmetry for these ground states. Conversely, for the excited $3^+1$ state in $^{18}$Ne/$^{18}$O and the $1/2^+_1$ state in $^{19}$Na/$^{19}$O, our GSM calculations showcase a deviation from this symmetry. To quantitatively examine this discrepancy, we introduce $\Delta E$ as the differential metric for significant isospin symmetry breaking. $\Delta E$ encapsulates the disparity between the GSM-1BC-2BC values in the state of the proton-rich nucleus and the energy calculated for the corresponding state in the neutron-rich mirror nucleus, which reads as $\Delta E = \langle\Psi$ correspond to the many-body wave functions of proton-rich and neutron-rich nuclei, respectively. The $\Delta E$ corresponds to the difference in the contribution of nuclear interactions in the mirror state.}}|H_{NN}|\Psi_{\text{proton}}\rangle - \langle\Psi_{\text{neutron}}|H_{NN}|\Psi_{\text{neutron}}\rangle$, where $\Psi_{\text{proton}}$ and $\Psi_{\text{neutron}

Our GSM calculations show that both $\Delta E$ and Coulomb interactions, including 1BC and 2BC, significantly influence the energy discrepancies observed in mirror states. Predominantly, the Coulomb interaction emerges as the dominant factor contributing to these differences. Illustrated in the lower panels of Figs. 5 and 6, we detail the $\Delta E$, 1BC, and 2BC contributions to the energy differences in the low-lying mirror states of $^{18}$Ne/$^{18}$O and $^{19}$Na/$^{19}$O mirror pairs. Our GSM results indicate that the energy differences in the ground states of $^{18}$Ne/$^{18}$O primarily stem from Coulomb interactions, with $\Delta E$ making a minimal contribution. For $^{19}$Na/$^{19}$O, the $\Delta E$ contribution is noted to be around 100 keV. Interestingly, we find varying contributions of $\Delta E$, 1BC, and 2BC across different mirror states within each pair. For instance, the $3^+_1$ states in $^{18}$Ne/$^{18}$O exhibit a higher $\Delta E$ contribution and lower Coulomb interactions relative to their ground states. The heightened $\Delta E$ values underscore the distinct nuclear interaction contributions to isospin symmetry breaking in these systems, showcasing the complex interplay of forces that shape the energy landscapes of mirror nuclei.

In evaluating the MED, the ground state energy of mirror nuclei serves as the baseline, with MED being determined by the discrepancy in excitation energies of corresponding mirror states. Adopting the energy difference of the ground states of mirror nuclei as a reference—illustrated by red dashed lines in Figs. 5 and 6—the difference between the values of ground and excited mirror states corresponds to the MED, highlighted by the red arrows in these figures. The results reveal that both the $\Delta E$ values and the Coulomb interaction exhibit significant variations across different mirror states, both contributing to the MED. Furthermore, to validate our conclusion, we performed Gamow shell model calculations using an optimized Hamiltonian fitted to reproduce a series of selected experimental data for sd-shell nuclei. This optimized Hamiltonian has also been employed to investigate low-lying states in $^{21}$Al \cite{66} and $^{22}$Si \cite{67}, as well as isospin symmetry breaking in those nuclei. While the calculated excitation energies exhibit slight differences, the MED results align with both the current GSM calculations and the established conclusions regarding the MED mechanism.

IV. Summary

Based on GSM calculations, in which both inter-nucleon correlation and continuum coupling are properly treated, we deduce that significant isospin symmetry breaking in mirror states, leading to large MED values, arises from the occupation of weakly bound or unbound $s_{1/2}$ partial waves in the proton-rich nucleus, while its counterpart in the neutron-rich nucleus remains deeply bound. This dichotomy culminates in a more expansive radial density distribution for states within the proton-rich nucleus, as opposed to their mirror counterparts. Additionally, the difference in radial density distributions between mirror states implies disparate contributions from nuclear interactions, underscored by significant $\Delta E$ values, which further highlight the presence of isospin symmetry breaking. Moreover, states with an extended radial density distribution tend to yield smaller Coulomb contributions compared to ground states characterized by more localized distributions. This factor chiefly accounts for the reduced excitation energies in states influenced by the Thomas-Ehrman shift effect, thereby engendering substantial negative MED values in mirror states. Our GSM calculations corroborate that both nuclear and Coulomb interactions play crucial roles in manifesting the significant isospin symmetry breaking associated with large MED values.

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