# Isospin-Symmetry Breaking within the Nuclear Shell Model: Present Status and Developments

## Abstract

**:**

## 1. Introduction

#### 1.1. Isospin Symmetry in Nuclear Structure

- For $\Delta T=1$ transitions (${T}_{f}={T}_{i}\pm 1$), the (reduced) matrix elements of analogue transitions in mirror nuclei or between respective analogue states should be identical, since they are governed only by the isovector term.
- In transitions between the states of the same isospin (${T}_{i}={T}_{f}=T$), both isoscalar and isovector terms contribute, and the matrix element for analogue transitions within an isobaric multiplet exhibits a linear trend as a function of ${M}_{T}$:$$\begin{array}{c}{\displaystyle \langle {J}_{f}{M}_{f};T{M}_{T}|{\widehat{O}}_{LM}|{J}_{i}{M}_{i};T{M}_{T}\rangle =\langle {J}_{f}{M}_{f}|{\widehat{O}}_{LM}^{\left(0\right)}|{J}_{i}{M}_{i}\rangle}\\ {\displaystyle +\frac{{M}_{T}}{\sqrt{T(T+1)(2T+1)}}\langle {J}_{f}{M}_{f};T||{\widehat{O}}_{LM}^{\left(1\right)}||{J}_{i}{M}_{i};T\rangle \phantom{\rule{0.166667em}{0ex}}.}\end{array}$$
- Another specific rule can be established for electric dipole operator. In the lowest order of the long-wavelength approximation, the electric-dipole ($E1$) operator is an isovector operator:$$\widehat{O}\left(E1\right)=\sum _{k=1}^{A}\mathrm{e}\left(k\right)\overrightarrow{r}\left(k\right)=\sum _{k=1}^{A}\left(\frac{1}{2}-{\widehat{t}}_{3}\left(k\right)\right)\mathrm{e}\overrightarrow{r}\left(k\right)\phantom{\rule{0.166667em}{0ex}}.$$Hence, $E1$ transitions between the states of the same isospin (${T}_{i}={T}_{f}=T$) in $N=Z$ nuclei are forbidden by the isospin symmetry because of the vanishing Clebsch–Gordan coefficient, $(T\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}}1\phantom{\rule{0.166667em}{0ex}}0|\phantom{\rule{0.166667em}{0ex}}T\phantom{\rule{0.166667em}{0ex}}0\phantom{\rule{0.166667em}{0ex}})=0$ (see Equation (11)).

#### 1.2. Isospin-Symmetry Breaking

- class I (${V}_{I}$) are charge-independent forces $\{1,\widehat{\mathbf{t}}\left(1\right)\xb7\widehat{\mathbf{t}}\left(2\right)\}$;
- class II (${V}_{II}$) are forces which break the charge independence, but preserve the charge symmetry of the two-nucleon system, $\left\{{\widehat{t}}_{3}\left(1\right){\widehat{t}}_{3}\left(2\right)\right\}$;
- class III (${V}_{III}$) are charge-symmetry breaking forces, which vanish in the neutron-proton system, $\{{\widehat{t}}_{3}\left(1\right)+{\widehat{t}}_{3}\left(2\right)\}$;
- class IV (${V}_{IV}$) are forces which do not conserve the isospin of the two-nucleon system: $\{\widehat{\mathbf{t}}\left(1\right)\times \widehat{\mathbf{t}}\left(2\right),{\widehat{t}}_{3}\left(1\right)-{\widehat{t}}_{3}\left(2\right)\}$.

## 2. Formalism

#### 2.1. Phenomenological Approaches

#### 2.2. Semi-Phenomenological Approaches

#### 2.3. Microscopic Approaches

## 3. Structure and Decay of Neutron-Deficient Nuclei

#### 3.1. IMME Coefficients for Masses and Excitation Spectra of Proton-Rich Nuclei

#### 3.2. Isospin-Forbidden Decays

#### 3.2.1. Isospin-Forbidden $\beta $-Decay

#### 3.2.2. Signatures of Isospin-Symmetry Breaking from Electromagnetic Transitions

#### 3.2.3. $\beta $-Delayed Proton, Diproton or $\alpha $ Emission

## 4. Theoretical Isospin-Symmetry Breaking Corrections to Weak Processes in Nuclei

#### 4.1. Superallowed Fermi $\beta $-Decay

#### 4.2. $\beta $-Decay between Mirror $T=1/2$ States

#### 4.3. Gamow–Teller Transitions in Mirror Nuclei

## 5. Astrophysical Applications

## 6. Conclusions and Perspectives

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

CD | charge-dependent |

CKM | Cabbibo-Kobayashi-Moskawa |

CNO | carbon-nitrogen-oxygen |

CVC | conserved vector current |

$E1$, $E2$ | electric-dipole, electric-quadrupole |

EFT | effective field theory |

F | Fermi |

GT | Gamow-Teller |

h.c. | hermitian congugate |

HF | Hartree–Fock |

IAS | isobaric analogue state |

IMME | isobaric-multiplet mass equation |

IMSRG | in-medium similarity-renormalization group |

INC | Isospin-nonconserving |

$M1$ | magnetic-dipole |

MED | mirror energy difference |

N${}^{3}$LO | next-to-next-to-next-to-leading |

$NN$ | nucleon–nucleon |

rms | root mean square |

TED | triplet energy difference |

TMBE | two-body matrix element |

V–A | vector–axial vector |

WS | Wood-Saxon |

USD | universal $sd$ shell |

$\chi $EFT | chiral effective field theory |

$3N$ | three-nucleon |

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**Figure 1.**Experimental (”Exp”) [5,76] and theoretical (”Theory”) IMME b coefficients for the lowest doublets (

**left**) and c coefficients for the lowest triplets (

**right**) in the $sd$ and $pf$ shells. The $sd$-shell results were quoted from Ref. [27], and $pf$-shell calculations were performed with GX1Acd interaction [77]. See text for details.

**Figure 2.**Experimental [5,76] (

**left**) and theoretical (

**right**) differences in IMME b coefficients (${\Delta}_{b}$) for the ground-state, first-excited and second-excited natural-parity $T=1/2$ multiplets in the $sd$ and $pf$ shells. The $sd$-shell results were obtained with the interaction from Ref. [27], and $pf$-shell calculations were performed with GX1Acd interaction [77].

**Figure 3.**Experimental [5,76] (

**left**) and theoretical (

**right**) IMME c coefficients for the lowest, first-excited and second-excited $T=1$ multiplets in the $sd$ and $pf$ shells. The $sd$-shell results were obtained with the interaction from Ref. [27], and $pf$-shell calculations were performed with GX1Acd interaction [77]. For $A=42$, the data are given for ${J}^{\pi}={0}^{+},{2}^{+},{4}^{+}$ states. See text for details.

**Figure 4.**Schematic picture of $\beta $-delayed p, $\gamma $, $2p$ and $\alpha $ emission from an IAS. See text for details.

**Figure 6.**Schematic picture of the Fermi strength distribution in the daughter nucleus due to the isospin-symmetry breaking effects, as can be viewed from the shell-model’s perspective.

**Left**: depletion of the Fermi strength from an IAS because of non-analogue transitions.

**Right**: insertion of the intermediate states to better constrain the radial part of the single-particle wave functions.

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Smirnova, N.A. Isospin-Symmetry Breaking within the Nuclear Shell Model: Present Status and Developments. *Physics* **2023**, *5*, 352-380.
https://doi.org/10.3390/physics5020026

**AMA Style**

Smirnova NA. Isospin-Symmetry Breaking within the Nuclear Shell Model: Present Status and Developments. *Physics*. 2023; 5(2):352-380.
https://doi.org/10.3390/physics5020026

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Smirnova, Nadezda A. 2023. "Isospin-Symmetry Breaking within the Nuclear Shell Model: Present Status and Developments" *Physics* 5, no. 2: 352-380.
https://doi.org/10.3390/physics5020026