*4.2. β-Decay between Mirror T* = 1/2 *States*

It was pointed also that *β*-decay between mirror states with *T* = 1/2, which are governed by both Fermi and Gamow–Teller operators, can also serve for the tests of the CVC hypothesis and extraction of *Vud*, once the GT component is eliminated [130,145]. To this end, an additional correlation coefficient has to be measured. Similarly to 0<sup>+</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> decay, the experimentally determined *f t* value has to be corrected for radiative effects and for isospin-symmetry breaking in decaying states. The shell-model framework relies on a similar expression of the realistic Fermi-matrix element, as discussed, with an intermediate state summation which involves a larger number of different states because of non-zero values of angular momenta involved. Currently achieved results are summarized in Ref. [129].

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

Another long-standing application is related to the asymmetry of Gamow–Teller *β*-decay rates in mirror nuclei, defined as

$$\delta = \left| \frac{M\_{\rm GT}^{+}}{M\_{\rm GT}^{-}} \right|^{2} - 1 \,, \tag{33}$$

where *M*± GT are reduced matrix elements for mirror GT *β*<sup>±</sup> transitions. The initial interest in the topic was due to the fact that the contribution to that asymmetry may be due to the presence of the induced tensor term (*gT*) in the axial-vector current; see Equation (28).

To pin down a possible manifestation of the induced tensor term, an accurate calculation of GT matrix elements, including isospin-symmetry breaking, is required. In the second-quantization formalism, the reduced matrix elements of the GT operator can be expressed as follows:

$$M\_{\rm GT}^{\pm} = \langle \Psi\_f || \hat{\mathcal{O}}\_{\rm GT}(\boldsymbol{\beta}^{\pm}) || \Psi\_i \rangle = \frac{1}{\sqrt{3}} \sum\_{a,b} \langle I\_f || [\hat{\varepsilon}\_a^{\dagger} \hat{\varepsilon}\_b]^{(1)} || I\_i \rangle \langle a, m\_{t\_a} || \hat{\sigma} \hat{t}\_+ || b, m\_{t\_b} \rangle \,, \tag{34}$$

where double bars denote reduction in angular momentum, and ˆ *<sup>c</sup>*˜*a*,*ma* = (−1)*ja*+*ma <sup>c</sup>*ˆ*a*,−*ma* . Again, realistic calculations should be ensured beyond the closure approximation, thereby inserting a complete sum over intermediate nucleus states. Several theoretical investigations have been performed from the 60s up to the present, without any indication on a possible application of the analysis to the weak-interaction problem because of high theoretical uncertainty of the nuclear wave functions (see Ref. [146,147] and references therein). Although experimental measurements of mirror transitions main an active field, the main impact of the results is on the structural aspects of the states involved in the decay. In this context, alternative constraints on the induced tensor current from *β*-*α* and *β*-*γ* angular correlation experiments tend to be much more advantageous [148,149].

#### **5. Astrophysical Applications**

One of the greatest motivations to explore the properties of nuclei is their need for nuclear astrophysics. Nuclear masses, half-lives, level densities, and nuclear, electromagnetic and weak-interaction reaction rates represent crucial ingredients for simulations and understanding of astrophysical processes [150]. In particular, the structure of neutrondeficient nuclei is important for comprehension of nucleosynthesis during stellar explosive hydrogen burning. Among the possible sites are X-ray bursts and novae outbursts.

Novae are understood as a result of thermonuclear runaway at the surface of a white dwarf within a binary star system. At high temperatures (∼ 108 K) and densities in O-Ne type novae, the break-out of the hot CNO (carbon-nitrogen-oxygen) cycle leads to nucleosynthesis of heavier elements *A* ≥ 20 by mainly (*p*, *γ*), in competition with (*α*, *p*) and inverse reactions, with the end point around Ca [151]. In X-ray bursts [152,153], based on a neutron star accreting hydrogen matter within a binary system, the temperatures are even higher (up to 2 × <sup>10</sup><sup>9</sup> K), and radiative proton capture reactions involve proton-rich nuclei towards the proton-drip line, being the most important reaction type in nucleosynthesis with up to roughly *A* ∼ 100 (*rp* process). Simulations of X-ray bursts exploit a huge set of nuclear reactions which have to be constrained.

For stable nuclei, the proton-capture reaction *Q*-values are relatively high, and the reaction rate may be approximated by a statistical model. For unstable (proton-rich) nuclei, *Q* values become small (in the order of a few MeV or less), and hence, the reaction rate is dominated by a few isolated resonances above the proton-emission threshold, together with a non-resonant reaction contribution in the energy range within a Gamow peak. In this case, accurate knowledge of resonance energies and decay widths is required.

Current state-of-the-art simulations are based on experimentally deduced information when it is available. If no data exist yet, then one can either deduce the missing information from mirror systems, assuming the isospin symmetry, or appeal to theory. Therefore, higher-precision theoretical calculations are important to reduce uncertainties.

Shell modeling is one of the approaches which can provide detailed information on nuclear states and transitions at low energies. The resonant part of a thermonuclear (*p*, *γ*) reaction rate for a single resonance can be expressed [154] as

$$N\_A \langle \sigma v \text{s.} \rangle\_{\varGamma} = 1.540 \times 10^{11} \left( \mu T\_\theta \right)^{-3/2} \omega \cdot \gamma \cdot \exp \left( \frac{-11.605 \, E\_\tau}{T\_\theta} \right) \, \text{cm}^3 \, \text{s}^{-1} \, \text{mol}^{-1}, \tag{35}$$

where *μ* = *ApA*/(*Ap* + *A*) is the reduced atomic mass number and *Er* is the resonance energy above the proton-emission threshold (in MeV), *T*<sup>9</sup> is the temperature in GK. The resonance strength *ω γ* (in MeV) depends on the spins of initial *Ji* and final (resonance) *Jf* states, the partial proton width Γ*<sup>p</sup>* for the entrance channel and gamma widths Γ*<sup>γ</sup>* for the exit channel:

$$
\omega \gamma = \frac{2J\_f + 1}{2(2J\_i + 1)} \frac{\Gamma\_p \Gamma\_\gamma}{\Gamma\_{tot}} \, , \tag{36}
$$

with Γ*tot* = Γ*<sup>p</sup>* + Γ*γ*. The proton decay width depends on the resonance energy via the proton width, which could be estimated from a single-particle potential model and the shell model's spectroscopic factor. In case of a few resonances, the resonant reaction rate represents a sum of single-resonance rates (35) over contributing final states.

The non-resonant part of (*p*, *γ*), the reaction rate is given by direct capture transitions to the ground or low-level states of the final nucleus.

A number of radiative proton-capture reaction rates have been evaluated with a good precision for *sd*-shell and *p f*-shell nuclei [155,156], since the shell model provides missing information on resonance states, proton and electromagnetic widths. The INC formalism is of particular interest for such problems. First, using theoretical IMME *b* coefficients, one can not only provide nuclear masses of proton-rich nuclei [28,91,92], but also determine positions of unknown resonances in a proton-rich nucleus, if the level scheme of a neutronrich mirror nucleus is known experimentally. The necessity to account for isospin-symmetry breaking to get more accurate results was demonstrated first in Ref. [155] and followed in numerous studies. A use of theoretical *c* coefficients may even be more advantageous for an *MT* = −1 nuclei if experimental information on *MT* = 0, 1 exists (see, e.g., [93,157,158].) The cross-shell *p*-*sd*-*p f* model space is necessary for the description of negative parity resonances in *sd*-shell nuclei (see, e.g., Refs. [159,160]).

A particularly interesting result was obtained a few years ago, indicating that the Thomas–Ehrman effect may significantly change values of theoretical spectroscopic factors [87]. More attention therefore has to be paid to small values of spectroscopic factors. This also indicates that results on spectroscopic factors from mirror systems should be accepted with caution.

#### **6. Conclusions and Perspectives**

The nuclear shell model provides a powerful formalism with which to deal with tiny breaking of isospin symmetry in nuclear states. Currently, the most accurate results are due to phenomenological treatment of nuclear wave functions and parametrization of the INC terms of the Hamiltonian. Although more work is required to have a better handle on large model spaces, extended applications to structure and decay proton-rich nuclei and nuclei along the *N* = *Z* line support experimental investigations. Important applications of that formalism exist, such as the calculation of isospin-symmetry breaking corrections for Fermi-matrix elements required to test the symmetries underlying the standard model. Finally, isospin-symmetry breaking is nowadays taken into account in the evaluation of thermonuclear reaction rates in proton-rich nuclei, which plays an important role in astrophysical simulations.

While phenomenological approaches still have to be pursued to assure solid support to experimental investigations, the eventual goal of nuclear theorists is to develop fundamental ab initio frameworks for many-body calculations towards a higher precision level that will be relevant for the isospin-symmetry breaking domain.

**Funding:** This research was funded by Master Projects *Isospin-symmetry breaking* (2017–2020) and *Exotic Nuclei, Fundamental Interactions and Astrophysics (ENFIA)* (2020–2023).

**Data Availability Statement:** The data presented can be found in the references cited.

**Acknowledgments:** The author acknowledges collaboration with N. Benouaret, B. Blank, B. A. Brown, Y. H. Lam, W. A. Richter, C. Volpe, and L. Xayavong on different topics related to isospinsymmetry breaking. Large-scale calculations have been performed at MCIA, University of Bordeaux.

**Conflicts of Interest:** The author declares no conflict of interest.
