3.2.1. Isospin-Forbidden *β*-Decay

To shed light on possible isospin impurities in nuclear states, one must appeal to isospin-forbidden transition probabilities. Let us remark that the only model-independent way to extract the amount of isospin-mixing from experiment is provided by Fermi *β*decay [49]. Since the Fermi operator (9) is given by the isospin ladder operators *T*ˆ <sup>±</sup>, its matrix element between members of an isobaric multiplet can be expressed as

$$|M\_\mathcal{F}^0| = |\langle T, M\_T \pm \mathbf{1} | \hat{T}\_\pm | T M\_T \rangle| = \sqrt{(T + M\_T)(T - M\_T + 1)}\,. \tag{20}$$

In isospin-symmetry limit, the whole strength would feed the IAS. A measured depletion of the Fermi strength from the IAS or observation of Fermi transitions to non-analogue states can bring information on the amount of isospin mixing in the IAS. In addition, if a *MT* > 0 nucleus *β*<sup>+</sup> (*β*−) decays, then the mixing is dominantly present in the parent (daughter) nucleus, and inversely for a *MT* < 0 nucleus. Then, the isospin-forbidden Fermi-matrix element in a non-analogue state can be estimated as |*MIF* <sup>F</sup> | = |*x*||*M*<sup>0</sup> F|.

Special cases of purely Fermi, non-analogue 0<sup>+</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> transitions are known, and they bring important information for the tests of the weak interaction in nuclear decays [104]. Distribution of the non-analogue Fermi strength, as experimentally measured recently in 62Ga [105], can shed light on the mixing matrix elements to cross-check the theory.

Transitions between states of the same (but non-zero) angular momentum, *<sup>J</sup><sup>π</sup>* <sup>→</sup> *<sup>J</sup><sup>π</sup>* (*J* = 0), are governed by both Fermi and Gamow–Teller components of the *β*-decay operator. A separation of the Gamow–Teller matrix element is an experimental challenge, bringing, however, direct knowledge on on isospin impurities, as elaborated in Refs. [49,106,107].

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

Observation of other isospin-forbidden decays requires theoretical calculations of corresponding nuclear processes for extraction of the mixing probability. For example, electromagnetic transitions which violate isospin selection rules propose another possibility to test the degree of isospin-symmetry breaking.

Electric dipole transitions play a special role in these explorations due to a specific isovector character of the operator; see Equation (12). In particular, in Section 1.1, it was mentioned that *E*1 transitions between the states of the same isospin in self-conjugate (*N* = *Z*) nuclei are forbidden by isospin symmetry. A few cases of observation of weak *E*1 transitions in *N* = *Z* nuclei between states of the same isospin have been reported [108,109]. This indicates breaking of isospin symmetry in the states involved in the decay. The shellmodel calculation of individual *E*1 transition rates is hampered by the fact that the model space should contain orbitals of different parities, which could also lead to a center-of-mass motion. Given that the center-of-mass separation is only approximate, it is a challenge to give a precise estimation of the *E*1 strength. Observed enhancements of *E*1 rates in *N* = *Z* nuclei and enhanced asymmetries of mirror *E*1 transitions can be related to the giant isovector monopole resonance [109].

An original idea of using *E*2/*M*1 (electric quadrupole/magnetic-dipole) mixing ratio of decays in a self-conjugate nucleus 54Co has been elaborated in Ref. [110] to pin down isospin impurities in a 4<sup>+</sup> doublet.

Electromagnetic transitions between isobaric analogue states provide other possibilities to test isospin selection rules. For example, linear dependence of the *E*2 matrix elements on *MT* in Δ*T* = 1 analogue transitions have been explored experimentally in a number of triplets (see Refs. [111,112] and references therein), and tests of equality of isovector matrix elements in mirror systems have been carried out [113,114].

An interesting idea to extract the amount of isospin mixing from *E*1 transition rates in mirror nuclei has been proposed and explored in Ref. [115].

Other possibilities to deduce isospin mixing in nuclear states from electromagnetic responses have been explored, e.g., in electron-scattering experiments [116] or via excitation of giant dipole resonance in *N* = *Z* nuclei, e.g., in Refs. [117–119].
