3.2.3. *β*-Delayed Proton, Diproton or *α* Emission

Nucleon(s) emission may serve as a stringent test of isospin purity [120]. Interesting cases are provided by *β*-delayed proton (or two-proton, *α*) emission when an IAS, populated in the *β*-decay, is located beyond the corresponding particle separation threshold [121,122]. As follows from a typical energy balance, in this case the proton (two-proton, alpha) emission from the IAS (*Jπ*, *T*), populated in the *β*-decay of a *MT* < 0 precursor, is forbidden by isospin symmetry (see Figure 4). Observation of such processes evidences the presence of isospin mixing, mainly, in the IAS which is surrounded by states of another isospin, (*Jπ*, *<sup>T</sup>* <sup>−</sup> 1). A large amount of mixing can be deduced from the missing Fermi strength. However, small amounts may be hidden by experimental uncertainties.

**Figure 4.** Schematic picture of *β*-delayed *p*, *γ*, 2*p* and *α* emission from an IAS. See text for details.

To deduce spectroscopic factors from isospin-forbidden proton emission on purely theoretical grounds is challenging [120,123]. Nevertheless, recently, it has been shown that one can deduce isospin mixing using experimental proton-*γ*branching ratios in the case of *β*-delayed *pγ*-emission [101,124] (two-proton or *α* emission were supposed to be absent or negligible in that study). Since the proton to *γ*-decay branching ratio for the IAS, *I*IAS *<sup>p</sup>* /*I*IAS *<sup>γ</sup>* , equals to the ratio of the corresponding widths, with the help of the theoretical electromagnetic width, ΓIAS *<sup>γ</sup>* , one can extract the proton width of the IAS as

$$
\Gamma\_p^{\text{LAS}} = \Gamma\_\gamma^{\text{LAS}} \frac{I\_p^{\text{LAS}}}{I\_\gamma^{\text{LAS}}} \,. \tag{21}
$$

Generally, the shell model provides a relatively robust description of electromagnetic widths, if experimental energies are used. Deduced proton widths are important in astrophysics applications. For example, radiative proton capture is an inverse process, where a nucleus capturing a proton gets excited to a specific level and is de-excited by *γ* emission. Proton and electromagnetic widths are thus essential ingredients with which to estimate the contribution of resonant capture.

In addition, if the angular momentum, *l*, of the proton is unambiguously determined (as in the decay from 0<sup>+</sup> state), from Equation (21) one can deduce the spectroscopic factor for an isospin-forbidden proton emission from the IAS. To this end, one can estimate theoretical single-particle proton width, ΓIAS *sp* , of the IAS and express the spectroscopic factor as

$$S\_p^{\text{LAS}} = \frac{\Gamma\_\gamma^{\text{LAS}}}{\Gamma\_{sp}^{\text{LAS}}} \frac{I\_p^{\text{LAS}}}{I\_\gamma^{\text{LAS}}} \,. \tag{22}$$

Let us remark that this estimation does not rely on the isospin mixing in the IAS, which would depend on energies of admixed states, but only on the experimental ratio of proton and gamma intensities and on the calculated width.

If a two-state mixing hypothesis approximately holds, for example, when the IAS is mostly mixed with a single nearby non-analogue state (*Jπ*, *<sup>T</sup>* <sup>−</sup> 1), then one can approximately estimate the amount of isospin mixing in the IAS. Namely, using the shell-model value for the spectroscopic factor of the admixed state, *S*T−<sup>1</sup> *<sup>p</sup>* , the probability of mixing can be expressed as *x*<sup>2</sup> = *S*IAS *<sup>p</sup>* / *S*T−<sup>1</sup> *<sup>p</sup>* . This procedure can be generalized to include isospinforbidden 2*p* or *α*-particle emission from the IAS.

For several measured proton branches that form the IAS, one can apply the above formalism to each of them separately, since relation (21) holds:

$$
\Gamma\_{p,i}^{\text{LAS}} = \Gamma\_{\gamma}^{\text{LAS}} \frac{I\_{p,i}^{\text{LAS}}}{I\_{\gamma}^{\text{LAS}}} \,. \tag{23}
$$

Proceed to extract spectroscopic factors and isospin mixing, if a two-level mixing model is applicable. This proposes an interesting possibility to cross-check the values of *x*<sup>2</sup> deduced from various branches. Such cases of *β*-delayed *pγ* emission from an odd *A* precursor have also been reported (see, e.g., Refs. [121,122,125]).

Actually, one can also determine an approximate value of isospin mixing in the IAS in a two-level mixing case, even if the set of quantum numbers (*nlj*) characterizing the emitted proton is not unique. In this case, the proton width is a sum of contributing partial widths corresponding to all allowed orbitals from a given model space, e.g., ΓIAS *<sup>p</sup>* = ∑*nlj S*IAS *<sup>p</sup>* (*nlj*)ΓIAS *sp* (*nlj*). Therefore, providing shell-model values of isospin-allowed spectroscopic factors, *ST*−<sup>1</sup> *<sup>p</sup>* (*nlj*), one can estimate the amount of isospin impurity of the IAS to be

$$\alpha^2 = \frac{\Gamma\_p^{\text{IAS}}}{\sum\_{nlj} S\_p^{T-1} (nlj) \Gamma\_{sp}^{\text{IAS}} (nlj)} \, \text{} \tag{24}$$

where ΓIAS *<sup>p</sup>* is deduced as in Equation (21) and the denominator is evaluated theoretically. Finally, individual spectroscopic factors (for each *nlj* channel) for isospin-forbidden proton emission can be obtained as *S*IAS *<sup>p</sup>* (*nlj*) = *x*2*ST*−<sup>1</sup> *<sup>p</sup>* (*nlj*) The uncertainty of theoretical estimation increases in this case, since a few theoretical quantities have to be used. In general, one should also remember that small spectroscopic factors (below 0.1) carry a significant theoretical uncertainty and this may prohibit extraction of the detailed information according to the proposed method.

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

At present, many-body calculations for nuclear structure are required to link experimental information on weak processes involving nuclei to the underlying theories of fundamental interactions. In this context, the nuclear shell model is among the most favorite tools to provide nuclear matrix elements necessary for the tests of the symmetries of the standard model and for the searches for physics beyond it. Those can be probed in nuclear *β*-decay, but also in charge–exchange reactions or, eventually, in muon capture experiments. Calculations allowing to account for isospin-symmetry breaking may become vital in studies of individual transitions involving proton-rich nuclei and nuclei along *N* = *Z* line.

The discussion below focuses on two activities related to the study of beta decay, which can be described by an effective axial-vecor and vector, *V*–*A*, interaction Hamiltonian,

$$
\hat{H}\_{V-A} = \frac{G\_V}{\sqrt{2}} f\_{\mu}^{\dagger} \hat{j}^{\mu} + \text{h.c.}\tag{25}
$$

where *J*† *<sup>μ</sup>* ((*j <sup>μ</sup>*) is hadronic (leptonic) weak current, the index *μ* represents the space-time 4-vector index and takes valuse 0 (time),1, 2, and 3 (space), "h.c." stays for "hermitian conjugate", and *GV* is the weak-interaction coupling constant responsible for this semileptonic decay. The most general form of a Lorentz-covariant form of the vector and axial-vector nucleon currents read

$$J^{+}\_{\mu} = \quad \mathcal{V}\_{\mu} + A\_{\mu} \tag{26}$$

$$\hat{\mathcal{V}}\_{\mu} \quad = \quad i\overline{\psi}\_{p} \left( g\_{V}(k^{2})\gamma\_{\mu} + \frac{g\_{W}(k^{2})}{2m\_{N}}\sigma\_{\mu\nu}k\_{\nu} + ig\_{S}(k^{2})k\_{\mu} \right) \psi\_{n} \tag{27}$$

$$\hat{A}\_{\mu} \quad = \quad i\overline{\psi}\_{p} \left( g\_{A}(k^{2})\gamma\_{\mu} + \frac{g\_{T}(k^{2})}{2m\_{N}}\sigma\_{\mu\nu}k\_{\nu} + ig\_{P}(k^{2})k\_{\mu} \right)\gamma\_{5}\psi\_{n} \tag{28}$$

where *ψ<sup>p</sup>* and *ψ<sup>n</sup>* are nucleon field operators, *mN* is the nucleon mass; *k<sup>μ</sup>* is the 4-momentum transferred from hadrons to leptons; *σμν* = [*γμ*, *γν*]/2*i* and *γμ* are Dirac matrices. The six form-factors are arbitrary real functions of Lorentz invariants of *k*2, to be compatible with time-reversal invariance. At low momentum transfer, they are known as the vector (*gV*), weak magnetism (*gW*), scalar (*gS*), axial-vector (*gA*), tensor (*gT*) and pseudo-scalar (*gP*) coupling constants.

The six terms have definite properties under the *G*ˆ-parity transformation, which is a combination of charge-conjugation (*C*ˆ) and rotation in isospin space over 180 degrees about the 2-axis (*G*ˆ = *C*ˆ exp (i*πT*ˆ 2)). Those which transform as leading-order vector and axialvector terms are called first-class currents, and those which have opposite transformation properties are called second-class currents. Of the latter type are the induced scalar term in the vector current and induced tensor term in the axial-vector current.

Various constraints on these coupling constants come from the symmetries underlying the standard model [126]. The most stringent condition is provided by the conserved vector current (CVC) hypothesis, which is based on the similarity in structure of the vector weak current and the isovector electromagnetic current. From CVC, it follows that the vector and weak-magnetism form factors are related to their electromagnetic counterparts, in particular, *gV*(*k*<sup>2</sup> → <sup>0</sup>) = 1. This symmetry also implies that the induced scalar term vanishes (*gS* = 0).

For the axial-vector current, only a partially conserved axial-vector current hypothesis exists, and it is less restrictive: it allows one to relate the main axial-vector coupling constant to the pion–nucleon coupling constant by famous Goldhaber–Trieman relations.

Nuclear *β*-decay experiments provide an excellent ground to test the structure of these currents and experimentally determine the magnitude of the coupling constants (see extensive reviews [127,128]). Two particular domains are described below, when theoretical calculation of nuclear matrix elements is required, along with an accurate treatment of isospin-symmetry breaking.
