*4.1. Superallowed Fermi β-Decay*

The most prominent application of the theoretical formalism exposed just above is the calculation of realistic Fermi-matrix elements for *β*-decay between 0<sup>+</sup> states or between the mirror states in *T* = 1/2 nuclei [129].

Indeed, Fermi type *β*-decay is governed uniquely by the vector part of the weak current. According to the CVC hypothesis, the absolute *Ft* values of such transitions in various emitters with a given isospin *T* should be the same. If this feature holds, from *Ft* one can deduce the vector's coupling constant, *GV*, that is responsible for this semi-leptonic decay (*<sup>u</sup>* <sup>→</sup> *de*+*νe*). Combining *GV* with the value of fundamental weak coupling constant *<sup>G</sup>*<sup>F</sup> obtained from a purely leptonic muon decay (*μ*<sup>+</sup> <sup>→</sup> *<sup>e</sup>*+*νeνμ*), one can determine the absolute value of the |*Vud*| = *GV*/*G*<sup>F</sup> matrix element of the Cabibbo–Kobayasi–Maskawa (CKM) quark-mixing matrix:

$$V\_{\rm CKM} = \begin{pmatrix} V\_{\rm uf} & V\_{\rm us} & V\_{\rm ub} \\ V\_{\rm cd} & V\_{\rm cs} & V\_{\rm cb} \\ V\_{\rm td} & V\_{\rm ts} & V\_{\rm tb} \end{pmatrix}.$$

Numerical values of CKM matrix elements are important for the unitarity tests, such as the normalization condition for its first row: |*Vud*| <sup>2</sup> + |*Vus*| <sup>2</sup> + |*Vub*| <sup>2</sup> = 1.

The absolute *Ft* value is obtained from the experimentally deduced *f t* value after incorporation of a few non-negligible theoretical corrections [130] as defined by the following equation:

$$\left| \,^c F t^{0^+ \to 0^+} \right| \equiv f t^{0^+ \to 0^+} \left( 1 + \delta\_R' \right) (1 + \delta\_{NS} - \delta\_{\mathbb{C}}) = \frac{K}{|M\_{\mathbb{F}}^0|^2 G\_V^2 (1 + \Delta\_{\mathbb{R}})} \,. \tag{29}$$

Here, *f* is the statistical rate function calculated from the decay energy, *t* is the partial half-life of the transitions, *<sup>K</sup>* = <sup>2</sup>*π*3*h*¯ ln <sup>2</sup>(*hc*¯ )6/(*mec*2)5, |*M*<sup>0</sup> <sup>F</sup>| is the Fermi-matrix element in the isospin-symmetry limit (20), *h*¯ is the reduced Planck's constant, *c* is the speed of light, and Δ*R*, *δ <sup>R</sup>* and *δNS* are transition-independent, transition-dependent and nuclear-structuredependent radiative corrections; and *δ<sup>C</sup>* is the isospin-symmetry breaking correction due to the lost analogue symmetry between the parent and the daughter nuclear states. The detailed description of the electroweak corrections and the current status in the field can be found in the latest survey by Hardy and Towner [130]. The most prominent feature discussed in recent years is an updated value of the transition independent term, Δ*R*, which was re-evaluated using the formalism of the effective field theory, and this brought fragility to the unitarity tests [131].

The present discussion focuses on the isospin-symmetry breaking correction, *δC*. This correction is defined as a deviation of the squared realistic Fermi-matrix element from its isospin-symmetry value: |*M*F| <sup>2</sup> = |*M*<sup>0</sup> F| <sup>2</sup>(<sup>1</sup> − *<sup>δ</sup>C*). Therefore, the estimation of *<sup>δ</sup><sup>C</sup>* requires an accurate calculation within a nuclear-structure model which can account for the broken isospin symmetry.

There have been lots of efforts within various theoretical approaches during a few decades already. Figure 5 summarizes predictions from different calculations for the 13 best known transitions (by now, the decay of 26Si has been added to this dataset).

**Figure 5.** Isospin-symmetry breaking correction, *δC*, from various theoretical approaches: SM-WS(2015) [130], SM-HF(1995) [132], RHF-RPA(2009) [39], RH-RPA(2009) [39], SV-DFT(2012) [42], SHZ2-DFT(2012) [42], Damgaard(1969) [133], IVMR(2009) [134]. Figure is adapted from Ref. [135].

The values obviously diverge. In addition, to note is that theoretical approaches assign important uncertainties to their values (those which present associated uncertainties). Currently, evaluation of *δ<sup>C</sup>* provides the largest contribution to the *Ft*-value uncertainty.

As has been demonstrated in Section 2 above, the INC shell model represents a wellsuited tool for the *δ<sup>C</sup>* calculation. By expressing the Fermi-matrix elements in the second quantization, one gets:

$$M\_{\rm F} = \left< \Psi\_f \middle| \hat{T}\_+ \middle| \Psi\_i \right> = \sum\_n \left< f \middle| \mathfrak{E}\_{a\_n}^\dagger \mathfrak{E}\_{a\_p} \middle| i \right> \left< a\_n \middle| \hat{\mathfrak{F}}\_+ \middle| a\_p \right> \,. \tag{30}$$

where *c*ˆ † *<sup>α</sup>* and *c*ˆ*<sup>α</sup>* are nucleon creation (destruction) operators; *α* denotes a full set of spherical quantum numbers, *α* = (*na*, *la*, *ja*, *ma*) ≡ (*a*, *ma*) and the two ingredients of Equation (30) are (i) one body-transition densities:

$$
\langle f|\mathfrak{E}\_{\mathfrak{a}\_n}^\dagger \mathfrak{E}\_{\mathfrak{a}\_p}|i\rangle \equiv \rho\_{\mathfrak{a}\_{\mathfrak{a}\_{\mathfrak{a}\_p}}} \tag{31}
$$

and isospin single particle matrix elements, given by overlap integrals:

$$<\langle \mathfrak{a}\_{\mathfrak{n}} | \hat{\mathfrak{f}}\_{+} | \mathfrak{a}\_{\mathfrak{p}} \rangle = \int\_{0}^{\infty} R\_{\mathfrak{a}\_{\mathfrak{n}}}(r) R\_{\mathfrak{a}\_{\mathfrak{p}}}(r) r^{2} dr \equiv \Omega\_{\mathfrak{a}}.\tag{32}$$

Here, *Rα* denotes the radial part of the single-particle wave function.

It has been pointed out by Miller and Schwenk [136,137] that the use of the exact isospin operator in Equation (30) would involve terms where the radial quantum number, *nα*, for of a proton state, *αp*, is different from that of a neutron state *αn*. Up till now, all shell-model work [132,135,138,139] has been done within an approximation that allows one to express the radial overlaps by Equation (32).

Calculation of realistic Fermi-matrix elements implies the use of one-body transition densities computed using many-body states obtained from the diagonalization of an INC Hamiltonian, and the use of radial wave functions, obtained from a realistic spherical single-particle potential, such as Wood–Saxon (WS) or Hartree–Fock (HF) potential, instead of the harmonic oscillator. Therefore,

$$M\_{\mathbb{F}} = \sum\_{\alpha} \rho\_{\alpha} \Omega\_{\alpha} \, .$$

and the model-independent value (20) can be obtained from one-body transitions densities in the isospin limit (*ρ*<sup>0</sup> *<sup>α</sup>*) and harmonic-oscillator radial overlaps (Ω<sup>0</sup> *<sup>α</sup>* = 1):

$$M\_{\mathcal{F}}^{0} = \sum\_{\alpha} \rho\_{\alpha}^{0} \Omega\_{\alpha}^{0} = \sum\_{\alpha} \rho\_{\alpha}^{0}$$

(the superscript "0" indicates that those quantities were calculated in the isospin limit). Therefore, there are two sources of isospin-symmetry breaking in the Fermi-matrix element: first comes from the difference in configuration mixing of the parent and daughter nuclei as obtained from the shell-model diagonalization of an INC Hamiltonian. The other is due to the deviation of the radial overlaps from unity, when calculated with realistic single-particle wave functions instead of the harmonic-oscillator ones. These deviations of one-body transitions densities and radial overlaps from their isospin-symmetry values are typically small. Keeping only linear terms in small quantities, one can express |*M*F| <sup>2</sup> as

$$|M\_{\rm F}|^2 \approx |M\_{\rm F}^0|^2 \left[1 - \underbrace{\frac{2}{M\_{\rm F}^0} \sum\_{a} \left(\rho\_a^0 - \rho\_a\right)}\_{\delta\_{\rm C1}} - \underbrace{\frac{2}{M\_{\rm F}^0} \sum\_{a} \rho\_a^0 (1 - \Omega\_a)}\_{\delta\_{\rm C2}}\right],$$

From this expression, it is seen that the correction splits into two terms according to the two sources of isospin-symmetry breaking mentioned above:

$$
\delta\_{\mathbf{C}} \approx \delta\_{\mathbf{C}1} + \delta\_{\mathbf{C}2} \cdot \mathbf{c}
$$

To get *δC*1, it is sufficient to perform calculations with INC interactions. As has been discussed in Section 3, the theoretical value for a depletion of the Fermi strength in the IAS is due to the mixing of the IAS with non-analogue states (see Figure 6 (left)). Therefore, the position of those states is vital.

**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.

To avoid this uncertainty, one may scale the strengths of individual transitions to non-analogue states with the energy difference between those states and the IAS [138]:

$$
\delta\_{\rm C1} = \delta\_{\rm C1}^{\rm th} \left( \frac{\Delta E\_{\rm th}}{\Delta E\_{\rm exp}} \right)^2
$$

.

Existing shell-model studies use various parametrizations of the INC Hamiltonian, ranging from realistic phenomenological fits [132,135,140] to individual parametrization of charge-dependent terms to each isobaric multiplet presented in Ref. [138,139]. Since this part of the correction is small, the results of both approaches are within typical uncertainties.

In addition to the isospin-symmetry breaking inside the model space, one has to replace harmonic-oscillator radial wave functions with realistic spherically symmetric wave functions from a WS or a HF potential, including Coulomb. This is the largest part of the correction; see Ref. [138] and references therein. A parametrization of a single-particle potential is crucial for the value of the correction. Due to this reason, potential parameters are adjusted to reproduce proton and neutron separation energies and nuclear charge radii. To achieve this goal, a calculation has to be done beyond the closure approximation. This means instead of inserts, a complete sum of intermediate nucleus states ({*π*}) in the Fermi-matrix element. Then, the radial-overlap correction can be expressed as

$$\delta\_{\rm C2} = \frac{2}{M\_{\rm F}^0} \sum\_{\alpha, \pi} \langle f | \mathcal{E}\_{\alpha\_n}^\dagger | \pi \rangle^0 \langle \pi | \mathcal{E}\_{\alpha\_p} | i \rangle^0 (1 - \Omega\_a^\pi) \dots$$

The two ingredients are the spectroscopic amplitudes, *f* |*c*ˆ † *<sup>α</sup><sup>n</sup>* |*π*0, obtained within the isospin-symmetry limit, and the radial-overlap integrals

$$
\Omega\_{\underline{a}}^{\pi} = \int\_0^{\infty} R\_{\alpha\_n}^{\pi}(r) R\_{\alpha\_p}^{\pi}(r) r^2 dr \,\underline{\pi}
$$

which now contain dependence on the excitation energy of the intermediate states *π*; see Figure 6 (right).

This opportunity to constrain theoretical calculations by experimental observables greatly helps to reduce uncertainty in the potential parameters and guaranties consistency of the results, as has been discussed in detail in Ref. [135]. In particular, the largest contribution to theoretical uncertainty on *δC*<sup>2</sup> is because of the experimental uncertainty on the nuclear charge radii.

Up till now, systematic calculations with the WS potential are the only ones who produce corrections consistent with the CVC hypothesis within a non-zero confidence limit [141]. The use of the HF wave functions, pioneered by Ormand and Brown [132,140,142], has been explored by a few other groups as well [138,143,144]. Self-consistent HF potentials are not immediately appropriate for calculations and have to also be adjusted to give rise to experimental proton and neutron separation energies. The procedures exploited by various authors are somewhat different, and, in general, led to smaller corrections than those obtained from a WS potential. This issue has recently been explored in detail in Ref. [144]. In particular, the authors examined the role of previously neglected effects, taking care of the approximate elimination of spurious isospin-mixing, two-body center-of-mass corrections, exact treatment of the exchange Coulomb term and many others. Moreover, INC terms have been added to the energy-density functional. Those corrective terms indeed explain some of the difference between the HF and WS results, allowing to suppose that the remaining part of the difference is due to the need for correlations beyond the HF approximation. Further efforts towards more sophisticated theories should be addressed in future studies.

In spite of these challenges in the computation of theoretical corrections, nuclear <sup>0</sup><sup>+</sup> <sup>→</sup> <sup>0</sup><sup>+</sup> *<sup>β</sup>*-decay provides the best opportunity to test the CVC and to extract the *Vud* value, among other ways (mirror transitions, neutron or pion beta decay) [130]. Therefore, it is reasonable to persist with efforts in improving theoretical modelization of the isospinsymmetry breaking correction.
