By studying isotopes with enhanced sensitivity to fundamental symmetries, nuclear physicists can test various aspects of the Standard Model in ways that are complementary to other sciences. For example, a possible explanation for the observed asymmetry between matter and anti-matter in the universe could be studied by searching for a permanent electric dipole moment larger than Standard Model predictions in heavy radioactive nuclei that have permanent octupole shapes. Likewise, the superallowed -decays of a handful of rare isotopes with similar numbers of protons and neutrons, in which both the parent and daughter nuclear states have zero angular momentum and positive parity, are the unique laboratory to study the strength of the weak force.
What makes these pure vector-current-mediated (Fermi) decays so useful for testing
the Standard Model is the hypothesis of
the conserved vector current (CVC), that is, independence of the vector
current on the nuclear medium. The consequence of the CVC hypothesis is that the product of the
statistical rate function and partial half-life
for the superallowed
Fermi
-decay
should be nucleus independent and equal to:
The relation (1) does not hold exactly and must be slightly amended
by introducing a set of radiative corrections to the -values,
and a correction to the nuclear matrix element due to isospin-symmetry breaking:
In spite of theoretical uncertainties in the evaluation of the radiative and
isospin-symmetry-breaking corrections, the superallowed -decay
is the most precise source of experimental information for determining the vector
coupling constant
, and provides us with a
stringent test of the CVC hypothesis. In turn, it is also the most precise
source of the matrix element
of the Cabibbo-Kobayashi-Maskawa
(CKM) three-generation quark mixing
matrix [8,9,3,10]. This is so because
the leptonic coupling constant,
GeV
, is well known from the muon decay [10].
The advantage of the superallowed -decay strategy results from the fact that, within the CVC
hypothesis,
can be extracted by averaging over several
transitions in different nuclei. For precise tests of the Standard Model, only these transitions that have
-values known with a relative
precision better than a fraction of a percent are acceptable.
Currently, 13 ``canonical" transitions spreading over a wide range of nuclei from
to
meet this criterion (have
-values measured with accuracy
of order of 0.3% or better) and are used to evaluate the values of
and
[3].
In this work we concentrate on the isospin-breaking (ISB) corrections
, which were already computed by various authors,
using a diverse set of nuclear
models [11,3,12,13,14,15,16,17,].
The standard in this field has been set by Towner and Hardy (HT) [3] who used
the nuclear shell-model to account for the configuration mixing effect, and the mean-field (MF) approach
to account for a radial mismatch of proton and neutron
single-particle (s.p.) wave functions caused by the Coulomb polarization. In this study, which constitutes an extension of our earlier work [16], we use the
isospin- and angular-momentum-projected density functional theory (DFT).
This method can account, in a rigorous quantum-mechanical way, for spontaneous symmetry-breaking (SSB)
effects, configuration mixing, and long-range Coulomb polarization effects.
Our paper is organized as follows. The model is described in Sec. 2.
The results of calculations for ISB corrections to the superallowed
Fermi transitions are summarized in Sec. 3.
The ISB corrections to the Fermi matrix elements
in mirror-symmetric
nuclei are discussed in Sec. 4. Section 5 studies a particular case of the Fermi decay of
Cl. Finally, the summary and perspectives are given in Sec. 6.