Atomic nucleus is a self-bound finite system composed of neutrons and
protons that interact by means of short-range, predominantly
isospin-symmetry-conserving strong force and long-range
isospin-symmetry-breaking Coulomb force. In studies of phenomena
related to the isospin-symmetry violation in nuclei, capturing a
delicate balance between these two forces is of utmost importance.
This is particularly true when evaluating the
isospin-symmetry-breaking (ISB) corrections to superallowed
-decays between isobaric analogue states,
[
.
Such -decays currently offer the most precise data that give
estimates of the vector coupling constant
and leading element
of the Cabibbo-Kobayashi-Maskawa (CKM) flavor-mixing
matrix [1,2]. The uncertainty of
extracted
from the superallowed
-decays is almost an order of magnitude
smaller than that from neutron or pion decays [3]. To test
the weak-interaction flavor-mixing sector of the Standard Model of
elementary particles, such precision is critical, because it allows
us to verify the unitarity of the CKM matrix, violation of which may
signal new physics beyond the Standard Model, see
Ref. [4] and references cited therein.
The isospin impurity of the nuclear wave function - a measure of the
ISB - is small. It varies from a fraction of a percent, in ground
states of even-even light nuclei, to about six percent in the
heaviest known
system,
Sn [5]. Nevertheless,
its microscopic calculation poses a real challenge to theory. The
reason is that the isospin impurity originates from the long-range
Coulomb force that polarizes the entire nucleus and can be,
therefore, calculated only within so-called no-core approaches. In
medium and heavy nuclei, it narrows the possible microscopic models to
those rooted within the nuclear density functional theory
(DFT) [6,7].
The absence of external binding requires that the nuclear DFT be
formulated in terms of intrinsic, and not laboratory densities. This,
in turn, leads to the spontaneous breaking of fundamental symmetries
of the nuclear Hamiltonian, including the rotational and isospin
symmetries, which in finite systems must be restored. Fully quantal
calculations of observables, such as matrix elements of
electromagnetic transitions or -decay rates, require symmetry
restoration. In most of practical applications, this is performed
with the aid of the generalized Wick's theorem [8]. Its
use, however, leads to the energy density functionals (EDFs) being
expressed in terms of the so-called transition densities, that is, to
a multi-reference (MR) DFT. Unfortunately, the resulting MR EDFs are,
in general, singular and require regularization, which still lacks
satisfactory and practical solution, see, e.g.,
Refs. [9,10,11]. An alternative way of building
a non-singular MR theory, the one that we use in the present work,
relies on employing the EDFs derived from a true interaction,
which then acquires a role of the EDF generator [12]. The
results presented here were obtained using in this role the
density-independent Skyrme interaction SV [13], augmented
by the tensor terms (SV
) [11].
Over the last few years we have developed the MR DFT approach based
on the angular-momentum and/or isospin projections of single Slater
determinants. The model, below referred to as static, was
specifically designed to treat rigorously the conserved rotational
symmetry and, at the same time, tackle the explicit Coulomb-force
mixing of good-isospin states. These unique approach allowed us to
determine the isospin impurities in
nuclei [5] and ISB corrections to superallowed
-decay matrix elements [14,15].
In this paper, following upon preliminary results announced at several conferences [16,17,18], we introduce a next-generation dynamic variant of the approach, which we call no-core configuration-interaction (NCCI) model. It constitutes a natural extension of the static MR DFT model, and allows for mixing states that are projected from different self-consistent Slater determinants representing low-lying (multi)particle-(multi)hole excitations. Technically, the model is analogous to the generator-coordinate-method (GCM) mixing of symmetry-projected states, see, e.g., Ref. [19]. However, the GCM pertains to mixing continuous sets of states, and thus builds collective states of the system, whereas NCCI involves mixing of discrete configurations. In quantum chemistry such a method is commonly known under the name of configuration interaction (CI), where the interaction means mixing of different electronic configurations. In nuclear physics, models of this type go by the name of the shell model, whereupon all configurations within a specific valence shell are considered. In recent years, in relatively light nuclei, a no-core variant of the shell model (NCSM) has been very successfully implemented [20]. Our approach combines the no-core aspect of the NCSM and the mixing aspect of the CI, and, by using sets of selected DFT configurations, it is not limited to light nuclei.
There are several cases when, to perform reliable calculations, the
NCCI approach is indispensable. One of the most important ones
relates to different possible shape-current orientations, which
within the static variant of the model appear in odd-odd
nuclei [15]. The configuration mixing is also needed
to resolve the issue of unphysical ISB corrections to the analogous
states of the isospin triplet [14,15].
The states that are mixed have good angular momenta and, at the same
time, include properly evaluated Coulomb isospin mixing; hence, the
extended model treats hadronic and Coulomb interactions on the same
footing. The model is based on a truncation scheme dictated by the
self-consistent deformed Hartree-Fock (HF) solutions, and can be used
to calculate spectra, transitions, and -decay rates in any
nucleus, irrespective of its even or odd neutron and proton numbers.
We begin by giving in Sec. 2 a short overview of the
theoretical framework of our NCCI model. In Sec. 3, a new
set of the ISB corrections to the canonical set of superallowed
-decay is presented. As compared to our previous
results [15], the new set includes mixing of reference
states corresponding to different shape-current orientations in
odd-odd
nuclei. In Sec. 4, applications involving
mixing of several low-energy (multi)particle-(multi)hole excitations
are discussed. Here, we determined low-spin energy spectra in
selected nuclei relevant to high-precision tests of the
weak-interaction flavor-mixing sector of the Standard Model. The
calculations were performed for:
Li and
Li nuclei
(Sec. 4.1),
=38 Ar, K, and Ca nuclei
(Sec. 4.2),
Sc and
Ca nuclei
(Sec. 4.3), and
Ga and
Zn nuclei
(Sec. 4.4. Summary and perspectives are given in
Sec. 5.
Jacek Dobaczewski 2016-03-05