W. Satula, J. Dobaczewski, W. Nazarewicz, and M. Rafalski
Nuclear decays provide us with the crucial information about the electroweak force and constraints on physics beyond the Standard Model [1,2]. Of particular importance are superallowed Fermi transitions between the members of an isospin multiplet that can be used to test the conserved vector current (CVC) hypothesis and provide the most restrictive test of the unitarity of the Cabibbo-Kobayashi-Maskawa (CKM) matrix. Under assumptions of zero energy transfer and pure isospin, the transition matrix elements for superallowed decays do not depend on nuclear structure.
For actual nuclei, however, small corrections to the Fermi matrix element of superallowed transitions must be applied (see Refs. [3,4,5,6] and Refs. quoted therein):
In spite of theoretical uncertainties in evaluation of radiative and isospin-breaking corrections, the superallowed -decays provide a stringent test of the CVC hypothesis. In turn, it is also the most precise source of information on the leading element of the CKM matrix [5,7]. Indeed, with the CVC hypothesis confirmed, can be extracted from the data by averaging over 13 precisely measured superallowed transitions spreading over a broad range of nuclei from to .
The main focus of this work is isospin-breaking corrections . This topic has been a subject of numerous theoretical studies using different techniques [5,8,9,10,11,12,13]. The standard in this field has been set by Hardy and Towner (HT) [14,4,5] who employed the nuclear shell model (SM) to account for configuration mixing and the mean-field approach to describe the radial mismatch of proton and neutron single-particle (s.p.) wave functions. Our approach to is based on the self-consistent isospin- and angular-momentum projected nuclear density functional theory (DFT) [15,16]. This framework can simultaneously describe various effects that profoundly impact matrix elements of the Fermi decay; namely, symmetry breaking, configuration mixing, and long-range Coulomb polarization. It should also be noted that our method is quantum-mechanically consistent (see discussion in Ref. ) and contains no adjustable free parameters.
The isospin- and angular-momentum projected DFT approach is based on self-consistent states which, in general, violate both rotational and isospin symmetries. While the rotational invariance is broken spontaneously [17,18], the isospin symmetry is broken both spontaneously (on DFT level) and directly by the Coulomb force. Consequently, the theoretical strategy is to restore the rotational invariance, remove the spurious isospin mixing present in the DFT wave function, and retain only the physical isospin mixing caused by the Coulomb interaction. This is achieved by the rediagonalization of the entire Hamiltonian, consisting the isospin-invariant kinetic energy and nuclear interaction (Skyrme) terms, and isospin-breaking Coulomb force, in a good-angular-momentum and good-isospin basis
The set of states (3) is, in general, overcomplete because the quantum number is not conserved. This difficulty is overcome by selecting first the subset of linearly independent states (collective space), which is spanned, for each and , by the natural states that are eigenstates of the overlap matrix [19,20]. Diagonalization of the Hamiltonian in the collective space yields the eigenfunctions:
As demonstrated in Ref. , in odd-odd nuclei, the isospin projection alone is not sufficient and a simultaneous angular-momentum projection is a must. Unfortunately, this leads to the appearance of singularities in the energy kernels , thus preventing us from using modern Skyrme energy density functionals (EDFs) as none of them is usable, whereas those depending on integer powers of the density, which are regularizable , are not yet developed. Hence, at present, the only practical option is to use the Hamiltonian-driven EDFs which, for Skyrme-type functionals, leaves only one option: the density-independent SV parametrization  supplemented by tensor terms.
The unusual form of SV impacts negatively its overall spectroscopic quality by impairing such key properties as the symmetry energy , level density, and level ordering. These deficiencies affect the calculated isospin mixing. For instance, for the case of Zr discussed above, SV yields the isospin mixing 2.8%, i.e., smaller than the mean isospin mixing 4.4% averaged over nine commonly used Skyrme EDFs, see Fig. 1 of Ref. . Of course, for the description of , of importance is not the absolute magnitude of isospin mixing but its difference between parent and daughter states . The lack of reasonable EDF is, admittedly, the weakest point of our current calculations; nevertheless, no significant improvement of this aspect can be expected in the near future.
The Fermi -decay proceeds between the ground state (g.s.) of the even-even nucleus and its isospin-analogue partner in the odd-odd nucleus. While the DFT state representing the even-even nucleus is unambiguously defined, the DFT state used to compute the wave function is the so-called anti-aligned configuration (or ), selected by placing the odd neutron and the odd proton in the lowest available time-reversed (or signature-reversed) s.p. orbits. The anti-aligned configurations manifestly break the isospin symmetry but they provide a way to reach the states in odd-odd nuclei . This situation creates additional technical problems. The anti-aligned configurations appear to be very difficult to converge in the symmetry-unrestricted DFT calculations. This can be traced back to time-odd components of the EDF. In fact, only in a few cases were we able to obtain symmetry-unrestricted self-consistent solutions. This forced us to impose the signature-symmetry on other DFT wave functions, which implied a specific s.p. angular-momentum alignment pattern .
The calculations presented here were done using the DFT solver HFODD (v2.48q) , which includes both the angular-momentum and isospin projection. The calculated values of depend on the basis size. In order to obtain converged result for with respect to basis truncation, we use 10 oscillator shells for nuclei, 12 oscillator shells for nuclei, and 14 oscillator shells for nuclei. The resulting systematic errors due the basis cut-off do not exceed 10%.
The equilibrium quadrupole deformations ( ) of the anti-aligned configurations in odd-odd nuclei are, in most cases, very close to those obtained for even-even isobaric analogs. Typical differences do not exceed and except for nearly spherical systems and , where the concept of static deformation is ill-defined, and for and pairs where odd-odd and even-even partners have fairly different shapes. As we shall see below, such deformation difference results in large values of .
All studied odd-odd nuclei, except for , 38, and 42, are deformed; thus, to carry out projections, we could use for them the unique lowest anti-aligned DFT states. Also for and 38, unique configurations based on the 1 and 2 subshells were used. A different approach was used to compute in near-spherical nuclei. In Sc, four possible anti-aligned DFT configurations built on the s.p. orbits originating from the spherical 1 subshells can be formed, and the corresponding DFT states differ slightly due to configuration-dependent polarizations . Consequently, to evaluate for we took an arithmetic mean over the values calculated for all anti-aligned configurations.
The unusually large correction % has been calculated for nuclei. Most likely, this is a consequence of incorrect shell structure predicted with SV. Specifically, as a result of incorrect balance between the spin-orbit and tensor terms in SV, the subshell is shifted up to the Fermi surface. This state is more sensitive to time-odd polarizations than other s.p. states around Ca core, see Table I in Ref. . Consequently, the K Ar transition has been excluded from our calculation of .
The predicted isospin-breaking corrections are listed in Table 1. All other ingredients needed to compute -values from Eq. (2), including empirical -values and radiative corrections and , were taken from the most recent compilation . In the error budget of in Table 1, apart from errors of and radiative corrections, we include 10% systematic uncertainty in the calculated due to basis truncation. The average value s was obtained using Gaussian-distribution-weighted formula to conform with standards set by HT. This leads to which coincides with both the HT result  and a central value obtained from the neutron decay . Combining the calculated with the values of and provided in Ref. , we obtain , which implies that unitarity of the CKM matrix is satisfied with precision of 0.1%.
While our value of is consistent with both HT and neutron-decay results, a question arises about its confidence level, especially in light of poor spectroscopic properties of SV. To this end, we carry out the confidence-level (CL) test proposed recently in Ref.  using variant including uncertainties on experiment, , and . The test is based on the assumption that the CVC hypothesis is valid to at least % precision, implying that a set of structure-dependent corrections should produce a statistically consistent set of values. Since only one set of calculated corrections exists , ``empirical" isospin-symmetry-breaking corrections can thus be defined by
In summary, the state-of-the-art isospin- and angular-momentum-projected DFT calculations have been performed to compute the isospin-breaking corrections to Fermi superallowed -decays. Our results for s and were found to be consistent with the recent HT value . While the CL of our values is low, primarily due to a poor spectroscopic quality of the EDF used, our framework contains no adjustable parameters and is capable of describing microscopically all elements of physics impacting . The results presented in this paper should thus be considered as a microscopic benchmark relative to which the further improvements (e.g., regularizable EDF and/or inclusion of pairing) will be assessed.
This work was supported in part by the Polish Ministry of Science under Contract Nos. N N202 328234 and N N202 239037, Academy of Finland and University of Jyväskylä within the FIDIPRO programme, and by the Office of Nuclear Physics, U.S. Department of Energy under Contract Nos. DE-FG02-96ER40963 (University of Tennessee) and DE-FC02-09ER41583 (UNEDF SciDAC Collaboration). We acknowledge the CSC - IT Center for Science Ltd, Finland for the allocation of computational resources.