Isospin-breaking corrections to superallowed Fermi $\beta$-decay in isospin- and angular-momentum-projected nuclear Density Functional Theory

W. Satula, J. Dobaczewski, W. Nazarewicz, and T.R. Werner

Abstract:

Background

The superallowed $\beta$-decay rates provide stringent constraints on physics beyond the Standard Model of particle physics. To extract crucial information about the electroweak force, small isospin-breaking corrections to the Fermi matrix element of superallowed transitions must be applied.

Purpose

We perform systematic calculations of isospin-breaking corrections to superallowed $\beta$-decays and estimate theoretical uncertainties related to the basis truncation, time-odd polarization effects related to the intrinsic symmetry of the underlying Slater determinants, and to the functional parametrization.

Methods

We use the self-consistent isospin- and angular-momentum-projected nuclear density functional theory employing two density functionals derived from the density independent Skyrme interaction. Pairing correlations are ignored. Our framework can simultaneously describe various effects that impact matrix elements of the Fermi decay: symmetry breaking, configuration mixing, and long-range Coulomb polarization.

Results

The isospin-breaking corrections to the $I=0^+,T=1\rightarrow I=0^+,T=1$ pure Fermi transitions are computed for nuclei from $A$=10 to $A$=98 and, for the first time, to the Fermi branch of the $I,T=1/2 \rightarrow I,T=1/2$ transitions in mirror nuclei from $A$=11 to $A$=49. We carefully analyze various model assumptions impacting theoretical uncertainties of our calculations and provide theoretical error bars on our predictions.

Conclusions

The overall agreement with empirical isospin-breaking corrections is very satisfactory. Using computed isospin-breaking corrections we show that the unitarity of the CKM matrix is satisfied with a precision better than 0.1%.