Background: 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, to 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: Isospin-breaking corrections to the $I=0^{+}$, $T=1\to I=0^{+}$, $T=1$ pure Fermi transitions are computed for nuclei from $A=10$ to $A=98$ and, for the first time, to the Fermi branch of the $I,T=1/2\to I$, $T=1/2$ 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 of better than 0.1$\%$.

• Received 4 October 2012

DOI:https://doi.org/10.1103/PhysRevC.86.054316