Errata-Publisher's Note: Electron Wave Functions in Beta-Decay Formulas Revisited (I): Gamow–Teller and Spin-Dipole Contributions to Allowed and First-Forbidden Transitions

We propose formulas of the nuclear beta-decay rate that are useful in a practical calculation. The decay rate is determined by the product of the lepton and hadron current densities. A widely used formula relies upon the fact that the low-energy lepton wave functions in a nucleus can be well approximated by a constant and are linear to the radius for the s-wave andp-wave wave functions, respectively. We find, however, that the deviation from such a simple approximation is evident for heavy nuclei with large Z by numerically solving the Dirac equation. In our proposed formulas, the neutrino wave function is treated exactly as a plane wave, while the electron wave function is obtained by iteratively solving the integral equation, thus we can control the uncertainty of the approximate wave function. The leading-order approximation gives a formula equivalent to the conventional one and overestimates the decay rate. We demonstrate that the next-to-leading-order formula reproduces well the exact result for a schematic transition density as well as a microscopic one obtained by a nuclear energy-density functional method.