Beta-decay formulas revisited (I): Gamow--Teller and spin-dipole contributions to allowed and first-forbidden transitions

TL;DR

This paper revisits nuclear beta-decay rate formulas, improving accuracy by exactly treating neutrino wave functions and iteratively solving electron wave functions, especially for heavy nuclei, and demonstrates the effectiveness of the next-to-leading-order approximation.

Contribution

The authors develop improved beta-decay rate formulas that account for deviations in lepton wave functions in heavy nuclei, enhancing precision over conventional approximations.

Findings

Next-to-leading-order formula accurately reproduces exact decay rates.

Exact treatment of neutrino wave functions improves calculation accuracy.

The new formulas outperform traditional approximations for heavy nuclei.

Abstract

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 linear to the radius for the $s$-wave and $p$-wave wave functions, respectively. We find, however, 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.