4.3 Article

Electron wave functions in beta-decay formulas revisited (I): Gamow-Teller and spin-dipole contributions to allowed and first-forbidden transitions

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OXFORD UNIV PRESS INC
DOI: 10.1093/ptep/ptab069

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  1. Japan Society for the Promotion of Science (JSPS) KAKENHI [JP18H01210, JP18H04569, JP18K03635, JP19H05104, JP19H05140, JP19K03824]
  2. Collaborative Research Program 2019-2021, Information Initiative Center, Hokkaido University
  3. JSPS/NRF/NSFC A3 Foresight Program Nuclear Physics in the 21st Century

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The paper proposes practical formulas for the nuclear beta-decay rate, finding a deviation from simple approximation for heavy nuclei. The next-to-leading-order formula performs well in predicting actual results.
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.

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