Symmetry restoration in the axially deformed proton-neutron quasiparticle random phase approximation for nuclear beta decay: The effect of angular-momentum projection

TL;DR

The paper addresses how rotational symmetry breaking in axially deformed pnQRPA affects nuclear $eta$ decay rates. By extending pnFAM to include exact angular-momentum projection in a PAV framework, the authors quantify changes in GT and Fermi strengths and compute realistic half-lives for neutron-rich Fe isotopes. The main findings show that symmetry restoration can reduce $eta$-decay half-lives by up to 60% compared to needle-approximation projections, with deformation and projection effects jointly altering transition strengths and phase-space factors. This work underscores the importance of rigorous symmetry restoration in EDF-based descriptions of weak transitions and outlines paths for further refinements, including particle-number projection.

Abstract

We examine the effects of symmetry restoration on nuclear beta decay within the axially deformed proton-neutron quasiparticle random phase approximation (QRPA). We employ the proton-neutron finite-amplitude method (pnFAM) to compute transition amplitudes, and perform angular-momentum projection both after variation and after the QRPA to restore rotational symmetry. Exact projection reduces the calculated beta decay half-lives from those that use the needle approximation by up to 60%, and even more when taking the effects of projection on the ground-state energy into account.

Figures (10)

• Figure 1: A schematic illustration of GT transitions from the ground state ($0^+_1$) of [64]Fe to a set of final states ($1^+_N$ ) with $K=0$ in the daughter nucleus [64]Co. The corresponding portion of the GT strength function $dB(\hat{F},\omega)/d\omega$ from Eq. (\ref{['eq:strength_function']}) as a function of the energy is also shown. The heights of the peaks are for illustration only. The shaded area contains transitions forbidden by energy conservation in $\beta$ decay. See main text for details.
• Figure 2: The energies of HFB states and those with projection onto angular momentum $J=0$ as a function of the quadrupole deformation $\beta_2$. All energies are normalized to those of the spherical states.
• Figure 3: GT transition strength for $^{64}$Fe calculated in the pnFAM, starting from the spherical HFB state, in spaces containing different numbers of harmonic oscillator shells.
• Figure 4: The distribution of Gamow-Teller transition strength $B^{\rm GT}_{K}(\Omega_N)=|\bra{N, 1^+(K)}\hat{\sigma}\tau_-\ket{0^+_1}|^2$ in [64]Fe as a function of energy, from calculations in the pnFAM and pnFAM+AMP, starting with HFB states with $\beta_2=0$ (a) and $\beta_2=0.12$ (b).
• Figure 5: The distribution of Fermi transition strength $B^{\rm Fermi}_K(\Omega_N) =|\bra{N, 0^+(0)}\tau_-\ket{0^+_i}|^2$ in [64]Fe as a function of energy, from calculations in pnFAM, with and without exact AMP, starting with the HFB state with $\beta_2=0.12$.